diff --git a/.codeclimate.yml b/.codeclimate.yml deleted file mode 100644 index 2264a52..0000000 --- a/.codeclimate.yml +++ /dev/null @@ -1,28 +0,0 @@ -version: "2" # required to adjust maintainability checks -checks: - argument-count: - config: - threshold: 8 - complex-logic: - config: - threshold: 4 - file-lines: - config: - threshold: 500 - method-complexity: - config: - threshold: 5 - method-count: - config: - threshold: 20 - method-lines: - config: - threshold: 50 - nested-control-flow: - config: - threshold: 4 - return-statements: - config: - threshold: 4 -exclude_patterns: - - docs/ diff --git a/.codecov.yml b/.codecov.yml new file mode 100644 index 0000000..704d9d7 --- /dev/null +++ b/.codecov.yml @@ -0,0 +1,13 @@ +codecov: + disable_default_path_fixes: false + +coverage: + status: + project: + default: + threshold: 1% + patch: + default: + target: auto +comment: + require_changes: true diff --git a/.deepsource.toml b/.deepsource.toml deleted file mode 100644 index 25bc3d7..0000000 --- a/.deepsource.toml +++ /dev/null @@ -1,8 +0,0 @@ -version = 1 - -[[analyzers]] -name = "python" -enabled = true - - [analyzers.meta] - runtime_version = "3.x.x" diff --git a/.github/workflows/Benchmarks.yml b/.github/workflows/Benchmarks.yml new file mode 100644 index 0000000..c09f1ce --- /dev/null +++ b/.github/workflows/Benchmarks.yml @@ -0,0 +1,40 @@ +name: Run benchmarks + +on: + push: + branches: [ main ] + pull_request: + +jobs: + test: + runs-on: ubuntu-latest + strategy: + matrix: + include: + - python-version: "3.8" + - python-version: "3.9" + - python-version: "3.10" + steps: + - uses: actions/checkout@v3 + - name: Set up Python ${{ matrix.python-version }} + uses: actions/setup-python@v4 + with: + python-version: ${{ matrix.python-version }} + cache: 'pip' + cache-dependency-path: setup.py + - name: Install dependencies + run: | + python -m pip install --upgrade pip + pip install .[dev] + + - name: Run performance benchmarks + run: | + cd Examples/Benchmarks + testflo -n 1 -v --benchmark . + python formatBenchmarkData.py + python CantileverValidation.py + + - name: Run validation benchmarks + run: | + cd Examples/Benchmarks + python CantileverValidation.py diff --git a/.github/workflows/Examples.yml b/.github/workflows/Examples.yml new file mode 100644 index 0000000..dcc3ff8 --- /dev/null +++ b/.github/workflows/Examples.yml @@ -0,0 +1,34 @@ +name: Run examples + +on: + push: + branches: [ main ] + pull_request: + +jobs: + test: + runs-on: ubuntu-latest + strategy: + matrix: + include: + - python-version: "3.8" + - python-version: "3.9" + - python-version: "3.10" + steps: + - uses: actions/checkout@v3 + - name: Set up Python ${{ matrix.python-version }} + uses: actions/setup-python@v4 + with: + python-version: ${{ matrix.python-version }} + cache: 'pip' + cache-dependency-path: setup.py + - name: Install dependencies + run: | + python -m pip install --upgrade pip + pip install . + + - name: Run examples + run: | + set -e + cd Examples + for f in *.py; do python "$f"; done diff --git a/.github/workflows/Tests.yml b/.github/workflows/Tests.yml index 5389c93..2443ab2 100644 --- a/.github/workflows/Tests.yml +++ b/.github/workflows/Tests.yml @@ -11,7 +11,6 @@ jobs: strategy: matrix: include: - - python-version: "3.7" - python-version: "3.8" - python-version: "3.9" - python-version: "3.10" @@ -22,6 +21,7 @@ jobs: with: python-version: ${{ matrix.python-version }} cache: 'pip' + cache-dependency-path: setup.py - name: Install dependencies run: | python -m pip install --upgrade pip @@ -36,12 +36,12 @@ jobs: run: | export NUMBA_DISABLE_JIT=1 testflo -n 1 -v --coverage --coverpkg FEMpy . + coverage xml - - name: Upload coverage to Code Climate + - name: Upload coverage to Codecov if: ${{ matrix.python-version == 3.10}} - uses: paambaati/codeclimate-action@v3.2.0 - env: - CC_TEST_REPORTER_ID: ${{ secrets.CODECLIMATE_COVERAGE_ID }} + uses: codecov/codecov-action@v3 with: - debug: true - coverageCommand: coverage xml + token: ${{ secrets.CODECOV_TOKEN }} + fail_ci_if_error: true + verbose: true diff --git a/.github/workflows/docs.yml b/.github/workflows/docs.yml index fac068e..22972a0 100644 --- a/.github/workflows/docs.yml +++ b/.github/workflows/docs.yml @@ -17,10 +17,10 @@ jobs: with: python-version: '3.9' cache: 'pip' + cache-dependency-path: setup.py - run: | python -m pip install --upgrade pip - python3 -m pip install -r ./requirements.txt - python3 -m pip install -e . + python3 -m pip install -e .[docs] - name: Build Docs run: mkdocs build -f docs/mkdocs.yml diff --git a/.gitignore b/.gitignore index 0fa03f2..114ac72 100644 --- a/.gitignore +++ b/.gitignore @@ -1,5 +1,21 @@ -# Gauss quadrature pkl files -Gauss*.pkl +# Ignore images, except those in the docs +*.png +*.pdf +!docs/docs/Images/**.png +!docs/docs/Images/**.pdf +**.autosave.ipe + +# Ignore examples results files +Examples/ExampleOutput +*.vtk +*.vtu +*.dat + +# Ignore testflo benchmark data +**benchmark_data.csv + +# Ignore flake 8 config +.flake8 # Profiling files *.prof diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml index 066a5bb..c838716 100644 --- a/.pre-commit-config.yaml +++ b/.pre-commit-config.yaml @@ -2,11 +2,12 @@ repos: - repo: https://github.com/pre-commit/pre-commit-hooks rev: v4.4.0 hooks: + - id: check-added-large-files + args: ['--maxkb=5000'] - id: trailing-whitespace - id: end-of-file-fixer - id: check-yaml - id: check-json - - id: check-added-large-files - id: mixed-line-ending - id: check-merge-conflict - id: debug-statements @@ -14,13 +15,15 @@ repos: rev: 22.10.0 hooks: - id: black - # It is recommended to specify the latest version of Python - # supported by your project here, or alternatively use - # pre-commit's default_language_version, see - # https://pre-commit.com/#top_level-default_language_version language_version: python3 - args: ["-l", "120", "--target-version", "py37", "--target-version", "py38", "--target-version", "py39", "--target-version", "py310"] + args: ["-l", "120", "--target-version", "py38", "--target-version", "py39", "--target-version", "py310"] - repo: https://github.com/PyCQA/flake8 rev: 6.0.0 hooks: - id: flake8 +- repo: https://github.com/econchick/interrogate + rev: 1.5.0 + hooks: + - id: interrogate + args: ["-I", "--fail-under=100", "--ignore-nested-functions", "-vv"] + exclude: ^(docs|setup.py|tests|Examples|.*__init__.py) diff --git a/Dockerfile b/Dockerfile deleted file mode 100644 index 7654a38..0000000 --- a/Dockerfile +++ /dev/null @@ -1,7 +0,0 @@ -FROM python:latest - -COPY requirements.txt /data/ - -WORKDIR /data - -RUN pip3 install -r requirements.txt diff --git a/Examples/Benchmarks/CantileverValidation.py b/Examples/Benchmarks/CantileverValidation.py new file mode 100644 index 0000000..92b93f0 --- /dev/null +++ b/Examples/Benchmarks/CantileverValidation.py @@ -0,0 +1,107 @@ +""" +============================================================================== +FEMpy cantilever beam validation case +============================================================================== +@Description : This file contains a simple 2D validation case where we compare +FEMpy against the analytical solution for a cantilevel beam subject to a uniform +load. +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +from numba import njit + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +import FEMpy as fp + + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True) +def warpFunc(x, y): + return x, 0.02 * y + + +# @njit( cache=True) +def createGridMesh(nx, ny, warpFunc=None): + xNodes = np.tile(np.linspace(0.0, 1.0, nx + 1), ny + 1) + yNodes = np.repeat(np.linspace(0.0, 1.0, ny + 1), nx + 1) + + # Warp the mesh + if warpFunc is not None: + xNodes, yNodes = warpFunc(xNodes, yNodes) + + # Create Connectivity matrix + numEl = nx * ny + conn = np.zeros((numEl, 4), dtype=int) + conn[:, 0] = np.tile(np.arange(nx), ny) + np.repeat((nx + 1) * np.arange(ny), nx) # Lower left + conn[:, 1] = conn[:, 0] + 1 # lower right + conn[:, 2] = conn[:, 1] + nx + 1 # upper right + conn[:, 3] = conn[:, 2] - 1 # upper left + + return np.array([xNodes, yNodes]).T, conn + + +if __name__ == "__main__": + refineVal = [1, 2, 4, 8, 16, 32, 64] + Error = [] + numDOF = [] + meshSize = [] + for refine in refineVal: + + # create constitutive model + con = fp.Constitutive.IsoPlaneStrain(E=70e9, nu=0.0, t=1.0, rho=2700.0) + + # create mesh + nodeCoords, conn = createGridMesh(10 * refine, 1 * refine, warpFunc=warpFunc) + conn = {"quad": conn} + + # create model + model = fp.FEMpyModel(con, nodeCoords=nodeCoords, connectivity=conn, options={"outputFormat": ".dat"}) + prob = model.addProblem("Static") + + # Find the right, left and top edges + rightEdgeNodeInds = np.argwhere(nodeCoords[:, 0] == 1.0).flatten() + topEdgeNodeInds = np.argwhere(nodeCoords[:, 1] == np.max(nodeCoords[:, 1])).flatten() + leftEdgeNodeInds = np.argwhere(nodeCoords[:, 0] == 0.0).flatten() + + # apply BCs + prob.addFixedBCToNodes("Fixed", leftEdgeNodeInds, dof=[0, 1], value=0.0) + prob.addLoadToNodes("Load", topEdgeNodeInds, dof=[1], value=-(10**3), totalLoad=True) + + # Calculate analytic displacement + momInertia = 1 / 12 * 0.02**3 + dmax_analytic = 10**3 / (8 * 70e9 * momInertia) + + # solve the problem + prob.solve() + prob.writeSolution(baseName="mesh%s.dat" % refine) + + # average the vertical displacements along the tip of the beam and compute the error + averageDisplacement = np.average(prob.states[rightEdgeNodeInds, 1]) + Error.append(np.abs(-averageDisplacement - dmax_analytic)) + numDOF.append(prob.numDOF) + meshSize.append(1 / np.sqrt(numDOF[-1])) + + # Print the results + Error = np.array(Error) + numDOF = np.array(numDOF) + meshSize = np.array(meshSize) + print("======================================") + print("Cantilever Beam Validation Results") + print("======================================") + print("| num DOF | Error |") + for i in range(len(Error)): + print(f"|{numDOF[i]:>10} | {Error[i]:11.6e} |") + print("=====================================") + + logFitCoeff = np.polyfit(np.log(meshSize[-3:]), np.log(Error[-3:]), 1) + print(f"Displacement error convergence Rate: {logFitCoeff[0]}") diff --git a/Examples/Benchmarks/LinAlg.py b/Examples/Benchmarks/LinAlg.py deleted file mode 100644 index a3aefa9..0000000 --- a/Examples/Benchmarks/LinAlg.py +++ /dev/null @@ -1,45 +0,0 @@ -""" -============================================================================== -Linear Algebra speed test -============================================================================== -@File : detTest.py -@Date : 2021/11/14 -@Author : Alasdair Christison Gray -@Description : Testing speed of computing determinants and inverses of 2x2 and 3x3 matrices with numpy's generic det function vs -manual implementations for 2x2 nd 3x3 matrices -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import timeit - -# ============================================================================== -# Extension modules -# ============================================================================== - -setupString = """ -import numpy as np -from FEMpy import det2, det3, inv2, inv3 - -A2 = np.random.rand(4, 2, 2) -A3 = np.random.rand(8, 3, 3) -det2(A2) -det3(A3) -inv2(A2) -inv3(A3) -""" -for func in ["det", "inv"]: - for size in [2, 3]: - print("\n") - for method in ["numpy", "numba"]: - if method == "numpy": - test = f"np.linalg.{func}(A{size})" - else: - test = f"{func}{size}(A{size})" - t = timeit.timeit(test, setup=setupString, number=50000) - print(f"Time to compute {size} x {size} {func}s with {method} = {t:7.11e} s") diff --git a/Examples/Benchmarks/PlotQuadMeshScaling.py b/Examples/Benchmarks/PlotQuadMeshScaling.py new file mode 100644 index 0000000..9912fb5 --- /dev/null +++ b/Examples/Benchmarks/PlotQuadMeshScaling.py @@ -0,0 +1,129 @@ +""" +============================================================================== +FEMpy 2D Quad Mesh benchmark case +============================================================================== +@File : QuadMeshScaling.py +@Date : 2021/05/04 +@Author : Jasmin Lim + Modeled by script written by: Alasdair Christison Gray +@Description : This file contains a simple 2D case using quad elements that is +used to benchmark the performance of FEMpy as part of a CI job. The case uses a +nxn mesh of 2D quad elements, comparing the run time for each +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import time +import unittest + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +import FEMpy as fp +from numba import njit + +import matplotlib.pyplot as plt +import niceplots + +niceplots.setRCParams() +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True) +def warpFunc(x, y): + return 2.0 * x, (2.0 - x) * y + + +# @njit( cache=True) +def createGridMesh(nx, ny, warpFunc=None): + xNodes = np.tile(np.linspace(0.0, 1.0, nx + 1), ny + 1) + yNodes = np.repeat(np.linspace(0.0, 1.0, ny + 1), nx + 1) + + # Warp the mesh + if warpFunc is not None: + xNodes, yNodes = warpFunc(xNodes, yNodes) + + # Create Connectivity matrix + numEl = nx * ny + conn = np.zeros((numEl, 4), dtype=int) + conn[:, 0] = np.tile(np.arange(nx), ny) + np.repeat((nx + 1) * np.arange(ny), nx) # Lower left + conn[:, 1] = conn[:, 0] + 1 # lower right + conn[:, 2] = conn[:, 1] + nx + 1 # upper right + conn[:, 3] = conn[:, 2] - 1 # upper left + + return np.array([xNodes, yNodes]).T, conn + + +def solve_problem(n): + con = fp.Constitutive.IsoPlaneStrain(E=70e9, nu=0.3, t=1.0, rho=2700.0) + nodeCoords, conn = createGridMesh(n, n, warpFunc=warpFunc) + conn = {"quad": conn} + model = fp.FEMpyModel(con, nodeCoords=nodeCoords, connectivity=conn) + prob = model.addProblem("Static", options={"printTiming": True}) + rightEdgeNodeInds = np.argwhere(nodeCoords[:, 0] == 2.0).flatten() + leftEdgeNodeInds = np.argwhere(nodeCoords[:, 0] == 0.0).flatten() + prob.addFixedBCToNodes("Fixed", leftEdgeNodeInds, dof=[0, 1], value=0.0) + prob.addLoadToNodes("Load", rightEdgeNodeInds, dof=[0], value=1e4) + + # This is just to make sure everything is JIT compiled before the timing starts + times = prob.solve() + prob.reset() + + return nodeCoords, times + + +if __name__ == "__main__": + numEl = [2, 2, 4, 8, 16, 32, 64, 128, 256, 512] + numReps = 3 + + forceIntTimeArray = np.ones(len(numEl)) * 1e12 + assemblyTimeArray = np.ones(len(numEl)) * 1e12 + solveTimeArray = np.ones(len(numEl)) * 1e12 + totalTimeArray = np.ones(len(numEl)) * 1e12 + numDOF = [] + + # run problem + for i in range(len(numEl)): + for j in range(2): + nodeCoords, times = solve_problem(numEl[i]) + + numNodes = np.shape(nodeCoords)[0] + if j == 0: + numDOF.append(2 * numNodes) + + resAssemblyTime = times["ResAssembled"] - times["Start"] + matAssemblyTime = times["JacAssembled"] - times["ResAssembled"] + solveTime = times["Solved"] - times["JacAssembled"] + totalTime = resAssemblyTime + matAssemblyTime + solveTime + + forceIntTimeArray[i] = min(forceIntTimeArray[i], resAssemblyTime) + assemblyTimeArray[i] = min(assemblyTimeArray[i], matAssemblyTime) + solveTimeArray[i] = min(solveTimeArray[i], solveTime) + totalTimeArray[i] = min(totalTimeArray[i], totalTime) + + # plot results + plotVars = [forceIntTimeArray, assemblyTimeArray, solveTimeArray, totalTimeArray] + plotVarNames = ["Residual Assembly", "Matrix Assembly", "Linear Solution", "Total"] + + fig, ax = plt.subplots(figsize=(12, 8)) + + ax.set_xlabel("DOF") + ax.set_xscale("log") + ax.set_ylabel("Time\n(s)", rotation="horizontal", ha="right") + ax.set_yscale("log") + niceplots.adjust_spines(ax, outward=True) + + for v, name in zip(plotVars, plotVarNames): + ax.plot(numDOF[1:], v[1:], "-o", markeredgecolor="w", label=name, clip_on=False) + + ax.set_xticks(numDOF[1:]) + ax.set_xticklabels(numDOF[1:]) + ax.legend(labelcolor="linecolor") + + fig.savefig("../../docs/docs/Images/QuadElScaling.png", dpi=400) + + np.savetxt("QuadMeshScaling.csv", plotVars, delimiter=",") diff --git a/Examples/Benchmarks/QuadMeshScaling.py b/Examples/Benchmarks/QuadMeshScaling.py deleted file mode 100644 index 7f381a4..0000000 --- a/Examples/Benchmarks/QuadMeshScaling.py +++ /dev/null @@ -1,163 +0,0 @@ -""" -============================================================================== -FEMpy scaling test -============================================================================== -@File : QuadMeshScaling.py -@Date : 2021/05/04 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== -import time - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -try: - from pypardiso import spsolve - - solverArgs = {"factorize": False} -except ModuleNotFoundError: - from scipy.sparse.linalg import spsolve - - solverArgs = {} -import FEMpy as fp -from numba import njit -import matplotlib.pyplot as plt -import niceplots - -niceplots.setRCParams() -# ============================================================================== -# Extension modules -# ============================================================================== - - -@njit(cache=True) -def tractionForce(x): - ft = np.zeros_like(x) - ft[:, 0] = 1e6 - return ft - - -@njit(cache=True) -def bodyForce(x): - fb = np.zeros_like(x) - fb[:, 1] = 1e6 - return fb - - -@njit(cache=True) -def warpFunc(x, y): - return 2.0 * x, (2.0 - x) * y - - -# @njit( cache=True) -def createGridMesh(nx, ny, warpFunc=None): - xNodes = np.tile(np.linspace(0.0, 1.0, nx + 1), ny + 1) - yNodes = np.repeat(np.linspace(0.0, 1.0, ny + 1), nx + 1) - - # Warp the mesh - if warpFunc is not None: - xNodes, yNodes = warpFunc(xNodes, yNodes) - - # Create Connectivity matrix - numEl = nx * ny - conn = np.zeros((numEl, 4), dtype=int) - conn[:, 0] = np.tile(np.arange(nx), ny) + np.repeat((nx + 1) * np.arange(ny), nx) # Lower left - conn[:, 1] = conn[:, 0] + 1 # lower right - conn[:, 2] = conn[:, 0] + nx + 1 # upper left - conn[:, 3] = conn[:, 2] + 1 # upper right - - return np.array([xNodes, yNodes]).T, conn - - -El = fp.QuadElement() -con = fp.isoPlaneStrain(E=70e9, nu=0.3, t=1.0) - -numEl = [2, 5, 10, 20, 40, 80, 160, 320] -forceIntTimeList = [] -assemblyTimeList = [] -solveTimeList = [] -totalTimeList = [] -numDOFList = [] - -leftEdgeDisp = 0.0 - -for e in numEl: - minforceIntTime = np.inf - minassemblyTime = np.inf - minsolveTime = np.inf - - nodeCoords, conn = createGridMesh(e, e, warpFunc=warpFunc) - nodeEls = fp.makeNodeElsMat(conn) - numNodes = np.shape(nodeCoords)[0] - numDOFList.append(2 * numNodes) - - leftEdgeNodes = np.argwhere(nodeCoords[:, 0] == 0.0).flatten() - knownDisp = np.empty(numNodes * 2) - knownDisp[:] = np.nan - knownDisp[leftEdgeNodes * 2] = leftEdgeDisp - knownDisp[leftEdgeNodes * 2 + 1] = leftEdgeDisp - - for _ in range(3): - - startTime = time.time() - - # --- Get elements and edge numbers associated with right hand side traction--- - rightEdgeNodes = np.argwhere(nodeCoords[:, 0] == 2.0).flatten() - edgeInds = [[0, 1], [1, 3], [3, 2], [2, 0]] - TractElems, TractEdges = fp.getEdgesfromNodes(rightEdgeNodes, conn, nodeEls, edgeInds) - - FTract = fp.assembleTractions(nodeCoords, conn, El, con, TractElems, TractEdges, tractionForce, knownDisp) - FBody = fp.assembleBodyForce(nodeCoords, conn, El, con, bodyForce, knownDisp, n=2) - forceIntTime = time.time() - startTime - - K, RHS = fp.assembleMatrix(nodeCoords, conn, El, con, knownDisp) - - assemblyTime = time.time() - forceIntTime - startTime - - u = spsolve(K, RHS + FTract + FBody, **solverArgs) - # u = u.reshape(len(u) // 2, 2) - - solveTime = time.time() - startTime - assemblyTime - forceIntTime - - minforceIntTime = min(minforceIntTime, forceIntTime) - minassemblyTime = min(minassemblyTime, assemblyTime) - minsolveTime = min(minsolveTime, solveTime) - - print("\n-------------------") - print(f"{e**2} Elements") - print(f"{numNodes} Nodes") - print(f"{numNodes*2} DOF") - print(f"Force integration: {forceIntTime:03e} s") - print(f"Matrix assembly: {assemblyTime:03e} s") - print(f"Linear System solution: {solveTime:03e} s") - print("-------------------") - forceIntTimeList.append(minforceIntTime) - assemblyTimeList.append(minassemblyTime) - solveTimeList.append(minsolveTime) - totalTimeList.append(minforceIntTime + minassemblyTime + minsolveTime) - -fig, ax = plt.subplots() -ax.set_xlabel("DOF") -ax.set_xscale("log") -ax.set_ylabel("Time\n(s)", rotation="horizontal", ha="right") -ax.set_yscale("log") -niceplots.adjust_spines(ax, outward=True) - -plotVars = [forceIntTimeList, assemblyTimeList, solveTimeList, totalTimeList] -plotVarNames = ["Force Assembly", "Matrix Assembly", "Linear Solution", "Total"] - -for v, name in zip(plotVars, plotVarNames): - ax.plot(numDOFList, v, "-o", markeredgecolor="w", label=name, clip_on=False) - -ax.set_xticks(numDOFList) -ax.set_xticklabels(numDOFList) -ax.legend(labelcolor="linecolor") -fig.savefig("../../docs/docs/Images/QuadElScaling.png", dpi=400) -plt.show() diff --git a/Examples/Benchmarks/benchmarkLinAlg.py b/Examples/Benchmarks/benchmarkLinAlg.py new file mode 100644 index 0000000..3b11f84 --- /dev/null +++ b/Examples/Benchmarks/benchmarkLinAlg.py @@ -0,0 +1,87 @@ +""" +============================================================================== +Linear Algebra speed test +============================================================================== +@File : detTest.py +@Date : 2021/11/14 +@Author : Alasdair Christison Gray +@Description : Testing speed of computing determinants and inverses of 2x2 and 3x3 matrices with numpy's generic det function vs +manual implementations for 2x2 nd 3x3 matrices +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import unittest +import parameterized +import time + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from FEMpy.LinAlg import det2, det3, inv2, inv3 + +# ============================================================================== +# Extension modules +# ============================================================================== + +npFuncMap = {"det": np.linalg.det, "inv": np.linalg.inv} +numbaFuncMap = {"det": {2: det2, 3: det3}, "inv": {2: inv2, 3: inv3}} + +test_params = [] +for func in ["det", "inv"]: + for size in [2, 3]: + for method in ["numpy", "numba"]: + test_params.append({"func": func, "size": size, "method": method, "func": func}) + + +def nameFunc(testcase_func, param_num, params): + return f"BenchmarkLinAlg-{params['method']}-{params['func']}-{params['size']}" + + +@parameterized.parameterized_class(test_params, class_name_func=nameFunc) +class BenchmarkLinalg(unittest.TestCase): + @classmethod + def setUpClass(cls): + """Call numba methods with a dummy array to compile them""" + A2 = np.random.rand(10, 2, 2) + A3 = np.random.rand(10, 3, 3) + + det2(A2) + det3(A3) + inv2(A2) + inv3(A3) + + def setUp(self): + # For some unknown reason, when running benchmarks, the non parameterized version of this class gets called, + # with none of the test parameters defined, in this case we just skip everything as we don't want to use this + # version of the class to test anything + if self.__class__.__name__ == "BenchmarkLinalg": + self.skip = True + return + else: + self.skip = False + self.numMats = int(1e5) + self.mats = np.random.rand(self.numMats, self.size, self.size) + self.N = 50 + + if self.method == "numpy": + self.testFunc = npFuncMap[self.func] + else: + self.testFunc = numbaFuncMap[self.func][self.size] + + def benchmark_LinAlg(self): + if not self.skip: + start = time.time() + for _ in range(self.N): + self.testFunc(self.mats) + end = time.time() + runTime = (end - start) / self.N + print( + f"\nTime to compute {self.numMats} {self.size} x {self.size} {self.func}s with {self.method} = {runTime:7.11e} s" + ) + + +if __name__ == "__main__": + unittest.main() diff --git a/Examples/Benchmarks/benchmarkQuadMeshScaling.py b/Examples/Benchmarks/benchmarkQuadMeshScaling.py new file mode 100644 index 0000000..917dc7f --- /dev/null +++ b/Examples/Benchmarks/benchmarkQuadMeshScaling.py @@ -0,0 +1,108 @@ +""" +============================================================================== +FEMpy 2D Quad Mesh benchmark case +============================================================================== +@File : QuadMeshScaling.py +@Date : 2021/05/04 +@Author : Alasdair Christison Gray +@Description : This file contains a simple 2D case using quad elements that is +used to benchmark the performance of FEMpy as part of a CI job. The case uses a +223x223 mesh of 2D quad elements that results in almost exactly 100k degrees of +freedom +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import time +import unittest + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +import FEMpy as fp +from numba import njit + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True) +def warpFunc(x, y): + return 2.0 * x, (2.0 - x) * y + + +# @njit( cache=True) +def createGridMesh(nx, ny, warpFunc=None): + xNodes = np.tile(np.linspace(0.0, 1.0, nx + 1), ny + 1) + yNodes = np.repeat(np.linspace(0.0, 1.0, ny + 1), nx + 1) + + # Warp the mesh + if warpFunc is not None: + xNodes, yNodes = warpFunc(xNodes, yNodes) + + # Create Connectivity matrix + numEl = nx * ny + conn = np.zeros((numEl, 4), dtype=int) + conn[:, 0] = np.tile(np.arange(nx), ny) + np.repeat((nx + 1) * np.arange(ny), nx) # Lower left + conn[:, 1] = conn[:, 0] + 1 # lower right + conn[:, 2] = conn[:, 1] + nx # upper right + conn[:, 3] = conn[:, 2] + 1 # upper left + + return np.array([xNodes, yNodes]).T, conn + + +class BenchmarkQuadMeshSolve(unittest.TestCase): + def setUp(self): + self.con = fp.Constitutive.IsoPlaneStrain(E=70e9, nu=0.3, t=1.0, rho=2700.0) + self.nodeCoords, self.conn = createGridMesh(317, 317, warpFunc=warpFunc) + self.conn = {"quad": self.conn} + self.model = fp.FEMpyModel(self.con, nodeCoords=self.nodeCoords, connectivity=self.conn) + self.prob = self.model.addProblem("Static", options={"printTiming": True}) + rightEdgeNodeInds = np.argwhere(self.nodeCoords[:, 0] == 2.0).flatten() + leftEdgeNodeInds = np.argwhere(self.nodeCoords[:, 0] == 0.0).flatten() + self.prob.addFixedBCToNodes("Fixed", leftEdgeNodeInds, dof=[0, 1], value=0.0) + self.prob.addLoadToNodes("Load", rightEdgeNodeInds, dof=[0], value=1e4) + + # This is just to make sure everything is JIT compiled before the timing starts + self.prob.solve() + self.prob.reset() + + self.numReps = 3 + + def benchmarkResidualAssembly(self): + """Benchmark the cost of assembling the residual""" + for _ in range(self.numReps): + self.prob.markResOutOfDate() + self.prob.updateResidual(applyBCs=True) + + def benchmarkResidualAssemblyNoBC(self): + """Benchmark the cost of assembling the residual without applying any boundary conditions""" + for _ in range(self.numReps): + self.prob.markResOutOfDate() + self.prob.updateResidual(applyBCs=False) + + def benchmarkJacobianAssembly(self): + """Benchmark the cost of assembling the residual jacobian""" + for _ in range(self.numReps): + self.prob.markJacOutOfDate() + self.prob.updateJacobian(applyBCs=True) + + def benchmarkJacobianAssemblyNoBC(self): + """Benchmark the cost of assembling the residual jacobian without applying any boundary conditions""" + for _ in range(self.numReps): + self.prob.markJacOutOfDate() + self.prob.updateJacobian(applyBCs=False) + + def benchmarkSolve(self): + """Benchmark the cost of assembling and solving the problem""" + for _ in range(self.numReps): + self.prob.markResOutOfDate() + self.prob.markJacOutOfDate() + self.prob.solve() + + +if __name__ == "__main__": + unittest.main() diff --git a/Examples/Benchmarks/formatBenchmarkData.py b/Examples/Benchmarks/formatBenchmarkData.py new file mode 100644 index 0000000..f36543a --- /dev/null +++ b/Examples/Benchmarks/formatBenchmarkData.py @@ -0,0 +1,48 @@ +""" +============================================================================== +Benchmark data formatting script +============================================================================== +@File : formatBenchmarkData.py +@Date : 2022/12/11 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import csv + +# ============================================================================== +# External Python modules +# ============================================================================== + +# ============================================================================== +# Extension modules +# ============================================================================== + +print( + """\n\n\n +FEMpy Benchmark Results: +============================================================================================== +| Case | Status | Time (s) | Memory (MB) | +==============================================================================================""" +) + +# 38 +filename = "benchmark_data.csv" + +with open(filename, "r") as file: + reader = csv.reader(file, delimiter=",") + for line in reader: + Casename = line[1].split(":")[-1] + Casename = Casename.replace("benchmark", "") + Casename = Casename.replace("Benchmark", "") + Casename = Casename.replace(".", " ") + status = line[2] + + time = float(line[3]) + mem = float(line[4]) + print(f"| {Casename:<48} | {status:^6} | {time:11.6e} | {mem:11.6e} |") + +print("==============================================================================================") diff --git a/Examples/LBracketExample.ipynb b/Examples/LBracketExample.ipynb new file mode 100644 index 0000000..380b4bd --- /dev/null +++ b/Examples/LBracketExample.ipynb @@ -0,0 +1,223 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# FEMpy Demonstration of L-Shaped Bracket\n", + "\n", + "In this demonstration, we have an L-Shaped Bracket that is fixed at the top and has applied loads to its right-hand side. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "First, we import `FEMpy` and any required packages:" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "import FEMpy as fp\n", + "import numpy as np" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Create an Isotropic Plane Stress Constitutive Model\n", + "\n", + "Lets consider 7000 series Aluminum wih elastic modulus $\\lambda = 71.7$ $GPa$, Poisson's ratio $\\nu = 0.33$, density $\\rho = 2780$ $\\frac{kg}{m^3}$, and thickness $t = 5$ $mm$.\n", + "\n", + "" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "λ, ν, ρ, t = 71.7e9, 0.33, 2780, 5e-3\n", + "constitutiveModel = fp.Constitutive.IsoPlaneStress(λ, ν, ρ, t)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Create `FEMpy` model \n", + "Load in mesh and add fixed boundary condition. We also specify the output directory name, the output format of the mesh, and Von-Mises-Stresses as a desired output quantity.\n", + "\n", + "The mesh is comprised of 8,184 quad elements\n", + "\n", + "" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "Welcome to\n", + " ______ ______ __ __\n", + "| ____| ____| \\/ |\n", + "| |__ | |__ | \\ / |_ __ _ _\n", + "| __| | __| | |\\/| | '_ \\| | | |\n", + "| | | |____| | | | |_) | |_| |\n", + "|_| |______|_| |_| .__/ \\__, |\n", + " | | __/ |\n", + " |_| |___/\n", + "\n", + "Version: 1.0.0\n", + "\n", + "\n" + ] + } + ], + "source": [ + "# load in mesh\n", + "options = {\"outputDir\": \"ExampleOutput\", \"outputFormat\": \".vtu\", \"outputFunctions\": [\"Von-Mises-Stress\"]}\n", + "model = fp.FEMpyModel(constitutiveModel, meshFileName=\"Meshes/LBracket.msh\", options=options)\n", + "\n", + "# add fixed boundary condition to top edge\n", + "nodeCoords = model.getCoordinates()\n", + "topEdgeNodeInds = np.argwhere(nodeCoords[:, 1] == np.max(nodeCoords[:, 1])).flatten()\n", + "\n", + "model.addFixedBCToNodes(name=\"Fixed\", nodeInds=topEdgeNodeInds, dof=[0, 1], value=0.0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Set up `FEMpy` problems\n", + "\n", + "Create two problem cases: Vertical Load and Horizonal Load of value $1\\times 10^5$\n", + "\n", + "" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "# add load cases to FEMpy model\n", + "verticalLoadCase = model.addProblem(\"Vertical-Load\", options={\"printTiming\": True})\n", + "horizontalLoadCase = model.addProblem(\"Horizontal-Load\", options={\"printTiming\": True})\n", + "\n", + "# get right-side node indices, add loads to each problem\n", + "rightEdgeNodeInds = np.argwhere(nodeCoords[:, 0] == np.max(nodeCoords[:, 0])).flatten()\n", + "\n", + "verticalLoadCase.addLoadToNodes(name=\"RightEdgeLoad\", nodeInds=rightEdgeNodeInds, dof=1, value=-1e5, totalLoad=True)\n", + "horizontalLoadCase.addLoadToNodes(name=\"RightEdgeLoad\", nodeInds=rightEdgeNodeInds, dof=0, value=-1e5, totalLoad=True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Solve each problem" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Updating Residual\n", + "Updating Jacobian\n", + "Factorising Jacobian\n", + "Solving linear system\n", + "\n", + "\n", + "+-----------------------------------------------------------------+\n", + " Timing information for FEMpy problem: Vertical-Load\n", + "+-----------------------------------------------------------------+\n", + "+ Residual Assembly: 1.81424e-01 s\n", + "+ Jacobian Assembly: 4.89288e-01 s\n", + "+ Linear Solution: 1.72724e-01 s\n", + "+-----------------------------------------------------------------+\n", + "\n", + "Updating Residual\n", + "Updating Jacobian\n", + "Factorising Jacobian\n", + "Solving linear system\n", + "\n", + "\n", + "+-----------------------------------------------------------------+\n", + " Timing information for FEMpy problem: Horizontal-Load\n", + "+-----------------------------------------------------------------+\n", + "+ Residual Assembly: 9.99920e-02 s\n", + "+ Jacobian Assembly: 4.54143e-01 s\n", + "+ Linear Solution: 1.76414e-01 s\n", + "+-----------------------------------------------------------------+\n", + "\n" + ] + } + ], + "source": [ + "for problem in model.problems:\n", + " problem.solve()\n", + " problem.writeSolution()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Results \n", + "The resulting mesh contains the quantities:\n", + "* X-displacements\n", + "* Y-displacements\n", + "* Von-Misen Stress\n", + "\n", + "The figures below show the resulting Von-Misen Stress, and the resulting displacements.\n", + "\n", + "" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.7" + }, + "vscode": { + "interpreter": { + "hash": "7ed29583e9d86b92ec0f5d4cbb62d571549f0b60bf3cf89b59ade42254393772" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Examples/LBracketExample.py b/Examples/LBracketExample.py new file mode 100644 index 0000000..b64ffaf --- /dev/null +++ b/Examples/LBracketExample.py @@ -0,0 +1,104 @@ +""" +============================================================================== +FEMpy example runscript +============================================================================== +@File : ExampleRunScript.py +@Date : 2022/11/21 +@Author : Alasdair Christison Gray and Jasmin Lim +@Description : This is an example of how I imagine a FEMpy runscript might look +The model used is an L shaped bracket that is fixed at the top and has loads +applied at the right hand edge +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import FEMpy as fp +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== + + +# --- Create constitutive model, 7000 series Aluminium --- +E = 71.7e9 +nu = 0.33 +rho = 2780 + +# This thickness value is a design variable, by default all elements will use this value, but we can change it later if +# we want +t = 5e-3 +constitutiveModel = fp.Constitutive.IsoPlaneStress(E, nu, rho, t) + +# ============================================================================== +# Create the FEMpy model by loading in a mesh +# ============================================================================== +options = {"outputDir": "ExampleOutput", "outputFormat": ".vtu", "outputFunctions": ["Von-Mises-Stress"]} +model = fp.FEMpyModel(constitutiveModel, meshFileName="Meshes/LBracket.msh", options=options) +# --- Define a boundary condition that will be applied in all problems, fixing the top edge of the bracket in x and y --- +# For now I will just manually find the nodes that are on the top edge of the mesh, maybe we can add better functionality for doing things like this in future +nodeCoords = model.getCoordinates() +topEdgeNodeInds = np.argwhere(nodeCoords[:, 1] == np.max(nodeCoords[:, 1])).flatten() +model.addFixedBCToNodes(name="Fixed", nodeInds=topEdgeNodeInds, dof=[0, 1], value=0.0) + +# ============================================================================== +# Setup FEMpy problems +# ============================================================================== +# --- Create 2 different problems for 2 different loading scenarios --- +problemOptions = {"printTiming": True} +verticalLoadCase = model.addProblem("Vertical-Load", options=problemOptions) +horizontalLoadCase = model.addProblem("Horizontal-Load", options=problemOptions) + +# --- Setup the vertical load loadcase --- +# Again I will just manually find the nodes that are on the right edge of the mesh +rightEdgeNodeInds = np.argwhere(nodeCoords[:, 0] == np.max(nodeCoords[:, 0])).flatten() + +# In the first case, we will add a downward force of 10^5N distributed across the right edge of the bracket +verticalLoadCase.addLoadToNodes(name="RightEdgeLoad", nodeInds=rightEdgeNodeInds, dof=1, value=-1e5, totalLoad=True) + +# And we will fix the right edge in the x-direction, so it can only move up and down +# verticalLoadCase.addFixedBCToNodes(name="RightEdgeRoller", nodeInds=rightEdgeNodeInds, dof=0, value=0.0) + +# Now add gravity as a body force +# verticalLoadCase.addBodyLoad(name="Gravity", loadingFunction=[0.0, -9.81 ]) + +# --- Add loads to the horizontal loadcase --- +# Add a total horizontal force of 1e5N to the right edge of the bracket +horizontalLoadCase.addLoadToNodes(name="RightEdgeLoad", nodeInds=rightEdgeNodeInds, dof=0, value=-1e5, totalLoad=True) + +# add gravity again +# horizontalLoadCase.addBodyLoad(name="Gravity", loadingFunction=[0.0, -9.81]) + +# ============================================================================== +# Solve the problems +# ============================================================================== + +for problem in model.problems: + problem.solve() + problem.writeSolution() + +# ============================================================================== +# Compute some functions +# ============================================================================== +for problem in model.problems: + mass = problem.computeFunction("Mass", elementReductionType="integrate", globalReductionType="sum") + maxStress = problem.computeFunction("Von-Mises-Stress", elementReductionType="mean", globalReductionType="max") + print(f"\nProblem: {problem.name}") + print(f"Max von Mises stress = {maxStress:11.6e}") + print(f"Total mass = {mass:11.6e}\n") + +# ============================================================================== +# Change the thickness design variables and solve again +# ============================================================================== +DVs = model.getDesignVariables() +DVs["Thickness"] = np.random.rand(len(DVs["Thickness"])) * 1e-2 + 1e-3 +model.setDesignVariables(DVs) + +for problem in model.problems: + problem.solve() + problem.writeSolution() diff --git a/Examples/Meshes/LBracket.geo b/Examples/Meshes/LBracket.geo new file mode 100644 index 0000000..24d8501 --- /dev/null +++ b/Examples/Meshes/LBracket.geo @@ -0,0 +1,77 @@ +// Gmsh project created on Mon Nov 14 20:29:56 2022 +//+ +Point(1) = {0, 1, 0, 1.0}; +//+ +Point(2) = {0, 0, 0, 1.0}; +//+ +Point(3) = {1, 0, 0, 1.0}; +//+ +Point(4) = {1, 0.4, 0, 1.0}; +//+ +Point(5) = {0.4, 0.4, 0, 1.0}; +//+ +Point(6) = {0.4, 1.0, 0, 1.0}; +//+ +Point(7) = {1.0, 0.36, 0, 1.0}; +//+ +Line(1) = {2, 3}; +//+ +Line(2) = {3, 7}; +//+ +Line(3) = {7, 4}; +//+ +Line(4) = {4, 5}; +//+ +Line(5) = {5, 6}; +//+ +Line(6) = {6, 1}; +//+ +Line(7) = {1, 2}; +//+ +Curve Loop(1) = {6, 7, 1, 2, 3, 4, 5}; +//+ +Plane Surface(1) = {1}; +//+ +Physical Curve("FixedBC", 8) = {6}; +//+ +Physical Curve("LoadedEdge", 9) = {3}; +//+ +Physical Surface("surf", 10) = {1}; +//+ +Physical Curve(" FixedBC", 8) -= {6}; +//+ +Physical Curve(" LoadedEdge", 9) -= {3}; +//+ +MeshSize {5} = 0.01; +//+ +MeshSize {5} = 0.01; +//+ +MeshSize {2} = 0.1; +//+ +MeshSize {2} = 10; +//+ +MeshSize {2} = 100; +//+ +MeshSize {2} = 1; +//+ +MeshSize {2} = 0.1; +//+ +MeshSize {2} = 1; +//+ +MeshSize {1, 2, 3} = 1; +//+ +MeshSize {6, 4} = 0.1; +//+ +MeshSize {2} = 1; +//+ +MeshSize {2} = 10; +//+ +MeshSize {6, 5, 4} = 0.1; +//+ +MeshSize {6, 5, 4} = 1; +//+ +MeshSize {2} = 10; +//+ +MeshSize {2} = 10; +//+ +MeshSize {5, 4, 6} = 0.1; diff --git a/Examples/Meshes/LBracket.msh b/Examples/Meshes/LBracket.msh new file mode 100644 index 0000000..4ae4fa7 --- /dev/null +++ b/Examples/Meshes/LBracket.msh @@ -0,0 +1,25001 @@ +$MeshFormat +4.1 0 8 +$EndMeshFormat +$PhysicalNames +1 +2 10 "surf" +$EndPhysicalNames +$Entities +7 7 1 0 +1 0 1 0 0 +2 0 0 0 0 +3 1 0 0 0 +4 1 0.4 0 0 +5 0.4 0.4 0 0 +6 0.4 1 0 0 +7 1 0.36 0 0 +1 0 0 0 1 0 0 0 2 2 -3 +2 1 0 0 1 0.36 0 0 2 3 -7 +3 1 0.36 0 1 0.4 0 0 2 7 -4 +4 0.4 0.4 0 1 0.4 0 0 2 4 -5 +5 0.4 0.4 0 0.4 1 0 0 2 5 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index 0000000..8d73f02 Binary files /dev/null and b/Examples/Presentation/VonMises_Vert.png differ diff --git a/FEMpy/Assembly.py b/FEMpy/Assembly.py deleted file mode 100644 index 9f9befd..0000000 --- a/FEMpy/Assembly.py +++ /dev/null @@ -1,239 +0,0 @@ -""" -============================================================================== -Matrix and Vector Assembly Procedures -============================================================================== -@File : Assembly.py -@Date : 2021/03/30 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== -from collections.abc import Iterable - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from scipy import sparse as sp - -# ============================================================================== -# Extension modules -# ============================================================================== - - -def sparseVec(entries, rows, length): - """generate a CSR sparse vector for COO sparse matrix type inputs - - Parameters - ---------- - entries : list - Non-zero entries of the vector - rows : list - Rows of the non-zero entries, duplicates allowed, in which case entries will be summed - length : int - Vector length] - - Returns - ------- - scipy.sparse.csr_matrix - Sparse length x 1 vector - """ - return sp.coo_matrix( - (entries, (rows, np.zeros_like(rows))), - shape=(length, 1), - ).tocsr() - - -def assembleMatrix(nodeCoords, conn, element, constitutive, knownStates, matType="stiffness"): - - numNodes = np.shape(nodeCoords)[0] - numEl = np.shape(conn)[0] - numDisp = element.numDisp - - MatRows = [] - MatColumns = [] - MatEntries = [] - RHSRows = [] - RHSEntries = [] - - localMat = np.zeros((element.numDOF, element.numDOF)) - RHSLocal = np.zeros(element.numDOF) - RHS = np.zeros(element.numDOF * numEl) - - if isinstance(constitutive, Iterable): - ConList = constitutive - else: - ConList = [constitutive] * numEl - - if matType.lower() == "stiffness": - matFunc = element.getStiffnessMat - elif matType.lower() == "mass": - matFunc = element.getMassMat - - for e in range(numEl): - elNodes = conn[e] - elDOF = (np.array([numDisp * elNodes]).T + np.arange(numDisp)).flatten() - elNodeCoords = nodeCoords[elNodes] - - # --- Compute the local matrix --- - localMat[:] = matFunc(elNodeCoords, ConList[e]) - - RHSLocal[:] = 0.0 - - # ============================================================================== - # Apply state BC's - # ============================================================================== - - # Find the local indices of any DOF in the current element that are restrained - localConstrainedDOF = np.argwhere(np.isfinite(knownStates[elDOF])).flatten() - localFreeDOF = np.argwhere(np.isnan(knownStates[elDOF])).flatten() - nonzeroConstrainedDOF = localConstrainedDOF[ - np.argwhere(knownStates[elDOF[localConstrainedDOF]] != 0.0).flatten() - ] - - # Add the force contribution of any nonzero fixed displacements - RHSLocal[localFreeDOF] = -np.sum( - knownStates[elDOF[nonzeroConstrainedDOF]] * localMat[np.ix_(localFreeDOF, nonzeroConstrainedDOF)], - axis=-1, - ) - - # For constrained DOF, set RHS equal to the fixed disp value - RHSLocal[nonzeroConstrainedDOF] = knownStates[elDOF[nonzeroConstrainedDOF]] - - # set all rows and cols of K corresponding to fixed DOF to zero, except diagonal entries which are 1 - localMat[:, localConstrainedDOF] = 0.0 - localMat[localConstrainedDOF, :] = 0.0 - localMat[localConstrainedDOF, localConstrainedDOF] = 1.0 - - # ============================================================================== - # Add local terms to arrays used to create global mat and RHS - # ============================================================================== - # For sparse matrix creation we need to create 3 lists which store, the row index, cloumn index and value of every nonzero term in the local stiffness matrix - nonzeroKInds = localMat.nonzero() - - MatRows += elDOF[nonzeroKInds[0]].tolist() - MatColumns += elDOF[nonzeroKInds[1]].tolist() - MatEntries += localMat[nonzeroKInds].tolist() - - RHSRows += elDOF[RHSLocal.nonzero()].tolist() - RHSEntries += RHSLocal[RHSLocal.nonzero()].tolist() - - # ============================================================================== - # Create global stiffness mat and RHS - # ============================================================================== - Mat = sp.coo_matrix( - (MatEntries, (MatRows, MatColumns)), - shape=(numNodes * numDisp, numNodes * numDisp), - ).tocsr() - RHS = sp.coo_matrix((RHSEntries, (RHSRows, np.zeros_like(RHSRows))), shape=(numNodes * numDisp, 1)).tocsr() - - return Mat, RHS - - -def assembleTractions( - nodeCoords, - conn, - element, - constitutive, - tractElems, - tractEdges, - tractFunc, - knownStates, -): - numNodes = np.shape(nodeCoords)[0] - numDisp = element.numDisp - - FRows = [] - FEntries = [] - FLocal = np.zeros(element.numDOF) - - if isinstance(constitutive, Iterable): - ConList = constitutive - else: - ConList = [constitutive] * len(tractElems) - - for i in range(len(tractElems)): - e = tractElems[i] - elNodes = conn[e] - elDOF = (np.array([numDisp * elNodes]).T + np.arange(numDisp)).flatten() - elNodeCoords = nodeCoords[elNodes] - FLocal[:] = element.integrateTraction(tractFunc, elNodeCoords, ConList[i], edges=tractEdges[i], n=2).flatten() - - # Add traction force entries that aren't zero and aren't at nodes where displacement is known - usefulDOF = np.argwhere(np.logical_and(np.isnan(knownStates[elDOF]), FLocal != 0.0)).flatten() - FRows += elDOF[usefulDOF].tolist() - FEntries += FLocal[usefulDOF].tolist() - - return sparseVec(FEntries, FRows, numNodes * numDisp) - - -def assembleBodyForce(nodeCoords, conn, element, constitutive, forceFunc, knownStates, n=2): - numNodes = np.shape(nodeCoords)[0] - numEl = np.shape(conn)[0] - numDisp = element.numDisp - - if isinstance(constitutive, Iterable): - ConList = constitutive - else: - ConList = [constitutive] * numEl - - FRows = [] - FEntries = [] - FLocal = np.zeros(element.numDOF) - for e in range(numEl): - elNodes = conn[e] - elDOF = (np.array([numDisp * elNodes]).T + np.arange(numDisp)).flatten() - elNodeCoords = nodeCoords[elNodes] - FLocal[:] = element.integrateBodyForce(forceFunc, elNodeCoords, ConList[e], n=n).flatten() - - # Add traction force entries that aren't zero and aren't at nodes where displacement is known - usefulDOF = np.argwhere(np.logical_and(np.isnan(knownStates[elDOF]), FLocal != 0.0)).flatten() - FRows += elDOF[usefulDOF].tolist() - FEntries += FLocal[usefulDOF].tolist() - - return sparseVec(FEntries, FRows, numNodes * numDisp) - - -def computeStresses(Element, ParamCoords, constitutive, nodeCoords, Conn, nodalDisp): - """Compute the stresses at a number of points inside all elements - - This function assumes that all elements have the same number of stress components - - Parameters - ---------- - Element : FEMpy Element object - [description] - ParamCoords : numPoint x numDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the stress at within - each element - constitutive : single or iterable of FEMpy constitutive classes - Constitutive class(es) for the elements, the D matrix is used for computing stresses - nodeCoords : numNode x numDim array - Element node real coordinates - Conn : 2D iterable - Mesh connectivity data, Conn[i][j] is the index of the jth node in the ith element - nodalDisp : numNode x numDim array - nodal displacements - - Returns - ------- - Stresses : numElement*numPoint x numStress array - Stresses at - """ - numEl = np.shape(Conn)[0] - - if isinstance(constitutive, Iterable): - ConList = constitutive - else: - ConList = [constitutive] * numEl - Stresses = np.zeros((4 * numEl, ConList[0].numStress)) - - for e in range(numEl): - Stresses[4 * e : 4 * (e + 1), :3] = Element.getStress( - ParamCoords, nodeCoords[Conn[e]], ConList[e], nodalDisp[Conn[e]] - ) - - return Stresses diff --git a/FEMpy/AxialBar.py b/FEMpy/AxialBar.py deleted file mode 100644 index 833cd74..0000000 --- a/FEMpy/AxialBar.py +++ /dev/null @@ -1,33 +0,0 @@ -""" -============================================================================== -Axial Bar Constitutive Class -============================================================================== -@File : AxialBar.py -@Date : 2021/03/30 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== -from .Constitutive import Constitutive - - -class AxialBar(Constitutive): - def __init__(self, E, A, rho): - super().__init__(numStrain=1, numDisp=1, numStress=1) - self.DMat = np.array([[E]]) - self.LMats[0] = np.array([[1]]) - self.E = E - self.A = A - self.rho = rho diff --git a/FEMpy/Basis/LagrangePoly.py b/FEMpy/Basis/LagrangePoly.py new file mode 100644 index 0000000..5b6fcca --- /dev/null +++ b/FEMpy/Basis/LagrangePoly.py @@ -0,0 +1,376 @@ +""" +============================================================================== +Lagrange polynomial interpolation +============================================================================== +@File : LagrangePoly.py +@Date : 2021/03/12 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from numba import njit + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True) +def LagrangePoly1d(x, n): + """Compute the values of the 1d Lagrange polynomials at a series of points + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + Points to compute polynomial values at, should be between -1 and 1 + n : int + Number of Lagrange polynomials, equal to the number of nodes along the edge, n points results in n-1 order + polynomials. + + Returns + ------- + N : nP x n array + Shape function values + """ + xp = x.flatten() + xi = np.linspace(-1.0, 1.0, n) + N = np.ones((len(xp), n), dtype=xp.dtype) + for m in range(n): + for i in list(range(m)) + list(range(m + 1, n)): + N[:, m] *= (xp - xi[i]) / (xi[m] - xi[i]) + # print("computing lagrange shape funcs") + return N + + +@njit(cache=True) +def LagrangePoly1dDeriv(x, n): + """Compute the derivatives of the 1d Lagrange polynomials at a series of points + + This method computes the derivates using the complex step method because I couldn't be bothered implementing the + analytical derivatives + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + Points to compute polynomial values at, should be between -1 and 1 + n : int + Number of Lagrange polynomials, equal to the number of nodes along the edge, n points results in n-1 order + polynomials. + + Returns + ------- + dNdx : nP x n array + Shape function derivative values + """ + xp = x.flatten() + 1j * 1e-200 + return 1e200 * np.imag(LagrangePoly1d(xp, n)) + + +@njit(cache=True) +def LagrangePoly2d(x, y, n): + """Compute the derivatives of the 2d Lagrange polynomials at a series of points in 2d space + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + x coordinates of points to compute polynomial values at, should be between -1 and 1 + y : array of length nP (0D, nPx1 or 1xnP) + y coordinates of points to compute polynomial values at, should be between -1 and 1 + n : int + Number of Lagrange polynomials, in 2d, there are n nodes in each direction, giving n^2, n-1 order polynomials + + Returns + ------- + N : nP x n array + Shape function values + """ + xp = x.flatten() + yp = y.flatten() + Nx = LagrangePoly1d(xp, n) + Ny = LagrangePoly1d(yp, n) + N = np.zeros((len(xp), n**2), dtype=xp.dtype) + for i in range(len(xp)): + for j in range(n): + for k in range(n): + N[i, j * n + k] = Nx[i, k] * Ny[i, j] + return N + + +@njit(cache=True) +def LagrangePoly2dDeriv(x, y, n): + """Compute the derivatives of the 2d Lagrange polynomials at a series of points in 2d space + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + x coordinates of points to compute polynomial values at, should be between -1 and 1 + y : array of length nP (0D, nPx1 or 1xnP) + y coordinates of points to compute polynomial values at, should be between -1 and 1s + z: array of length nP (0D, nPx1 or 1xnP) + yzcoordinates of points to compute polynomial values at, should be between -1 and 1s + n : int + Number of Lagrange polynomials, in 2d, there are n nodes in each direction, giving n^2, n-1 order polynomials + + Returns + ------- + dNdx : nP x 2 x n array + Shape function derivative values + """ + xp = x.flatten() + yp = y.flatten() + Nx = LagrangePoly1d(xp, n) + Ny = LagrangePoly1d(yp, n) + dNdx = LagrangePoly1dDeriv(xp, n) + dNdy = LagrangePoly1dDeriv(yp, n) + N = np.zeros((len(xp), 2, n**2)) + for i in range(len(xp)): + for j in range(n): + for k in range(n): + N[i, 0, j * n + k] = dNdx[i, k] * Ny[i, j] + N[i, 1, j * n + k] = Nx[i, k] * dNdy[i, j] + return N + + +@njit(cache=True) +def LagrangePoly3d(x, y, z, n): + """Compute the derivatives of the 3d Lagrange polynomials at a series of points in 3d space + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + x coordinates of points to compute polynomial values at, should be between -1 and 1 + y : array of length nP (0D, nPx1 or 1xnP) + y coordinates of points to compute polynomial values at, should be between -1 and 1 + z : array of length nP (0D, nPx1 or 1xnP) + z coordinates of points to compute polynomial values at, should be between -1 and 1 + n : int + Number of Lagrange polynomials, in 3d, there are n nodes in each direction, giving n^3, n-1 order polynomials + + Returns + ------- + N : nP x n array + Shape function values + """ + xp = x.flatten() + yp = y.flatten() + zp = z.flatten() + Nx = LagrangePoly1d(xp, n) + Ny = LagrangePoly1d(yp, n) + Nz = LagrangePoly1d(zp, n) + N = np.zeros((len(xp), n**3), dtype=xp.dtype) + for pointInd in range(len(xp)): + for zInd in range(n): + for yInd in range(n): + for xInd in range(n): + flatInd = zInd * n**2 + yInd * n + xInd + N[pointInd, flatInd] = Nx[pointInd, xInd] * Ny[pointInd, yInd] * Nz[pointInd, zInd] + return N + + +@njit(cache=True) +def LagrangePoly3dDeriv(x, y, z, n): + """Compute the derivatives of the 3d Lagrange polynomials at a series of points in 3d space + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + x coordinates of points to compute polynomial values at, should be between -1 and 1 + y : array of length nP (0D, nPx1 or 1xnP) + y coordinates of points to compute polynomial values at, should be between -1 and 1s + n : int + Number of Lagrange polynomials, in 3d, there are n nodes in each direction, giving n^2, n-1 order polynomials + + Returns + ------- + dNdx : nP x 2 x n array + Shape function derivative values + """ + xp = x.flatten() + yp = y.flatten() + zp = z.flatten() + Nx = LagrangePoly1d(xp, n) + Ny = LagrangePoly1d(yp, n) + Nz = LagrangePoly1d(zp, n) + dNdx = LagrangePoly1dDeriv(xp, n) + dNdy = LagrangePoly1dDeriv(yp, n) + dNdz = LagrangePoly1dDeriv(zp, n) + N = np.zeros((len(xp), 3, n**3)) + for pointInd in range(len(xp)): + for zInd in range(n): + for yInd in range(n): + for xInd in range(n): + flatInd = zInd * n**2 + yInd * n + xInd + N[pointInd, 0, flatInd] = dNdx[pointInd, xInd] * Ny[pointInd, yInd] * Nz[pointInd, zInd] + N[pointInd, 1, flatInd] = Nx[pointInd, xInd] * dNdy[pointInd, yInd] * Nz[pointInd, zInd] + N[pointInd, 2, flatInd] = Nx[pointInd, xInd] * Ny[pointInd, yInd] * dNdz[pointInd, zInd] + return N + + +@njit(cache=True) +def LagrangePolyTri(x, y, n): + """Compute the values of the Lagrangian polynomial for a triangular basis up to order 3 + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + x coordinates of points to compute polynomial values at, should be between 0 and 1 + y : array of length nP (0D, nPx1 or 1xnP) + y coordinates of points to compute polynomial values at, should be between 0 and 1 + n : int + Polynomial/element order + Returns + ------- + N : nP x (n+1)*(n+2)/2 array + Shape function values + """ + xp = x.flatten() + yp = y.flatten() + N = np.zeros((len(xp), (n + 1) * (n + 2) // 2), dtype=xp.dtype) + + if n == 1: + N[:, 0] = 1.0 - xp - yp + N[:, 1] = xp + N[:, 2] = yp + + elif n == 2: + x2 = xp**2 + y2 = yp**2 + xy = xp * yp + N[:, 0] = 2 * x2 + 4 * xy - 3 * xp + 2 * y2 - 3 * y + 1 + N[:, 1] = 2 * x2 - xp + N[:, 2] = 2 * y2 - yp + N[:, 3] = -4 * (x2 + xy - xp) + N[:, 4] = 4 * xy + N[:, 5] = -4 * (xy + y2 - yp) + + elif n == 3: + x2 = xp**2 + y2 = yp**2 + x3 = xp**3 + y3 = yp**3 + xy = xp * yp + x2y = x2 * yp + xy2 = xp * y2 + N[:, 0] = ( + -4.5 * x3 + - 0.5 * 27.0 * x2y + + 9 * x2 + - 0.5 * 27.0 * xy2 + + 18.0 * xy + - 5.5 * xp + - 4.5 * y3 + + 9.0 * y2 + - 5.5 * yp + + 1.0 + ) + N[:, 1] = 0.5 * (9.0 * x3 - 9.0 * x2 + 2.0 * xp) + N[:, 2] = 0.5 * (9.0 * y3 - 9.0 * y2 + 2.0 * yp) + N[:, 3] = 4.5 * (3.0 * x3 + 6.0 * x2y - 5.0 * x2 + 3.0 * xy2 - 5.0 * xy + 2.0 * xp) + N[:, 4] = 4.5 * (-3.0 * x3 - 3.0 * x2y + 4.0 * x2 + xy - xp) + N[:, 5] = 4.5 * (3.0 * x2y - xy) + N[:, 6] = 4.5 * (3.0 * xy2 - xy) + N[:, 7] = 4.5 * (-3.0 * xy2 + xy - 3.0 * y3 + 4.0 * y2 - yp) + N[:, 8] = 4.5 * (3.0 * x2y + 6 * xy2 - 5.0 * xy + 3.0 * y3 - 5.0 * y2 + 2.0 * yp) + N[:, 9] = -27.0 * (x2y + xy2 - xy) + return N + + +@njit(cache=True) +def LagrangePolyTriDeriv(x, y, n): + """Compute the derivatives of the triangular basis Lagrange polynomials at a series of points in 2d space + + Parameters + ---------- + x : array of length nP (0D, nPx1 or 1xnP) + x coordinates of points to compute polynomial values at, should be between 0 and 1 + y : array of length nP (0D, nPx1 or 1xnP) + y coordinates of points to compute polynomial values at, should be between 0 and 1s + n : int + Number of Lagrange polynomials, in 2d, there are n nodes in each direction, giving n^2, n-1 order polynomials + Returns + ------- + dNdx : nP x 2 x (n+1)*(n+2)/2 array + Shape function derivative values + """ + + xp = x.flatten() + yp = y.flatten() + dNdx = np.zeros((len(xp), 2, (n + 1) * (n + 2) // 2)) + + if n == 1: + dNdx[:, :, 0] = -1.0 + dNdx[:, 0, 1] = 1.0 + dNdx[:, 1, 2] = 1.0 + + elif n == 2: + dNdx[:, 0, 0] = 4.0 * xp + 4.0 * yp - 3.0 + dNdx[:, 1, 0] = 4.0 * xp + 4.0 * yp - 3.0 + dNdx[:, 0, 1] = 4.0 * xp - 1.0 + dNdx[:, 1, 2] = 4.0 * yp - 1.0 + dNdx[:, 0, 3] = -4.0 * (2.0 * xp + yp - 1.0) + dNdx[:, 1, 3] = -4.0 * xp + dNdx[:, 0, 4] = 4.0 * yp + dNdx[:, 1, 4] = 4.0 * xp + dNdx[:, 0, 5] = -4.0 * yp + dNdx[:, 1, 5] = -4.0 * (xp + 2.0 * yp - 1.0) + + elif n == 3: + x2 = xp**2 + y2 = yp**2 + xy = xp * yp + dNdx[:, 0, 0] = -13.5 * x2 - 27.0 * xy + 18 * xp - 0.5 * 27.0 * y2 + 18.0 * yp - 5.5 + dNdx[:, 1, 0] = -0.5 * 27.0 * x2 - 27.0 * xy + 18.0 * xp - 13.5 * y2 + 18.0 * yp - 5.5 + + dNdx[:, 0, 1] = 0.5 * (27.0 * x2 - 18.0 * xp + 2.0) + + dNdx[:, 1, 2] = 0.5 * (27.0 * y2 - 18.0 * yp + 2.0) + + dNdx[:, 0, 3] = 4.5 * (9.0 * x2 + 12.0 * xy - 10.0 * xp + 3.0 * y2 - 5.0 * yp + 2.0) + dNdx[:, 1, 3] = 4.5 * (6.0 * x2 + 6.0 * xy - 5.0 * xp) + + dNdx[:, 0, 4] = 4.5 * (-9.0 * x2 - 6.0 * xy + 8.0 * xp + yp - 1.0) + dNdx[:, 1, 4] = 4.5 * (xp - 3.0 * x2) + + dNdx[:, 0, 5] = 4.5 * (6.0 * xy - yp) + dNdx[:, 1, 5] = 4.5 * (3.0 * x2 - xp) + + dNdx[:, 0, 6] = 4.5 * (3.0 * y2 - yp) + dNdx[:, 1, 6] = 4.5 * (6.0 * xy - xp) + + dNdx[:, 0, 8] = 4.5 * (6.0 * xy + 6.0 * y2 - 5.0 * yp) + dNdx[:, 1, 8] = 4.5 * (3.0 * x2 + 12.0 * xy - 5.0 * xp + 9.0 * y2 - 10.0 * yp + 2.0) + + dNdx[:, 0, 7] = 4.5 * (-3.0 * y2 + yp) + dNdx[:, 1, 7] = 4.5 * (-6.0 * xy + xp - 9.0 * y2 + 8.0 * yp - 1.0) + + dNdx[:, 0, 9] = -27.0 * (2.0 * xy + y2 - yp) + dNdx[:, 1, 9] = -27.0 * (x2 + 2.0 * xy - xp) + + return dNdx + + +if __name__ == "__main__": + import time + + x = np.array([np.linspace(-1.0, 1.0, 4)]) + y = -np.ones_like(x) + # Call functions so they get jit compiled and we can check their output + print(LagrangePoly1d(x, 3), "\n") + print(LagrangePoly1dDeriv(x, 3), "\n") + print(LagrangePoly2d(x, y, 3), "\n") + print(LagrangePoly2dDeriv(x, y, 3), "\n\n\n") + startTime = time.time() + for _ in range(1000): + LagrangePoly1d(x, 3) + LagrangePoly1dDeriv(x, 3) + LagrangePoly2d(x, y, 3) + LagrangePoly2dDeriv(x, y, 3) + print(time.time() - startTime) diff --git a/FEMpy/Basis/__init__.py b/FEMpy/Basis/__init__.py new file mode 100644 index 0000000..224aa89 --- /dev/null +++ b/FEMpy/Basis/__init__.py @@ -0,0 +1 @@ +from .LagrangePoly import * diff --git a/FEMpy/Constitutive.py b/FEMpy/Constitutive.py deleted file mode 100644 index d9657b8..0000000 --- a/FEMpy/Constitutive.py +++ /dev/null @@ -1,32 +0,0 @@ -""" -============================================================================== -Constitutive Relation Class -============================================================================== -@File : Constitutive.py -@Date : 2021/03/12 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== - - -class Constitutive: - def __init__(self, numStrain, numDisp, numStress=None, numDim=None) -> None: - self.numStrain = numStrain - self.numStress = numStress if numStress is not None else numStrain - self.numDim = numDim if numDim is not None else numDisp - self.numDisp = numDisp - # self.DMat = np.zeros((self.numStrain, self.numStress)) - self.LMats = np.zeros((self.numDim, self.numStrain, self.numDisp)) diff --git a/FEMpy/Constitutive/ConstitutiveModel.py b/FEMpy/Constitutive/ConstitutiveModel.py new file mode 100644 index 0000000..7da5563 --- /dev/null +++ b/FEMpy/Constitutive/ConstitutiveModel.py @@ -0,0 +1,460 @@ +""" +============================================================================== +FEMpy Constitutive model base class +============================================================================== +@File : NewConstitutive.py +@Date : 2022/11/28 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import abc + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from numba import njit, prange + +# ============================================================================== +# Extension modules +# ============================================================================== + + +class ConstitutiveModel: + """The base class for all FEMpy constitutive models + + The constitutive model defines the underlying PDE being solved. Currently, this base class is defined for + solid mechanics problems, but in future it may be extended for other PDE types. + + It contains information on: + + - The number of spatial dimensions the model is valid for + - The number and names of the PDE states + - The number and names of the stresses and strains for this model + - The number and names of the design variables associated with the PDE + - The names of functions which can be computed for this constitutive model (e.g mass, Von Mises stress etc) + + And contains methods to: + + - Given the coordinates, state value, state gradient, and design variables at a point, compute: + - The strain components + - The sensitivities of the strain components + - The stress components + - The sensitivities of the stress components + - The pointwise mass + - The volume integral scaling parameter (e.g thickness for 2D plane models or $2 \pi r$ for 2D axisymmetric problems) + - The weak form residual + - The weak form residual Jacobian + - Other arbitrary output values (e.g failure criterion) + """ + + def __init__(self, numDim, stateNames, strainNames, stressNames, designVars, functionNames, linear=True) -> None: + """_summary_ + + + + Parameters + ---------- + numDim : int + Number of spatial dimensions the model is valid for + stateNames : list of str + Names for each state variable + strainNames : list of str + Names for each strain component + stressNames : list of str + Names for each stress component + designVars : dict + A nested dictionary of design variables, with the key being the name of the design variable and the value + being a dictionary that contains various pieces of information about that DV, including: + - "defaultValue" : The default value of that DV + functionNames : list of str + The names of functions that can be computed with this constitutive model + linear : bool, optional + Whether the constitutive model is linear or not, a.k.a whether the weak residual is a linear function of the states/state gradients, by default True + """ + self.numDim = numDim + + self.stateNames = stateNames + self.numStates = len(stateNames) + + self.strainNames = strainNames + self.numStrains = len(strainNames) + if len(stressNames) != self.numStrains: + raise ValueError("Number of strains must equal number of stresses") + self.stressNames = stressNames + + self.designVariables = designVars + self.numDesignVariables = len(designVars) + + self.functionNames = functionNames + + self.lowerCaseFuncNames = [func.lower() for func in self.functionNames] + + self.isLinear = linear + + # ============================================================================== + # Abstract methods: To be implemented by derived classes + # ============================================================================== + @abc.abstractmethod + def computeStrains(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the strains at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + numPoints x numStrains array + Strain components at each point + """ + raise NotImplementedError + + @abc.abstractmethod + def computeStrainStateGradSens(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the + sensitivity of the strains to the state gradient at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + raise NotImplementedError + + @abc.abstractmethod + def computeStresses(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the stresses at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + numPoints x numStresses array + Stress components at each point + """ + raise NotImplementedError + + @abc.abstractmethod + def computeStressStrainSens(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the sensitivity of the stresses to the strains at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + sens : numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + raise NotImplementedError + + @abc.abstractmethod + def computeVolumeScaling(self, coords, dvs): + """Given the coordinates and design variables at a bunch of points, compute the volume scaling parameter at each one + + The volume scaling parameter is used to scale functions that are integrated over the element to get a true + volume integral. For example, in a 2D plane stress model, we need to multiply by the thickness of the element + to get a true volume integral. In a 2D axisymmetric model, we need to multiply by 2*pi*r to get a true volume + integral. + + Parameters + ---------- + coords : numPoints x numDim array + Coordinates of each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + numPoints length array + Volume scaling parameter at each point + """ + raise NotImplementedError + + @abc.abstractmethod + def getFunction(self, name): + """Return a function that can be computed for this constitutive model + + Parameters + ---------- + name : str + Name of the function to compute + + Returns + ------- + callable + A function that can be called to compute the desired function at a bunch of points with the signature, + `f(states, stateGradients, coords, dvs)`, where: + + - states is a numPoints x numStates array + - stateGradients is a numPoints x numStates x numDim array + - coords is a numPoints x numDim array + - dvs is a dictionary of numPoints length arrays + """ + + raise NotImplementedError + + # ============================================================================== + # Public methods + # ============================================================================== + def computeWeakResiduals(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the weak residual integrand + + For a solid mechanics problem, the weak residual, derived from the virtual work principle is: + + $R = \int r dV = \int (du'/dq)^T (d\epsilon/du')^T \sigma \\theta d\Omega$ + + Where: + + - $du'/dq$ is the sensitivity of the state gradient to the nodal state values, this is handled by the element + - $d\epsilon/du'$ is the sensitivity of the strain to the state gradient + - $\sigma$ are the stresses + - $\\theta$ is the volume scaling parameter + - $\Omega$ is the element + + This function computes $(de/du')^T * \sigma * \\theta$ at each point + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + residuals : numPoints x self.numDim x self.numStates array + Weak residual integrand at each point + """ + numPoints = coords.shape[0] + strain = self.computeStrains(states, stateGradients, coords, dvs) + stress = self.computeStresses(strain, dvs) + scale = self.computeVolumeScaling(coords, dvs) + + strainSens = self.computeStrainStateGradSens(states, stateGradients, coords, dvs) + + residuals = np.zeros((numPoints, self.numDim, self.numStates)) + _computeWeakResidualProduct(strainSens, stress, scale, residuals) + return residuals + + # return np.einsum("pesd,pe,p->pds", strainSens, stress, scale, optimize=["einsum_path", (0, 1), (0, 1)]) + + def computeWeakResidualJacobian(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the + weak residual jacobian integrand + + $j = (d\epsilon/du')^T \\times d\sigma/d\epsilon \\times d\epsilon/du'$ + + Where: + + - $d\epsilon/du'$ is the sensitivity of the strain to the state gradient + - $d\sigma/d\epsilon$ the sensitivity of the stress to the strain gradient + + This function computes `de/du'^T * sigma * scale` at each point + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : dictionary of len(numpoints) array + design variable values at each point + + Returns + ------- + Jacobians : numPoints x numDim x numStates x numStates x numDim array + The sensibility of the weak residual integrand components to the state gradients at each point + """ + strainSens = self.computeStrainStateGradSens(states, stateGradients, coords, dvs) + strain = self.computeStrains(states, stateGradients, coords, dvs) + stressSens = self.computeStressStrainSens(strain, dvs) + scale = self.computeVolumeScaling(coords, dvs) + numPoints = states.shape[0] + # strainSens = strainSens.reshape(numPoints, self.numStrains, self.numStates * self.numDim) + # Jacobian = _computeWeakJacobianProduct(strainSens, stressSens, scale) + # points = p + # strains = e + # stress = o + # states = s + # dim = d + # Jacobians = np.einsum( + # "posd,poe,peSD,p->pdsSD", + # strainSens, + # stressSens, + # strainSens, + # scale, + # optimize=["einsum_path", (1, 3), (0, 2), (0, 1)], + # ) + Jacobians = np.zeros((numPoints, self.numDim, self.numStates, self.numStates, self.numDim)) + _computeWeakJacobianProduct(strainSens, stressSens, scale, Jacobians) + + return Jacobians + + # ============================================================================== + # Private methods + # ============================================================================== + + +@njit(cache=True, fastmath=True, parallel=True) +def _computeWeakResidualProduct(strainSens, stress, scale, result): + """_summary_ + + Equivalent to :: + + np.einsum("pesd,pe,p->pds", strainSens, stress, scale, optimize=["einsum_path", (0, 1), (0, 1)]) + + Complete contraction: pesd,pe,p->pds + Naive scaling: 4 + Optimized scaling: 4 + Naive FLOP count: 1.844e+8 + Optimized FLOP count: 1.298e+8 + Theoretical speedup: 1.421e+0 + Largest intermediate: 1.025e+7 elements + -------------------------------------------------------------------------------- + scaling BLAS current remaining + -------------------------------------------------------------------------------- + 2 0 p,pe->pe pesd,pe->pds + 4 0 pe,pesd->pds pds->pds + + Parameters + ---------- + strainSens : _type_ + _description_ + stress : _type_ + _description_ + scale : _type_ + _description_ + result : _type_ + _description_ + """ + numPoints = strainSens.shape[0] + numStrains = strainSens.shape[1] + numStates = strainSens.shape[2] + numDim = strainSens.shape[3] + for p in prange(numPoints): + for d in range(numDim): + for s in range(numStates): + for e in range(numStrains): + result[p, d, s] += strainSens[p, e, s, d] * stress[p, e] + result[p] *= scale[p] + + +@njit(parallel=True, cache=True, fastmath=True) +def _computeWeakJacobianProduct(strainSens, stressSens, scale, jacs): + """Computes the following product which is necessary for compute the weak residual Jacobian: + + $$j = (d\epsilon/du')^T \\times d\sigma/d\epsilon \\times d\epsilon/du'$$ + + Equivalent to:: + + np.einsum( + "posd,poe,peSD,p->pdsSD", + strainSens, + stressSens, + strainSens, + scale, + optimize=["einsum_path", (1, 3), (0, 2), (0, 1)], + out = jacs + ) + + # points = p + # strains = e + # stress = o + # states = s + # dim = d + + Complete contraction: posd,poe,peSD,p->pdsSD + Naive scaling: 7 + Optimized scaling: 6 + Naive FLOP count: 1.866e+10 + Optimized FLOP count: 2.650e+9 + Theoretical speedup: 7.043e+0 + Largest intermediate: 1.296e+8 elements + -------------------------------------------------------------------------------- + scaling BLAS current remaining + -------------------------------------------------------------------------------- + 3 0 p,poe->poe posd,peSD,poe->pdsSD + 5 0 poe,posd->pesd peSD,pesd->pdsSD + 6 0 pesd,peSD->pdsSD pdsSD->pdsSD + + Parameters + ---------- + strainSens : _type_ + _description_ + stressSens : _type_ + _description_ + scale : _type_ + _description_ + jacs : _type_ + _description_ + """ + numPoints = jacs.shape[0] + numDim = jacs.shape[1] + numStrains = stressSens.shape[1] + numStates = jacs.shape[2] + + # poe,posd,p->pesd + intProd1 = np.zeros((numPoints, numStrains, numStates, numDim)) + for p in prange(numPoints): + for e in range(numStrains): + for s in range(numStates): + for d in range(numDim): + for o in range(numStrains): + intProd1[p, e, s, d] += stressSens[p, o, e] * strainSens[p, o, s, d] + intProd1[p] *= scale[p] + + # pesd,peSD->pdsSD + for p in prange(numPoints): + for d in range(numDim): + for s in range(numStates): + for S in range(numStates): + for D in range(numDim): + for e in range(numStrains): + jacs[p, d, s, S, D] += intProd1[p, e, s, d] * strainSens[p, e, S, D] diff --git a/FEMpy/Constitutive/Iso3D.py b/FEMpy/Constitutive/Iso3D.py new file mode 100644 index 0000000..4e1cb33 --- /dev/null +++ b/FEMpy/Constitutive/Iso3D.py @@ -0,0 +1,216 @@ +""" +============================================================================== +3D Isotropic elasticity constitutive model +============================================================================== +@File : Iso3D.py +@Date : 2022/12/11 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Constitutive.StrainModels import strain3D, strain3DSens +from FEMpy.Constitutive.StressModels import iso3DStress, iso3DStressStrainSens, vonMises3D +from FEMpy.Constitutive import ConstitutiveModel + + +class Iso3D(ConstitutiveModel): + """A constitutive model for a 3D isotropic material + + Inherits + ---------- + ConstitutiveModel : _type_ + The base class for FEMpy constitutive models + """ + + def __init__(self, E, nu, rho, linear=True): + """Create an isotropic plane stress constitutive model + + + + Parameters + ---------- + E : float + Elastic Modulus + nu : float + Poisson's ratio + rho : float + Density + linear : bool, optional + Whether to use the linear kinematic equations for strains, by default True + """ + # --- Design variables --- + # This model has no design variables + designVars = {} + + # --- States --- + stateNames = ["X-Displacement", "Y-Displacement", "Z-Displacement"] + + # --- Strains --- + strainNames = ["e_xx", "e_yy", "e_zz", "gamma_xy", "gamma_xz", "gamma_yz"] + + # --- Stresses --- + stressNames = ["sigma_xx", "sigma_yy", "sigma_zz", "tau_xy", "tau_xz", "tau_yz"] + + # --- Functions --- + functionNames = ["Mass", "Von-Mises-Stress"] + + # --- Material properties --- + self.E = E + self.nu = nu + self.rho = rho + + numDim = 3 + + super().__init__(numDim, stateNames, strainNames, stressNames, designVars, functionNames, linear) + + def computeStrains(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the strains at each one + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStrains array + Strain components at each point + """ + return strain3D(UPrime=stateGradients, nonlinear=not self.isLinear) + + def computeStrainStateGradSens(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the + sensitivity of the strains to the state gradient at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + return strain3DSens(UPrime=stateGradients, nonlinear=not self.isLinear) + + def computeStresses(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the stresses at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStresses array + Stress components at each point + """ + return iso3DStress(strains, E=self.E, nu=self.nu) + + def computeStressStrainSens(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the sensitivity of the stresses to the strains at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : _type_ + _description_ + + Returns + ------- + sens : numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + return iso3DStressStrainSens(strains, E=self.E, nu=self.nu) + + def computeVolumeScaling(self, coords, dvs): + """Given the coordinates and design variables at a bunch of points, compute the volume scaling parameter at each one + + For this 2D model, the volume scaling is just the thickness + + Parameters + ---------- + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints length array + Volume scaling parameter at each point + """ + return np.ones(coords.shape[0]) + + def getFunction(self, name): + """Return a function that can be computed for this constitutive model + + Parameters + ---------- + name : str + Name of the function to compute + + Returns + ------- + callable + A function that can be called to compute the desired function at a bunch of points with the signature, f(states, stateGradients, coords, dvs) + where: + states is a numPoints x numStates array + stateGradients is a numPoints x numStates x numDim array + coords is a numPoints x numDim array + dvs is a dictionary of numPoints length arrays + """ + lowerCaseFuncNames = [func.lower() for func in self.functionNames] + if name.lower() not in self.lowerCaseFuncNames: + raise ValueError( + f"{name} is not a valid function name for this constitutive model, valid choices are {self.functionNames}" + ) + + if name.lower() == "mass": + + def func(states, stateGradients, coords, dvs): + return np.ones(states.shape[0]) * self.rho + + if name.lower() == "von-mises-stress": + + def func(states, stateGradients, coords, dvs): + strains = self.computeStrains(states, stateGradients, coords, dvs) + stresses = self.computeStresses(strains, dvs) + return vonMises3D(stresses) + + return func diff --git a/FEMpy/Constitutive/IsoPlaneStrain.py b/FEMpy/Constitutive/IsoPlaneStrain.py new file mode 100644 index 0000000..2e02628 --- /dev/null +++ b/FEMpy/Constitutive/IsoPlaneStrain.py @@ -0,0 +1,223 @@ +""" +============================================================================== +2D Isotropic plane strain constitutive model +============================================================================== +@File : NewIsoPlaneStrain.py +@Date : 2022/11/30 +@Author : Alasdair Christison Gray and M.A. Saja A.Kaiyoom +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Constitutive.StrainModels import strain2D, strain2DSens +from FEMpy.Constitutive.StressModels import isoPlaneStrainStress, isoPlaneStrainStressStrainSens, vonMises2DPlaneStrain +from FEMpy.Constitutive import ConstitutiveModel + + +class IsoPlaneStrain(ConstitutiveModel): + """A constitutive model for the 2D isotropic plane strain equations + + + + Inherits + ---------- + ConstitutiveModel : _type_ + The base class for FEMpy constitutive models + """ + + def __init__(self, E, nu, rho, t, linear=True): + """Create an isotropic plane stress constitutive model + + + + Parameters + ---------- + E : float + Elastic Modulus + nu : float + Poisson's ratio + rho : float + Density + t : float + Thickness, this will be used as the initial thickness value for all elements but can be changed later by calling setDesignVariables in the model + linear : bool, optional + Whether to use the linear kinematic equations for strains, by default True + """ + # --- Design variables --- + # This model has only one design variable, the thickness of the material + designVars = {} + designVars["Thickness"] = {"defaultValue": t} + + # --- States --- + stateNames = ["X-Displacement", "Y-Displacement"] + + # --- Strains --- + strainNames = ["e_xx", "e_yy", "gamma_xy"] + + # --- Stresses --- + stressNames = ["sigma_xx", "sigma_yy", "tau_xy"] + + # --- Functions --- + functionNames = ["Mass", "Von-Mises-Stress"] + + # --- Material properties --- + self.E = E + self.nu = nu + self.rho = rho + + numDim = 2 + + super().__init__(numDim, stateNames, strainNames, stressNames, designVars, functionNames, linear) + + def computeStrains(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the strains at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStrains array + Strain components at each point + """ + return strain2D(UPrime=stateGradients, nonlinear=not self.isLinear) + + def computeStrainStateGradSens(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the + sensitivity of the strains to the state gradient at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + return strain2DSens(UPrime=stateGradients, nonlinear=not self.isLinear) + + def computeStresses(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the stresses at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStresses array + Stress components at each point + """ + return isoPlaneStrainStress(strains, E=self.E, nu=self.nu) + + def computeStressStrainSens(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the sensitivity of the stresses to the strains at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : _type_ + _description_ + + Returns + ------- + sens : numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + return isoPlaneStrainStressStrainSens(strains, E=self.E, nu=self.nu) + + def computeVolumeScaling(self, coords, dvs): + """Given the coordinates and design variables at a bunch of points, compute the volume scaling parameter at each one + + For this 2D model, the volume scaling is just the thickness + + Parameters + ---------- + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints length array + Volume scaling parameter at each point + """ + return dvs["Thickness"] + + def getFunction(self, name): + """Return a function that can be computed for this constitutive model + + Parameters + ---------- + name : str + Name of the function to compute + + Returns + ------- + callable + A function that can be called to compute the desired function at a bunch of points with the signature, f(states, stateGradients, coords, dvs) + where: + states is a numPoints x numStates array + stateGradients is a numPoints x numStates x numDim array + coords is a numPoints x numDim array + dvs is a dictionary of numPoints length arrays + """ + lowerCaseFuncNames = [func.lower() for func in self.functionNames] + if name.lower() not in self.lowerCaseFuncNames: + raise ValueError( + f"{name} is not a valid function name for this constitutive model, valid choices are {self.functionNames}" + ) + + if name.lower() == "mass": + + def func(states, stateGradients, coords, dvs): + return np.ones(states.shape[0]) * self.rho + + if name.lower() == "von-mises-stress": + + def func(states, stateGradients, coords, dvs): + strains = self.computeStrains(states, stateGradients, coords, dvs) + stresses = self.computeStresses(strains, dvs) + return vonMises2DPlaneStrain(stresses, self.nu) + + return func diff --git a/FEMpy/Constitutive/IsoPlaneStress.py b/FEMpy/Constitutive/IsoPlaneStress.py new file mode 100644 index 0000000..3db0bc0 --- /dev/null +++ b/FEMpy/Constitutive/IsoPlaneStress.py @@ -0,0 +1,223 @@ +""" +============================================================================== +2D Isotropic plane stress constitutive model +============================================================================== +@File : NewIsoPlaneStress.py +@Date : 2022/11/30 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Constitutive.StrainModels import strain2D, strain2DSens +from FEMpy.Constitutive.StressModels import isoPlaneStressStress, isoPlaneStressStressStrainSens, vonMises2DPlaneStress +from FEMpy.Constitutive import ConstitutiveModel + + +class IsoPlaneStress(ConstitutiveModel): + """A constitutive model for the 2D isotropic plane stress equations + + + + Inherits + ---------- + ConstitutiveModel : + The base class for FEMpy constitutive models + """ + + def __init__(self, E, nu, rho, t, linear=True): + """Create an isotropic plane stress constitutive model + + + + Parameters + ---------- + E : float + Elastic Modulus + nu : float + Poisson's ratio + rho : float + Density + t : float + Thickness, this will be used as the initial thickness value for all elements but can be changed later by calling setDesignVariables in the model + linear : bool, optional + Whether to use the linear kinematic equations for strains, by default True + """ + # --- Design variables --- + # This model has only one design variable, the thickness of the material + designVars = {} + designVars["Thickness"] = {"defaultValue": t} + + # --- States --- + stateNames = ["X-Displacement", "Y-Displacement"] + + # --- Strains --- + strainNames = ["e_xx", "e_yy", "gamma_xy"] + + # --- Stresses --- + stressNames = ["sigma_xx", "sigma_yy", "tau_xy"] + + # --- Functions --- + functionNames = ["Mass", "Von-Mises-Stress"] + + # --- Material properties --- + self.E = E + self.nu = nu + self.rho = rho + + numDim = 2 + + super().__init__(numDim, stateNames, strainNames, stressNames, designVars, functionNames, linear) + + def computeStrains(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the strains at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStrains array + Strain components at each point + """ + return strain2D(UPrime=stateGradients, nonlinear=not self.isLinear) + + def computeStrainStateGradSens(self, states, stateGradients, coords, dvs): + """Given the coordinates, state value, state gradient, and design variables at a bunch of points, compute the + sensitivity of the strains to the state gradient at each one + + + + Parameters + ---------- + states : numPoints x numStates array + State values at each point + stateGradients : numPoints x numStates x numDim array + State gradients at each point + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + return strain2DSens(UPrime=stateGradients, nonlinear=not self.isLinear) + + def computeStresses(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the stresses at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints x numStresses array + Stress components at each point + """ + return isoPlaneStressStress(strains, E=self.E, nu=self.nu) + + def computeStressStrainSens(self, strains, dvs): + """Given the strains and design variables at a bunch of points, compute the sensitivity of the stresses to the strains at each one + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strain components at each point + dvs : _type_ + _description_ + + Returns + ------- + sens : numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + return isoPlaneStressStressStrainSens(strains, E=self.E, nu=self.nu) + + def computeVolumeScaling(self, coords, dvs): + """Given the coordinates and design variables at a bunch of points, compute the volume scaling parameter at each one + + For this 2D model, the volume scaling is just the thickness + + Parameters + ---------- + coords : numPoints x numDim array + Coordinates of each point + dvs : _type_ + _description_ + + Returns + ------- + numPoints length array + Volume scaling parameter at each point + """ + return dvs["Thickness"] + + def getFunction(self, name): + """Return a function that can be computed for this constitutive model + + Parameters + ---------- + name : str + Name of the function to compute + + Returns + ------- + callable + A function that can be called to compute the desired function at a bunch of points with the signature, f(states, stateGradients, coords, dvs) + where: + states is a numPoints x numStates array + stateGradients is a numPoints x numStates x numDim array + coords is a numPoints x numDim array + dvs is a dictionary of numPoints length arrays + """ + + if name.lower() not in self.lowerCaseFuncNames: + raise ValueError( + f"{name} is not a valid function name for this constitutive model, valid choices are {self.functionNames}" + ) + + if name.lower() == "mass": + + def func(states, stateGradients, coords, dvs): + return np.ones(states.shape[0]) * self.rho + + if name.lower() == "von-mises-stress": + + def func(states, stateGradients, coords, dvs): + strains = self.computeStrains(states, stateGradients, coords, dvs) + stresses = self.computeStresses(strains, dvs) + return vonMises2DPlaneStress(stresses) + + return func diff --git a/FEMpy/Constitutive/StrainModels/Strain2D.py b/FEMpy/Constitutive/StrainModels/Strain2D.py new file mode 100644 index 0000000..629cc7a --- /dev/null +++ b/FEMpy/Constitutive/StrainModels/Strain2D.py @@ -0,0 +1,110 @@ +""" +============================================================================== +Functions for computing strains in 2D +============================================================================== +@File : Strain2D.py +@Date : 2022/11/30 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from numba import njit + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True, fastmath=True, parallel=True) +def strain2D(UPrime, nonlinear=False): + """Compute 2D strains from the displacement gradient + + Parameters + ---------- + UPrime : numPoints x 2 x 2 array + Displacement gradients at each point + nonlinear : bool, optional + Whether to compute the nonlinear Greene strain, by default False + + Returns + ------- + strains : numPoints x 3 array + Strains at each point, these are the engineering strains [e_xx, e_yy, gamma_xy] + """ + numPoints = UPrime.shape[0] + strains = np.zeros((numPoints, 3), dtype=UPrime.dtype) + + # e_xx = du_x/dx + strains[:, 0] = UPrime[:, 0, 0] + + # e_yy = du_y/dy + strains[:, 1] = UPrime[:, 1, 1] + + # gamma_xy = du_x/dy + du_y/dx + strains[:, 2] = UPrime[:, 0, 1] + UPrime[:, 1, 0] + + if nonlinear: + # e_xx = du_x/dx + 0.5 * (du_x/dx^2 + du_y/dx^2) + strains[:, 0] += 0.5 * (UPrime[:, 0, 0] ** 2 + UPrime[:, 1, 0] ** 2) + + # e_yy = du_y/dy + 0.5 * (du_x/dy^2 + du_y/dy^2) + strains[:, 1] += 0.5 * (UPrime[:, 0, 1] ** 2 + UPrime[:, 1, 1] ** 2) + + # gamma_xy = du_x/dy + du_y/dx + (du_x/dx * du_x/dy + du_y/dx * du_y/dy) + strains[:, 2] += UPrime[:, 0, 0] * UPrime[:, 0, 1] + UPrime[:, 1, 0] * UPrime[:, 1, 1] + + return strains + + +@njit(cache=True, fastmath=True, parallel=True) +def strain2DSens(UPrime, nonlinear=False): + """Compute the sensitivity of the 2D strains to the displacement gradients + + Parameters + ---------- + UPrime : numPoints x 2 x 2 array + Displacement gradients at each point + nonlinear : bool, optional + Whether to use the nonlinear Greene strain, by default False + + Returns + ------- + strainSens : numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + numPoints = UPrime.shape[0] + strainSens = np.zeros((numPoints, 3, 2, 2), dtype=UPrime.dtype) + + # e_xx = du_x/dx + strainSens[:, 0, 0, 0] = 1.0 + # e_yy = du_y/dy + strainSens[:, 1, 1, 1] = 1.0 + # gamma_xy = du_x/dy + du_y/dx + strainSens[:, 2, 0, 1] = 1.0 + strainSens[:, 2, 1, 0] = 1.0 + + if nonlinear: + # e_xx = du_x/dx + 0.5 * (du_x/dx^2 + du_y/dx^2) + strainSens[:, 0, 0, 0] += UPrime[:, 0, 0] + strainSens[:, 0, 1, 0] += UPrime[:, 1, 0] + + # e_yy = du_y/dy + 0.5 * (du_x/dy^2 + du_y/dy^2) + strainSens[:, 1, 0, 1] += UPrime[:, 0, 1] + strainSens[:, 1, 1, 1] += UPrime[:, 1, 1] + + # gamma_xy = du_x/dy + du_y/dx + (du_x/dx * du_x/dy + du_y/dx * du_y/dy) + strainSens[:, 2, 0, 0] += UPrime[:, 0, 1] + strainSens[:, 2, 0, 1] += UPrime[:, 0, 0] + + strainSens[:, 2, 1, 0] += UPrime[:, 1, 1] + strainSens[:, 2, 1, 1] += UPrime[:, 1, 0] + + return strainSens diff --git a/FEMpy/Constitutive/StrainModels/Strain3D.py b/FEMpy/Constitutive/StrainModels/Strain3D.py new file mode 100644 index 0000000..33758ea --- /dev/null +++ b/FEMpy/Constitutive/StrainModels/Strain3D.py @@ -0,0 +1,174 @@ +""" +============================================================================== +Functions for computing strains in 3D +============================================================================== +@File : Strain3D.py +@Date : 2022/11/30 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from numba import njit + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True, fastmath=True, parallel=True) +def strain3D(UPrime, nonlinear=False): + """Compute 3D strains from the displacement gradient + + Parameters + ---------- + UPrime : numPoints x 3 x 3 array + Displacement gradients at each point + nonlinear : bool, optional + Whether to compute the nonlinear strain, by default False + + Returns + ------- + strains : numPoints x 6 array + Strains at each point, these are the engineering strains [e_xx, e_yy, e_zz, gamma_xy, gamma_xz, gamma_yz] + """ + numPoints = UPrime.shape[0] + strains = np.zeros((numPoints, 6), dtype=UPrime.dtype) + + # e_xx = du_x/dx + strains[:, 0] = UPrime[:, 0, 0] + + # e_yy = du_y/dy + strains[:, 1] = UPrime[:, 1, 1] + + # e_zz = du_z/dz + strains[:, 2] = UPrime[:, 2, 2] + + # gamma_xy = du_x/dy + du_y/dx + strains[:, 3] = UPrime[:, 0, 1] + UPrime[:, 1, 0] + + # gamma_xz = du_x/dz + du_z/dx + strains[:, 4] = UPrime[:, 0, 2] + UPrime[:, 2, 0] + + # gamma_yz = du_y/dz + du_z/dy + strains[:, 5] = UPrime[:, 1, 2] + UPrime[:, 2, 1] + + if nonlinear: + # e_xx = du_x/dx + 0.5 * (du_x/dx^2 + du_y/dx^2+ du_z/dx^2) + strains[:, 0] += 0.5 * (UPrime[:, 0, 0] ** 2 + UPrime[:, 1, 0] ** 2 + UPrime[:, 2, 0] ** 2) + + # e_yy = du_y/dy + 0.5 * (du_x/dy^2 + du_y/dy^2 + du_z/dy^2) + strains[:, 1] += 0.5 * (UPrime[:, 0, 1] ** 2 + UPrime[:, 1, 1] ** 2 + UPrime[:, 2, 1] ** 2) + + # e_zz = du_z/dz + 0.5 * (du_x/dz^2 + du_y/dz^2 + du_z/dz^2) + strains[:, 2] += 0.5 * (UPrime[:, 0, 2] ** 2 + UPrime[:, 1, 2] ** 2 + UPrime[:, 2, 2] ** 2) + + # gamma_xy = du_x/dy + du_y/dx + (du_x/dx * du_x/dy + du_y/dx * du_y/dy + du_z/dx * du_z/dy) + strains[:, 3] += ( + UPrime[:, 0, 0] * UPrime[:, 0, 1] + UPrime[:, 1, 0] * UPrime[:, 1, 1] + UPrime[:, 2, 0] * UPrime[:, 2, 1] + ) + + # gamma_xz = du_x/dz + du_z/dx + (du_x/dx * du_x/dz + du_y/dx * du_y/dz + du_z/dx * du_z/dz) + strains[:, 4] += ( + UPrime[:, 0, 0] * UPrime[:, 0, 2] + UPrime[:, 1, 0] * UPrime[:, 1, 2] + UPrime[:, 2, 0] * UPrime[:, 2, 2] + ) + + # gamma_yz = du_y/dz + du_z/dy + (du_x/dy * du_x/dz + du_y/dy * du_y/dz + du_z/dy * du_z/dz) + strains[:, 5] += ( + UPrime[:, 0, 1] * UPrime[:, 0, 2] + UPrime[:, 1, 1] * UPrime[:, 1, 2] + UPrime[:, 2, 1] * UPrime[:, 2, 2] + ) + + return strains + + +@njit(cache=True, fastmath=True, parallel=True) +def strain3DSens(UPrime, nonlinear=False): + """Compute the sensitivity of the 3D strains to the displacement gradients + + Parameters + ---------- + UPrime : numPoints x 3 x 3 array + Displacement gradients at each point + nonlinear : bool, optional + Whether to use the nonlinear Greene strain, by default False + + Returns + ------- + strainSens : numPoints x numStrains x numStates x numDim array + Strain sensitivities, sens[i,j,k,l] is the sensitivity of strain component j at point i to state gradient du_k/dx_l + """ + numPoints = UPrime.shape[0] + strainSens = np.zeros((numPoints, 6, 3, 3), dtype=UPrime.dtype) + + # e_xx = du_x/dx + strainSens[:, 0, 0, 0] = 1.0 + + # e_yy = du_y/dy + strainSens[:, 1, 1, 1] = 1.0 + + # e_zz = du_z/dz + strainSens[:, 2, 2, 2] = 1.0 + + # gamma_xy = du_x/dy + du_y/dx + strainSens[:, 3, 0, 1] = 1.0 + strainSens[:, 3, 1, 0] = 1.0 + + # gamma_xz = du_x/dz + du_z/dx + strainSens[:, 4, 0, 2] = 1.0 + strainSens[:, 4, 2, 0] = 1.0 + + # gamma_yz = du_y/dz + du_z/dy + strainSens[:, 5, 1, 2] = 1.0 + strainSens[:, 5, 2, 1] = 1.0 + + if nonlinear: + # e_xx = du_x/dx + 0.5 * (du_x/dx^2 + du_y/dx^2+ du_z/dx^2) + strainSens[:, 0, 0, 0] += UPrime[:, 0, 0] + strainSens[:, 0, 1, 0] += UPrime[:, 1, 0] + strainSens[:, 0, 2, 0] += UPrime[:, 2, 0] + + # e_yy = du_y/dy + 0.5 * (du_x/dy^2 + du_y/dy^2 + du_z/dy^2) + strainSens[:, 1, 0, 1] += UPrime[:, 0, 1] + strainSens[:, 1, 1, 1] += UPrime[:, 1, 1] + strainSens[:, 1, 2, 1] += UPrime[:, 2, 1] + + # e_zz = du_z/dz + 0.5 * (du_x/dz^2 + du_y/dz^2 + du_z/dz^2) + strainSens[:, 2, 0, 2] += UPrime[:, 0, 2] + strainSens[:, 2, 1, 2] += UPrime[:, 1, 2] + strainSens[:, 2, 2, 2] += UPrime[:, 2, 2] + + # gamma_xy = du_x/dy + du_y/dx + (du_x/dx * du_x/dy + du_y/dx * du_y/dy + du_z/dx * du_z/dy) + strainSens[:, 3, 0, 0] += UPrime[:, 0, 1] + strainSens[:, 3, 1, 0] += UPrime[:, 1, 1] + strainSens[:, 3, 2, 0] += UPrime[:, 2, 1] + + strainSens[:, 3, 0, 1] += UPrime[:, 0, 0] + strainSens[:, 3, 1, 1] += UPrime[:, 1, 0] + strainSens[:, 3, 2, 1] += UPrime[:, 2, 0] + + # gamma_xz = du_x/dz + du_z/dx + (du_x/dx * du_x/dz + du_y/dx * du_y/dz + du_z/dx * du_z/dz) + strainSens[:, 4, 0, 0] += UPrime[:, 0, 2] + strainSens[:, 4, 1, 0] += UPrime[:, 1, 2] + strainSens[:, 4, 2, 0] += UPrime[:, 2, 2] + + strainSens[:, 4, 0, 2] += UPrime[:, 0, 0] + strainSens[:, 4, 1, 2] += UPrime[:, 1, 0] + strainSens[:, 4, 2, 2] += UPrime[:, 2, 0] + + # gamma_yz = du_y/dz + du_z/dy + (du_x/dy * du_x/dz + du_y/dy * du_y/dz + du_z/dy * du_z/dz) + strainSens[:, 5, 0, 1] += UPrime[:, 0, 2] + strainSens[:, 5, 1, 1] += UPrime[:, 1, 2] + strainSens[:, 5, 2, 1] += UPrime[:, 2, 2] + + strainSens[:, 5, 0, 2] += UPrime[:, 0, 1] + strainSens[:, 5, 1, 2] += UPrime[:, 1, 1] + strainSens[:, 5, 2, 2] += UPrime[:, 2, 1] + + return strainSens diff --git a/FEMpy/Constitutive/StrainModels/__init__.py b/FEMpy/Constitutive/StrainModels/__init__.py new file mode 100644 index 0000000..0936654 --- /dev/null +++ b/FEMpy/Constitutive/StrainModels/__init__.py @@ -0,0 +1,2 @@ +from .Strain2D import * +from .Strain3D import * diff --git a/FEMpy/Constitutive/StressModels/IsoStress.py b/FEMpy/Constitutive/StressModels/IsoStress.py new file mode 100644 index 0000000..5de09b9 --- /dev/null +++ b/FEMpy/Constitutive/StressModels/IsoStress.py @@ -0,0 +1,256 @@ +""" +============================================================================== +Stress equations for isotropic materials +============================================================================== +@File : IsoStress.py +@Date : 2022/11/30 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== + + +def planeStressMat(E, nu): + """Get constitutive (stress-strain) matrix for plane stress + + Parameters + ---------- + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + 3x3 array + Constitutive matrix for plane stress + """ + return ( + E + / (1.0 - nu**2) + * np.array( + [ + [1.0, nu, 0.0], + [nu, 1.0, 0.0], + [0.0, 0.0, (1.0 - nu) / 2.0], + ] + ) + ) + + +def planeStrainMat(E, nu): + """Get constitutive (stress-strain) matrix for plane stress + + Parameters + ---------- + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + 3x3 array + Constitutive matrix for plane stress + """ + return ( + E + / ((1.0 + nu) * (1.0 - 2.0 * nu)) + * np.array( + [ + [1.0 - nu, nu, 0.0], + [nu, 1.0 - nu, 0.0], + [0.0, 0.0, (1.0 - 2.0 * nu) / 2.0], + ] + ) + ) + + +def threeDMat(E, nu): + """Get constitutive (stress-strain) matrix for a 3D stress state + + Parameters + ---------- + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + 3x3 array + Constitutive matrix for plane stress + """ + return ( + E + / ((1.0 + nu) * (1.0 - 2.0 * nu)) + * np.array( + [ + [1.0 - nu, nu, nu, 0.0, 0.0, 0.0], + [nu, 1.0 - nu, nu, 0.0, 0.0, 0.0], + [nu, nu, 1.0 - nu, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, (1.0 - 2.0 * nu) / 2.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, (1.0 - 2.0 * nu) / 2.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, (1.0 - 2.0 * nu) / 2.0], + ] + ) + ) + + +# ============================================================================== +# 2D plane stress +# ============================================================================== +def isoPlaneStressStress(strains, E, nu): + """Compute stress from strains for a 2D plane stress state + + Parameters + ---------- + strains : numPoints x numStrains array + Strains at each point (e_xx, e_yy, gamma_xy) + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + stresses : numPoints x numStresses array + Stresses at each point (sigma_xx, sigma_yy, tau_xy) + """ + + mat = planeStressMat(E, nu) + stress = np.einsum("as,pa->ps", mat, strains) + return stress + + +def isoPlaneStressStressStrainSens(strains, E, nu): + """Compute the sensitivity of the stress with respect to the strain for a 2D plane stress state + + Parameters + ---------- + strains : numPoints x numStrains array + Strains at each point (e_xx, e_yy, gamma_xy) + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + numPoints x numStresses x numStrains array + Sensitivity of each stress to each strain at each point + """ + mat = planeStressMat(E, nu) + strainSens = np.tile(mat, (strains.shape[0], 1, 1)) + return strainSens + + +# ============================================================================== +# 2D plane strain +# ============================================================================== + + +def isoPlaneStrainStress(strains, E, nu): + """Compute stress from strains for a 2D plane strain state + + Parameters + ---------- + strains : numPoints x numStrains array + Strains at each point (e_xx, e_yy, gamma_xy) + E : _type_ + _description_ + nu : _type_ + _description_ + + Returns + ------- + stresses : numPoints x numStresses array + Stresses at each point (sigma_xx, sigma_yy, tau_xy) + """ + + mat = planeStrainMat(E, nu) + stress = np.einsum("as,pa->ps", mat, strains) + return stress + + +def isoPlaneStrainStressStrainSens(strains, E, nu): + """Compute the sensitivity of the stress with respect to the strain + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strains at each point (e_xx, e_yy, gamma_xy) + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + numPoints x numStresses x numStrains array + Sensitivity of each stress to each strain at each point + """ + mat = planeStrainMat(E, nu) + strainSens = np.tile(mat, (strains.shape[0], 1, 1)) + return strainSens + + +# ============================================================================== +# 3D +# ============================================================================== + + +def iso3DStress(strains, E, nu): + """Compute the 3D stress state in a 3D isotropic material from strains + + Parameters + ---------- + strains : numPointsx numStrains array + Strains at each point (e_xx, e_yy, e_zz, gamma_xy, gamm_xz, gamma_yz) + E : float + Elastic modulus + nu : float + Poisson's ratio + """ + + mat = threeDMat(E, nu) + stress_3D = np.einsum("as,pa->ps", mat, strains) + return stress_3D + + +def iso3DStressStrainSens(strains, E, nu): + """Compute the sensitivity of the stress with respect to the strain + + + + Parameters + ---------- + strains : numPoints x numStrains array + Strains at each point (e_xx, e_yy, e_zz, gamma_xy, gamm_xz, gamma_yz) + E : float + Elastic modulus + nu : float + Poisson's ratio + + Returns + ------- + numPoints x numStresses x numStrains array + Sensitivity of each stress to each strain at each point + """ + mat = threeDMat(E, nu) + strainSens = np.tile(mat, (strains.shape[0], 1, 1)) + return strainSens diff --git a/FEMpy/Constitutive/StressModels/VonMises.py b/FEMpy/Constitutive/StressModels/VonMises.py new file mode 100644 index 0000000..900a89e --- /dev/null +++ b/FEMpy/Constitutive/StressModels/VonMises.py @@ -0,0 +1,91 @@ +""" +============================================================================== +Functions for computing the von Mises stress +============================================================================== +@File : VonMises.py +@Date : 2022/11/30 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from numba import njit + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True) +def vonMises2DPlaneStress(stresses): + """Compute the von Mises stress for a 2d plane stress state + + Parameters + ---------- + stresses : numPoints x 3 array + Stresses in the form [sigma_xx, sigma_yy, tau_xy] + + Returns + ------- + array of length numPoints + Von Mises stress at each point + """ + s11 = stresses[:, 0] + s22 = stresses[:, 1] + s12 = stresses[:, 2] + return np.sqrt(s11**2 - s11 * s22 + s22**2 + 3 * s12**2) + + +@njit(cache=True) +def vonMises2DPlaneStrain(stresses, nu): + """Compute the von Mises stress for a 2d plane strain state + + Parameters + ---------- + stresses : numPoints x 3 array + Stresses in the form [sigma_xx, sigma_yy, tau_xy] + nu : float + Poisson's ratio + + Returns + ------- + array of length numPoints + Von Mises stress at each point + """ + s11 = stresses[:, 0] + s22 = stresses[:, 1] + s12 = stresses[:, 2] + s33 = nu * (s11 + s22) + return np.sqrt(0.5 * ((s11 - s22) ** 2 + (s22 - s33) ** 2 + (s33 - s11) ** 2) + 3 * s12**2) + + +@njit(cache=True) +def vonMises3D(stresses): + """Compute the von Mises stress for a 3d stress state + + Parameters + ---------- + stresses : numPoints x 6 array + Stresses in the form [sigma_xx, sigma_yy, sigma_zz, tau_xy, tau_xz, tau_yz] + + Returns + ------- + array of length numPoints + Von Mises stress at each point + """ + s11 = stresses[:, 0] + s22 = stresses[:, 1] + s33 = stresses[:, 2] + s12 = stresses[:, 3] + s13 = stresses[:, 4] + s23 = stresses[:, 5] + return np.sqrt( + 0.5 * ((s11 - s22) ** 2 + (s22 - s33) ** 2 + (s33 - s11) ** 2) + 3 * (s12**2 + s13**2 + s23**2) + ) diff --git a/FEMpy/Constitutive/StressModels/__init__.py b/FEMpy/Constitutive/StressModels/__init__.py new file mode 100644 index 0000000..9f5e7b4 --- /dev/null +++ b/FEMpy/Constitutive/StressModels/__init__.py @@ -0,0 +1,9 @@ +from .IsoStress import ( + isoPlaneStressStress, + isoPlaneStressStressStrainSens, + isoPlaneStrainStress, + isoPlaneStrainStressStrainSens, + iso3DStress, + iso3DStressStrainSens, +) +from .VonMises import * diff --git a/FEMpy/Constitutive/__init__.py b/FEMpy/Constitutive/__init__.py new file mode 100644 index 0000000..8be949e --- /dev/null +++ b/FEMpy/Constitutive/__init__.py @@ -0,0 +1,6 @@ +from .ConstitutiveModel import ConstitutiveModel +from .IsoPlaneStress import IsoPlaneStress +from .IsoPlaneStrain import IsoPlaneStrain +from .Iso3D import Iso3D +from . import StrainModels +from . import StressModels diff --git a/FEMpy/Dynamics.py b/FEMpy/Dynamics.py deleted file mode 100644 index 44a6ac7..0000000 --- a/FEMpy/Dynamics.py +++ /dev/null @@ -1,103 +0,0 @@ -""" -============================================================================== -Dynamic Analysis Functions -============================================================================== -@File : Dynamics.py -@Date : 2021/03/31 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from scipy.sparse.linalg import factorized - -# ============================================================================== -# Extension modules -# ============================================================================== - - -def NewmarkExplicit(MMat, KMat, Force, tStep, tf, t0=0.0, u0=None, uDot0=None): - """Perform a transient analysis using the explicit central difference version of the Newmark-beta method - - At each timestep: - u_n+1 = u_n + dt*uDot_n + 1/2*uDDot_n*dt**2 - uDDot_n+1 = M^-1(f_n+1 - K*u_n+1) - uDot_n+1 = uDot_n + dt/2*(uDDot_n + uDDot_n+1) - - Stability condition is w*dt<=2, where w is the highest natural frequency of the structure.py - - For more details see `the wikipedia page for the method. `_ - - Parameters - ---------- - MMat : numpy array or Scipy sparse matrix - Mass matrix - KMat : numpy array or Scipy sparse matrix - Stiffness matrix - Force : Function or vector - Force vector, can provide a vector if force is constant or a function which takes a single time argument and - returns a force array for a time varying force - tStep : float - Timestep size - tf : float - Final analysis time - t0 : float, optional - Initial analysis time, by default 0. - u0 : array, optional - Inital states, by default zero - uDot0 : array, optional - Initial velocities, by default zero - - Returns - ------- - u : nState x nTimestep array - State history - uDot : nState x nTimestep array - Velocity history - uDDot : nState x nTimestep array - Acceleration history - t : array of length nTimestep - Time history - """ - # --- Convert force to function of time if it isn't --- - ForceFunc = (lambda t: Force) if not callable(Force) else Force - # --- Generate time steps --- - t = np.concatenate((np.arange(t0, tf, tStep), np.array([tf]))) - dt = t[1:] - t[:-1] - - # --- Generate history matrices --- - F0 = ForceFunc(t0) - numStates = np.shape(F0)[0] - u = np.zeros((numStates, len(t))) - u[:, 0] = 0.0 if u0 is None else u0 - uDot = np.zeros((numStates, len(t))) - uDot[:, 0] = 0.0 if uDot0 is None else uDot0 - uDDot = np.zeros((numStates, len(t))) - - # --- Factorise the mass matrix --- - MSolve = factorized(MMat) - - # --- Compute the initial acceleration --- - Res = F0 - KMat @ u[:, 0] - uDDot[:, 0] = MSolve(Res) - print( - f"Iteration 0, t = 0, max U = {np.linalg.norm(u[:, 0], ord = np.inf):.04e}, max UDot = {np.linalg.norm(uDot[:, 0], ord = np.inf):.04e}, max UDDot = {np.linalg.norm(uDDot[:, 0], ord = np.inf):.04e}" - ) - - for i in range(len(dt)): - u[:, i + 1] = u[:, i] + dt[i] * uDot[:, i] + 0.5 * dt[i] ** 2 * uDDot[:, i] - Res[:] = ForceFunc(t[i + 1]) - KMat @ u[:, i + 1] - uDDot[:, i + 1] = MSolve(Res) - uDot[:, i + 1] = uDot[:, i] + 0.5 * dt[i] * (uDDot[:, i + 1] + uDDot[:, i]) - print( - f"Iteration {i+1}, t = {t[i + 1]:.03e}, max U = {np.linalg.norm(u[:, i+1], ord = np.inf):.04e}, max UDot = {np.linalg.norm(uDot[:, i+1], ord = np.inf):.04e}, max UDDot = {np.linalg.norm(uDDot[:, i+1], ord = np.inf):.04e}" - ) - - return u, uDot, uDDot, t diff --git a/FEMpy/Element.py b/FEMpy/Element.py deleted file mode 100644 index fa26807..0000000 --- a/FEMpy/Element.py +++ /dev/null @@ -1,492 +0,0 @@ -""" -============================================================================== -Element Class -============================================================================== -@File : Element.py -@Date : 2021/03/11 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== -import abc - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from numba import njit -from scipy.optimize import root - -# ============================================================================== -# Extension modules -# ============================================================================== -from .GaussQuad import gaussQuad1d, gaussQuad2d, gaussQuad3d -from .LinAlg import det1, det2, det3, inv1, inv2, inv3 - - -@njit(cache=True) -def _makeBMat(NPrime, LMats, numStrain, numDim, numNodes): - numPoints = np.shape(NPrime)[0] - BMat = np.zeros((numPoints, numStrain, numDim * numNodes)) - for p in range(numPoints): - for n in range(numNodes): - for d in range(numDim): - BMat[p, :, n * numDim : (n + 1) * numDim] += LMats[d] * NPrime[p, d, n] - return BMat - - -@njit(cache=True) -def _makeNMat(N, numDim): - s = np.shape(N) - numPoints = s[0] - numShapeFunc = s[1] - NMat = np.zeros((numPoints, numDim, numDim * numShapeFunc)) - for p in range(numPoints): - for n in range(numShapeFunc): - NMat[p, :, numDim * n : numDim * (n + 1)] = N[p, n] * np.eye(numDim) - return NMat - - -@njit(cache=True) -def _computeNTFProduct(F, N): - # Compute N^T fb at each point, it's complicated because things are not the right shape - nP = np.shape(F)[0] - nD = np.shape(F)[1] - nN = np.shape(N)[1] - Fb = np.zeros((nP, nN, nD)) - for p in range(nP): - for d in range(nD): - Fb[p, :, d] = (F[p, d] * N[p]).T - return Fb - - -class Element: - def __init__(self, numNodes, numDimensions, numDisplacements=None): - """Instantiate an Element object - - Parameters - ---------- - numNodes : int - Number of nodes in each element - numDimensions : int - Number of spatial dimensions the element models - numDisplacements : int, optional - Number of displacements at each node, by default this is set equal to the number of spatial dimensions - """ - self.numNodes = numNodes - self.numDim = numDimensions - self.numDisp = numDimensions if numDisplacements is None else numDisplacements - self.numDOF = numNodes * self.numDisp - self.name = f"{self.numNodes}Node-{self.numDisp}Disp-{self.numDim}D-Element" - - # --- Define fast jacobian determinant function based on number of dimensions --- - if self.numDim == 1: - self.jacDet = det1 - self.jacInv = inv1 - elif self.numDim == 2: - self.jacDet = det2 - self.jacInv = inv2 - elif self.numDim == 3: - self.jacDet = det3 - self.jacInv = inv3 - - def getRealCoord(self, paramCoords, nodeCoords): - """Compute the real coordinates of a point in isoparametric space - - - Parameters - ---------- - paramCoords : n x numDim array - isoparametric coordinates, one row for each point in isoparametric space to be converted - nodeCoords : numNode x numDim array - Element node real coordinates - - Returns - ------- - coords : n x nD array - Point coordinates in real space - """ - N = self.getShapeFunctions(paramCoords) - - return N[:, : self.numNodes] @ nodeCoords - - def getParamCoord(self, realCoords, nodeCoords, maxIter=10, tol=1e-8): - """Find the parametric coordinates within an element corresponding to a point in real space - - Note this function only currently works for finding the parametric coordinates of one point inside one element - - Parameters - ---------- - realCoords : array of length numDim - Real coordinates to find the paranmetric coordinates of the desired point - nodeCoords : numNode x numDim array - Element node real coordinates - maxIter : int, optional - Maximum number of search iterations, by default 4 - - Returns - ------- - x : array of length numDim - Parametric coordinates of the desired point - """ - # x = np.zeros(self.numDim) - # for i in range(maxIter): - # res = realCoords - self.getRealCoord(np.array([x]), nodeCoords).flatten() - # if np.max(np.abs(res)) < tol: - # break - # else: - # jacT = self.getJacobian(np.array([x]), nodeCoords)[0].T - # x += np.linalg.solve(jacT, res) - # return x - - def resFunc(x): - return realCoords - self.getRealCoord(np.array([x]), nodeCoords).flatten() - - sol = root(resFunc, np.zeros(self.numDim), method="krylov", tol=tol) - return sol.x - - def getJacobian(self, paramCoords, nodeCoords): - """Get the element Jacobians at a set of parametric coordinates - - - Parameters - ---------- - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - nodeCoords : numNode x numDim array - Element node real coordinates - - Returns - ------- - Jac : n x numDim x numDim array - The Jacobians at each point - """ - return self.getShapeFunctionDerivs(paramCoords) @ nodeCoords - - @abc.abstractmethod - def getShapeFunctions(self, paramCoords): - """Compute shape function values at a set of parametric coordinates - - This function must be implemented in any child classes - - Parameters - ---------- - paramCoords : n x nDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - N : n x numNode array - Shape function values, N[i][j] is the value of the jth shape function at the ith point - """ - raise NotImplementedError - - @abc.abstractmethod - def getShapeFunctionDerivs(self, paramCoords): - """Compute shape function derivatives at a set of parametric coordinates - - These are the derivatives of the shape functions with respect to the parametric coordinates (si, eta, gamma) - - This function must be implemented in any child classes - - Parameters - ---------- - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - NPrimeParam : n x numDim x numNode array - Shape function values, N[i][j][k] is the value of the kth shape function at the ith point w.r.t the kth - parametric coordinate - """ - raise NotImplementedError - - def getNPrime(self, paramCoords, nodeCoords): - """Compute shape function derivatives at a set of parametric coordinates - - These are the derivatives of the shape functions with respect to the real coordinates (x,y,z) - - Parameters - ---------- - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - nodeCoords : numNode x numDim array - Element node real coordinates - - Returns - ------- - NPrime : n x numDim x numNode array - [description] - """ - NPrimeParam = self.getShapeFunctionDerivs(paramCoords) - # The Jacobian is NPrimeParam * nodeCoords so we don't need to waste time recomputing NPrimeParam inside the - # getJacobian function - return self.jacInv(NPrimeParam @ nodeCoords) @ NPrimeParam - - def getBMat(self, paramCoords, nodeCoords, constitutive): - """Compute the element B matrix at a set of parametric coordinates - - The B matrix is the matrix that converts nodal DOF's to strains - - strain = B*q - - K = int (B^T * D * B) dv - - Parameters - ---------- - - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - nodeCoords : numNode x numDim array - Element node real coordinates - - Returns - ------- - B : n x numStrain x (numNode*numDim) array - The B matrices, B[i] returns the 2D B matrix at the ith parametric point - """ - NPrime = self.getNPrime(paramCoords, nodeCoords) - return self._makeBMat(NPrime, constitutive) - - def getStrain(self, paramCoords, nodeCoords, constitutive, uNodes): - BMat = self.getBMat(paramCoords, nodeCoords, constitutive) - return BMat @ uNodes.flatten() - - def getStress(self, paramCoords, nodeCoords, constitutive, uNodes): - return self.getStrain(paramCoords, nodeCoords, constitutive, uNodes) @ constitutive.DMat - - def getU(self, paramCoords, uNodes): - """Compute the displacements at a set of parametric coordinates - - - Parameters - ---------- - paramCoords : n x numDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - uNodes : numNode x numDim array - Nodal displacements - - Returns - ------- - u : n x numDim array - Array of displacement values, u[i,j] is the jth displacement component at the ith point - """ - N = self.getShapeFunctions(paramCoords) - return N @ uNodes - - def getUPrime(self, paramCoords, nodeCoords, uNodes): - """Compute the displacement derivatives at a set of parametric coordinates - - Parameters - ---------- - paramCoords : n x numDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - nodeCoords : numNode x numDim array - Element node real coordinates - uNodes : numNode x numDim array - Nodal displacements - - Returns - ------- - NPrime : n x numDim x numNode array - [description] - """ - NPrime = self.getNPrime(paramCoords, nodeCoords) - return NPrime @ uNodes - - def getStiffnessIntegrand(self, paramCoords, nodeCoords, constitutive): - B = self.getBMat(paramCoords, nodeCoords, constitutive) - J = self.getJacobian(paramCoords, nodeCoords) - detJ = self.jacDet(J) - BDB = np.swapaxes(B, 1, 2) @ constitutive.DMat @ B - return (BDB.T * detJ).T - - def getStiffnessMat(self, nodeCoords, constitutive, n=None): - if n is None: - n = self.order + 1 - if self.numDim == 1: - f = lambda x1: self.getStiffnessIntegrand(np.array([x1]).T, nodeCoords, constitutive) # noqa: E731 - return gaussQuad1d(f=f, n=n) - if self.numDim == 2: - f = lambda x1, x2: self.getStiffnessIntegrand(np.array([x1, x2]).T, nodeCoords, constitutive) # noqa: E731 - return gaussQuad2d(f=f, n=n) - if self.numDim == 3: - f = lambda x1, x2, x3: self.getStiffnessIntegrand( # noqa: E731 - np.array([x1, x2, x3]).T, nodeCoords, constitutive - ) - return gaussQuad3d(f, n) - - def getMassMat(self, nodeCoords, constitutive, n=None): - if n is None: - n = self.order + 1 - if self.numDim == 1: - f = lambda x1: self.getMassIntegrand(np.array([x1]).T, nodeCoords, constitutive) # noqa: E731 - return gaussQuad1d(f=f, n=n) - if self.numDim == 2: - f = lambda x1, x2: self.getMassIntegrand(np.array([x1, x2]).T, nodeCoords, constitutive) # noqa: E731 - return gaussQuad2d(f=f, n=n) - if self.numDim == 3: - f = lambda x1, x2, x3: self.getMassIntegrand( # noqa: E731 - np.array([x1, x2, x3]).T, nodeCoords, constitutive - ) - return gaussQuad3d(f, n) - - def getMassIntegrand(self, paramCoords, nodeCoords, constitutive): - N = self.getShapeFunctions(paramCoords) - NMat = self._makeNMat(N) - J = self.getJacobian(paramCoords, nodeCoords) - detJ = self.jacDet(J) - NTN = np.swapaxes(NMat, 1, 2) @ NMat * constitutive.rho - return (NTN.T * detJ).T - - def integrateBodyForce(self, f, nodeCoords, n=1): - """Compute equivalent nodal forces due to body forces through numerical integration - - - Parameters - ---------- - f : Body force function - Should accept an nP x numDim array as input and output a nP x numDisp array, ie f(x)[i] returns the body - force components at the ith point queried - nodeCoords : numNode x numDim array - Element node real coordinates - n : int, optional - Number of integration points, can be a single value or a list with a value for each direction, by default 1 - - Returns - ------- - Fb : numNode x numDisp array - Equivalent nodal loads due to body force - """ - if self.numDim == 1: - bodyForceFunc = lambda x1: self.bodyForceIntegrad(f, np.array([x1]).T, nodeCoords) # noqa: E731 - return gaussQuad1d(bodyForceFunc, n) - if self.numDim == 2: - bodyForceFunc = lambda x1, x2: self.bodyForceIntegrad(f, np.array([x1, x2]).T, nodeCoords) # noqa: E731 - return gaussQuad2d(bodyForceFunc, n) - if self.numDim == 3: - bodyForceFunc = lambda x1, x2, x3: self.bodyForceIntegrad( # noqa: E731 - f, np.array([x1, x2, x3]).T, nodeCoords - ) - return gaussQuad3d(bodyForceFunc, n) - - def _computeNTFProduct(self, F, N): - """A basic wrapper for the jit compiled function _computeNTFProduct""" - return _computeNTFProduct(F, N) - - def _makeNMat(self, N): - """A basic wrapper for the jit compiled function _makeNMat""" - return _makeNMat(N, self.numDim) - - def _makeBMat(self, NPrime, constitutive): - """A basic wrapper for the jit compiled function _makeBMat""" - return _makeBMat( - NPrime, - constitutive.LMats, - constitutive.numStrain, - self.numDim, - self.numNodes, - ) - - def bodyForceIntegrad(self, f, paramCoord, nodeCoords): - # Compute shape functions and Jacobian determinant at parametric coordinates - N = self.getShapeFunctions(paramCoord) - J = self.getJacobian(paramCoord, nodeCoords) - detJ = self.jacDet(J) - # Transform parametric to real coordinates in order to compute body force components - realCoord = self.getRealCoord(paramCoord, nodeCoords) - F = f(realCoord) - Fb = self._computeNTFProduct(F, N) - return (Fb.T * detJ).T - - # ============================================================================== - # Functions for testing element implementations - # ============================================================================== - - def getRandParamCoord(self, n=1): - """Generate a set of random parametric coordinates - - By default this method assumes that the valid range for all parametric coordinates is [-1, 1]. - For elements where this is not the case, this method should be reimplemented. - - Parameters - ---------- - n : int, optional - number of points to generate, by default 1 - - Returns - ------- - paramCoords : n x numDim array - isoparametric coordinates, one row for each point - """ - return np.atleast_2d(np.random.rand(n, self.numDim)) - - @abc.abstractmethod - def getRandomNodeCoords(self): - """Generate a random, but valid, set of node coordinates for an element - - This method should be implemented for each element. - - Returns - ------- - nodeCoords : numNode x numDim array - Node coordinates - """ - raise NotImplementedError - - def testGetParamCoord(self, n=10, maxIter=40, tol=1e-10): - """Test the getParamCoord method - - This test works by generating a set of random parametric coordinates, converting them to real coordinates, and - then checking that the parametric coordinates returned by getParamCoord match the original random values. - - Parameters - ---------- - n : int, optional - Number of random coordinates to generate, by default 10 - """ - paramCoord = self.getRandParamCoord(n) - nodeCoords = self.getRandomNodeCoords() - realCoords = self.getRealCoord(paramCoord, nodeCoords) - error = np.zeros_like(realCoords) - for i in range(n): - error[i] = paramCoord[i] - self.getParamCoord(realCoords[i], nodeCoords, maxIter=maxIter, tol=tol) - return error - - def testShapeFunctionDerivatives(self, n=10): - """Test the implementation of the shape function derivatives using the complex-step method - - Parameters - ---------- - n : int, optional - Number of random coordinates to generate, by default 10 - """ - paramCoords = self.getRandParamCoord(n) - coordPert = np.zeros_like(paramCoords, dtype="complex128") - dN = self.getShapeFunctionDerivs(paramCoords) - dNApprox = np.zeros_like(dN) - for i in range(self.numDim): - np.copyto(coordPert, paramCoords) - coordPert[:, i] += 1e-200 * 1j - dNApprox[:, i, :] = 1e200 * np.imag(self.getShapeFunctions(coordPert)) - return dN - dNApprox - - def testShapeFunctionSum(self, n=10): - """Test the basic property that shape function values should sum to 1 everywhere within an element - - Parameters - ---------- - n : int, optional - Number of points to test at, by default 10 - """ - paramCoords = self.getRandParamCoord(n) - N = self.getShapeFunctions(paramCoords) - return np.sum(N, axis=1) - - # TODO: Tests to add - # - Complex step validation of jacobian - # - Validate stiffness matrix against resdiual (would need to implement a residual assembly method) diff --git a/FEMpy/Elements/Element.py b/FEMpy/Elements/Element.py new file mode 100644 index 0000000..b92057c --- /dev/null +++ b/FEMpy/Elements/Element.py @@ -0,0 +1,1196 @@ +""" +============================================================================== +FEMpy - New Element Class +============================================================================== +@File : NewElement.py +@Date : 2022/11/17 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import abc + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from numba import guvectorize, float64, njit, prange +from scipy.optimize import minimize, Bounds, LinearConstraint +from scipy.spatial.transform import Rotation + + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.LinAlg import det1, det2, det3, inv1, inv2, inv3 +from FEMpy.Utils.KSAgg import ksAgg + + +class Element: + """The base class from which all other elements are derived.""" + + def __init__(self, numNodes, numDim, quadratureOrder, numStates=None): + """Instantiate an element object + + Parameters + ---------- + numNodes : int + Number of nodes in the element + numDim : int + Number of spatial dimensions the element lives in + quadratureOrder : int + Integration quadrature order + numStates : int, optional + Number of states in the underlying PDE, a.k.a the number of DOF per node, by default uses numDim + """ + self.numNodes = numNodes + self.numDim = numDim + self.numStates = numStates if numStates is not None else numDim + self.numDOF = self.numNodes * self.numStates + self.quadratureOrder = quadratureOrder + self.name = f"{self.numNodes}Node-{self.numStates}Disp-{self.numDim}D-Element" + + # --- Parametric coordinate bounds --- + # By default it is assumed that the parametric coordinates are in the range [-1, 1] in each dimension, for + # elements where this is not true (e.g a 2d triangular element), these attributes should be overwritten + self.paramCoordLowerBounds = -np.ones(self.numDim) + self.paramCoordUpperBounds = np.ones(self.numDim) + self.paramCoordLinearConstaintMat = None + self.paramCoordLinearConstaintUpperBounds = None + self.paramCoordLinearConstaintLowerBounds = None + + # --- Define fast jacobian determinant function based on number of dimensions --- + if self.numDim == 1: + self.jacDet = det1 + self.jacInv = inv1 + elif self.numDim == 2: + self.jacDet = det2 + self.jacInv = inv2 + elif self.numDim == 3: + self.jacDet = det3 + self.jacInv = inv3 + + if self.numDim not in [1, 2, 3]: + raise ValueError(f"Sorry, FEMpy doesn't support {self.numDim}-dimensional problems") + + # ============================================================================== + # Abstract methods: Must be implemented by derived classes + # ============================================================================== + + @abc.abstractmethod + def computeShapeFunctions(self, paramCoords): + """Compute the shape function values at a given set of parametric coordinates + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape functions at + + Returns + ------- + N: numPoint x numNodes array + Array of shape function values at the given parametric coordinates, N[i][j] is the value of the jth shape function at the ith parametric point + """ + raise NotImplementedError + + @abc.abstractmethod + def computeShapeFunctionGradients(self, paramCoords): + """Compute the derivatives of the shape functions with respect to the parametric coordinates at a given set of parametric coordinates + + + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape function gradients at + + Returns + ------- + NPrimeParam: numPoint x numDim x numNodes array + Shape function gradient values, NPrimeParam[i][j][k] is the derivative of the kth shape function at the ith point w.r.t the jth parametric coordinate + """ + raise NotImplementedError + + @abc.abstractmethod + def getIntegrationPointWeights(self, order=None): + """Compute the integration point weights for a given quadrature order on this element + + Parameters + ---------- + order : int, optional + Integration order + + Returns + ------- + array of length numIntpoint + Integration point weights + """ + raise NotImplementedError + + @abc.abstractmethod + def getIntegrationPointCoords(self, order=None): + """Compute the integration point parameteric coordinates for a given quadrature order on this element + + Parameters + ---------- + order : int, optional + Integration order + + Returns + ------- + numIntpoint x numDim array + Integration point coordinates + """ + raise NotImplementedError + + @abc.abstractmethod + def getReferenceElementCoordinates(self): + """Get the node coordinates for the reference element, a.k.a the element on which the shape functions are defined + + Returns + ------- + numNodes x numDim array + Element node coordinates + """ + raise NotImplementedError + + # ============================================================================== + # Implemented methods + # ============================================================================== + def computeFunction(self, nodeCoords, nodeStates, elementDVs, function, elementReductionType, intOrder=None): + """Given a function that can depend on true coordinates, the state, state gradients and some design variables, compute the value of that function over the element + + + + Parameters + ---------- + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + nodeStates : numElements x numNodes x numStates array + State values at the nodes of each element + elementDVs : dict of numElement length arrays + Design variable values for each element + function : callable + Function to evaluate at each point within each element, must have signature f(x, u, u', dvs), where: + x is an n x numDim array of coordinates + u is an n x numStates array of state values + u' is an n x (numStates*numDim) array of state gradients + dvs is an n x numDVs array of design variable values + elementReductionType : _type_ + Type of reduction to do to get a single value for each element, can be: + - 'sum' : sum all values + - 'mean' : average all values + - `integrate` : integrate the function over the element + - 'max' : take the maximum value + - 'min' : take the minimum value + - 'ksmax' : Compute a smooth approximation of the maximum value using KS aggregation + - 'ksmin' : Compute a smooth approximation of the minimum value using KS aggregation + + Returns + ------- + values : numElements array + Value of the function for each element + """ + + numElements = nodeCoords.shape[0] + nodeCoords = np.ascontiguousarray(nodeCoords) + nodeStates = np.ascontiguousarray(nodeStates) + + # - Get integration point parametric coordinates and weights (same for all elements of same type) + intOrder = self.quadratureOrder if intOrder is None else intOrder + intPointWeights = self.getIntegrationPointWeights(intOrder) # numIntPoints + intPointParamCoords = self.getIntegrationPointCoords(intOrder) # numIntPoints x numDim + numIntPoints = len(intPointWeights) + + # Get the quantities we need for the weak residual evaluation at the integration points for each element + pointQuantities = self._computeFunctionEvaluationQuantities( + intPointParamCoords, + nodeStates, + nodeCoords, + elementDVs, + quantities=["Coord", "State", "StateGrad", "DVs", "JacDet"], + ) + values = function( + pointQuantities["State"], pointQuantities["StateGrad"], pointQuantities["Coord"], pointQuantities["DVs"] + ) + values = values.reshape((numElements, numIntPoints)) + + # perform element reduction if specified + if elementReductionType is not None: + assert elementReductionType.lower() in [ + "sum", + "mean", + "integrate", + "min", + "max", + "ksmax", + "ksmin", + ], "elementReductionType not valid" + + # Simple numpy reductions + if elementReductionType.lower() in ["sum", "mean", "min", "max"]: + + if elementReductionType.lower() == "sum": + reductFunc = np.sum + + if elementReductionType.lower() == "mean": + reductFunc = np.average + + if elementReductionType.lower() == "min": + reductFunc = np.min + + if elementReductionType.lower() == "max": + reductFunc = np.max + + return reductFunc(values, axis=1) + + if elementReductionType.lower() == "integrate": + # compute integration using weighted sum of w*values*detJ over each set of element points + pointQuantities["JacDet"] = pointQuantities["JacDet"].reshape((numElements, numIntPoints)) + + return np.einsum( + "ep,ep,p->e", + values, + pointQuantities["JacDet"], + intPointWeights, + optimize=["einsum_path", (0, 1), (0, 1)], + ) + + if elementReductionType.lower() in ["ksmax", "ksmin"]: + reducedValues = np.zeros(numElements) + for i in range(numElements): + reducedValues[i] = ksAgg(values[i, :], "max") + return reducedValues + + if elementReductionType.lower() == "ksmin": + reducedValues = np.zeros(numElements) + for i in range(numElements): + reducedValues[i] = ksAgg(values[i, :], "min") + return reducedValues + + return values + + def _computeFunctionEvaluationQuantities(self, paramCoords, nodeStates, nodeCoords, designVars, quantities): + """Compute a series of values that are used for evaluating functions at multiple points over multiple elements + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate quantities at + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + nodeStates : numElements x numNodes x numStates array + State values at the nodes of each element + designVars : dict of numElements arrays + Design variable values for each element + quantities : list of strings, optional + List of quantities to compute, by default None are computed, valid quatities are: + - "N" : Shape function values + - "NPrimeParam" : Shape function gradients + - "Coord" : True coordinates + - "State" : State values + - "StateGrad" : State gradients + - "DVs" : Design variable values + - "Jac" : Jacobian of the transformation from parametric to true coordinates + - "JacInv" : Inverse of the Jacobian of the transformation from parametric to true coordinates + - "JacDet" : Determinant of the Jacobian of the transformation from parametric to true coordinates + - "StateGradSens" : Sensitivity of the state gradients to the nodal states + """ + lowerCaseQuantities = [q.lower() for q in quantities] + outputs = {} + numElements = nodeCoords.shape[0] + numPoints = paramCoords.shape[0] + numPointsTotal = numElements * numPoints + # - Get shape functions N (du/dq) and their gradients in parametric coordinates at points + # (same for all elements of same type) + if any([q in lowerCaseQuantities for q in ["n", "coords", "state"]]): + N = self.computeShapeFunctions(paramCoords) # numPoints x numNodes + if "n" in lowerCaseQuantities: + outputs["N"] = N + + if any( + [q in lowerCaseQuantities for q in ["nprimeparam", "stategrad", "jac", "jacinv", "jacdet", "stategradsens"]] + ): + NPrimeParam = self.computeShapeFunctionGradients(paramCoords) # numPoints x numDim x numNodes + if "nprimeparam" in lowerCaseQuantities: + outputs["NPrimeParam"] = NPrimeParam + + # - Compute real coordinates at points (different for each element) + if "coord" in lowerCaseQuantities: + pointCoords = _interpolationProduct(N[:, : self.numNodes], nodeCoords) # numElements x numPoints x numDim + outputs["Coord"] = pointCoords + + # - Compute states at points (different for each element) + if "state" in lowerCaseQuantities: + pointStates = _interpolationProduct(N, nodeStates) # numElements x numPoints x numStates + outputs["State"] = pointStates + + # - Compute Jacobians, their inverses, and their determinants at points (different for each element) + if any([q in lowerCaseQuantities for q in ["jac", "jacinv", "jacdet", "stategrad", "stategradsens"]]): + pointJacs = np.zeros((numElements, numPoints, self.numDim, self.numDim)) + _computeNPrimeCoordProduct(NPrimeParam, nodeCoords, pointJacs) + if "jac" in lowerCaseQuantities: + outputs["Jac"] = pointJacs + if any([q in lowerCaseQuantities for q in ["jacinv", "stategrad", "stategradsens"]]): + pointJacInvs = self.jacInv(pointJacs) + if "jacInv" in lowerCaseQuantities: + outputs["JacInv"] = pointJacInvs + if "jacdet" in lowerCaseQuantities: + outputs["JacDet"] = self.jacDet(pointJacs) # numElements x numPoints + + # - Compute du'/dq at points (different for each element) + if "stategradsens" in lowerCaseQuantities: + pointDUPrimedq = np.zeros((numElements, numPoints, self.numDim, self.numNodes)) + _computeDUPrimeDqProduct(pointJacInvs, NPrimeParam, pointDUPrimedq) + outputs["StateGradSens"] = pointDUPrimedq + + # - Compute u' at points (different for each element) + if "stategrad" in lowerCaseQuantities: + pointStateGradients = np.zeros((numElements, numPoints, self.numStates, self.numDim)) + _computeUPrimeProduct(pointJacInvs, NPrimeParam, nodeStates, pointStateGradients) + outputs["StateGrad"] = pointStateGradients + + # Currently everything is in numElements x numPoints x ... arrays, but the constitutive model doesn't care about + # the distinction between different elements, so we need to flatten the first two dimensions of each array, so + # they're all (numElements x numPoints) x ... + for key in outputs: + outputs[key] = np.ascontiguousarray(outputs[key].reshape(numPointsTotal, *outputs[key].shape[2:])) + + # For the DVs it's a bit different, we have one DV value per element, so we actually need to expand them so that + # we have one value per point + if "dvs" in lowerCaseQuantities: + outputs["DVs"] = {} + for dvName, dvValues in designVars.items(): + outputs["DVs"][dvName] = np.ascontiguousarray(np.repeat(dvValues, numPoints)) + + return outputs + + def computeResiduals(self, nodeStates, nodeCoords, designVars, constitutiveModel, intOrder=None): + """Compute the local residual for a series of elements + + Parameters + ---------- + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + nodeStates : numElements x numNodes x numStates array + State values at the nodes of each element + designVars : dict of numElements arrays + Design variable values for each element + constitutiveModel : FEMpy constitutive model object + The constitutive model of the element + + Returns + ------- + numElement x numNodes x numStates array + The local residuals for each element + """ + numElements = nodeCoords.shape[0] + nodeCoords = np.ascontiguousarray(nodeCoords) + nodeStates = np.ascontiguousarray(nodeStates) + + # - Get integration point parametric coordinates and weights (same for all elements of same type) + intOrder = self.quadratureOrder if intOrder is None else intOrder + intPointWeights = self.getIntegrationPointWeights(intOrder) # numIntPoints + intPointParamCoords = self.getIntegrationPointCoords(intOrder) # numIntPoints x numDim + numIntPoints = len(intPointWeights) + + # Get the quantities we need for the weak residual evaluation at the integration points for each element + pointQuantities = self._computeFunctionEvaluationQuantities( + intPointParamCoords, + nodeStates, + nodeCoords, + designVars, + quantities=["Coord", "State", "StateGrad", "JacDet", "StateGradSens", "DVs"], + ) + + weakRes = constitutiveModel.computeWeakResiduals( + pointQuantities["State"], pointQuantities["StateGrad"], pointQuantities["Coord"], pointQuantities["DVs"] + ) + + # - Compute r = du'/dq^T * f + r = np.zeros((numElements * numIntPoints, self.numNodes, self.numStates)) + _transformResidual(pointQuantities["StateGradSens"], weakRes, r) + r = r.reshape((numElements, numIntPoints, self.numNodes, self.numStates)) + pointQuantities["JacDet"] = pointQuantities["JacDet"].reshape((numElements, numIntPoints)) + + # - Compute R, weighted sum of w * r * detJ over each set of integration points + # R = np.einsum( + # "epns,ep,p->ens", r, pointQuantities["JacDet"], intPointWeights, optimize=["einsum_path", (1, 2), (0, 1)] + # ) + R = np.zeros((numElements, self.numNodes, self.numStates)) + _integrate(r, pointQuantities["JacDet"], intPointWeights, R) + return R + + def computeResidualJacobians(self, nodeStates, nodeCoords, designVars, constitutiveModel, intOrder=None): + """Given node coordinates and states, design variable values, and a constitutive model, compute the residual Jacobian for a bunch of elements + + Parameters + ---------- + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + nodeStates : numElements x numNodes x numStates array + State values at the nodes of each element + designVars : dict of numElements arrays + Design variable values for each element + constitutiveModel : FEMpy constitutive model object + The constitutive model of the element + + Returns + ------- + numElement x (numNodes * numStates) x (numNodes * numStates) array + The local residual Jacobian matrix for each element + """ + numElements = nodeCoords.shape[0] + nodeCoords = np.ascontiguousarray(nodeCoords) + nodeStates = np.ascontiguousarray(nodeStates) + + # - Get integration point parametric coordinates and weights (same for all elements of same type) + intOrder = self.quadratureOrder if intOrder is None else intOrder + intPointWeights = self.getIntegrationPointWeights(intOrder) # numIntPoints + intPointParamCoords = self.getIntegrationPointCoords(intOrder) # numIntPoints x numDim + numIntPoints = len(intPointWeights) + + # Get the quantities we need for the weak residual evaluation at the integration points for each element + pointQuantities = self._computeFunctionEvaluationQuantities( + intPointParamCoords, + nodeStates, + nodeCoords, + designVars, + quantities=["Coord", "State", "StateGrad", "JacDet", "StateGradSens", "DVs"], + ) + + # Compute the weak residual Jacobians, dr/du' + weakJacs = constitutiveModel.computeWeakResidualJacobian( + pointQuantities["State"], pointQuantities["StateGrad"], pointQuantities["Coord"], pointQuantities["DVs"] + ) + + # Compute dr/dq = du'/dq^T * dr/du' * du'/dq + # Jacs = np.einsum( + # "pdn,pdsSD,pDN->pnsNS", + # pointQuantities["StateGradSens"], + # weakJacs, + # pointQuantities["StateGradSens"], + # optimize=["einsum_path", (0, 2), (0, 1)], + # ) + Jacs = np.zeros((numElements * numIntPoints, self.numNodes, self.numStates, self.numNodes, self.numStates)) + _transformResidualJacobians(pointQuantities["StateGradSens"], weakJacs, Jacs) + Jacs = Jacs.reshape((numElements, numIntPoints, self.numNodes, self.numStates, self.numNodes, self.numStates)) + pointQuantities["JacDet"] = pointQuantities["JacDet"].reshape((numElements, numIntPoints)) + + # - Compute R, weighted sum of w * r * detJ over each set of integration points + dRdq = np.zeros((numElements, self.numNodes, self.numStates, self.numNodes, self.numStates)) + _integrate(Jacs, pointQuantities["JacDet"], intPointWeights, dRdq) + + return dRdq.reshape((numElements, self.numDOF, self.numDOF)) + + def computeStates(self, paramCoords, nodeStates): + """Given nodal DOF, compute the state at given parametric coordinates within the element + + This function is vectorised both across multiple elements, and multiple points within each element, + but the parametric coordinates are assumed to be the same across all elements + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to compute state at + nodeStates : numElements x numNodes x numStates array + State values at the nodes of each element + + Returns + ------- + states : numElements x numPoint x numStates array + State values at the given parametric coordinates for each element + """ + + # Compute shape functions at the given parametric coordinates + N = self.computeShapeFunctions(paramCoords) + + # Then for each element, compute the states at the points, the einsum below is equivalent to: + # product = np.zeros((numElements, numPoints, numStates)) + # for ii in range(numElements): + # product[ii] = N @ nodeStates[ii] + return _interpolationProduct(N, nodeStates) + + def computeCoordinates(self, paramCoords, nodeCoords): + """Given nodal coordinates, compute the real coordinates at given parametric coordinates within the element + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to compute real coordinates of + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + """ + # Compute shape functions at the given parametric coordinates + N = self.computeShapeFunctions(paramCoords) + + # Then for each element, compute the states at the points, the einsum below is equivalent to: + # product = np.zeros((numElements, numPoints, numStates)) + # for ii in range(numElements): + # product[ii] = N[:, : self.numNodes] @ nodeStates[ii] + return _interpolationProduct(N[:, : self.numNodes], nodeCoords) + + def computeJacobians(self, paramCoords, nodeCoords): + """Compute the Jacobian at a set of parametric coordinates within a set of elements + + This function is vectorised both across multiple elements, and multiple points within each element, + but the parametric coordinates are assumed to be the same across all elements + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to compute Jacobians at + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + + Returns + ------- + Jac : numElements x numPoints x numDim x numDim array + The Jacobians at each point in each element + """ + NPrimeParam = self.computeShapeFunctionGradients(paramCoords) # numPoints x numDim x numNodes + numElements = nodeCoords.shape[0] + numPoints = paramCoords.shape[0] + Jac = np.zeros((numElements, numPoints, self.numDim, self.numDim)) + nodeCoords = np.ascontiguousarray(nodeCoords) + + # The function call below does the following: + # for ii in range(numElements): + # for jj in range(numPoints): + # Jac[ii, jj] = NPrimeParam[jj] @ nodeCoords[ii] + _computeNPrimeCoordProduct(NPrimeParam, nodeCoords, Jac) + return Jac + + def computeStateGradients(self, paramCoords, nodeStates, nodeCoords): + """Given nodal DOF, compute the gradient of the state at given parametric coordinates within the element + + The gradient of the state at each point in each element is a numStates x numDim array. + + This function is vectorised both across multiple elements, and multiple points within each element, + but the parametric coordinates are assumed to be the same across all elements + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to compute state at + nodeStates : numElements x numNodes x numStates array + State values at the nodes of each element + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + + Returns + ------- + stateGradients : numElements x numPoint x numStates x numDim array + The gradient of the states at each point in each element, stateGradient[i, j, k, l] is the value of + $du_k/dx_l$ at the $j^{th}$ point in the $i^{th}$ element + """ + + numElements = nodeCoords.shape[0] + numPoints = paramCoords.shape[0] + NPrimeParam = self.computeShapeFunctionGradients(paramCoords) + Jac = np.zeros((numElements, numPoints, self.numDim, self.numDim)) + _computeNPrimeCoordProduct(NPrimeParam, nodeCoords, Jac) + JacInv = self.jacInv(np.reshape(Jac, (numElements * numPoints, self.numDim, self.numDim))) + JacInv = np.reshape(JacInv, (numElements, numPoints, self.numDim, self.numDim)) + UPrime = np.zeros((numElements, numPoints, self.numStates, self.numDim)) + # The function call below is equivalent to the following + # for ii in range(numElements): + # for jj in range(numPoints): + # result[ii, jj] = (JacInv[ii, jj] @ NPrimeParam[jj] @ nodeStates[ii]).T + _computeUPrimeProduct(JacInv, NPrimeParam, np.ascontiguousarray(nodeStates), UPrime) + return UPrime + + # Given a function that can depend on true coordinates, the state, state gradients and some design variables, compute the value of that function over the element + + def getClosestPoints(self, nodeCoords, point, **kwargs): + """Given real coordinates of a point, find the parametric coordinates of the closest point on a series of + elements to that point + + Computing the closest point is an optimization problem of the form: + + min ||X(x) - P||^2 + + s.t Ax <= b + lb <= x <= ub + + Where X are the real coordinates of a point in the element, x the parametric coordinates of that point, and P is + the target point. lb <= x <= ub and Ax <= b are a set of bounds and linear constraints on the parametric + coordinates that encode the bounds of the element. + + Parameters + ---------- + nodeCoords : numElements x numNodes x numDim array + The coordinates of the elements + point : array of length numDim + Target point coordinates + + Returns + ------- + closestParamCoords : numElements x numDim array + The parametric coordinates of the closest point in each element + closestDistances : numElements array + The distances from the closest point in each element to the target point + """ + numElements = nodeCoords.shape[0] + closestDistances = np.zeros(numElements) + closestParamCoords = np.zeros((numElements, self.numDim)) + + paramCoordBounds = Bounds(lb=self.paramCoordLowerBounds, ub=self.paramCoordUpperBounds) + if self.paramCoordLinearConstaintMat is not None: + paramCoordLinearConstraints = LinearConstraint( + self.paramCoordLinearConstaintMat, + self.paramCoordLinearConstaintLowerBounds, + self.paramCoordLinearConstaintUpperBounds, + keep_feasible=True, + ) + else: + paramCoordLinearConstraints = None + + for ii in range(numElements): + closestParamCoords[ii], closestDistances[ii] = self._getClosestPoint( + nodeCoords[ii], point, paramCoordBounds, paramCoordLinearConstraints, **kwargs + ) + + return closestParamCoords, closestDistances + + def _getClosestPoint(self, nodeCoords, point, paramCoordBounds, paramCoordLinearConstraints, **kwargs): + """Find the closest point on a single element to a given point + + Parameters + ---------- + nodeCoords : numNodes x numDim array + The coordinates of the element nodes + point : array of length numDim + Target point coordinates + paramCoordBounds : scipy.optimize.Bounds object + Parametric coordinate bounds + paramCoordLinearConstraints : scipy.optimize.LinearConstraint object + Any linear constraints required to enforce the parametric coordinate bounds + + Returns + ------- + array of length numDim + The parametric coordinates of the closest point in the element + float + The distance from the closest point in the element to the target point + """ + nodeCoordCopy = np.zeros((1, self.numNodes, self.numDim)) + nodeCoordCopy[0] = nodeCoords + + def r(xParam): + xTrue = self.computeCoordinates(np.atleast_2d(xParam), nodeCoordCopy).flatten() + error = np.linalg.norm(xTrue - point) + print(f"{xParam=}, {xTrue=}, {point=}, {error=}") + return error + + def drdxParam(xParam): + xTrue = self.computeCoordinates(np.atleast_2d(xParam), nodeCoordCopy).flatten() + Jac = self.computeJacobians(np.atleast_2d(xParam), nodeCoordCopy) + return 2 * (xTrue - point) @ Jac[0, 0].T + + if "tol" not in kwargs: + kwargs["tol"] = 1e-10 + + maxAttempts = 10 + closestPointFound = False + for _ in range(maxAttempts): + x0 = self.getRandParamCoord(1)[0] + sol = minimize( + r, + x0, + jac=drdxParam, + bounds=paramCoordBounds, + constraints=paramCoordLinearConstraints, + method="trust-constr", + **kwargs, + ) + closestPointFound = sol.fun < 1e-4 + if closestPointFound: + break + + return sol.x, sol.fun + + def getRandomElementCoordinates(self, rng=None): + """Compute random node coordinates for an element + + The random node coordinates are computed by taking the reference element coordinates and then applying: + - Random perturbations to each node + - Random translation in each dimension + - Random scalings in each dimension + - Random rotations around each available axis + + Parameters + ---------- + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + if rng is None: + rng = np.random.default_rng() + coords = self.getReferenceElementCoordinates() # numNodes x numDim array + + # Perturb coordinates by up to 10% of the maximum distance between any two nodes + maxDistance, _ = _computeMaxMinDistance(coords) + coords += rng.random(coords.shape) * 0.1 * maxDistance + + # Apply random translation + for ii in range(self.numDim): + translation = rng.random() * 2 * maxDistance - maxDistance + coords[:, ii] += translation + + # Scale each dimension by a random factor between 0.1 and 10 + for dim in range(self.numDim): + scalingPower = rng.random() * 2 - 1 + coords[:, dim] *= 10**scalingPower + + # Rotate the element around each axis by a random angle + if self.numDim == 2: + angle = rng.random() * 4 * np.pi - 2 * np.pi + c, s = np.cos(angle), np.sin(angle) + R = np.array(((c, s), (-s, c))) + coords = coords @ R.T + elif self.numDim == 3: + R = Rotation.random(random_state=rng) + coords = coords @ R.as_matrix().T + + return coords + + # ============================================================================== + # Testing methods + # ============================================================================== + + def getRandParamCoord(self, n, rng=None): + """Get a random set of parametric coordinates within the element + + By default this method assumes the the valid parametric coordinates are between -1 and 1 in each direction. If this is not the case for a particular element then that element should reimplemnt this method. + + Parameters + ---------- + n : int + Number of points to generate + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + if rng is None: + rng = np.random.default_rng() + return rng.random((n, self.numDim)) * 2 - 1 + + def testShapeFunctionDerivatives(self, n=10, rng=None): + """Test the implementation of the shape function derivatives using the complex-step method + + Parameters + ---------- + n : int, optional + Number of random coordinates to generate, by default 10 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + paramCoords = self.getRandParamCoord(n, rng=rng) + coordPert = np.zeros_like(paramCoords, dtype="complex128") + dN = self.computeShapeFunctionGradients(paramCoords) + dNApprox = np.zeros_like(dN) + for i in range(self.numDim): + np.copyto(coordPert, paramCoords) + coordPert[:, i] += 1e-200 * 1j + dNApprox[:, i, :] = 1e200 * np.imag(self.computeShapeFunctions(coordPert)) + return dN - dNApprox + + def testShapeFunctionSum(self, n=10, rng=None): + """Test the basic property that shape function values should sum to 1 everywhere within an element + + Parameters + ---------- + n : int, optional + Number of points to test at, by default 10 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + paramCoords = self.getRandParamCoord(n, rng=rng) + N = self.computeShapeFunctions(paramCoords) + return np.sum(N, axis=1) + + def testInterpolation(self, n=10, rng=None): + """Validate that, when the element geometry matches the reference element exactly, the parametric and real coordinates are the same + + Parameters + ---------- + n : int, optional + Number of points to test at, by default 10 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + nodeCoords = np.zeros((1, self.numNodes, self.numDim)) + nodeCoords[0] = self.getReferenceElementCoordinates() + paramCoords = self.getRandParamCoord(n, rng=rng) + error = np.zeros((n, self.numDim)) + x = self.computeCoordinates(paramCoords, nodeCoords) + error = x - paramCoords + return error + + def testIdentityJacobian(self, n=10, rng=None): + """Validate that, when the element geometry matches the reference element exactly, the mapping Jacobian is the identity matrix everywhere. + + Parameters + ---------- + n : int, optional + Number of points to test at, by default 10 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + nodeCoords = np.zeros((1, self.numNodes, self.numDim)) + nodeCoords[0] = self.getReferenceElementCoordinates() + paramCoords = self.getRandParamCoord(n, rng=rng) + + # The expected Jacobians are a stack of n identity matrices + expectedJacs = np.tile(np.eye(self.numDim), (1, n, 1, 1)) + Jacs = self.computeJacobians(paramCoords, nodeCoords) + return Jacs - expectedJacs + + def testStateGradient(self, n=10, rng=None): + """Test that the state gradient is correctly reconstructed within the element + + This test works by generating random node coordinates, then computing the states at each node using the + following equation: + + u_i = a_i * x + b_i * y + c_i * z + d_i + + This field has a gradient, du/dx, of [a_i, b_i, c_i] everywhere in the element, which should be exactly + reproduced by the state gradient computed by the element. + + Parameters + ---------- + n : int, optional + _description_, by default 10 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + if rng is None: + rng = np.random.default_rng() + nodeCoords = np.zeros((1, self.numNodes, self.numDim)) + nodeCoords[0] = self.getRandomElementCoordinates(rng=rng) + paramCoords = self.getRandParamCoord(n, rng=rng) + + randStateGradient = rng.random((self.numStates, self.numDim)) + ExpectedStateGradients = np.tile(randStateGradient, (1, n, 1, 1)) + + nodeStates = np.zeros((1, self.numNodes, self.numStates)) + for ii in range(self.numNodes): + for jj in range(self.numStates): + nodeStates[:, ii, jj] = np.dot(nodeCoords[0, ii], randStateGradient[jj]) + + stateGradient = self.computeStateGradients(paramCoords, nodeStates, nodeCoords) + + return stateGradient - ExpectedStateGradients + + def testGetClosestPoints(self, n=10, tol=1e-10, rng=None): + """Test the getClosestPoints method + + This test works by generating a set of random parametric coordinates, converting them to real coordinates, and + then checking that the parametric coordinates returned by getClosestPoints match the original random values. + + Parameters + ---------- + n : int, optional + Number of random coordinates to generate, by default 10 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + """ + nodeCoords = np.zeros((1, self.numNodes, self.numDim)) + nodeCoords[0] = self.getRandomElementCoordinates(rng=rng) + paramCoords = self.getRandParamCoord(n, rng=rng) + realCoords = self.computeCoordinates(paramCoords, nodeCoords) + error = np.zeros_like(realCoords) + for i in range(n): + coords, _ = self.getClosestPoints(nodeCoords, realCoords[0, i], tol=tol) + error[0, i] = coords - paramCoords[i] + return error + + +@njit(cache=True, fastmath=True, parallel=True) +def _interpolationProduct(N, nodeValues): + """Compute the product of the interpolation matrix and a set of node values + + Parameters + ---------- + N : numPoints x numNodes array + Shape function values at each point + nodeValues : numElements x numNodes x numValues array + Nodal values for each element + + Returns + ------- + numElements x numPoints x numStates array + Interpolated values for each element + """ + numElements, _, numStates = nodeValues.shape + numPoints = N.shape[0] + product = np.zeros((numElements, numPoints, numStates)) + for ii in prange(numElements): + product[ii] = N @ nodeValues[ii] + return product + + +@guvectorize( + [(float64[:, :, ::1], float64[:, :, ::1], float64[:, :, :, ::1])], + "(p,d,n),(e,n,d)->(e,p,d,d)", + nopython=True, + cache=True, + fastmath=True, + boundscheck=False, + target="parallel", +) +def _computeNPrimeCoordProduct(NPrimeParam, nodeCoords, Jac): + """This function computes a nasty product of two 3d arrays that is required when computing element mapping Jacobians + + Given the shape function derivatives at each point NPrimeParam (a numPoints x numDim x numNodes array), and the node + coordinates at each element (a numElements x numNodes x numDim array), we want to compute: + + `NPrimeParam[jj] @ nodeCoords[ii]` + + For each element ii and parametric point jj, this function does this, but is vectorized by numba to make it fast. + + NOTE: It's very important to make sure that the arrays passed into this function are memory contiguous, + if you're not sure if they are then run numpy's `ascontiguousarray` function on them before passing them in + + Parameters + ---------- + NPrimeParam : numPoints x numDim x numNodes array + Shape function gradient values, NPrimeParam[i][j][k] is the derivative of the kth shape function at the ith point w.r.t the jth + parametric coordinate + nodeCoords : numElements x numNodes x numDim array + Node coordinates for each element + Jac : numElements x numPoints x numDim x numDim array + The Jacobians at each point in each element + """ + numElements = Jac.shape[0] + numPoints = Jac.shape[1] + for ii in prange(numElements): + for jj in range(numPoints): + Jac[ii, jj] = NPrimeParam[jj] @ nodeCoords[ii] + + +@guvectorize( + [(float64[:, :, :, ::1], float64[:, :, ::1], float64[:, :, ::1], float64[:, :, :, ::1])], + "(e,p,d,d),(p,d,n),(e,n,s)->(e,p,s,d)", + nopython=True, + cache=True, + fastmath=True, + boundscheck=False, + target="parallel", +) +def _computeUPrimeProduct(JacInv, NPrimeParam, nodeStates, result): + """Compute the nasty product of 3 and 4d arrays required when computing state gradients at multiple points within + multiple elements + + Given the shape function derivatives in the parametric coordinates at each point, NPrimeParam (a numPoints x numDim + x numNodes array), the inverses of the element mapping Jacobians at each point in each element, JacInv + (a numElements x numPoints x numDim x numDim array) and the nodal state values for each element, nodeStates + (a numElement x numNodes x numStates), we want to compute: + + `UPrime[ii, jj] = (JacInv[ii, jj] @ NPrimeParam[jj] @ nodeStates[ii]).T` + + For each element ii and parametric point jj, this function does this, but is vectorized by numba to make it fast. + + Parameters + ---------- + JacInv : numElements x numPoints x numDim x numDim + Inverse element mapping Jacobians at each point in each element + NPrimeParam : numPoints x numDim x numNodes array + Shape function gradients in the parametric coordinates at each point + nodeStates : numElements x numNodes x numStates array + Nodal state values for each element + result : numElements x numPoints x numStates x numDim array + The state gradient at each point in each element + """ + numElements = JacInv.shape[0] + numPoints = JacInv.shape[1] + for ii in prange(numElements): + for jj in range(numPoints): + result[ii, jj] = (JacInv[ii, jj] @ NPrimeParam[jj] @ nodeStates[ii]).T + + +@guvectorize( + [(float64[:, :, :, ::1], float64[:, :, ::1], float64[:, :, :, ::1])], + "(e,p,d,d),(p,d,n)->(e,p,d,n)", + nopython=True, + cache=True, + fastmath=True, + boundscheck=False, + target="parallel", +) +def _computeDUPrimeDqProduct(JacInv, NPrimeParam, result): + """_summary_ + + + + Parameters + ---------- + JacInv : numElements x numPoints x numDim x numDim + Inverse element mapping Jacobians at each point in each element + NPrimeParam : numPoints x numDim x numNodes array + Shape function gradients in the parametric coordinates at each point + result : numElements x numPoints x numDim x numNodes array + result[i, j, k, l] contains the sensitvity of du/dx_k at the jth point in the ith element to the state at the lth node + """ + numElements = JacInv.shape[0] + numPoints = JacInv.shape[1] + for ii in prange(numElements): + for jj in range(numPoints): + result[ii, jj] = JacInv[ii, jj] @ NPrimeParam[jj] + + +@njit(parallel=True, cache=True, fastmath=True) +def _transformResidualJacobians(stateGradSens, weakJacs, jacs): + """ + When computing the residual jacobians, we first compute the weak residual jacobians from the constitutive model, + which are the sensitivities of the weak residuals to the state gradient at each point. Then, we need to transform + those sensitivities to be with respect to the nodal states, to do that we need to compute the product: + $$dr/dq = du'/dq^T * dr/du' * du'/dq$$ + Where: + + - $q$ are the nodal states + - $u'$ are the state gradients at each point + - $r$ are the weak residuals + + This function computes this product, it is essentially an explicitly written version of the following numpy call :: + + Jacs = np.einsum( + "pdn,pdsSD,pDN->pnsNS", + pointQuantities["StateGradSens"], + weakJacs, + pointQuantities["StateGradSens"], + optimize=["einsum_path", (0, 1), (0, 1)], + ) + + Where the optimized path was computed by opt_einsum, opt_einsum's path summary is below:: + + Complete contraction: pdn,pdsSD,pDN->pnsNS + Naive scaling: 7 + Optimized scaling: 6 + Naive FLOP count: 2.212e+10 + Optimized FLOP count: 6.758e+9 + Theoretical speedup: 3.273e+0 + Largest intermediate: 8.192e+8 elements + -------------------------------------------------------------------------------- + scaling BLAS current remaining + -------------------------------------------------------------------------------- + 6 0 pdsSD,pdn->psSDn pDN,psSDn->pnsNS + 6 0 psSDn,pDN->pnsNS pnsNS->pnsNS + """ + numPoints = jacs.shape[0] + numDim = stateGradSens.shape[1] + numNodes = stateGradSens.shape[2] + numStates = jacs.shape[-1] + + intProd1 = np.zeros((numPoints, numStates, numStates, numDim, numNodes)) + # This loop does the pdsSD,pdn->psSDn bit + for p in prange(numPoints): + for s in prange(numStates): # + for S in range(numStates): + for D in range(numDim): + for n in range(numNodes): + for i in range(numDim): + intProd1[p, s, S, D, n] += weakJacs[p, i, s, S, D] * stateGradSens[p, i, n] + + # And this one does the psSDn,pDN->pnsNS bit + for p in prange(numPoints): + for n in prange(numNodes): + for s in range(numStates): + for N in range(numNodes): + for S in range(numStates): + for i in range(numDim): + jacs[p, n, s, N, S] += intProd1[p, s, S, i, n] * stateGradSens[p, i, N] + + +@guvectorize( + [(float64[:, :, ::1], float64[:, :, ::1], float64[:, :, ::1])], + "(p,d,n),(p,d,s)->(p,n,s)", + nopython=True, + cache=True, + fastmath=True, + boundscheck=False, + target="parallel", +) +def _transformResidual(dUPrimedq, weakRes, result): + """Compute a nasty product of high dimensional arrays to compute integration point residuals + + The weak residual computed by the constitutive model is essentially the derivative of the energy at each point with + respect to the state gradients, this function transforms this to the derivative of the energy with respect to the + nodal DOF, which is what we need to compute the element residual: + + r = dE/dq^T = dU'/dq^T @ dE/dU'^T + + Parameters + ---------- + dUPrimedq : (numElements * numPoints) x numDim x numNodes array + Sensitivity of the state gradients to nodal DOFs + weakRes : (numElements * numPoints) x numStates x numDim array + Weak residual values + result : (numElements * numPoints) x numNodes x numStates array + Integration point residual contributions + """ + numPoints = dUPrimedq.shape[0] + + for ii in prange(numPoints): + result[ii] = dUPrimedq[ii].T @ weakRes[ii] + + +@njit(cache=True, fastmath=True, parallel=True) +def _integrate(intPointQuantities, jacDets, intPointWeights, result): + """Given a set of quantities (scalar, vector, tensor) at each integration point in a set of elements, + integrate them to get a quantity for each element + + Parameters + ---------- + intPointQuantities : numElement x numIntPoint x ... array + The quantities to integrate + jacDets : numElement x numIntPoint array + Element mapping jacobian determinatnts at each integration point in each element + intPointWeights : array of length numIntPoint + Integration point weights, same for every element + result : numElement x ... array + Integrated quantities + """ + numElements = intPointQuantities.shape[0] + numPoints = intPointQuantities.shape[1] + + for e in prange(numElements): + for p in range(numPoints): + result[e] += intPointQuantities[e, p] * jacDets[e, p] * intPointWeights[p] + + +@njit(cache=True, fastmath=True, boundscheck=False) +def _computeMaxMinDistance(coords): + """Compute the maximum distance between any two points in a set of coordinates + + Parameters + ---------- + coords : numPoints x numDim array + The coordinates to compute the distances between + + Returns + ------- + maxDistance : float + The maximum distance between any two points in the set of coordinates + minDistance : float + The minimum distance between any two points in the set of coordinates + """ + numPoints = coords.shape[0] + maxDistance = 0.0 + minDistance = np.inf + for ii in range(numPoints): + for jj in range(ii + 1, numPoints): + distance = np.linalg.norm(coords[ii] - coords[jj]) + maxDistance = max(maxDistance, distance) + maxDistance = max(maxDistance, distance) + return maxDistance, minDistance diff --git a/FEMpy/Elements/HexElement3D.py b/FEMpy/Elements/HexElement3D.py new file mode 100644 index 0000000..15c73fa --- /dev/null +++ b/FEMpy/Elements/HexElement3D.py @@ -0,0 +1,188 @@ +""" +============================================================================== + +============================================================================== +@File : HexElement3D.py +@Date : 2022/12/06 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Basis import LagrangePoly as LP +from FEMpy.Elements.Element import Element +from FEMpy.Quadrature import getGaussQuadWeights, getGaussQuadPoints + + +class HexElement3D(Element): + """An "arbitrary order" 3d hexahedral finite element + + Like the QuadElement2D, the arbitrary order bit is in quotes because I have not figured out how to do the node + reordering from the shape function ordering to the node ordering used by MeshIO for anything more than 2nd order + hex elements yet + + Inherits from + ------------- + Element : FEMpy Element class + The base FEMpy element class + """ + + def __init__(self, order=1, numStates=None, quadratureOrder=None): + """Create a new 3d hexahedral finite element object + + Parameters + ---------- + order : int, optional + Element order, a first order hex has 6 nodes, 2nd order 27 etc, currently only orders 1-3 are + supported, by default 1 + numStates : int, optional + Number of states in the underlying PDE, by default 3 + quadratureOrder : int, optional + Quadrature order to use for numerical integration, by default None, in which case a valid order for the + chosen element order is used + + Raises + ------ + ValueError + Raises error if order is not 1, 2 or 3 + """ + if order not in [1, 2]: + raise ValueError("Hex elements only support orders 1 and 2") + self.order = order + numNodes = (order + 1) ** 3 + if quadratureOrder is None: + shapeFuncOrder = 2 * order + quadratureOrder = int(np.ceil((shapeFuncOrder + 1) / 2)) + super().__init__(numNodes, numDim=3, quadratureOrder=quadratureOrder, numStates=numStates) + + self.name = f"Order{self.order}-LagrangeHex" + + self.shapeFuncToNodeOrder = self._getNodeReordering(self.order) + + # ============================================================================== + # Public methods + # ============================================================================== + + def computeShapeFunctions(self, paramCoords): + """Compute the shape function values at a given set of parametric coordinates + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape functions at + + Returns + ------- + N: numPoint x numNodes array + Array of shape function values at the given parametric coordinates, N[i][j] is the value of the jth shape function at the ith parametric point + """ + N = LP.LagrangePoly3d(paramCoords[:, 0], paramCoords[:, 1], paramCoords[:, 2], self.order + 1) + return np.ascontiguousarray(N[:, self.shapeFuncToNodeOrder]) + + def computeShapeFunctionGradients(self, paramCoords): + """Compute the derivatives of the shape functions with respect to the parametric coordinates at a given set of parametric coordinates + + + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape function gradients at + + Returns + ------- + NGrad: numPoint x numDim x numNodes array + Shape function gradient values, NGrad[i][j][k] is the value of the kth shape function at the ith point w.r.t the kth + parametric coordinate + """ + NPrimeParam = LP.LagrangePoly3dDeriv(paramCoords[:, 0], paramCoords[:, 1], paramCoords[:, 2], self.order + 1) + return np.ascontiguousarray(NPrimeParam[:, :, self.shapeFuncToNodeOrder]) + + def getIntegrationPointWeights(self, order=None): + """Compute the integration point weights for a given quadrature order on this element + + Parameters + ---------- + order : int + Integration order + + Returns + ------- + array of length numIntpoint + Integration point weights + """ + if order is None: + order = self.quadratureOrder + return getGaussQuadWeights(self.numDim, order) + + def getIntegrationPointCoords(self, order=None): + """Compute the integration point parameteric coordinates for a given quadrature order on this element + + Parameters + ---------- + order : int + Integration order + + Returns + ------- + numIntpoint x numDim array + Integration point coordinates + """ + if order is None: + order = self.quadratureOrder + return getGaussQuadPoints(self.numDim, order) + + def getReferenceElementCoordinates(self): + """Get the node coordinates for the reference element, a.k.a the element on which the shape functions are defined + + For the quad element, the nodes of the reference element are simply in a order+1 x order+1 grid over the range [-1, 1] in both x and y, reordered as described by _getNodeReordering + + Returns + ------- + numNodes x numDim array + Element node coordinates + """ + numPoints = self.order + 1 + p = np.linspace(-1, 1, numPoints) + x = np.tile(p, numPoints**2) + y = np.tile(np.repeat(p, numPoints), numPoints) + z = np.repeat(p, numPoints**2) + return np.vstack((x[self.shapeFuncToNodeOrder], y[self.shapeFuncToNodeOrder], z[self.shapeFuncToNodeOrder])).T + + # ============================================================================== + # Private methods + # ============================================================================== + + @staticmethod + def _getNodeReordering(order): + """Compute the reordering required between shape functions and nodes + + The 23d lagrange polynomial shape functions are ordered left to right, front to back and then bottom to top, but the node ordering is defined differently in finite element meshes. This method computes the reordering required to map the shape functions to the correct node ordering. As of now I have simply manually implemented this for the first few orders, but it should be possible to compute this for any order with some sort of recursion. + + Parameters + ---------- + order : int + Quad element order + + Returns + ------- + np.array + Reordering array, array[i] = j indicates that the ith shape function should be reordered to the jth node + """ + if order == 1: + return np.array([0, 1, 3, 2, 4, 5, 7, 6]) + if order == 2: + return np.array( + [0, 2, 8, 6, 18, 20, 26, 24, 1, 5, 7, 3, 19, 23, 25, 21, 9, 11, 17, 15, 12, 14, 10, 16, 4, 22, 13] + ) diff --git a/FEMpy/Elements/QuadElement2D.py b/FEMpy/Elements/QuadElement2D.py new file mode 100644 index 0000000..5c4bb7f --- /dev/null +++ b/FEMpy/Elements/QuadElement2D.py @@ -0,0 +1,186 @@ +""" +============================================================================== +FEMpy: 2d Quad Element +============================================================================== +@File : newQuadElement.py +@Date : 2022/11/27 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Elements import Element +from FEMpy.Basis import LagrangePoly as LP +from FEMpy.Quadrature import getGaussQuadWeights, getGaussQuadPoints + + +class QuadElement2D(Element): + """An "arbitrary order" 2d quadrilateral finite element + + Arbitrary order is in quotes at the moment because although the shape functions can in theory be computed for an + arbitrary with the current LagrangePoly implementation, I have not figured out how to element the node reordering + required to reorder the shape functions into the node ordering used by MeshIO yet for anything more than 3rd order + elements. + + Inherits from + ------------- + Element : FEMpy.Element + The FEMpy element base class + """ + + def __init__(self, order=1, numStates=None, quadratureOrder=None): + """Create a new 2d quadrilateral finite element object + + Parameters + ---------- + order : int, optional + Element order, a first order quad has 4 nodes, 2nd order 9, 3rd order 16 etc, currently only orders 1-3 are + supported, by default 1 + numStates : int, optional + Number of states in the underlying PDE, by default 2 + quadratureOrder : int, optional + Quadrature order to use for numerical integration, by default None, in which case a valid order for the + chosen element order is used + + Raises + ------ + ValueError + Raises error if order is not 1, 2 or 3 + """ + if order not in [1, 2, 3]: + raise ValueError("Quad elements only support orders 1, 2 and 3") + self.order = order + numNodes = (order + 1) ** 2 + if quadratureOrder is None: + shapeFuncOrder = 2 * order + quadratureOrder = int(np.ceil((shapeFuncOrder + 1) / 2)) + super().__init__(numNodes, numDim=2, quadratureOrder=quadratureOrder, numStates=numStates) + + self.name = f"Order{self.order}-LagrangeQuad" + + self.shapeFuncToNodeOrder = self._getNodeReordering(self.order) + + # ============================================================================== + # Public methods + # ============================================================================== + + def computeShapeFunctions(self, paramCoords): + """Compute the shape function values at a given set of parametric coordinates + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape functions at + + Returns + ------- + N: numPoint x numNodes array + Array of shape function values at the given parametric coordinates, N[i][j] is the value of the jth shape function at the ith parametric point + """ + N = LP.LagrangePoly2d(paramCoords[:, 0], paramCoords[:, 1], self.order + 1) + return np.ascontiguousarray(N[:, self.shapeFuncToNodeOrder]) + + def computeShapeFunctionGradients(self, paramCoords): + """Compute the derivatives of the shape functions with respect to the parametric coordinates at a given set of parametric coordinates + + + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape function gradients at + + Returns + ------- + NGrad: numPoint x numDim x numNodes array + Shape function gradient values, NGrad[i][j][k] is the value of the kth shape function at the ith point w.r.t the kth + parametric coordinate + """ + NPrimeParam = LP.LagrangePoly2dDeriv(paramCoords[:, 0], paramCoords[:, 1], self.order + 1) + return np.ascontiguousarray(NPrimeParam[:, :, self.shapeFuncToNodeOrder]) + + def getIntegrationPointWeights(self, order=None): + """Compute the integration point weights for a given quadrature order on this element + + Parameters + ---------- + order : int + Integration order + + Returns + ------- + array of length numIntpoint + Integration point weights + """ + if order is None: + order = self.quadratureOrder + return getGaussQuadWeights(self.numDim, order) + + def getIntegrationPointCoords(self, order=None): + """Compute the integration point parameteric coordinates for a given quadrature order on this element + + Parameters + ---------- + order : int + Integration order + + Returns + ------- + numIntpoint x numDim array + Integration point coordinates + """ + if order is None: + order = self.quadratureOrder + return getGaussQuadPoints(self.numDim, order) + + def getReferenceElementCoordinates(self): + """Get the node coordinates for the reference element, a.k.a the element on which the shape functions are defined + + For the quad element, the nodes of the reference element are simply in a order+1 x order+1 grid over the range [-1, 1] in both x and y, reordered as described by _getNodeReordering + + Returns + ------- + numNodes x numDim array + Element node coordinates + """ + x = np.tile(np.linspace(-1, 1, self.order + 1), self.order + 1) + y = np.repeat(np.linspace(-1, 1, self.order + 1), self.order + 1) + return np.vstack((x[self.shapeFuncToNodeOrder], y[self.shapeFuncToNodeOrder])).T + + # ============================================================================== + # Private methods + # ============================================================================== + + @staticmethod + def _getNodeReordering(order): + """Compute the reordering required between shape functions and nodes + + The 2d lagrange polynomial shape functions are ordered left to right, bottom to top, but the node ordering is defined differently, (e.g for a four node element it is bottom left, bottom right, top left, top right). This method computes the reordering required to map the shape functions to the correct node ordering. As of now I have simply manually implemented this for the first few orders, but it should be possible to compute this for any order with some sort of recursion. + + Parameters + ---------- + order : int + Quad element order + + Returns + ------- + np.array + Reordering array, array[i] = j indicates that the ith shape function should be reordered to the jth node + """ + if order == 1: + return np.array([0, 1, 3, 2]) + if order == 2: + return np.array([0, 2, 8, 6, 1, 5, 7, 3, 4]) + if order == 3: + return np.array([0, 3, 15, 12, 1, 2, 7, 11, 14, 13, 8, 4, 5, 6, 10, 9]) diff --git a/FEMpy/Elements/TriElement2D.py b/FEMpy/Elements/TriElement2D.py new file mode 100644 index 0000000..ef27f46 --- /dev/null +++ b/FEMpy/Elements/TriElement2D.py @@ -0,0 +1,223 @@ +""" +============================================================================== +FEMpy 2D Tri element +============================================================================== +@File : TriElement2D.py +@Date : 2022/12/04 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from .Element import Element +from FEMpy.Basis import LagrangePoly as LP +from FEMpy.Quadrature import getTriQuadPoints, getTriQuadWeights + + +class TriElement2D(Element): + """ + An up-to-3rd-order 2d triangular element with 3, 6 or 10 nodes respectively. + + The node numbering follows the meshio conventions shown below: + + 1st order element:: + + 2 + |\\ + | \\ + | \\ + | \\ + | \\ + 0-----1 + + 2nd order element:: + + 2 + |\\ + | \\ + 5 4 + | \\ + | \\ + 0--3--1 + + 3rd Order:: + + 2 + |\\ + | \\ + 7 6 + | \\ + | \\ + 8 9 5 + | \\ + | \\ + 0--3--4--1 + + """ + + def __init__(self, order=1, numStates=None, quadratureOrder=None): + """Create a new 2d triangular finite element object + + Parameters + ---------- + order : int, optional + Element order, a first order quad has 4 nodes, 2nd order 9, 3rd order 16 etc, currently only orders 1-3 are + supported, by default 1 + numStates : int, optional + Number of states in the underlying PDE, by default 2 + quadratureOrder : int, optional + Quadrature order to use for numerical integration, by default None, in which case a valid order for the + chosen element order is used + + Raises + ------ + ValueError + Raises error if order is not 1, 2 or 3 + """ + if order not in [1, 2, 3]: + raise ValueError("Triangular elements only support orders 1, 2 and 3") + self.order = order + numNodes = (order + 1) * (order + 2) // 2 + if quadratureOrder is None: + shapeFuncOrder = order + quadratureOrder = int(np.ceil((shapeFuncOrder + 1) / 2)) + + super().__init__(numNodes, numDim=2, quadratureOrder=quadratureOrder, numStates=numStates) + + self.name = f"Order{self.order}-LagrangeTri" + + # --- Define parametric coordinate bounds --- + self.paramCoordLowerBounds = -np.zeros(self.numDim) + self.paramCoordLinearConstaintMat = np.array([1.0, 1.0]) + self.paramCoordLinearConstaintUpperBounds = 1.0 + self.paramCoordLinearConstaintLowerBounds = -np.inf + + def computeShapeFunctions(self, paramCoords): + """Compute the shape function values at a given set of parametric coordinates + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape functions at + + Returns + ------- + N: numPoint x numNodes array + Array of shape function values at the given parametric coordinates, N[i][j] is the value of the jth shape function at the ith parametric point + """ + return LP.LagrangePolyTri(paramCoords[:, 0], paramCoords[:, 1], self.order) + + def computeShapeFunctionGradients(self, paramCoords): + """Compute the derivatives of the shape functions with respect to the parametric coordinates at a given set of parametric coordinates + + Parameters + ---------- + paramCoords : numPoint x numDim array + Array of parametric point coordinates to evaluate shape function gradients at + + Returns + ------- + NGrad: numPoint x numDim x numNodes array + Shape function gradient values, NGrad[i][j][k] is the value of the kth shape function at the ith point w.r.t the kth + parametric coordinate + """ + return LP.LagrangePolyTriDeriv(paramCoords[:, 0], paramCoords[:, 1], self.order) + + def getIntegrationPointWeights(self, order=None): + """Compute the integration point weights for a given quadrature order on this element + + Parameters + ---------- + order : int + Integration order + + Returns + ------- + array of length numIntpoint + Integration point weights + """ + if order is None: + order = self.quadratureOrder + return getTriQuadWeights(order) + + def getIntegrationPointCoords(self, order=None): + """Compute the integration point parameteric coordinates for a given quadrature order on this element + + Parameters + ---------- + order : int + Integration order + + Returns + ------- + numIntpoint x numDim array + Integration point coordinates + """ + if order is None: + order = self.quadratureOrder + return getTriQuadPoints(order) + + def getReferenceElementCoordinates(self): + """Get the node coordinates for the reference element, a.k.a the element on which the shape functions are defined + + Returns + ------- + numNodes x numDim array + Element node coordinates + """ + if self.order == 1: + return np.array([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]]) + elif self.order == 2: + return np.array([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [0.5, 0.0], [0.5, 0.5], [0.0, 0.5]]) + elif self.order == 3: + return np.array( + [ + [0.0, 0.0], + [1.0, 0.0], + [0.0, 1.0], + [1 / 3, 0.0], + [2 / 3, 0.0], + [2 / 3, 1 / 3], + [1 / 3, 2 / 3], + [0.0, 2 / 3], + [0.0, 1 / 3], + [1 / 3, 1 / 3], + ] + ) + + def getRandParamCoord(self, n, rng=None): + """Generate a set of random parametric coordinates + For a tri element we need u and v in range [0,1] and u + v <= 1, we can generate these points by generating + random points in a square on the domain [0,1] and then reflecting any points outside the triangle to the inside. + Parameters + ---------- + n : int, optional + number of points to generate, by default 1 + rng : numpy random Generator, optional + Random number generator to use, useful for creating consistent test behaviour, by default None, in which + case a new one is created for this call + + Returns + ------- + paramCoords : n x numDim array + isoparametric coordinates, one row for each point + """ + if rng is None: + rng = np.random.default_rng() + coords = np.atleast_2d(rng.random((n, 2))) + for i in range(n): + coordSum = coords[i, 0] + coords[i, 1] + if coordSum > 1: + coords[i] = 1 - coords[i] + return coords diff --git a/FEMpy/Elements/__init__.py b/FEMpy/Elements/__init__.py new file mode 100644 index 0000000..787243e --- /dev/null +++ b/FEMpy/Elements/__init__.py @@ -0,0 +1,4 @@ +from .Element import Element +from .QuadElement2D import QuadElement2D +from .TriElement2D import TriElement2D +from .HexElement3D import HexElement3D diff --git a/FEMpy/GaussQuad.py b/FEMpy/GaussQuad.py deleted file mode 100644 index 8d47089..0000000 --- a/FEMpy/GaussQuad.py +++ /dev/null @@ -1,188 +0,0 @@ -""" -============================================================================== -Gauss Quadrature Points and Weights -============================================================================== -@File : guassQuad.py -@Date : 2021/03/10 -@Author : Alasdair Christison Gray -@Description : This file contains the coordinates and weights (to far too many decimal places) for numerical integration -on the domain (-1, 1) using Gauss Quadrature. Weights and coordinates are given to 256 decimal places for up to 64 point -integration which can integrate up to 127th order polynomials exactly. These values were taken from: - -https://pomax.github.io/bezierinfo/legendre-gauss.html - -To use these values include `from guassQuad import getGaussPoints, getgaussWeights` in your code and then use -`getGaussPoints(n)`, `getGaussPoints(n)` to get lists of the points and weights for n-point integration. -""" - -import numpy as np -import pickle -import os - -# --- Load the Gauss quadrature weights and coordinates --- -dataDir = os.path.dirname(os.path.abspath(__file__)) -with open(os.path.join(dataDir, "GaussQuadWeights.pkl"), "rb") as f: - gaussWeights = pickle.load(f) -with open(os.path.join(dataDir, "GaussQuadCoords.pkl"), "rb") as f: - gaussCoords = pickle.load(f) - - -def getgaussWeights(n): - """Get the weights for n-point numerical integration using Gauss Quadrature - - Gauss Quadrature integration with n points will exactly integrate polynomials of order <= 2n-1 - - Parameters - ---------- - n : int - number of integration points - - Returns - ------- - array - array of n point weights - """ - return gaussWeights[n - 1] - - -def getGaussPoints(n): - """Get the coordinates for n-point numerical integration using Gauss Quadrature on the interval (-1, 1)) - - Gauss Quadrature integration with n points will exactly integrate polynomials of order <= 2n-1 - - Parameters - ---------- - n : int - number of integration points - - Returns - ------- - array - array of n point coordinates - """ - return gaussCoords[n - 1] - - -def gaussQuad1d(f, n, a=-1.0, b=1.0): - """Perform a one dimensional integration using Gauss Quadrature - - Parameters - ---------- - f : function - The function to be integrated, should be able to accept vectorised input - n : int - Number of points to use - a : float, optional - Lower limit of integration, by default -1.0 - b : float, optional - Upper limit of integration, by default 1.0 - - Returns - ------- - return type of f - The integrand - """ - scale = 0.5 * (b - a) - offset = 0.5 * (b + a) - xGauss = getGaussPoints(n) - xReal = scale * xGauss + offset - W = getgaussWeights(n) - - return scale * np.sum(f(xReal).T * W, axis=-1).T - - -def gaussQuad2d(f, n, a=-1.0, b=1.0): - """Perform a two dimensional integration using Gauss Quadrature - - Parameters - ---------- - f : function - The function to be integrated, should be able to accept vectorised input in each dimension - n : int or list of ints - Number of gauss quadrature points, can be different in each dimension - a : float or list of floats, optional - Lower integration limit can be different in each dimension, by default -1.0 for all dimensions - b : float or lists of floats, optional - Upper integration limit can be different in each dimension, by default 1.0 for all dimensions - - Returns - ------- - return type of f - The integrand - """ - if isinstance(n, int): - n = [n] * 2 - if isinstance(a, float): - a = [a] * 2 - if isinstance(b, float): - b = [b] * 2 - - Eta = [] - W = [] - scale = [] - offset = [] - for i in range(1): - Eta.append(np.array(getGaussPoints(n[i]))) - W.append(np.array(getgaussWeights(n[i]))) - scale.append(np.array(0.5 * (b[i] - a[i]))) - offset.append(np.array(0.5 * (a[i] + b[i]))) - - # --- Initialize the integrand, doesn't matter if function returns an array as scalar + array = array --- - intF = 0.0 - - for i in range(n[0]): - x1 = Eta[0][i] * scale[0] + offset[0] - func = lambda x2: f(x1 * np.ones_like(x2), x2) # noqa: E731 - intF += scale[0] * W[0][i] * gaussQuad1d(func, n[-1], a[-1], b[-1]) - - return intF - - -def gaussQuad3d(f, n, a=-1.0, b=1.0): - """Perform a two dimensional integration using Gauss Quadrature - - Parameters - ---------- - f : function - The function to be integrated, should be able to accept vectorised input in each dimension - n : int or list of ints - Number of gauss quadrature points, can be different in each dimension - a : float or list of floats, optional - Lower integration limit can be different in each dimension, by default -1.0 for all dimensions - b : float or lists of floats, optional - Upper integration limit can be different in each dimension, by default 1.0 for all dimensions - - Returns - ------- - return type of f - The integrand - """ - if isinstance(n, int): - n = [n] * 3 - if isinstance(a, float): - a = [a] * 3 - if isinstance(b, float): - b = [b] * 3 - - Eta = [] - W = [] - scale = [] - offset = [] - for i in range(2): - Eta.append(np.array(getGaussPoints(n[i]))) - W.append(np.array(getgaussWeights(n[i]))) - scale.append(np.array(0.5 * (b[i] - a[i]))) - offset.append(np.array(0.5 * (a[i] + b[i]))) - totalScale = scale[0] * scale[1] - - # --- Initialize the integrand, doesn't matter if function returns an array as scalar + array = array --- - intF = 0.0 - - for i in range(n[0]): - x1 = Eta[0][i] * scale[0] + offset[0] - for j in range(n[1]): - x2 = Eta[1][j] * scale[1] + offset[1] - func = lambda x3: f(x1 * np.ones_like(x3), x2 * np.ones_like(x3), x3) # noqa: E731 - intF += totalScale * W[0][i] * W[1][j] * gaussQuad1d(func, n[-1], a[-1], b[-1]) - - return intF diff --git a/FEMpy/Lagrange1dElement.py b/FEMpy/Lagrange1dElement.py deleted file mode 100644 index 1b7a892..0000000 --- a/FEMpy/Lagrange1dElement.py +++ /dev/null @@ -1,99 +0,0 @@ -""" -============================================================================== -1D Lagrangian Element -============================================================================== -@File : Lagrange1dElement.py -@Date : 2021/03/30 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== -from FEMpy.Element import Element -from .LagrangePoly import LagrangePoly1d, LagrangePoly1dDeriv - - -class Lagrange1dElement(Element): - """An arbitrary order 1D finite element using Lagrange polynomial shape functions""" - - def __init__(self, order): - """Initialise an arbitrary order 1D finite element - - Parameters - ---------- - order : int - Shape function polynomial order - """ - self.order = order - super().__init__(numNodes=order + 1, numDimensions=1) - self.name = f"Order{self.order}-Lagrange1D" - - def getShapeFunctions(self, paramCoords): - """Compute shape function values at a set of parametric coordinates - - This function returns a zero array and should be re-implemented in any child classes - - Parameters - ---------- - paramCoords : n x nDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - N : n x numNode array - Shape function values, N[i][j] is the value of the jth shape function at the ith point - """ - return LagrangePoly1d(paramCoords, n=self.order + 1) - - def getShapeFunctionDerivs(self, paramCoords): - """Compute shape function derivatives at a set of parametric coordinates - - These are the derivatives of the shape functions with respect to the parametric coordinates (si, eta, gamma) - - This function returns a zero array and should be re-implemented in any child classes - - Parameters - ---------- - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - NPrime : n x numDim x numNode array - Shape function values, N[i][j] is the value of the jth shape function at the ith point - """ - NPrime = np.zeros((np.shape(paramCoords)[0], np.shape(paramCoords)[1], self.numNodes)) - NPrime[:, 0, :] = LagrangePoly1dDeriv(paramCoords, n=self.order + 1) - return NPrime - - def getStiffnessMat(self, nodeCoords, constitutive, n=None): - return super().getStiffnessMat(nodeCoords, constitutive, n=n) * constitutive.A - - def getMassMat(self, nodeCoords, constitutive, n=None): - return super().getMassMat(nodeCoords, constitutive, n=n) * constitutive.A - - def getRandomNodeCoords(self): - """Generate a random, but valid, set of node coordinates for an element - - For a 1D element, we simply create evenly spaced points from 0 to one, add some slight random noise and then - apply a random scaling factor - - Returns - ------- - nodeCoords : numNode x numDim array - Node coordinates - """ - nodeCoords = np.random.rand(self.numNodes, 1) * 0.1 - nodeCoords[:, 0] += np.linspace(0, 1, self.numNodes) - return nodeCoords * np.random.rand(1) diff --git a/FEMpy/LagrangePoly.py b/FEMpy/LagrangePoly.py deleted file mode 100644 index 4c27146..0000000 --- a/FEMpy/LagrangePoly.py +++ /dev/null @@ -1,162 +0,0 @@ -""" -============================================================================== -Lagrange polynomial interpolation -============================================================================== -@File : LagrangePoly.py -@Date : 2021/03/12 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from numba import njit - -# ============================================================================== -# Extension modules -# ============================================================================== - - -@njit(cache=True) -def LagrangePoly1d(x, n): - """Compute the values of the 1d Lagrange polynomials at a series of points - - - - Parameters - ---------- - x : array of length nP (0D, nPx1 or 1xnP) - Points to compute polynomial values at, should be between -1 and 1 - n : int - Number of Lagrange polynomials, equal to the number of nodes along the edge, n points results in n-1 order - polynomials. - - Returns - ------- - N : nP x n array - Shape function values - """ - xp = x.flatten() - xi = np.linspace(-1.0, 1.0, n) - N = np.ones((len(xp), n), dtype=xp.dtype) - for m in range(n): - for i in list(range(m)) + list(range(m + 1, n)): - N[:, m] *= (xp - xi[i]) / (xi[m] - xi[i]) - # print("computing lagrange shape funcs") - return N - - -@njit(cache=True) -def LagrangePoly1dDeriv(x, n): - """Compute the derivatives of the 1d Lagrange polynomials at a series of points - - This method computes the derivates using the complex step method because I couldn't be bothered implementing the - analytical derivatives - - Parameters - ---------- - x : array of length nP (0D, nPx1 or 1xnP) - Points to compute polynomial values at, should be between -1 and 1 - n : int - Number of Lagrange polynomials, equal to the number of nodes along the edge, n points results in n-1 order - polynomials. - - Returns - ------- - dNdx : nP x n array - Shape function derivative values - """ - xp = x.flatten() + 1j * 1e-200 - return 1e200 * np.imag(LagrangePoly1d(xp, n)) - - -@njit(cache=True) -def LagrangePoly2d(x, y, n): - """Compute the derivatives of the 2d Lagrange polynomials at a series of points in 2d space - - - - Parameters - ---------- - x : array of length nP (0D, nPx1 or 1xnP) - x coordinates of points to compute polynomial values at, should be between -1 and 1 - y : array of length nP (0D, nPx1 or 1xnP) - y coordinates of points to compute polynomial values at, should be between -1 and 1 - n : int - Number of Lagrange polynomials, in 2d, there are n nodes in each direction, giving n^2, n-1 order polynomials - - Returns - ------- - N : nP x n array - Shape function values - """ - xp = x.flatten() - yp = y.flatten() - Nx = LagrangePoly1d(xp, n) - Ny = LagrangePoly1d(yp, n) - N = np.zeros((len(xp), n**2), dtype=xp.dtype) - for i in range(len(xp)): - for j in range(n): - for k in range(n): - N[i, j * n + k] = Nx[i, k] * Ny[i, j] - return N - - -@njit(cache=True) -def LagrangePoly2dDeriv(x, y, n): - """Compute the derivatives of the 2d Lagrange polynomials at a series of points in 2d space - - - - Parameters - ---------- - x : array of length nP (0D, nPx1 or 1xnP) - x coordinates of points to compute polynomial values at, should be between -1 and 1 - y : array of length nP (0D, nPx1 or 1xnP) - y coordinates of points to compute polynomial values at, should be between -1 and 1s - n : int - Number of Lagrange polynomials, in 2d, there are n nodes in each direction, giving n^2, n-1 order polynomials - - Returns - ------- - dNdx : nP x 2 x n array - Shape function derivative values - """ - xp = x.flatten() - yp = y.flatten() - Nx = LagrangePoly1d(xp, n) - Ny = LagrangePoly1d(yp, n) - dNdx = LagrangePoly1dDeriv(xp, n) - dNdy = LagrangePoly1dDeriv(yp, n) - N = np.zeros((len(xp), 2, n**2)) - for i in range(len(xp)): - for j in range(n): - for k in range(n): - N[i, 0, j * n + k] = dNdx[i, k] * Ny[i, j] - N[i, 1, j * n + k] = Nx[i, k] * dNdy[i, j] - return N - - -if __name__ == "__main__": - import time - - x = np.array([np.linspace(-1.0, 1.0, 4)]) - y = -np.ones_like(x) - # Call functions so they get jit compiled and we can check their output - print(LagrangePoly1d(x, 3), "\n") - print(LagrangePoly1dDeriv(x, 3), "\n") - print(LagrangePoly2d(x, y, 3), "\n") - print(LagrangePoly2dDeriv(x, y, 3), "\n\n\n") - startTime = time.time() - for _ in range(1000): - LagrangePoly1d(x, 3) - LagrangePoly1dDeriv(x, 3) - LagrangePoly2d(x, y, 3) - LagrangePoly2dDeriv(x, y, 3) - print(time.time() - startTime) diff --git a/FEMpy/LinAlg.py b/FEMpy/LinAlg/LinAlg.py similarity index 56% rename from FEMpy/LinAlg.py rename to FEMpy/LinAlg/LinAlg.py index 0dc9932..687a90e 100644 --- a/FEMpy/LinAlg.py +++ b/FEMpy/LinAlg/LinAlg.py @@ -1,6 +1,55 @@ -from numba import njit +""" +============================================================================== +FEMpy linear algebra module +============================================================================== +@File : LinAlg.py +@Date : 2022/12/07 +@Author : Alasdair Christison Gray +@Description : This module contains function for computing determinants and +inverses of 1x1, 2x2 and 3x3 matrices waaaaay faster than numpy. This is very +useful inside the element classes for mapping between the real and reference +elements. +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +from numba import njit, prange import numpy as np +# ============================================================================== +# Extension modules +# ============================================================================== + + +@njit(cache=True) +def convertTo3D(A): + """Convert a numpy array from (n+2)D to 3D by flattening the first n dimensions. + + Parameters + ---------- + A : (n+2)D numpy array + Array to convert + + Returns + ------- + 3D numpy array + The converted 3D array + int + The new flattened first dimension + """ + origShape = A.shape + n = 1 + for ii in range(len(origShape) - 2): + n *= origShape[ii] + Anew = A.reshape(n, *origShape[-2:]) + + return Anew, n + def det1(A): """Compute the determinants of a series of 1x1 matrices. @@ -15,53 +64,56 @@ def det1(A): dets : array of length n Matrix determinants """ - return A.flatten() + return np.ascontiguousarray(A.reshape(A.shape[:-2])) -@njit(cache=True) +@njit(cache=True, fastmath=True, parallel=True) def det2(A): """Compute the determinants of a series of 2x2 matrices. Parameters ---------- - A : nx2x2 array_like - Arrays to compute determinants of + A : nx...x2x2 array_like + Multidimensional array of 2x2 arrays to compute determinants of Returns ------- - dets : array of length n + dets : nxmx... array Matrix determinants """ - n = np.shape(A)[0] + origShape = A.shape + A, n = convertTo3D(A) + dets = np.zeros(n) - for i in range(n): + for i in prange(n): dets[i] = A[i, 0, 0] * A[i, 1, 1] - A[i, 0, 1] * A[i, 1, 0] - return dets + return np.ascontiguousarray(dets.reshape(origShape[:-2])) -@njit(cache=True) +@njit(cache=True, fastmath=True, parallel=True) def det3(A): """Compute the determinants of a series of 3x3 matrices. Parameters ---------- - A : nx3x3 array_like - Arrays to compute determinants of + A : nx...x2x2 array_like + Multidimensional array of 3x3 arrays to compute determinants of Returns ------- dets : array of length n Matrix determinants """ - n = np.shape(A)[0] + origShape = A.shape + A, n = convertTo3D(A) dets = np.zeros(n) - for i in range(n): + for i in prange(n): dets[i] = ( A[i, 0, 0] * (A[i, 1, 1] * A[i, 2, 2] - A[i, 1, 2] * A[i, 2, 1]) - A[i, 0, 1] * (A[i, 1, 0] * A[i, 2, 2] - A[i, 1, 2] * A[i, 2, 0]) + A[i, 0, 2] * (A[i, 1, 0] * A[i, 2, 1] - A[i, 1, 1] * A[i, 2, 0]) ) - return dets + return np.ascontiguousarray(dets.reshape(origShape[:-2])) def inv1(A): @@ -80,49 +132,51 @@ def inv1(A): return 1.0 / A -@njit(cache=True) +@njit(cache=True, fastmath=True, parallel=True) def inv2(A): """Compute the inverses of a series of 2x2 matrices. Parameters ---------- - A : nx2x2 array_like - Arrays to compute inverses of + A : nx...x2x2 array_like + Multidimensional array of 2x2 arrays to compute inverses of Returns ------- dets : nx2x2 array Matrix inverses """ + origShape = A.shape + A, n = convertTo3D(A) invdets = 1.0 / det2(A) - n = len(invdets) invs = np.zeros((n, 2, 2)) - for i in range(n): + for i in prange(n): invs[i, 0, 0] = invdets[i] * A[i, 1, 1] invs[i, 1, 1] = invdets[i] * A[i, 0, 0] invs[i, 0, 1] = -invdets[i] * A[i, 0, 1] invs[i, 1, 0] = -invdets[i] * A[i, 1, 0] - return invs + return np.ascontiguousarray(invs.reshape(origShape)) -@njit(cache=True) +@njit(cache=True, fastmath=True, parallel=True) def inv3(A): """Compute the inverses of a series of 3x3 matrices. Parameters ---------- - A : nx3x3 array_like - Arrays to compute inverses of + A : nx...x2x2 array_like + Multidimensional array of 3x3 arrays to compute inverses of Returns ------- dets : nx3x3 array Matrix inverses """ + origShape = A.shape + A, n = convertTo3D(A) invdets = 1.0 / det3(A) - n = len(invdets) invs = np.zeros((n, 3, 3)) - for i in range(n): + for i in prange(n): invs[i, 0, 0] = invdets[i] * (A[i, 1, 1] * A[i, 2, 2] - A[i, 1, 2] * A[i, 2, 1]) invs[i, 0, 1] = -invdets[i] * (A[i, 0, 1] * A[i, 2, 2] - A[i, 0, 2] * A[i, 2, 1]) invs[i, 0, 2] = invdets[i] * (A[i, 0, 1] * A[i, 1, 2] - A[i, 0, 2] * A[i, 1, 1]) @@ -132,7 +186,7 @@ def inv3(A): invs[i, 2, 0] = invdets[i] * (A[i, 1, 0] * A[i, 2, 1] - A[i, 1, 1] * A[i, 2, 0]) invs[i, 2, 1] = -invdets[i] * (A[i, 0, 0] * A[i, 2, 1] - A[i, 0, 1] * A[i, 2, 0]) invs[i, 2, 2] = invdets[i] * (A[i, 0, 0] * A[i, 1, 1] - A[i, 0, 1] * A[i, 1, 0]) - return invs + return np.ascontiguousarray(invs.reshape(origShape)) if __name__ == "__main__": diff --git a/FEMpy/LinAlg/__init__.py b/FEMpy/LinAlg/__init__.py new file mode 100644 index 0000000..ca3053b --- /dev/null +++ b/FEMpy/LinAlg/__init__.py @@ -0,0 +1 @@ +from .LinAlg import * diff --git a/FEMpy/Mesh.py b/FEMpy/Mesh.py deleted file mode 100644 index 803f802..0000000 --- a/FEMpy/Mesh.py +++ /dev/null @@ -1,126 +0,0 @@ -""" -============================================================================== -Mesh Utilities -============================================================================== -@File : Mesh.py -@Date : 2021/04/08 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from numba import njit - -# ============================================================================== -# Extension modules -# ============================================================================== - - -def makeNodeElsMat(Conn): - """Generate data structure listing elements connected to each node - - - - Parameters - ---------- - Conn : 2D iterable - Mesh connectivity data, Conn[i][j] is the index of the jth node in the ith element - - Returns - ------- - nodeEls : list of lists - List containing indices of each element a node is used by, i.e nodeEls[i] is a list of the elements using node i - """ - numEl = np.shape(Conn)[0] - numNode = np.max(Conn) + 1 - nodeEls = [[] for _ in range(numNode)] - for row in range(numEl): - for n in Conn[row]: - nodeEls[n].append(row) - return nodeEls - - -def getEdgesfromNodes(nodes, conn, nodeEls, edgeInds): - """Compute the edges connecting a set of nodes - - Currently assumes all elements are same type - - Parameters - ---------- - nodes : list or array - node indices - conn : 2d array - Element connectivity matrix - nodeEls : list of lists - Node element data structure, generated by `makeNodeElsMat` - edgeInds : list of lists - edgeInds[i] contains the local indices of the nodes that make up the ith edge of the element type, for example, - for a 4 node quad where the nodes are ordered LL, LR, UL, UR, then the bottom, right, upper and left edges are - given by: edgeInds = [[0, 1], [1, 3], [3, 2], [2, 0]] - - Returns - ------- - Elements : list - Indices of elements containing edges described by the node set - Edges : list of lists - Edges[i] contains the indices of the edges of Elements[i] in set - """ - # --- Loop through nodes and create dict with element numbers as keys and element nodes in set as values --- - elNodes = {} - for node in nodes: - els = nodeEls[node] - for el in els: - if el in elNodes: - elNodes[el].append(node) - else: - elNodes[el] = [node] - - # --- Now loop through each of the elements we just found that have at least one node in the set and see if the nodes from the set in that element make up any edges --- - elEdges = {} - for e in elNodes: - for j in range(len(edgeInds)): - locEdgeNodes = edgeInds[j] - edgeNodes = conn[e][locEdgeNodes] - if all(i in nodes for i in edgeNodes): - if e in elEdges: - elEdges[e].append(j) - else: - elEdges[e] = [j] - Elements = [] - Edges = [] - for el, edge in elEdges.items(): - Elements.append(el) - Edges.append(edge) - return Elements, Edges - - -@njit(cache=True) -def computeElementCentroids(nodeCoords, conn): - """Compute element centroids - - - - Parameters - ---------- - nodeCoords : numNode x numDim array - Element node real coordinates - conn : 2D iterable - Mesh connectivity data, Conn[i][j] is the index of the jth node in the ith element - - Returns - ------- - centroids : numElement x numDim array - Element centroid coordinates - """ - centroids = np.zeros((conn.shape[0], np.shape(nodeCoords)[1])) - for i in range(conn.shape[0]): - els = conn[i] - centroids[i] = 1.0 / len(els) * np.sum(nodeCoords[els], axis=0) - return centroids diff --git a/FEMpy/Model.py b/FEMpy/Model.py new file mode 100644 index 0000000..cec3aec --- /dev/null +++ b/FEMpy/Model.py @@ -0,0 +1,509 @@ +""" +============================================================================== +FEMpy Model Class +============================================================================== +@File : Model.py +@Date : 2022/11/14 +@Author : Alasdair Christison Gray +@Description : This file contains the FEMpy model class, this is the main object +that the user interfaces with to read in a mesh and setup a finite element model. +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import os +from typing import Iterable, Union, Optional, Dict, Any +import copy +import warnings + +# ============================================================================== +# External Python modules +# ============================================================================== +import meshio +import numpy as np +from numba import njit +from baseclasses.solvers import BaseSolver + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy import Elements, FEMpyProblem +from FEMpy import __version__ + + +class FEMpyModel(BaseSolver): + """_summary_ + + A FEMpy model contains overall information about the finite element model, including but not limited to: + - The node coordinates + - The element-node associativity + - The constitutive model + - The constitutive model design variables + + It also contains methods for: + - Setting/Getting node coordinates + - Setting/Getting design variable values + """ + + # ============================================================================== + # Public methods + # ============================================================================== + + def __init__( + self, + constitutiveModel, + meshFileName: Optional[str] = None, + nodeCoords: Optional[np.ndarray] = None, + connectivity: Optional[Dict[str, np.ndarray]] = None, + options: Optional[dict] = None, + ) -> None: + """Create a FEMpy model, either from a mesh file or from node coordinate and connectivity arrays + + Parameters + ---------- + constitutiveModel : FEMpy constitutive model object + The constitutive model to use for the finite element model + meshFileName : str, optional + Filename of the mesh file to load + nodeCoords : numNodes x numDim array, optional + Array of node coordinates + connectivity : dictionary, optional + Dictionary of element-node connectivity arrays, e.g {"elementType1": array(numElements x numNodesPerElement), + "elementType2": array()}, where `elementType1` and `elementType2` are meshio element type names + options : dict, optional + Dictionary of options to pass to the model, by default None, for a list of options see the documentation + """ + + # --- Set solver options by getting the defaults and updating them with any that the user supplied --- + defaultOptions = self._getDefaultOptions() + if options is None: + options = {} + + # instantiate the solver + super().__init__("FEMpy", "FEA Solver", defaultOptions=defaultOptions, options=options) + + self._printWelcomeMessage() + + self.constitutiveModel = constitutiveModel + + if meshFileName is not None: + # --- Read in the mesh using meshio --- + mesh = meshio.read(meshFileName) + self.nodeCoords = copy.deepcopy(mesh.points) + self.connectivity = copy.deepcopy(mesh.cells_dict) + else: + if nodeCoords is None or connectivity is None: + raise Exception("Must supply either meshFileName or nodeCoords and connectivity") + self.nodeCoords = np.zeros((nodeCoords.shape[0], 3)) + self.nodeCoords[:, : nodeCoords.shape[1]] = nodeCoords + self.connectivity = copy.deepcopy(connectivity) + + # Extract mesh coordinates, detect whether the mesh is 1D, 2D or 3D and only keep the active dimensions + self.numNodes = self.nodeCoords.shape[0] + self.activeDimensions = [] + + # Keep track of the inactive coordinates so we can always convert back to full 3D coordinates if needed + self.inactiveDimensions = [] + for ii in range(3): + if self.nodeCoords[:, ii].max() != self.nodeCoords[:, ii].min(): + self.activeDimensions.append(ii) + else: + self.inactiveDimensions.append(ii) + + if self.numDim != len(self.activeDimensions): + raise ValueError( + f"You have chosen a {self.numDim}D constitutive model model but the mesh is {len(self.activeDimensions)}D" + ) + + self.nodeCoords = self.nodeCoords[:, self.activeDimensions] + + # --- Set the consitutive model --- + self.numDOF = self.numNodes * self.numStates + + # --- For each element type in the mesh, we need to assign a FEMpy element object --- + self.elements = {} + self.numElements = 0 + for elType in self.connectivity: + elObject = self._getElementObject(elType) + if elObject is None: + warnings.warn(f"Element type {elType} is not supported by FEMpy and will be ignored") + else: + self.elements[elType] = {} + self.elements[elType]["connectivity"] = self.connectivity[elType] + self.elements[elType]["DOF"] = self.getDOFfromNodeInds(self.elements[elType]["connectivity"]) + self.elements[elType]["elementObject"] = elObject + self.elements[elType]["numElements"] = self.elements[elType]["connectivity"].shape[0] + + # Create element indices for this set of elements, starting from the current max element index + self.elements[elType]["elementIDs"] = np.arange( + self.numElements, self.numElements + self.elements[elType]["numElements"] + ) + + # Update the element count + self.numElements += self.elements[elType]["numElements"] + + # --- Store dersign variable values for each element set --- + # The constitutive model has a certain number of design variables, each of which has a name, we store + # the values of each design variable + self.dvs = {} + + for dvName in self.constitutiveModel.designVariables: + defaultVal = self.constitutiveModel.designVariables[dvName]["defaultValue"] + self.dvs[dvName] = defaultVal * np.ones(self.numElements) + + # --- List for keeping track of all problems associated with this model --- + self.problems = [] + + # --- Dictionary of global boundary conditions --- + self.BCs = {} + + @property + def problemNames(self) -> Iterable[str]: + """Get a list of the names of all problems associated with this model""" + return [problem.name for problem in self.problems] + + @property + def numDim(self): + """Number of active dimensions in the model""" + return self.constitutiveModel.numDim + + @property + def numStates(self): + """Number of states in the model""" + return self.constitutiveModel.numStates + + def createOutputData(self, nodeValues={}, elementValues={}): + """Create the meshio data structure for writing out results + + + + Parameters + ---------- + nodeValues : dictionary + {"Temperature": array(one value per node), "Pressure": array()} + elementValues : _type_ + {"elementType1":{"VariableName1": array(one value per element), "VariableName2": array()}, + "elementType2":{"VariableName1": array(one value per element), "VariableName2": array()}} + + Returns + ------- + meshio mesh object + Mesh object containing the results + """ + if nodeValues: # if dictionary is not empty + for varName in nodeValues: + # check arrays are correct length + assert ( + nodeValues[varName].shape[0] == self.numNodes + ), f"nodeValues array for variable '{varName}' must be length of number of nodes" + + elementData = {} + if elementValues: # if dictionary is not empty + for elType in elementValues: + for varName in elementValues[elType]: + # first, check arrays are the correct length + assert ( + elementValues[elType][varName].shape[0] == self.elements[elType]["numElements"] + ), f"elementValues array of element type '{elType}' for variable '{varName}' must be length number of '{elType}' elements" + + # store values in meshio element data format + if varName in elementData: + elementData[varName].append(elementValues[elType][varName]) + else: + elementData[varName] = [elementValues[elType][varName]] + + outputMesh = meshio.Mesh( + self.getCoordinates(force3D=True), self.connectivity, point_data=nodeValues, cell_data=elementData + ) + return outputMesh + + def getCoordinates(self, force3D: Optional[bool] = False) -> np.ndarray: + """Get the current node coordinates + + Parameters + ---------- + force3D : bool, optional + If True then 3D coordinates will be returned, even if the model is 1D or 2D, by default False + + Returns + ------- + numNodes x numDim array + Node coordinates + """ + currentCoords = np.copy(self.nodeCoords) + if not force3D or self.numDim == 3: + return currentCoords + else: + coords = np.zeros((self.numNodes, 3)) + coords[:, self.activeDimensions] = currentCoords + return coords + + def setCoordinates(self, nodeCoords: np.ndarray) -> None: + """Set the current node coordinates + + Parameters + ---------- + nodeCoords : numNodes x numDim array + Node coordinates + """ + size = nodeCoords.shape[1] + if size == self.numDim: + return np.copy(nodeCoords) + if self.numDim != 3 and size == 3: + self.nodeCoords = nodeCoords[:, self.activeDimensions] + else: + raise ValueError( + f"Invalid number of coordinate dimensions, problem is {self.numDim}D but {size}D coordinates were supplied" + ) + + def setDesignVariables(self, dvs: Dict[str, np.ndarray]) -> None: + """Set the design variables for the model + + Parameters + ---------- + dvs : dictionary + {"VariableName1": array(one value per element), "VariableName2": array()} + """ + for dvName in dvs: + if dvs[dvName].shape == self.dvs[dvName].shape: + self.dvs[dvName] = np.copy(dvs[dvName]) + else: + raise ValueError( + f"Invalid shape for design variable '{dvName}', expected {self.dvs[dvName].shape} but got {dvs[dvName].shape}" + ) + + for prob in self.problems: + prob.markResOutOfDate() + prob.markJacOutOfDate() + + def getDesignVariables(self) -> Dict[str, np.ndarray]: + """Get the design variables for the model + + Returns + ------- + dictionary + Design variables dictionary, e.g ``{"VariableName1": array(one value per element), "VariableName2": array()}`` + """ + return copy.deepcopy(self.dvs) + + def getElementCoordinates(self, elementType: str) -> np.ndarray: + """Get the coordinates of the nodes for all elements of the specified type + + Parameters + ---------- + elementType : str + Name of the element type to get the coordinates for + + Returns + ------- + numElements x numNodes x numDim array + Node coordinates + """ + numElements = self.elements[elementType]["numElements"] + element = self.elements[elementType]["elementObject"] + nodeCoords = np.zeros((numElements, element.numNodes, self.numDim)) + for ii in range(numElements): + nodeInds = self.elements[elementType]["connectivity"][ii] + nodeCoords[ii] = self.nodeCoords[nodeInds] + return nodeCoords + + def getElementDVs(self, elementType: str): + """Get the design variable values for a given element set + + + + Parameters + ---------- + elementType : str + Element type name + + Returns + ------- + dict of arrays + Dictionary of design variable values for this element sets, one array per design variable + """ + elementDVs = {} + elementInds = self.elements[elementType]["elementIDs"] + for dv in self.dvs: + elementDVs[dv] = self.dvs[dv][elementInds] + + return elementDVs + + def addFixedBCToNodes( + self, + name, + nodeInds: Union[int, Iterable[int]], + dof: Union[int, Iterable[int]], + value: Union[float, Iterable[float]], + ) -> None: + """Add a boundary condition that is applied to all problems associated with this model + + Parameters + ---------- + name : str + Name for this boundary condition + nodeInds : int or iterable of ints + Indicies of nodes to apply the boundary condition to + dof : int or iterable of ints + Degrees of freedom to apply this boundary condition to + value : float or iterable of floats + Value to fix states at, if a single value is supplied then this value is applied to all specified degrees of freedom + """ + # store the BC + self.BCs[name] = {} + dofNodes = [] + valDOF = [] + + if isinstance(nodeInds, int): + nodeInds = [nodeInds] + + if isinstance(dof, int): + dof = [dof] + + if isinstance(value, float) or isinstance(value, int): + value = [value] * len(dof) + elif len(value) != len(dof): + raise Exception("value should be a single entry or a list of values the same length as the DOF list.") + + for i in range(len(nodeInds)): + for j in range(len(dof)): + dofNodes.append(nodeInds[i] * self.numStates + dof[j]) + valDOF.append(value[j]) + self.BCs[name]["DOF"] = dofNodes + self.BCs[name]["Value"] = valDOF + + def addProblem(self, name, options: Optional[Dict[str, Any]] = None): + """Create a new FEMpy problem that uses this model + + Parameters + ---------- + name : str + Name of the problem to create + """ + problem = FEMpyProblem(name, self, options) + self.problems.append(problem) + + return problem + + def getDOFfromNodeInds(self, nodeIndices): + """Convert an array of node indices to an array of DOF indices + + Parameters + ---------- + index : numpy array of ints + array of node indices in arbitrary dimensions + + Returns + ------- + numpy array of ints + array of DOF indices, has same dimensions as input array, except that the last dimension is larger by a + factor of the number of DOF per node + """ + # Figure out the output shape + shape = list(nodeIndices.shape) + shape[-1] *= self.numStates + + # Flatten the input array and convert to a flattened array of DOF indices + nodeIndices = nodeIndices.flatten() + # dofIndices = [] + dofIndices = np.zeros(len(nodeIndices) * self.numStates, dtype=np.int64) + for i in range(len(nodeIndices)): + # ind = range(nodeIndices[i] * self.numStates, (nodeIndices[i] + 1) * (self.numStates)) + ind = np.arange(nodeIndices[i] * self.numStates, (nodeIndices[i] + 1) * (self.numStates)) + dofIndices[i * self.numStates : (i + 1) * self.numStates] = ind + + return np.reshape(dofIndices, shape) + + # ============================================================================== + # Private methods + # ============================================================================== + @staticmethod + def _getDefaultOptions(): + """Return the default FEMpy model options""" + defaultOptions = { + "outputDir": [str, "./"], + "outputFormat": [str, "vtk"], + "outputFunctions": [list, []], + } + return defaultOptions + + def _getElementObject(self, meshioName): + """Given the meshio name for an element type return the corresponding FEMpy element object + + + + Parameters + ---------- + meshioName : str + meshio element type name + + Returns + ------- + An instantiated FEMpy element object + """ + elName = meshioName.lower() + elObject = None + + # TODO: Add more element types once they're implemented + + # --- 2D Quad elements --- + if elName[:4] == "quad": + if elName == "quad": + elObject = Elements.QuadElement2D(order=1, numStates=self.numStates) + else: + numNodes = int(elName[4:]) + if numNodes == 8: + pass + # elObject = Elements.serendipityQuadElement(numStates=self.numStates) + else: + # If the sqrt of the number of nodes is a whole number then this is a valid quad element + order = np.sqrt(numNodes) - 1 + if int(order) == order: + elObject = Elements.QuadElement2D(order=int(order), numStates=self.numStates) + + # --- 2D Triangle elements --- + if elName == "triangle": + elObject = Elements.TriElement2D(order=1, numStates=self.numStates) + if elName == "triangle6": + elObject = Elements.TriElement2D(order=2, numStates=self.numStates) + if elName == "triangle10": + elObject = Elements.TriElement2D(order=3, numStates=self.numStates) + + # --- 3D Hex elements --- + if "hexahedron" in elName: + if elName == "hexahedron": + elObject = Elements.HexElement3D(order=1, numStates=self.numStates) + else: + # If the cube root of the number of nodes is a whole number then this is a valid hex element + order = np.cbrt(numNodes) - 1 + if int(order) == order: + elObject = Elements.HexElement3D(order=int(order), numStates=self.numStates) + + # # --- 1D Line Elements --- + # if elName[:4] == "line": + # if elName == "line": + # elObject = Elements.QuadElement(order=1, numStates=self.numStates) + # else: + # numNodes = int(elName[4:]) + # order = numNodes - 1 + # elObject = Elements.QuadElement(order=order, numStates=self.numStates) + + return elObject + + def _printWelcomeMessage(self) -> None: + """Print a welcome message to the console""" + self.pp("\nWelcome to") + self.pp( + """ ______ ______ __ __ +| ____| ____| \/ | +| |__ | |__ | \ / |_ __ _ _ +| __| | __| | |\/| | '_ \| | | | +| | | |____| | | | |_) | |_| | +|_| |______|_| |_| .__/ \__, | + | | __/ | + |_| |___/ +""" + ) + self.pp("Version: " + __version__) + self.pp("") diff --git a/FEMpy/OrthoPlaneStress.py b/FEMpy/OrthoPlaneStress.py deleted file mode 100644 index 8b8540d..0000000 --- a/FEMpy/OrthoPlaneStress.py +++ /dev/null @@ -1,70 +0,0 @@ -""" -============================================================================== -Orthotropic Plane Stress Constitutive Class -============================================================================== -@File : isoPlaneStress.py -@Date : 2021/04/16 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -import pyComposite as pc - -# ============================================================================== -# Extension modules -# ============================================================================== -from .Constitutive import Constitutive - - -class orthoPlaneStress(Constitutive): - def __init__(self, E1, E2, nu12, G12, t, theta) -> None: - """Initialise an orthotropic plane stress constitutive object - - This class models a thin, orthotropic material, e.g a single composite lamina - - Parameters - ---------- - E1 : float - Longitudinal modulus - E2 : float - Transverse Modulus - nu12 : float - Major Poisson's ratio - G12 : float - Shear modulus - t : float - Thickness - theta : float - Material orientation, in radians, theta = 0 means the material 1 axis is aligned with x - """ - super().__init__(numStrain=3, numDisp=2, numStress=3) - self.LMats[0] = np.array([[1, 0], [0, 0], [0, 1]]) - self.LMats[1] = np.array([[0, 0], [0, 1], [1, 0]]) - self.matProps = { - "E1": E1, - "E2": E2, - "G12": G12, - "v12": nu12, - } - self.t = t - self.theta = theta - self.lamina = pc.lamina(self.matProps) - - @property - def DMat(self): - return self.lamina.getQBar(self.theta) - - -if __name__ == "__main__": - OPS = orthoPlaneStress(E1=145.88e6, E2=13.312e6, G12=4.386e6, nu12=0.263, t=1.0, theta=0.0) - print(OPS.DMat) - OPS.theta = np.deg2rad(45.0) - print(OPS.DMat) diff --git a/FEMpy/Problem.py b/FEMpy/Problem.py new file mode 100644 index 0000000..818fd80 --- /dev/null +++ b/FEMpy/Problem.py @@ -0,0 +1,683 @@ +""" +============================================================================== +FEMpy Problem Class +============================================================================== +@File : Problem.py +@Date : 2022/11/21 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +from typing import Iterable, Union, Callable, Optional, Dict +import copy +import time +import os + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np +from scipy.sparse import csr_array, coo_array +from baseclasses import BaseSolver + +try: + from pypardiso import factorized +except ModuleNotFoundError: + from scipy.sparse.linalg import factorized + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Utils import AssemblyUtils +from FEMpy.Utils.KSAgg import ksAgg + + +class FEMpyProblem(BaseSolver): + """The FEMpy problem class represents a single finite element problem, many such problems may be associated with a + single FEMpy model to represent different loading and boundary conditions. + + The problem class contains: + - The state vector + - The residual vector + - The Jacobian matrix + + And contains methods for: + - Assembling local vectors and matrices into global vectors and matrices + - Solving the problem + - Writing the solution to a file + """ + + def __init__(self, name, model, options=None) -> None: + """Create a new problem object + + It is usually a better idea to use the ``addProblem`` method of the model object to create a new problem than + to call this method directly + + Parameters + ---------- + name : str + A descriptive name for the problem + model : FEMpyModel object + The FEMpyModel object that this problem is associated with + options : dict, optional + The options for this problem, many of the options related to file output are inherited from the model if not + specified for the problem, by default None + """ + self.model = model + self.states = np.zeros((self.numNodes, self.numStates)) + self.Jacobian = None + self.jacUpToDate = False + self.factorizedJac = None + self.Res = np.zeros(self.numNodes * self.numStates) + self.resUpToDate = False + + self.solveCounter = 0 + + # --- Dictionary of boundary conditions --- + self.BCs = {} + self.loads = {} + + # --- Set problem options by getting the defaults and updating them with any that the user supplied --- + defaultOptions = self._getDefaultOptions() + if options is None: + options = {} + + # instantiate the solver + super().__init__(name, "Finite Element Problem", defaultOptions=defaultOptions, options=options) + + # --- Take some options from the model if they are not defined for this problem --- + inheritableOptions = ["outputDir", "outputFormat", "outputFunctions"] + for option in inheritableOptions: + if self.getOption(option) is None: + self.setOption(option, self.model.getOption(option)) + + @property + def constitutiveModel(self): + """The constitutive model object associated with this problem""" + return self.model.constitutiveModel + + @property + def isLinear(self) -> bool: + """Whether the problem is linear""" + return self.constitutiveModel.isLinear + + @property + def numDim(self) -> int: + """The number of dimensions of the problem""" + return self.model.numDim + + @property + def numNodes(self) -> int: + """The number of nodes in the model""" + return self.model.numNodes + + @property + def numDOF(self) -> int: + """The number of degrees of freedom in the problem""" + return self.model.numDOF + + @property + def numStates(self) -> int: + """The number of states + + This is not the total number of states in the problem, but the number of states associated with the problem's + constitutive model, e.g for a heat transfer problem there is a single state (Temperature), and for a 2D + elasticity problem there are 2 states (u and v displacements) + """ + return self.constitutiveModel.numStates + + @property + def elements(self): + """The element data structure""" + return self.model.elements + + def solve(self): + """Solve the finite element problem + + The solution is stored in the ``states`` attribute of the problem object + + Returns + ------- + Dictionary of timing data + """ + + times = {} + times["Start"] = time.time() + # --- Assemble the residual --- + self.updateResidual(applyBCs=True) + times["ResAssembled"] = time.time() + + # --- Assemble the stiffness matrix --- + self.updateJacobian(applyBCs=True) + times["JacAssembled"] = time.time() + + # --- Solve for an update in the state --- + update = self.solveJacLinear(-self.Res) + times["Solved"] = time.time() + + # --- Apply the state update --- + self.incrementState(update) + + if self.getOption("printTiming"): + self._printTiming(times) + + self.solveCounter += 1 + + return times + + def solveJacLinear(self, RHS: np.ndarray) -> np.ndarray: + """Solve the linearised residual equations with a right hand side + + This function solves the linearised residual equations with a right hand side, i.e. it solves the equation + + ``Jacobian @ update = RHS`` + + Some checks are first made as to whether the Jacobian needs to be recomputed before the solve, and the + factorized Jacobian matrix is stored for future solves. + + Parameters + ---------- + RHS : np.ndarray + The right hand side of the linear system + + Returns + ------- + np.ndarray + The solution to the linear system + """ + if not self.jacUpToDate: + self.updateJacobian() + if self.factorizedJac is None or not self.jacIsFactorized: + print("Factorising Jacobian") + self.factorizedJac = factorized(self.Jacobian) + self.jacIsFactorized = True + print("Solving linear system") + return self.factorizedJac(RHS) + + def incrementState(self, update: np.ndarray) -> None: + """Add an increment to the state vector, effectively equivalent to ``self.states += update`` + + This method should be used instead of ``self.states += update`` so that the problem can keep track of whether + the residual and Jacobian are up to date. + + Parameters + ---------- + update : _type_ + _description_ + """ + if update.shape == self.states.shape: + self.states += update + else: + self.states += update.reshape(self.states.shape) + self.markResOutOfDate() + if not self.isLinear: + self.markJacOutOfDate() + + def updateState(self, update: np.ndarray) -> None: + """Update the state vector + + Parameters + ---------- + update : np.ndarray + The update to the state vector + """ + if update.shape == self.states.shape: + self.states[:] = update + else: + self.states[:] = update.reshape(self.states.shape) + self.markResOutOfDate() + if not self.isLinear: + self.markJacOutOfDate() + + def reset(self): + """Reset the problem to its initial state""" + self.updateState(np.zeros((self.numNodes, self.numStates))) + + def updateResidual(self, applyBCs: bool = True): + """Update the residual based on the current state + + Parameters + ---------- + applyBCs : bool, optional + Whether to apply the boundary conditions, by default True + """ + if not self.resUpToDate: + print("Updating Residual") + self.Res = self._assembleResidual(self.states, applyBCs=applyBCs) + self.markResUpToDate() + + def updateJacobian(self, applyBCs: bool = True): + """Update the residual jacobian based on the current state + + Parameters + ---------- + applyBCs : bool, optional + Whether to apply the boundary conditions, by default True + """ + if not self.jacUpToDate: + print("Updating Jacobian") + self.Jacobian = self._assembleMatrix(self.states, applyBCs=applyBCs) + self.markJacUpToDate() + self.jacIsFactorized = False + + def markResOutOfDate(self): + """Mark the status of the residual as out of date""" + self.resUpToDate = False + + def markJacOutOfDate(self): + """Mark the status of the jacobian as out of date""" + self.jacUpToDate = False + + def markResUpToDate(self): + """Mark the status of the residual as up to date""" + self.resUpToDate = True + + def markJacUpToDate(self): + """Mark the status of the jacobian as up to date""" + self.jacUpToDate = True + + def computeFunction(self, name, elementReductionType, globalReductionType=None): + """Compute a function over the whole model + + Parameters + ---------- + name : str, optional + Name of the function to compute, this must be one of the function names of the problem's constitutive model, by default "" + elementReductionType : _type_, optional + Type of reduction to do over each element (sum, mean, integrate, max, min, ksmax, ksmin), by default None + globalReductionType : _type_, optional + Type of reduction to do over all elements (sum, mean, max, min, ksmax, ksmin), by default None + + Returns + ------- + float + for global reduction + dict + for element reduction + """ + functionValues = {} + + elementDvs = self.model.dvs # need to confirm size + eval_func = self.model.constitutiveModel.getFunction(name) + for elType in self.model.elements: + elObject = self.model.elements[elType]["elementObject"] + nodeCoords = self.model.getElementCoordinates(elType) + nodeStates = self.getElementStates(elType) + elementDvs = self.model.getElementDVs(elType) + + functionValues[elType] = elObject.computeFunction( + nodeCoords, nodeStates, elementDvs, eval_func, elementReductionType + ) + + # perform global reduction if specified + if globalReductionType is not None: + assert globalReductionType.lower() in [ + "sum", + "mean", + "min", + "max", + "ksmax", + "ksmin", + ], "globalReductionType not valid" + + # create reduction function + if globalReductionType.lower() == "mean": + reductionFunc = np.average + if globalReductionType.lower() == "sum": + reductionFunc = np.sum + if globalReductionType.lower() == "min": + reductionFunc = np.min + if globalReductionType.lower() == "max": + reductionFunc = np.max + if globalReductionType.lower() == "ksmax": + reductionFunc = lambda values: ksAgg(values, "max") + if globalReductionType.lower() == "ksmin": + reductionFunc = lambda values: ksAgg(values, "min") + + globalValues = np.zeros(len(self.model.elements)) + for i, elType in enumerate(functionValues): + globalValues[i] = reductionFunc(functionValues[elType]) + + return reductionFunc(globalValues) + + return functionValues + + def addFixedBCToNodes( + self, + name: str, + nodeInds: Union[int, Iterable[int]], + dof: Union[int, Iterable[int]], + value: Union[float, Iterable[float]], + ) -> None: + """Add a fixed boundary condition to a set of nodes + + For example ``addFixedBCToNodes("BCName", [0, 1, 2], [0,1], 0.0)`` would fix DOF 0 and 1 of nodes 0, 1 and 2 to 0.0 + + Parameters + ---------- + name : str + Name for this boundary condition + nodeInds : int or iterable of ints + Indicies of nodes to apply the boundary condition to + dof : int or iterable of ints + Degrees of freedom to apply this boundary condition to, the same DOF are fixed at every node in ``nodeInds`` + values : float or iterable of floats + Values to fix states at, if a single value is supplied then this value is applied to all specified degrees of freedom + """ + + # store the BC + self.BCs[name] = {} + dofNodes = [] + valDOF = [] + + if isinstance(nodeInds, int): + nodeInds = [nodeInds] + + if isinstance(dof, int): + dof = [dof] + + if isinstance(value, float) or isinstance(value, int): + value = [value] * len(dof) + elif len(value) != len(dof): + raise Exception("value should be a single entry or a list of values the same length as the DOF list.") + # value = np.ones(len(nodeInds)) * value + + for i in range(len(nodeInds)): + for j in range(len(dof)): + dofNodes.append(nodeInds[i] * self.numStates + dof[j]) + valDOF.append(value[j]) + self.BCs[name]["DOF"] = dofNodes + self.BCs[name]["Value"] = valDOF + + def addLoadToNodes( + self, + name: str, + nodeInds: Iterable[int], + dof: Union[int, Iterable[int]], + value: Union[float, Iterable[float]], + totalLoad: bool = False, + ) -> None: + """Add a load to a set of nodes + + Parameters + ---------- + name : str + Name for this load + nodeInds : int or iterable of ints + Indicies of nodes to apply the load to + dof : int or iterable of ints + Degrees of freedom to apply this load to at each node + values : float or iterable of floats + Load values, if a single value is supplied then this value is applied to all specified degrees of freedom + totalLoad : bool, optional + If true then the `values` are treated as total loads and split uniformly between the nodes, by default False, in which case the `values` are applied at each node + """ + + # store the BC + scale = 1.0 + if totalLoad: + scale = float(len(nodeInds)) + + self.loads[name] = {} + dofNodes = [] + valDOF = [] + + if isinstance(nodeInds, int): + nodeInds = [nodeInds] + + if isinstance(dof, int): + dof = [dof] + + if isinstance(value, float) or isinstance(value, int): + value = [value] * len(dof) + elif len(value) != len(dof): + raise Exception("value should be a single entry or a list of values the same length as the DOF list.") + + for i in range(len(nodeInds)): + for j in range(len(dof)): + dofNodes.append(nodeInds[i] * self.numStates + dof[j]) + valDOF.append((value[j] / scale)) + self.loads[name]["DOF"] = dofNodes + self.loads[name]["Value"] = valDOF + + def addBodyLoad(self, name: str, loadingFunction: Union[Callable, Iterable[float]]) -> None: + """Add a volumetric forcing term, commonly known as a "body force" + + + + Parameters + ---------- + name : str + Name for the load + loadingFunction : function or array of length numDim + Pass an array to define a uniform field, otherwise, pass a function with the signature + `F = loadingFunction(coord)` where coord is an n x numDim array of coordinates and F is an n x numDim array + of loads at each point. + """ + raise NotImplementedError("Body loads are not yet implemented, sorry :(") + + def getBCs(self) -> Dict[str, Dict[str, Union[Iterable[int], Iterable[float]]]]: + """Get the full set of boundary conditions for this problem + + This combines the BCs defined specifically for this problem with those defined at the model level + + Returns + ------- + Dict[str, Dict[str, Union[Iterable[int], Iterable[float]]]] + Dictionary of boundary conditions, each entry is a dictionary with keys "DOF" and "Value" + """ + BCs = copy.deepcopy(self.model.BCs) + BCs.update(self.BCs) + return BCs + + def getElementStates(self, elementType: str) -> np.ndarray: + """Get the states of the nodes for all elements of the specified type + + Parameters + ---------- + elementType : str + Name of the element type to get the coordinates for + + Returns + ------- + numElements x numNodes x numStates array + Node coordinates + """ + numElements = self.elements[elementType]["numElements"] + element = self.elements[elementType]["elementObject"] + nodeStates = np.zeros((numElements, element.numNodes, self.numStates)) + for ii in range(numElements): + nodeInds = self.elements[elementType]["connectivity"][ii] + nodeStates[ii] = self.states[nodeInds] + return nodeStates + + def writeSolution(self, baseName: Optional[str] = None, fileFormat: Optional[str] = None): + """Write the current solution to a file + + Parameters + ---------- + baseName : str, optional + Filename, extension will be ignored if included, by default uses ``f"{self.name}_{self.solveCounter:04d}"`` + fileFormat : str, optional + File format/extension, can be any format supported by meshio, by default uses the format defined by the + model or problem ``outputFormat`` option + """ + # Get nodal state values + nodeValues = {} + for ii, state in enumerate(self.constitutiveModel.stateNames): + nodeValues[state] = self.states[:, ii] + + # Get element DV values + elementValues = {} + for elementType in self.elements: + elementValues[elementType] = {} + elementDVs = self.model.getElementDVs(elementType) + for name, values in elementDVs.items(): + elementValues[elementType][name] = values + + # Get element function values + for funcName in self.getOption("outputFunctions"): + for reduction_type in ["mean", "min", "max"]: + outName = f"{funcName}_{reduction_type}" + funcVals = self.computeFunction( + name=funcName, elementReductionType=reduction_type, globalReductionType=None + ) + for elementType in self.elements: + elementValues[elementType][outName] = funcVals[elementType] + + outputMesh = self.model.createOutputData(nodeValues=nodeValues, elementValues=elementValues) + + if baseName is None: + baseName = f"{self.name}_{self.solveCounter:04d}" + else: + # Get filename without extension + baseName = os.path.splitext(baseName)[0] + + if fileFormat is None: + fileFormat = self.getOption("outputFormat") + if fileFormat[0] != ".": + fileFormat = "." + fileFormat + + # --- Create the output directory if it doesn't already exist --- + outDir = self.getOption("outputDir") + if not os.path.exists(outDir): + os.makedirs(outDir) + + outFileName = os.path.join(self.getOption("outputDir"), baseName + fileFormat) + outputMesh.write(outFileName) + + # ============================================================================== + # Private methods + # ============================================================================== + + def _assembleMatrix(self, states: np.ndarray, applyBCs: Optional[bool] = True) -> csr_array: + """Assemble the global residual Jacobian matrix for the problem (a.k.a the stiffness matrix) + + Parameters + ---------- + stateVector : numNodes x numStates array + The current system states + applyBCs : bool, optional + Whether to modify the matrix to include the boundary conditions, by default True + + Returns + ------- + scipy csr_array + The residual Jacobian + """ + + matRows = [] + matColumns = [] + matEntries = [] + + # - For each element type: + # - Get the node coordinates, node states and design variable values for all elements of that type + # - Compute the local matrices for all elements of that type + # - Convert to COO row, col, value lists + # - Add to the global lists + for elementType, elementData in self.elements.items(): + element = elementData["elementObject"] + nodeCoords = self.model.getElementCoordinates(elementType) + nodeStates = self.getElementStates(elementType) + elementDVs = self.model.getElementDVs(elementType) + + localMats = element.computeResidualJacobians(nodeStates, nodeCoords, elementDVs, self.constitutiveModel) + row_inds, col_inds, values = AssemblyUtils.localMatricesToCOOArrays(localMats, elementData["DOF"]) + matRows.append(row_inds) + matColumns.append(col_inds) + matEntries.append(values) + matRows = np.concatenate(matRows) + matColumns = np.concatenate(matColumns) + matEntries = np.concatenate(matEntries) + + # Apply boundary conditions to COO data + if applyBCs: + BCs = self.getBCs() + BCDOF, _ = AssemblyUtils.convertBCDictToLists(BCs) + matRows, matColumns, matEntries = AssemblyUtils.applyBCsToMatrix(matRows, matColumns, matEntries, BCDOF) + + # create and return sparse matrix, we need to create a coo array first as this correctly handle duplicate + # entries, then we convert to a csr array as that's what the scipy sparse linear solver likes to work with + mat = coo_array((matEntries, (matRows, matColumns)), shape=(self.numDOF, self.numDOF)) + + return csr_array(mat) + + def _assembleResidual(self, states: np.ndarray, applyBCs: Optional[bool] = True): + """Assemble the global residual for the problem + + + + Parameters + ---------- + states : numNodes x numStates array + The current system states + applyBCs : bool, optional + Whether to modify the vector to include the boundary conditions, by default True + + Returns + ------- + numDOF array + The residual vector + """ + # - For each element type: + # - Get the node coordinates, node states and design variable values for all elements of that type + # - Compute the local residual for all elements of that type + # - Scatter local residuals into global residual vector + # - Add loads to residual to the global residual vector + # - Apply boundary conditions to global residual vector + + globalResidual = np.zeros(self.numDOF) + + for elementType, elementData in self.elements.items(): + element = elementData["elementObject"] + nodeCoords = self.model.getElementCoordinates(elementType) + nodeStates = self.getElementStates(elementType) + elementDVs = self.model.getElementDVs(elementType) + + elementResiduals = element.computeResiduals(nodeStates, nodeCoords, elementDVs, self.constitutiveModel) + AssemblyUtils.scatterLocalResiduals(elementResiduals, elementData["connectivity"], globalResidual) + + # Add external loads to the residual + globalResidual -= AssemblyUtils.convertLoadsDictToVector(self.loads, self.numDOF) + + # Apply boundary conditions + if applyBCs: + BCs = self.getBCs() + BCDOF, BCValues = AssemblyUtils.convertBCDictToLists(BCs) + AssemblyUtils.applyBCsToVector(globalResidual, states.flatten(), BCDOF, BCValues) + + return globalResidual + + @staticmethod + def _getDefaultOptions(): + """Return the default FEMpy model options""" + defaultOptions = { + "printTiming": [bool, False], + "outputDir": [(str, type(None)), None], + "outputFormat": [(str, type(None)), None], + "outputFunctions": [(list, type(None)), None], + } + return defaultOptions + + def _printTiming(self, times): + """Print some nicely formatted timing information about a solution + + Parameters + ---------- + times : dict + Dictionary of timing information + """ + resAssemblyTime = times["ResAssembled"] - times["Start"] + matAssemblyTime = times["JacAssembled"] - times["ResAssembled"] + solveTime = times["Solved"] - times["JacAssembled"] + print("\n") + print("+-----------------------------------------------------------------+") + print(f" Timing information for FEMpy problem: {self.name}") + print("+-----------------------------------------------------------------+") + print("Residual Assembly: {:11.5e} s".format(resAssemblyTime)) + print("Jacobian Assembly: {:11.5e} s".format(matAssemblyTime)) + print("Linear Solution: {:11.5e} s".format(solveTime)) + print("+-----------------------------------------------------------------+\n") diff --git a/FEMpy/QuadElement.py b/FEMpy/QuadElement.py deleted file mode 100644 index c642603..0000000 --- a/FEMpy/QuadElement.py +++ /dev/null @@ -1,196 +0,0 @@ -""" -============================================================================== -Q4 Element -============================================================================== -@File : QuadElement.py -@Date : 2021/03/11 -@Author : Alasdair Christison Gray -@Description : This file contains a class that implements a 2D, 4-Node quadrilateral finite element -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== -from .Element import Element -from . import LagrangePoly as LP -from .GaussQuad import gaussQuad1d - - -class QuadElement(Element): - def __init__(self, order=1, numDisplacements=2): - """Instantiate an arbitrary order 2d quadrilateral finite element - - Note that this element does not use the typical CCW node ordering, to make it simpler to work with arbitrary - element orders, the nodes are ordered left to right followed by bottom to top, so for a 1st order element the - ordering is: - - 1) Bottom left - 2) Bottom Right - 3) Top left - 4) Top right - - Parameters - ---------- - order : int, optional - Element order, by default 1 - numDisplacements : int, optional - Number of variables at each node, by default 2 - """ - self.order = order - nodes = (order + 1) ** 2 - super().__init__(numNodes=nodes, numDimensions=2, numDisplacements=numDisplacements) - - # bottom, right, top, left order, - # 0 means psi varies along edge, 1 means eta varies along edge - self.edgeFreeCoord = [0, 1, 0, 1] - # value of the fixed coordinate on each edge - self.edgeFixedCoord = [-1.0, 1.0, 1.0, -1.0] - - self.name = f"Order{self.order}-LagrangeQuad" - - def getShapeFunctions(self, paramCoords): - """Compute shape function values at a set of parametric coordinates - - - Parameters - ---------- - paramCoords : n x nDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - N : n x numNode array - Shape function values, N[i][j] is the value of the jth shape function at the ith point - """ - return LP.LagrangePoly2d(paramCoords[:, 0], paramCoords[:, 1], self.order + 1) - - def getShapeFunctionDerivs(self, paramCoords): - """Compute shape function derivatives at a set of parametric coordinates - - These are the derivatives of the shape functions with respect to the parametric coordinates (si, eta, gamma) - - Parameters - ---------- - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - NPrime : n x numDim x numNode array - Shape function values, N[i][j][k] is the value of the kth shape function at the ith point w.r.t the kth - parametric coordinate - """ - return LP.LagrangePoly2dDeriv(paramCoords[:, 0], paramCoords[:, 1], self.order + 1) - - def getStiffnessMat(self, nodeCoords, constitutive, n=None): - # TODO: this function should not presume the constitutive object has a thickness, what if you want to use this - # quad element for something else, like 2D heat transfer where there's no thickness - return super().getStiffnessMat(nodeCoords, constitutive, n=n) * constitutive.t - - def integrateBodyForce(self, f, nodeCoords, constitutive, n=1): - # TODO: this function should not presume the constitutive object has a thickness, what if you want to use this - # quad element for something else, like 2D heat transfer where there's no thickness - return super().integrateBodyForce(f, nodeCoords, n=n) * constitutive.t - - def integrateTraction(self, f, nodeCoords, constitutive, edges=None, n=1): - """Compute equivalent nodal forces due to body forces through numerical integration - - - Parameters - ---------- - f : Body force function - Should accept an nP x numDim array as input and output a nP x numDisp array, ie f(x)[i] returns the body force components at the ith point queried - nodeCoords : numNode x numDim array - Element node real coordinates - n : int, optional - Number of integration points, can be a single value or a list with a value for each direction, by default 1 - constitutive : [type] - [description] - edges : list, optional - [description], by default [1,2,3,4] - - Returns - ------- - Fb : numNode x numDisp array - Equivalent nodal loads due to traction forces force - """ - if edges is None: - edges = [0, 1, 2, 3] - if isinstance(edges, (int, np.integer)): - edges = [edges] - Ft = np.zeros((self.numNodes, self.numDisp)) - for e in edges: - if self.edgeFreeCoord[e] == 0: - func = lambda x1: self.tractionIntegrand( # noqa: E731 - f, - np.array([x1, self.edgeFixedCoord[e] * np.ones_like(x1)]).T, - nodeCoords, - e, - ) - else: - func = lambda x2: self.tractionIntegrand( # noqa: E731 - f, - np.array([self.edgeFixedCoord[e] * np.ones_like(x2), x2]).T, - nodeCoords, - e, - ) - Ft += gaussQuad1d(func, n) - return Ft * constitutive.t - - def tractionIntegrand(self, f, paramCoord, nodeCoords, edgeNum): - # Compute shape functions and Jacobian determinant at parametric coordinates - N = self.getShapeFunctions(paramCoord) - J = self.getJacobian(paramCoord, nodeCoords) - detJStar = np.linalg.norm(J[:, self.edgeFreeCoord[edgeNum], :], axis=-1) - - # Transform parametric to real coordinates in order to compute body force components - realCoord = self.getRealCoord(paramCoord, nodeCoords) - F = f(realCoord) - - # Compute N^T fb at each point - Fb = self._computeNTFProduct(F, N) - return (Fb.T * detJStar).T - - def getRandomNodeCoords(self): - """Generate a random, but valid, set of node coordinates for an element - - For the Quad element, we simply create a grid of evenly spaced points then add some random noise to each point - before applying a random scaling and rotation - - Returns - ------- - nodeCoords : numNode x numDim array - Node coordinates - """ - xy = np.linspace(0, 1, self.order + 1) - nodeCoords = np.random.rand((self.order + 1) ** 2, self.numDim) * 0.1 - nodeCoords[:, 0] += np.tile(xy, self.order + 1) - nodeCoords[:, 1] += np.repeat(xy, self.order + 1) - nodeCoords *= np.random.rand(1) - theta = np.random.rand(1) * np.pi - R = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])[:, :, 0] - return (R @ nodeCoords.T).T - - -if __name__ == "__main__": - - QuadElem = QuadElement() - - # Test that we get the expected output when the original element matches the isoparametric element - nodecoords = np.array([[-1.0, -1.0], [1.0, -1.0], [-1.0, 1.0], [1.0, 1.0]]) - uNodes = np.array([[-1.0, -1.0], [1.0, -1.0], [-1.0, 1.0], [1.0, 1.0]]) - paramCoords = np.array([[-1.0, -1.0], [0.0, 0.0], [1.0, 1.0]]) - print(QuadElem.getShapeFunctions(paramCoords), "\n") - print(QuadElem.getRealCoord(paramCoords, nodecoords), "\n") - print(QuadElem.getU(paramCoords, uNodes), "\n") - print(QuadElem.getUPrime(paramCoords, nodecoords, uNodes), "\n") - print(QuadElem.getJacobian(paramCoords, nodecoords), "\n\n\n") diff --git a/FEMpy/Quadrature/GaussQuad.py b/FEMpy/Quadrature/GaussQuad.py new file mode 100644 index 0000000..f8f2c86 --- /dev/null +++ b/FEMpy/Quadrature/GaussQuad.py @@ -0,0 +1,122 @@ +""" +============================================================================== +Gauss Quadrature Points and Weights +============================================================================== +@File : guassQuad.py +@Date : 2021/03/10 +@Author : Alasdair Christison Gray +@Description : Methods for computing Gauss Quadrature points and weights +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +from functools import lru_cache + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@lru_cache(maxsize=None) +def getGaussQuadWeights(numDim, order): + """Compute arbitrary order Gauss Quadrature weights in up to 3 dimensions + + Parameters + ---------- + numDim : int + Number of dimensions to compute weights for + order : int + Order of the quadrature rule + + Returns + ------- + np.array of length order ** numDim + Gauss quadrature weights + + Raises + ------ + ValueError + If requested number of dimensions is not supported + """ + _, W = _getGaussQuad1dData(order) + if numDim == 1: + return W + elif numDim >= 2: + W1 = np.repeat(W, order) # Rows + W2 = np.tile(W, order) # Columns + if numDim == 2: + return W1 * W2 + elif numDim == 3: + W1 = np.tile(W1, order) + W2 = np.tile(W2, order) + W3 = np.repeat(W, order**2) # 3rd dimension + return W1 * W2 * W3 + else: + raise ValueError( + f"Gauss quadrature weights only computable for 1, 2 or 3 dimensions, you asked for {numDim} dimensions" + ) + + +@lru_cache(maxsize=None) +def getGaussQuadPoints(numDim, order): + """Compute arbitrary order Gauss Quadrature point coordinates in up to 3 dimensions + + Parameters + ---------- + numDim : int + Number of dimensions to compute points for + order : int + Order of the quadrature rule + + Returns + ------- + (order ** numDim) x numDim np.array + Gauss quadrature point coordinates + + Raises + ------ + ValueError + If requested number of dimensions is not supported + """ + points, _ = _getGaussQuad1dData(order) + if numDim == 1: + return points + elif numDim >= 2: + X = np.repeat(points, order) # Rows + Y = np.tile(points, order) # Columns + if numDim == 2: + return np.vstack((X, Y)).T + elif numDim == 3: + X = np.tile(X, order) + Y = np.tile(Y, order) + Z = np.repeat(points, order**2) # 3rd dimension + return np.vstack((X, Y, Z)).T + else: + raise ValueError( + f"Gauss quadrature weights only computable for 1, 2 or 3 dimensions, you asked for {numDim} dimensions" + ) + + +@lru_cache(maxsize=None) +def _getGaussQuad1dData(order): + """Just a wrapper around numpy's leggauss function that caches the results + + Parameters + ---------- + order : int + integration order + + Returns + ------- + 1d array + 1D gauss quadrature point coordinates + 1d array + 1D gauss quadrature weights + """ + return np.polynomial.legendre.leggauss(order) diff --git a/FEMpy/Quadrature/GaussQuadCoords.pkl b/FEMpy/Quadrature/GaussQuadCoords.pkl new file mode 100644 index 0000000..6119bf2 Binary files /dev/null and b/FEMpy/Quadrature/GaussQuadCoords.pkl differ diff --git a/FEMpy/Quadrature/GaussQuadWeights.pkl b/FEMpy/Quadrature/GaussQuadWeights.pkl new file mode 100644 index 0000000..12bd028 Binary files /dev/null and b/FEMpy/Quadrature/GaussQuadWeights.pkl differ diff --git a/FEMpy/Quadrature/TriQuad.py b/FEMpy/Quadrature/TriQuad.py new file mode 100644 index 0000000..4be716b --- /dev/null +++ b/FEMpy/Quadrature/TriQuad.py @@ -0,0 +1,86 @@ +""" +============================================================================== +Triangular quadrature rules +============================================================================== +@File : TriQuad.py +@Date : 2022/12/04 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +from functools import lru_cache + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== + + +@lru_cache(maxsize=None) +def getTriQuadPoints(order: int) -> np.ndarray: + """Get integration points for a triangular element + + Only 1st 2nd and 3rd order quadrature are supported + + Parameters + ---------- + order : int + Order of the quadrature rule + + Returns + ------- + numIntPoints x numDim array + Gauss quadrature point parametric coordinates + + Raises + ------ + ValueError + If requested order is not supported + """ + if order == 1: + return np.array([[1 / 3, 1 / 3]]) + elif order == 2: + return np.array([[0.5, 0.5], [0.0, 0.5], [0.5, 0.0]]) + elif order == 3: + return np.array([[1 / 3, 1 / 3], [0.6, 0.2], [0.2, 0.6], [0.2, 0.2]]) + else: + raise ValueError(f"Triangular quadrature of order {order} not supported") + + +@lru_cache(maxsize=None) +def getTriQuadWeights(order: int) -> np.ndarray: + """Get integration point weightss for a triangular element + + Only 1st 2nd and 3rd order quadrature are supported + + Parameters + ---------- + order : int + Order of the quadrature rule + + Returns + ------- + array of numIntPoints length + Gauss quadrature point weights + + Raises + ------ + ValueError + If requested order is not supported + """ + if order == 1: + w = np.array([1.0]) + elif order == 2: + w = np.ones(3) / 3 + elif order == 3: + w = np.array([-27 / 28, 25 / 48, 25 / 48, 25 / 48]) + else: + raise ValueError(f"Triangular quadrature of order {order} not supported") + return 0.5 * w diff --git a/FEMpy/Quadrature/__init__.py b/FEMpy/Quadrature/__init__.py new file mode 100644 index 0000000..8a08079 --- /dev/null +++ b/FEMpy/Quadrature/__init__.py @@ -0,0 +1,2 @@ +from .GaussQuad import * +from .TriQuad import * diff --git a/FEMpy/SerendipityQuad.py b/FEMpy/SerendipityQuad.py deleted file mode 100644 index fd155c2..0000000 --- a/FEMpy/SerendipityQuad.py +++ /dev/null @@ -1,201 +0,0 @@ -""" -============================================================================== -8 Node Serendipity Quad Element -============================================================================== -@File : SerendipityQuad.py -@Date : 2021/03/30 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from numba import njit - -# ============================================================================== -# Extension modules -# ============================================================================== -from .QuadElement import QuadElement - - -@njit(cache=True) -def serendipityShapeFuncs(x, y): - """Compute the shape functions of an 8 node serendipity Quad element - - - - Parameters - ---------- - x : array of length nP (0D, nPx1 or 1xnP) - x coordinates of points to compute polynomial values at, should be between -1.0 and 1.0 - y : array of length nP (0D, nPx1 or 1xnP) - y coordinates of points to compute polynomial values at, should be between -1.0 and 1.0 - - Returns - ------- - N : nP x n array - Shape function values - """ - psi = x.flatten() - eta = y.flatten() - N = np.zeros((len(psi), 8), dtype=x.dtype) - - N[:, 0] = 0.25 * (1.0 - psi) * (1.0 - eta) * (-psi - eta - 1.0) - N[:, 1] = 0.25 * (1.0 + psi) * (1.0 - eta) * (psi - eta - 1.0) - N[:, 2] = 0.25 * (1.0 + psi) * (1.0 + eta) * (psi + eta - 1.0) - N[:, 3] = 0.25 * (1.0 - psi) * (1.0 + eta) * (-psi + eta - 1.0) - N[:, 4] = 0.5 * (1.0 - psi**2) * (1.0 - eta) - N[:, 5] = 0.5 * (1.0 + psi) * (1.0 - eta**2) - N[:, 6] = 0.5 * (1.0 - psi**2) * (1.0 + eta) - N[:, 7] = 0.5 * (1.0 - psi) * (1.0 - eta**2) - - return N - - -@njit(cache=True) -def serendipityShapeFuncDerivs(x, y): - """Compute the derivatives of the shape functions of an 8 node serendpity Quad element at a series of points in 2d space - - - - Parameters - ---------- - x : array of length nP (0D, nPx1 or 1xnP) - x coordinates of points to compute polynomial values at, should be between -1 and 1 - y : array of length nP (0D, nPx1 or 1xnP) - y coordinates of points to compute polynomial values at, should be between -1 and 1s - - Returns - ------- - dNdx : nP x 2 x n array - Shape function derivative values - """ - psi = x.flatten() - eta = y.flatten() - dNdxy = np.zeros((len(psi), 2, 8)) - - # --- dNdPsi --- - dNdxy[:, 0, 0] = 0.25 * (1.0 - eta) * (2.0 * psi + eta) - dNdxy[:, 0, 1] = 0.25 * (1.0 - eta) * (2.0 * psi - eta) - dNdxy[:, 0, 2] = 0.25 * (1.0 + eta) * (2.0 * psi + eta) - dNdxy[:, 0, 3] = 0.25 * (1.0 + eta) * (2.0 * psi - eta) - dNdxy[:, 0, 4] = -psi * (1.0 - eta) - dNdxy[:, 0, 5] = 0.5 * (1 - eta**2) - dNdxy[:, 0, 6] = -psi * (1.0 + eta) - dNdxy[:, 0, 7] = -0.5 * (1 - eta**2) - - # --- dNdEta --- - dNdxy[:, 1, 0] = 0.25 * (1.0 - psi) * (2 * eta + psi) - dNdxy[:, 1, 1] = 0.25 * (1.0 + psi) * (2 * eta - psi) - dNdxy[:, 1, 2] = 0.25 * (1.0 + psi) * (2 * eta + psi) - dNdxy[:, 1, 3] = 0.25 * (1.0 - psi) * (2 * eta - psi) - dNdxy[:, 1, 4] = -0.5 * (1.0 - psi**2) - dNdxy[:, 1, 5] = -eta * (1.0 + psi) - dNdxy[:, 1, 6] = 0.5 * (1.0 - psi**2) - dNdxy[:, 1, 7] = -eta * (1.0 - psi) - - return dNdxy - - -class serendipityQuadElement(QuadElement): - """An 8 noded quadratic quad element, also known as a seredipity quad""" - - def __init__(self, numDisplacements=2): - # --- Initialise a 9 noded quad and then change the number of nodes to 8 --- - super().__init__(order=2, numDisplacements=numDisplacements) - self.numNodes = 8 - self.numDOF = self.numNodes * numDisplacements - self.name = "SerendipityQuad" - - def getShapeFunctions(self, paramCoords): - """Compute shape function values at a set of parametric coordinates - - Parameters - ---------- - paramCoords : n x nDim array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - N : n x numNode array - Shape function values, N[i][j] is the value of the jth shape function at the ith point - """ - return serendipityShapeFuncs(paramCoords[:, 0], paramCoords[:, 1]) - - def getShapeFunctionDerivs(self, paramCoords): - """Compute shape function derivatives at a set of parametric coordinates - - These are the derivatives of the shape functions with respect to the parametric coordinates (si, eta, gamma) - - Parameters - ---------- - paramCoords : n x nD array - isoparametric coordinates, one row for each point in isoparametric space to compute the Jacobian at - - Returns - ------- - NPrime : n x numDim x numNode array - Shape function derivative values, N[i][j][k] is the value of the derivative of the kth shape function in the - jth direction at the ith point - """ - return serendipityShapeFuncDerivs(paramCoords[:, 0], paramCoords[:, 1]) - - def getRandomNodeCoords(self): - """Generate a random, but valid, set of node coordinates for an element - - Here we simply call the getRandomNodeCoords method of the parent 2nd order QuadElement class and then remove - the central point, the points need to be reordered too because the 2nd order QuadElement points are not in CCW - order as the Serendipity quad points are - - Returns - ------- - nodeCoords : numNode x numDim array - Node coordinates - """ - nodeCoords = super().getRandomNodeCoords() - return nodeCoords[[0, 2, 8, 6, 1, 5, 7, 3]] - - -if __name__ == "__main__": - import time - import niceplots - import matplotlib.pyplot as plt - - niceplots.setRCParams() - # ============================================================================== - # Speed test - # ============================================================================== - x = np.array([np.linspace(-1.0, 1.0, 3)]) - y = -np.ones_like(x) - # Call functions so they get jit compiled - serendipityShapeFuncs(x, y) - serendipityShapeFuncDerivs(x, y) - startTime = time.time() - for _ in range(1000): - serendipityShapeFuncs(x, y) - # print("\n") - serendipityShapeFuncDerivs(x, y) - # print("\n\n\n") - print(time.time() - startTime) - - # ============================================================================== - # Plot shape functions - # ============================================================================== - n = 101 - psi, eta = np.meshgrid(np.linspace(-1.0, 1.0, n), np.linspace(-1.0, 1.0, n)) - N = serendipityShapeFuncs(psi.flatten(), eta.flatten()) - - fig, axes = plt.subplots(nrows=2, ncols=4, subplot_kw={"projection": "3d"}) - axes = axes.flatten() - for i in range(8): - ax = axes[i] - Ni = N[:, i].reshape((n, n)) - ax.plot_surface(psi, eta, Ni, cmap=niceplots.parula_map) - ax.set_title(f"N{i}") - plt.show() diff --git a/FEMpy/Smoothing.py b/FEMpy/Smoothing.py deleted file mode 100644 index 8d1c8c6..0000000 --- a/FEMpy/Smoothing.py +++ /dev/null @@ -1,80 +0,0 @@ -""" -============================================================================== -Smoothing -============================================================================== -@File : Smoothing.py -@Date : 2021/04/08 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -from scipy.sparse.linalg import factorized -from scipy.sparse import csc_matrix - -# ============================================================================== -# Extension modules -# ============================================================================== - - -def getSmoother(intPointParamCoords, element, conn): - """Generate a function which converts integration point values to nodal values using global smoothing - - Given integration point values f_int, the smoother solves an over/underdetermined system of equations to find the - nodal values which would best recover f_int through interpolation: - - M * f_node = f_int - - The least squares solution is: - - f_node = (M^T * M)^-1 * M^T * F_int - - Parameters - ---------- - intPointParamCoords : [type] - [description] - - Returns - ------- - smoother : function - A function which performs integration point to nodal value smoothing using a pre-factorized least-squares solution (a.k.a should be very very fast) - MTM : Scipy CSC sparse matrix - The matrix M^T * M which is used to compute the least squares solution, this output is not required - MT : Scipy CSC sparse matrix - The matrix M^T which is used to compute the least squares solution, this output is not required - """ - numIntPoints = np.shape(intPointParamCoords)[0] - numNode = np.max(conn) + 1 - numEl = np.shape(conn)[0] - # Compute the shape function values at the integration points - N = element.getShapeFunctions(intPointParamCoords) - - entries = [] - rows = [] - columns = [] - - for e in range(numEl): - elNodes = conn[e].tolist() - for ni in range(numIntPoints): - entries += N[ni].tolist() - rows += numIntPoints * [numIntPoints * e + ni] - columns += elNodes - - # --- Somewhat confusingly, actually compute the transpose of M as it saves some transpose operations later --- - MT = csc_matrix((entries, (columns, rows)), shape=(numNode, numIntPoints * numEl)) - - # --- Create the matrix for the least squares solution and then factorise it --- - MTM = MT @ MT.T - smoothingSolver = factorized(MTM) - - def smoother(x): - return smoothingSolver(MT @ x) - - return smoother, MTM, MT diff --git a/FEMpy/TecplotIO.py b/FEMpy/TecplotIO.py deleted file mode 100644 index f9ba8e6..0000000 --- a/FEMpy/TecplotIO.py +++ /dev/null @@ -1,150 +0,0 @@ -""" -============================================================================== -Tecplot I/O Functions -============================================================================== -@File : TecplotIO.py -@Date : 2021/04/02 -@Author : Alasdair Christison Gray -@Description : Functions for reading and writing tecplot files -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== -import re - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== - - -def readTecplot(filename): - """Read a text based tecplot file - - This function assumes the following file structure: - - TITLE = - VARIABLES = "" "" "" .... - ZONE N=, E=, .... - ... - . - . - . - ... - . - . - . - - Parameters - ---------- - filename : str - File name - - Returns - ------- - data : dictionary - A dictionary containing a "title" field and a separate array for each each nodal variable - conn : 2d array - Element connectivity matrix - """ - # --- Open file and read lines --- - with open(filename) as file: - lines = file.readlines() - - data = {} - - # --- Read title --- - TitleMatch = re.match(r"TITLE = (.*$)", lines[0], re.M | re.I) - data["title"] = TitleMatch[1].rstrip() - - # --- Read number of nodes and elements --- - nums = re.match(r"ZONE N=([0-9]*), E=([0-9]*)", lines[2], re.M | re.I) - numNodes = int(nums[1]) - numEl = int(nums[2]) - - # --- Read variable names --- - varMatch = re.match(r"VARIABLES = (.*$)", lines[1], re.M | re.I) - varNames = varMatch[1].rstrip().split() - for i in range(len(varNames)): - varNames[i] = varNames[i].replace('"', "") - data[varNames[i]] = np.zeros(numNodes) - - # --- Read nodal data values --- - for i in range(3, 3 + numNodes): - line = lines[i].rstrip().split() - for col in range(len(varNames)): - data[varNames[col]][i - 3] = float(line[col]) - - # --- Read connectivity data, for now I just assume all elements have same number of nodes --- - connLine = lines[3 + numNodes].rstrip().split() - numCon = len(connLine) - conn = np.zeros((numEl, numCon), dtype=np.int) - - for i in range(3 + numNodes, len(lines)): - line = lines[i].rstrip().split() - for col in range(numCon): - conn[i - (3 + numNodes), col] = int(line[col]) - 1 - - return data, conn - - -def writeTecplot(nodeCoords, data: dict, conn, filename: str, elementType=None, title=None): - """Write a text based tecplot file - - Currently only works for meshes with one element type - - Parameters - ---------- - nodeCoords : numNode x numDim array - Node coordinates - data : [type] - [description] - conn : [type] - [description] - filename : [type] - [description] - """ - numEl = np.shape(conn)[0] - Conn = np.array(conn) if isinstance(conn, list) else conn - numNode = np.max(Conn) + 1 - dirs = ["X", "Y", "Z"] - varNames = dirs[: np.shape(nodeCoords)[1]] - - # --- Combine all data into single array --- - dataArray = np.copy(nodeCoords) - for name, arr in data.items(): - varNames.append(name) - dataArray = np.hstack((dataArray, arr.reshape((numNode, 1)))) - - if elementType is None: - nodePerEl = np.shape(Conn)[1] - if nodePerEl == 4: - elementType = "QUADRILATERAL" - elif nodePerEl == 3: - elementType = "TRIANGLE" - else: - raise ValueError( - f"Unknown element type with {nodePerEl} nodes, please supply an elementType input argument" - ) - - with open(filename, "w") as file: - # --- Write file header --- - if title is None: - title = "FEMpy result" - file.write(f"TITLE = {title}\n") - file.write("VARIABLES = ") - for name in varNames: - file.write(f'"{name}" ') - file.write("\n") - file.write(f"ZONE N={numNode}, E={numEl}, F=FEPOINT, ET={elementType}\n") - - # --- Now write the nodal variable values, can write all in one go since we combined into a single array --- - np.savetxt(file, dataArray) - - # --- Finally write the element connectivity --- - np.savetxt(file, Conn + 1, fmt="%i") diff --git a/FEMpy/Utils/AssemblyUtils.py b/FEMpy/Utils/AssemblyUtils.py new file mode 100644 index 0000000..e56d5ef --- /dev/null +++ b/FEMpy/Utils/AssemblyUtils.py @@ -0,0 +1,258 @@ +""" +============================================================================== +FEMpy Assembly utility functions +============================================================================== +@File : AssemblyUtils.py +@Date : 2022/11/23 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== + +# ============================================================================== +# External Python modules +# ============================================================================== +from numba import njit +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== + + +def applyBCsToMatrix(rows, cols, values, bcDOF): + """Given the COO data for a sparse matrix, modify the data to apply a set of fixed state boundary conditions + + In the original matrix, the ith row of the linear system looks something like: + + ``[A_i1, ..., A_ii, ..., A_in] @ [du_1, ..., du_i, ..., du_n].T = [b_i]`` + + To enforce the boundary condition ``u_i = c``, we alter the ith row of the matrix to represent this equation: + + ``[0, ..., 1, ..., 0] @ [du_1, ..., du_i, ..., du_n].T = [u_i - c]`` + + This function performs the modification of the matrix data, the modification of the right hand side vector is + performed by the function :func:`applyBCsToVector`. + + Parameters + ---------- + rows : array of int + Row indicies of the matrix entries + cols : array of int + Column indicies of the matrix entries + values : array of float + Values of the matrix entries + bcDOF : array of int + Degrees of freedom to apply boundary conditions to + + Returns + ------- + rowInds : + """ + indsToDelete = np.nonzero(np.in1d(rows, bcDOF))[0] + values[indsToDelete] = 0.0 + + rows = np.append(rows, bcDOF) + cols = np.append(cols, bcDOF) + values = np.append(values, np.ones(len(bcDOF))) + + return rows, cols, values + + +def applyBCsToVector(vector, state, bcDOF, bcValues): + """Given the current state and RHS vectors, modify the data to apply a set of fixed state boundary conditions + + Take a linear system that we are solving to find an update in the state: K @ du = b + In this system of equation, the ith row looks something like: + + ``[A_i1, ..., A_ii, ..., A_in] @ [du_1, ..., du_i, ..., du_n].T = [b_i]`` + + To enforce the boundary condition ``u_i = c``, we alter the ith row of the matrix like this: + + ``[0, ..., 1, ..., 0] @ [du_1, ..., du_i, ..., du_n].T = [c - u_i]`` + + This function performs the modification of the RHS vector, the modification of the matrix data is performed by the + function :func:`applyBCsToMatrix`. + + Parameters + ---------- + vector : numpy array + Vector to apply boundary conditions to + state : numpy array + The current state vector + bcDOF : array of int + Degrees of freedom to apply boundary conditions to + vector : numpy array + Vector to apply boundary conditions to + bcValues : array of float + Values to fix degrees of freedom at + """ + vector[bcDOF] = state[bcDOF] - bcValues + return vector + + +def convertBCDictToLists(bcDict): + """Convert a dictionary of boundary conditions to lists of DOF and values + + The boundary condition dictionary stores boundary conditions in this form:: + + BCDict = { + "BC1Name": {"DOF": [0, 1, 2], "Value": [0, 0, 0]}, + "BC2Name": {"DOF": [13, 46, 1385], "Value": [1.0, 1.0, -1.0]}, + "BC3Name": {"DOF": [837, 25], "Value": [1.0, 1.0]}, + } + + + Parameters + ---------- + bcDict : dict + Dictionary of boundary conditions + + Returns + ------- + bcDOF : array of int + Degrees of freedom to apply boundary conditions to + bcValues : array of float + Values to fix degrees of freedom at + + Examples + -------- + >>> BCDict = { + ... "BC1Name": {"DOF": [0, 1, 2], "Value": [0, 0, 0]}, + ... "BC2Name": {"DOF": [13, 46, 1385], "Value": [1.0, 1.0, -1.0]}, + ... "BC3Name": {"DOF": [837, 25], "Value": [1.0, 1.0]}, + ... } + ... convertBCDictToLists(BCDict) + ([0, 1, 2, 13, 46, 1385, 837, 25], [0, 0, 0, 1.0, 1.0, -1.0, 1.0, 1.0]) + + """ + bcDOF = [] + bcValues = [] + for key in bcDict: + bcDOF += bcDict[key]["DOF"] + bcValues += bcDict[key]["Value"] + return bcDOF, bcValues + + +def convertLoadsDictToVector(loadsDict, numDOF: int): + """Convert a dictionary of loads to a global load vector + + The loads dictionary stores loads in this form:: + + loadsDict = { + "Load1Name": {"DOF": [0, 1, 2], "Value": [0, 0, 0]}, + "Load2Name": {"DOF": [13, 46, 1385], "Value": [1.0, 1.0, -1.0]}, + "Load3Name": {"DOF": [837, 25], "Value": [1.0, 1.0]}, + } + + + Parameters + ---------- + loadsDict : dict + Dictionary of loads + + Returns + ------- + loads : array of float + global load vector, with loads applied to the correct DOF + """ + loadForce = np.zeros(numDOF) + for key in loadsDict: + loadForce[loadsDict[key]["DOF"]] += loadsDict[key]["Value"] + return loadForce + + +@njit(cache=True) +def localMatricesToCOOArrays(localMats, localDOF): + """Convert a set of local matrices for a set of elements to COO format data for a global matrix + + Parameters + ---------- + localMats : numElements x n x n array + Local matrices for each element + localDOF : numElement x n array + Global DOF indices for each element + + Returns + ------- + rows : list of int + Row indicies of the global matrix entries + cols : list of int + Column indicies of the global matrix entries + values : list of float + Values of the global matrix entries + + """ + numElements = localMats.shape[0] + dofPerElement = localDOF.shape[1] + rows = np.zeros(numElements * dofPerElement**2, dtype=np.int64) + cols = np.zeros(numElements * dofPerElement**2, dtype=np.int64) + values = np.zeros(numElements * dofPerElement**2) + + for ii in range(numElements): + # Figure out where in the COO arrays to write the entires from this element + startInd = ii * dofPerElement**2 + + for rowInd in range(dofPerElement): + rowStartInd = startInd + rowInd * dofPerElement + for colInd in range(dofPerElement): + flatInd = rowStartInd + colInd + rows[flatInd] = localDOF[ii, rowInd] + cols[flatInd] = localDOF[ii, colInd] + values[flatInd] = localMats[ii, rowInd, colInd] + + # Now we have all the data, return only the non-zero entries + nonzeroEntries = np.flatnonzero(values) + + rows = rows[nonzeroEntries] + cols = cols[nonzeroEntries] + values = values[nonzeroEntries] + + return rows, cols, values + + +@njit(cache=True) +def scatterLocalResiduals(localResiduals, connectivity, globalResidual): + """Scatter local element residuals into the global residual vector + + Parameters + ---------- + localResiduals : numElements x numNodes x numStates array + Element local residuals + connectivity : numElements x numNodes array + Element connectivity matrix, each row contains the nodeIDs for an element + globalResidual : numpy array + The global residual vector + """ + numStates = localResiduals.shape[2] + numElements = localResiduals.shape[0] + numNodes = localResiduals.shape[1] + + for elNum in range(numElements): + for localNodeID in range(numNodes): + nodeID = connectivity[elNum, localNodeID] + for stateNum in range(numStates): + globalResidual[nodeID * numStates + stateNum] += localResiduals[elNum, localNodeID, stateNum] + + +if __name__ == "__main__": + + rows = np.array([0, 0, 1, 2], dtype=np.int64) + cols = np.array([1, 2, 2, 0], dtype=np.int64) + values = np.array([1, 0.5, 2, 3]) + bcDOF = np.array([1, 2], dtype=np.int64) + bcValues = np.array([10, 15]) + rows, cols, values = applyBCsToMatrix(rows, cols, values, bcDOF) + print(rows, cols, values) + + BCDict = { + "BC1Name": {"DOF": [0, 1, 2], "Value": [0, 0, 0]}, + "BC2Name": {"DOF": [13, 46, 1385], "Value": [1.0, 1.0, -1.0]}, + "BC3Name": {"DOF": [837, 25], "Value": [1.0, 1.0]}, + } + bcDOF, bcValues = convertBCDictToLists(BCDict) + + print(bcDOF, bcValues) diff --git a/FEMpy/Utils.py b/FEMpy/Utils/KSAgg.py similarity index 72% rename from FEMpy/Utils.py rename to FEMpy/Utils/KSAgg.py index a62f2a7..0ed627f 100644 --- a/FEMpy/Utils.py +++ b/FEMpy/Utils/KSAgg.py @@ -24,7 +24,7 @@ @njit(cache=True) -def ksAgg(g, rho=100.0): +def ksAgg(g, ksType, rho=100.0): """Compute a smooth approximation to the maximum of a set of values us KS agregation @@ -32,7 +32,9 @@ def ksAgg(g, rho=100.0): Parameters ---------- g : 1d array - Values to approximate the maximum of + Values to perform KS aggregation + ksType : string + determines which type of KS aggregation is performed ["min", "max"] rho : float, optional KS Weight parameter, larger values give a closer but less smooth approximation of the maximum, by default 100.0 @@ -41,6 +43,17 @@ def ksAgg(g, rho=100.0): float The KS agregated value """ + assert ksType.lower() in ["min", "max"], "KS aggregation type not valid" + ng = len(g) + if ksType.lower() == "min": + g = -1 * g + maxg = np.max(g) - return maxg + 1.0 / rho * np.log(1.0 / ng * np.sum(np.exp(rho * (g - maxg)))) + ks = maxg + 1.0 / rho * np.log(1.0 / ng * np.sum(np.exp(rho * (g - maxg)))) + + if ksType.lower() == "max": + return ks + + if ksType.lower() == "min": + return -1 * ks diff --git a/FEMpy/Utils/__init__.py b/FEMpy/Utils/__init__.py new file mode 100644 index 0000000..b03a900 --- /dev/null +++ b/FEMpy/Utils/__init__.py @@ -0,0 +1,2 @@ +from .KSAgg import * +from . import AssemblyUtils diff --git a/FEMpy/__init__.py b/FEMpy/__init__.py index ffc4d20..cfcf834 100644 --- a/FEMpy/__init__.py +++ b/FEMpy/__init__.py @@ -1,20 +1,11 @@ -__version__ = "0.0.1" +__version__ = "1.0.0" -from .Element import * -from .QuadElement import * -from .SerendipityQuad import * -from .Lagrange1dElement import * -from .Constitutive import * -from .isoPlaneStrain import * -from .isoPlaneStress import * -from .OrthoPlaneStress import * -from .AxialBar import * -from .LagrangePoly import * -from .GaussQuad import * -from .Assembly import * -from .Dynamics import * -from .TecplotIO import * -from .Mesh import * -from .Smoothing import getSmoother -from .Utils import * -from .LinAlg import * + +from . import LinAlg +from . import Quadrature +from . import Constitutive +from . import Basis +from . import Elements +from . import Utils +from .Problem import FEMpyProblem +from .Model import FEMpyModel diff --git a/FEMpy/isoPlaneStrain.py b/FEMpy/isoPlaneStrain.py deleted file mode 100644 index 31b5f6b..0000000 --- a/FEMpy/isoPlaneStrain.py +++ /dev/null @@ -1,47 +0,0 @@ -""" -============================================================================== -Isotropic Plane Strain Constitutive Class -============================================================================== -@File : isoPlaneStress.py -@Date : 2021/03/19 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== -from .Constitutive import Constitutive - - -class isoPlaneStrain(Constitutive): - def __init__(self, E, nu, t) -> None: - super().__init__(numStrain=3, numDisp=2, numStress=3) - self.LMats[0] = np.array([[1, 0], [0, 0], [0, 1]]) - self.LMats[1] = np.array([[0, 0], [0, 1], [1, 0]]) - self.E = E - self.nu = nu - self.t = t - - @property - def DMat(self): - return ( - self.E - / ((1.0 - 2.0 * self.nu) * (1.0 + self.nu)) - * np.array( - [ - [1.0 - self.nu, self.nu, 0.0], - [self.nu, 1.0 - self.nu, 0.0], - [0.0, 0.0, 0.5 * (1.0 - 2.0 * self.nu)], - ] - ) - ) diff --git a/FEMpy/isoPlaneStress.py b/FEMpy/isoPlaneStress.py deleted file mode 100644 index 949c4b1..0000000 --- a/FEMpy/isoPlaneStress.py +++ /dev/null @@ -1,47 +0,0 @@ -""" -============================================================================== -Isotropic Plane Stress Constitutive Class -============================================================================== -@File : isoPlaneStress.py -@Date : 2021/03/19 -@Author : Alasdair Christison Gray -@Description : -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np - -# ============================================================================== -# Extension modules -# ============================================================================== -from .Constitutive import Constitutive - - -class isoPlaneStress(Constitutive): - def __init__(self, E, nu, t) -> None: - super().__init__(numStrain=3, numDisp=2, numStress=3) - self.LMats[0] = np.array([[1, 0], [0, 0], [0, 1]]) - self.LMats[1] = np.array([[0, 0], [0, 1], [1, 0]]) - self.E = E - self.nu = nu - self.t = t - - @property - def DMat(self): - return ( - self.E - / (1.0 - self.nu**2.0) - * np.array( - [ - [1.0, self.nu, 0.0], - [self.nu, 1.0, 0.0], - [0.0, 0.0, 0.5 * (1.0 - self.nu)], - ] - ) - ) diff --git a/README.md b/README.md index b66697a..48962d9 100644 --- a/README.md +++ b/README.md @@ -2,19 +2,19 @@ [![Docs](https://github.com/A-CGray/FEMpy/actions/workflows/docs.yml/badge.svg)](https://A-CGray.github.io/FEMpy/) [![Unit Tests](https://github.com/A-CGray/FEMpy/actions/workflows/Tests.yml/badge.svg)](https://A-CGray.github.io/FEMpy/) +[![Test Coverage](https://codecov.io/gh/A-CGray/FEMpy/branch/main/graph/badge.svg?token=VLBE5HHP68)](https://codecov.io/gh/A-CGray/FEMpy) [![Code Formatting](https://github.com/A-CGray/FEMpy/actions/workflows/Formatting.yml/badge.svg)](https://A-CGray.github.io/FEMpy/) [![Code style: black](https://img.shields.io/badge/code%20style-black-000000.svg)](https://github.com/psf/black) -[![Test Coverage](https://api.codeclimate.com/v1/badges/ba60cf87e121a05dfe5d/test_coverage)](https://codeclimate.com/github/A-CGray/FEMpy/test_coverage) -[![Maintainability](https://api.codeclimate.com/v1/badges/ba60cf87e121a05dfe5d/maintainability)](https://codeclimate.com/github/A-CGray/FEMpy/maintainability) -FEMpy is my attempt to implement a basic object-oriented finite element method in python. -![Pretty Colours](docs/docs/Images/PrettyColours.png) +FEMpy is my attempt to implement a basic object-oriented finite element method in python that's as fast and flexible as possible. -FEMpy uses scipy's sparse matrix implementation to enable scaling to problems with many (>100k) degrees of freedom. -Wherever possible, operations use numpy vectorisation or numba JIT compiling for speed, there's still plenty of room for improvement though! +![Pretty Colours](docs/docs/Images/VonMises_Vert.png) -![FEMpy can easily handle problems with 100,000 degrees of freedom](docs/docs/Images/QuadElScaling.png) +FEMpy uses Numba JIT compilation extensively to perform fast, multithread operations over many elements at once. +It uses scipy's sparse matrix implementation and can use the super-fast [Paradiso solver](https://github.com/haasad/PyPardisoProject) from Intel's MKL to assemble and solve problems with 500,000 degrees of freedom in a few seconds. + +![FEMpy can easily handle problems with 500,000 degrees of freedom](docs/docs/Images/QuadElScaling.png) ## How to install Inside the FEMpy root directory run: @@ -23,23 +23,15 @@ pip install . ``` Or, if you want to make changes to the code: ```shell -pip install -e . -``` -If you want to build documentation locally, or run the unit tests, make sure to install the necessary dependencies: -```shell -pip install -e .[docs, dev] -``` -And then run: -```shell -make build +pip install -e .[dev] +pre-commit install ``` -To build the docs locally, or: +To run unit tests. ```shell testflo -n 1 -v . ``` -To run unit tests. ## Documentation -View the documentation (still under construction) [here](https://A-CGray.github.io/FEMpy/) +View the documentation [here](https://A-CGray.github.io/FEMpy/) diff --git a/docs/docs/ConstitutiveModels/ConstitutiveModel.md b/docs/docs/ConstitutiveModels/ConstitutiveModel.md new file mode 100644 index 0000000..7c52bcb --- /dev/null +++ b/docs/docs/ConstitutiveModels/ConstitutiveModel.md @@ -0,0 +1,3 @@ +# Constitutive Model Base Class + +::: FEMpy.Constitutive.ConstitutiveModel.ConstitutiveModel diff --git a/docs/docs/ConstitutiveModels/IsoPlaneStrain.md b/docs/docs/ConstitutiveModels/IsoPlaneStrain.md new file mode 100644 index 0000000..44f63f1 --- /dev/null +++ b/docs/docs/ConstitutiveModels/IsoPlaneStrain.md @@ -0,0 +1,3 @@ +# Isotropic Plane Strain + +::: FEMpy.Constitutive.IsoPlaneStrain.IsoPlaneStrain diff --git a/docs/docs/ConstitutiveModels/IsoPlaneStress.md b/docs/docs/ConstitutiveModels/IsoPlaneStress.md new file mode 100644 index 0000000..53e0356 --- /dev/null +++ b/docs/docs/ConstitutiveModels/IsoPlaneStress.md @@ -0,0 +1,3 @@ +# Isotropic Plane Stress + +::: FEMpy.Constitutive.IsoPlaneStress.IsoPlaneStress diff --git a/docs/docs/DevelopmentGuide/DevelopmentGuide.md b/docs/docs/DevelopmentGuide/DevelopmentGuide.md index 77df6b0..4cb07a9 100644 --- a/docs/docs/DevelopmentGuide/DevelopmentGuide.md +++ b/docs/docs/DevelopmentGuide/DevelopmentGuide.md @@ -2,10 +2,11 @@ ## Code formatting and Linting -The following checks will be run on any PR using [pre-commit](pre-commit.com) before it can be merged into the main branch: +The following checks will be run on any PR using [pre-commit](https://www.pre-commit.com) before it can be merged into the main branch: -- Black code formatter, with 120 column line width and target python versions 3.7-3.10 -- Flake8 linter, using the [MDO Lab's configuration file](https://raw.githubusercontent.com/mdolab/.github/master/.flake8) +- [Black](black.readthedocs.io/en/stable) code formatter, with 120 column line width and target python versions 3.8-3.10 +- [Flake8](https://flake8.pycqa.org/en/latest) linter, using the [MDO Lab's configuration file](https://raw.githubusercontent.com/mdolab/.github/master/.flake8) +- [Interrogate](https://interrogate.readthedocs.io) docstring check - `check-yaml` - checks yaml files for parseable syntax. - `check-json` - checks json files for parseable syntax. - `check-added-large-files` - prevents giant files from being committed. diff --git a/docs/docs/Elements/Element.md b/docs/docs/Elements/Element.md index ccc3f94..f8bc14a 100644 --- a/docs/docs/Elements/Element.md +++ b/docs/docs/Elements/Element.md @@ -1,3 +1,3 @@ -# Element Parent Class +# Element Base Class -::: FEMpy.Element.Element +::: FEMpy.Elements.Element.Element diff --git a/docs/docs/Elements/Hex3D.md b/docs/docs/Elements/Hex3D.md new file mode 100644 index 0000000..e1ea1b6 --- /dev/null +++ b/docs/docs/Elements/Hex3D.md @@ -0,0 +1,3 @@ +# 3D Hex Element + +::: FEMpy.Elements.HexElement3D.HexElement3D diff --git a/docs/docs/Elements/Lagrange1dElement.md b/docs/docs/Elements/Lagrange1dElement.md deleted file mode 100644 index a535156..0000000 --- a/docs/docs/Elements/Lagrange1dElement.md +++ /dev/null @@ -1,3 +0,0 @@ -# 1D Line Element - -::: FEMpy.Lagrange1dElement.Lagrange1dElement diff --git a/docs/docs/Elements/Quad2D.md b/docs/docs/Elements/Quad2D.md new file mode 100644 index 0000000..538a628 --- /dev/null +++ b/docs/docs/Elements/Quad2D.md @@ -0,0 +1,3 @@ +# 2D Quad Element + +::: FEMpy.Elements.QuadElement2D.QuadElement2D diff --git a/docs/docs/Elements/QuadElement.md b/docs/docs/Elements/QuadElement.md deleted file mode 100644 index eb741b7..0000000 --- a/docs/docs/Elements/QuadElement.md +++ /dev/null @@ -1,3 +0,0 @@ -# 2D Quad Element - -::: FEMpy.QuadElement.QuadElement diff --git a/docs/docs/Elements/SerendipityQuad.md b/docs/docs/Elements/SerendipityQuad.md deleted file mode 100644 index 76f88aa..0000000 --- a/docs/docs/Elements/SerendipityQuad.md +++ /dev/null @@ -1,3 +0,0 @@ -# Serendipity Quad Element - -::: FEMpy.SerendipityQuad.serendipityQuadElement diff --git a/docs/docs/Elements/Tri2D.md b/docs/docs/Elements/Tri2D.md new file mode 100644 index 0000000..3116338 --- /dev/null +++ b/docs/docs/Elements/Tri2D.md @@ -0,0 +1,3 @@ +# 2D Triangle Element + +::: FEMpy.Elements.TriElement2D.TriElement2D diff --git a/docs/docs/Images/1D-Element.pdf b/docs/docs/Images/1D-Element.pdf new file mode 100644 index 0000000..82c33a2 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l +-0.5 0.5 l +h + + +-0.6 -0.6 m +0.6 -0.6 l +0.6 0.6 l +-0.6 0.6 l +h +-0.4 -0.4 m +0.4 -0.4 l +0.4 0.4 l +-0.4 0.4 l +h + + + + + + +-0.43 -0.57 m +0.57 0.43 l +0.43 0.57 l +-0.57 -0.43 l +h + + +-0.43 0.57 m +0.57 -0.43 l +0.43 -0.57 l +-0.57 0.43 l +h + + + + + +0 0 m +-1 0.333 l +-1 -0.333 l +h + + + + +0 0 m +-1 0.333 l +-0.8 0 l +-1 -0.333 l +h + + + + +0 0 m +-1 0.333 l +-0.8 0 l +-1 -0.333 l +h + + + + +-1 0.333 m +0 0 l +-1 -0.333 l + + + + +0 0 m +-1 0.333 l +-1 -0.333 l +h +-1 0 m +-2 0.333 l +-2 -0.333 l +h + + + + +0 0 m +-1 0.333 l +-1 -0.333 l +h +-1 0 m +-2 0.333 l +-2 -0.333 l +h + + + + +0.5 0 m +-0.5 0.333 l +-0.5 -0.333 l +h + + + + +0.5 0 m +-0.5 0.333 l +-0.5 -0.333 l +h + + + + +0.5 0 m +-0.5 0.333 l +-0.3 0 l +-0.5 -0.333 l +h + + + + +0.5 0 m +-0.5 0.333 l +-0.3 0 l +-0.5 -0.333 l +h + + + + +1 0 m +0 0.333 l +0 -0.333 l +h +0 0 m +-1 0.333 l +-1 -0.333 l +h + + + + +1 0 m +0 0.333 l +0 -0.333 l +h +0 0 m +-1 0.333 l +-1 -0.333 l +h + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +\vec{x}_1 +\vec{x}_2 +\vec{x}_3 +u_1 = [\alpha_1, \beta_1, \gamma_1] +u_2 = [\alpha_2, \beta_2, \gamma_2] +u_3 = [\alpha_3, \beta_3, \gamma_3] +u(\tilde{x}) = [\alpha(\tilde{x}), \beta(\tilde{x}), \gamma(\tilde{x})], \\ +u'(\tilde{x}) = \frac{du}{d\vec{x}} = \begin{bmatrix} +\frac{d\alpha}{dx} & \frac{d\alpha}{dy} & \frac{d\alpha}{dz} \\ +\frac{d\beta}{dx} & \frac{d\beta}{dy} & \frac{d\beta}{dz} \\ +\frac{d\gamma}{dx} & \frac{d\gamma}{dy} & \frac{d\gamma}{dz} \\ +\end{bmatrix} +\vec{x}(\tilde{x}) + +139.463 460.761 m +162.021 508.884 l + +q = \begin{bmatrix} +\alpha_1 & \beta_1 & \gamma_1 \\ +\alpha_2 & \beta_2 & \gamma_2 \\ +\alpha_3 & \beta_3 & \gamma_3 \\ +\end{bmatrix} + +195.481 719.797 m +198.113 638.59 l + + + diff --git a/docs/docs/Images/Assembly.pdf b/docs/docs/Images/Assembly.pdf new file mode 100644 index 0000000..3fb5401 Binary files /dev/null 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determinant or inverse of the element's Jacobian. -Since these Jacobians are small (max 3x3) FEMpy uses the manual implementations below for these inverse and determinant calculations. -With the help of numba, these routines are approximately 1-2 orders of magnitude quicker than numpy's `linalg.det` and `linalg.inv` functions. - -::: FEMpy.LinAlg diff --git a/docs/docs/Theory/Theory.md b/docs/docs/Theory/Theory.md new file mode 100644 index 0000000..992e010 --- /dev/null +++ b/docs/docs/Theory/Theory.md @@ -0,0 +1,7 @@ +# Finite Element Theory (or FEMpy's version of it at least) + +## Isoparametric Elements + +In FEMpy, elements defining the mapping between the discrete and continuous. +An element has a certain number of nodes which have coordinates in real space (reffered to as $\vec{x}$), and at which the state, $u$ is defined. +Depending on the PDE being modelled, the state can be a scalar or a vector, (a single temperature value, or a 3-component displacement). diff --git a/docs/docs/index.md b/docs/docs/index.md index 79e5bf1..f7f95c1 100644 --- a/docs/docs/index.md +++ b/docs/docs/index.md @@ -1,12 +1,13 @@ # FEMpy -FEMpy is my attempt to implement a basic object-oriented finite element method in python -![Pretty Colours](Images/PrettyColours.png) +FEMpy is my attempt to implement a basic object-oriented finite element method in python that's as fast and flexible as possible. -FEMpy uses scipy's sparse matrix implementation to enable scaling to problems with many (>100k) degrees of freedom. -Wherever possible, operations use numpy vectorisation or numba JIT compiling for speed, there's still plenty of room for improvement though! +![Pretty Colours](Images/VonMises_Vert.png) -![FEMpy can easily handle problems with 100,000 degrees of freedom](Images/QuadElScaling.png) +FEMpy uses Numba JIT compilation extensively to perform fast, multithread operations over many elements at once. +It uses scipy's sparse matrix implementation and can use the super-fast [Paradiso solver](https://github.com/haasad/PyPardisoProject) from Intel's MKL to assemble and solve problems with 500,000 degrees of freedom in a few seconds. + +![FEMpy can easily handle problems with 500,000 degrees of freedom](Images/QuadElScaling.png) ## How to install Inside the FEMpy root directory run: @@ -17,11 +18,8 @@ Or, if you want to make changes to the code: ```shell pip install -e . ``` -If you want to build documentation locally, or run the unit tests, make sure to install the necessary dependencies: -```shell -pip install -e .[docs, dev] -``` -And then run: + +To run unit tests. ```shell -make build +testflo -n 1 -v . ``` diff --git a/docs/mkdocs.yml b/docs/mkdocs.yml index 85e925f..634dd3a 100644 --- a/docs/mkdocs.yml +++ b/docs/mkdocs.yml @@ -29,13 +29,16 @@ theme: nav: - Home: index.md - Elements: - Generic Element: Elements/Element.md - 2D Quad: Elements/QuadElement.md - Serendipity Quad: Elements/SerendipityQuad.md - 1D Line Element: Elements/Lagrange1dElement.md - - Utilities: - Linear Algebra: LinAlg/LinAlg.md + Element Base Class: Elements/Element.md + 2D Quad: Elements/Quad2D.md + 2D Tri: Elements/Tri2D.md + 3D Hex: Elements/Hex3D.md + - Constitutive Models: + Constitutive Model Base Class: ConstitutiveModels/ConstitutiveModel.md + Isotropic Plane Stress: ConstitutiveModels/IsoPlaneStress.md + Isotropic Plane Strain: ConstitutiveModels/IsoPlaneStrain.md - Contributing: DevelopmentGuide/DevelopmentGuide.md + - Theory: Theory/Theory.md markdown_extensions: - pymdownx.superfences - pymdownx.progressbar @@ -56,6 +59,10 @@ markdown_extensions: # set to false to turn off pygments_style noclasses: False - pymdownx.tasklist + - pymdownx.arithmatex: + generic: true + + extra_javascript: - js/flowchart-loader.js - js/details.js @@ -63,6 +70,9 @@ extra_javascript: - https://cdnjs.cloudflare.com/ajax/libs/underscore.js/1.8.3/underscore-min.js - https://cdnjs.cloudflare.com/ajax/libs/js-sequence-diagrams/1.0.6/sequence-diagram-min.js - https://cdnjs.cloudflare.com/ajax/libs/flowchart/1.8.0/flowchart.min.js + - javascripts/mathjax.js + - https://polyfill.io/v3/polyfill.min.js?features=es6 + - https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js extra_css: - css/progress_bar.css - css/details.css @@ -77,6 +87,3 @@ plugins: docstring_style: 'numpy' rendering: show_if_no_docstring: True -google_analytics: - - G-CE4BXN9PR8 - - auto diff --git a/requirements.txt b/requirements.txt deleted file mode 100644 index 08f78fb..0000000 --- a/requirements.txt +++ /dev/null @@ -1,7 +0,0 @@ -mkdocs -livereload -pymdown-extensions -mkdocs-material -mkdocstrings -pytkdocs[numpy-style] -Jinja2<3.0,>=2.11 diff --git a/setup.py b/setup.py index cc765dc..defa23a 100644 --- a/setup.py +++ b/setup.py @@ -1,4 +1,4 @@ -from setuptools import setup +from setuptools import setup, find_packages from setuptools.command.develop import develop from setuptools.command.install import install import re @@ -11,41 +11,6 @@ )[0] -def computeGaussQuadValues(n, outdir=None): - from numpy.polynomial.legendre import leggauss - import pickle - - if outdir is None: - outdir = "" - gaussWeights = {} - gaussCoords = {} - for i in range(1, n + 1): - gaussCoords[i - 1], gaussWeights[i - 1] = leggauss(i) - with open(os.path.join(outdir, "FEMpy/GaussQuadWeights.pkl"), "wb") as f: - pickle.dump(gaussWeights, f, protocol=-1) - with open(os.path.join(outdir, "FEMpy/GaussQuadCoords.pkl"), "wb") as f: - pickle.dump(gaussCoords, f, protocol=-1) - - -class installWrapper(install): - """wrapper around setuptools' install method that will run a post install script""" - - def run(self): - from distutils.sysconfig import get_python_lib - - install.run(self) - computeGaussQuadValues(64, outdir=get_python_lib()) - - -class developWrapper(develop): - """wrapper around setuptools' develop method that will run a post install script""" - - def run(self): - develop.run(self) - computeGaussQuadValues(64) - subprocess.run("pre-commit install", shell=True) - - setup( name="FEMpy", version=__version__, @@ -55,21 +20,19 @@ def run(self): author_email="", url="https://github.com/A-Gray-94/FEMpy", license="Apache License Version 2.0", - packages=["FEMpy"], + packages=find_packages(where="."), install_requires=[ + "mdolab-baseclasses", + "meshio", "numpy", "numba", - "scipy", - "pyComposite @ git+https://github.com/A-Gray-94/pyComposite.git", + "scipy>=1.8.0", ], extras_require={ "docs": [ - "mkdocs", - "pymdown-extensions", "mkdocs-material", - "mkdocstrings", + "mkdocstrings[python]", "pytkdocs[numpy-style]", - "Jinja2<3.0,>=2.11", ], "dev": ["parameterized", "testflo", "black==22.12.0", "flake8==3.9.2", "pre-commit"], }, @@ -77,5 +40,4 @@ def run(self): "Operating System :: OS Independent", "Programming Language :: Python", ], - cmdclass={"install": installWrapper, "develop": developWrapper}, ) diff --git a/testRequirements.txt b/testRequirements.txt deleted file mode 100644 index e7ab444..0000000 --- a/testRequirements.txt +++ /dev/null @@ -1 +0,0 @@ -.[dev] diff --git a/tests/testAssembleutils.py b/tests/testAssembleutils.py new file mode 100644 index 0000000..065c3c6 --- /dev/null +++ b/tests/testAssembleutils.py @@ -0,0 +1,86 @@ +""" +============================================================================== +FEMpy Model class unit tests +============================================================================== +@File : testModel.py +@Date : 2022/11/14 +@Author : M.A. Saja A.Kaiyoom and Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import unittest +import os +import numpy as np + +# ============================================================================== +# External Python modules +# ============================================================================== + +# ============================================================================== +# Extension modules +# ============================================================================== +# from FEMpy import FEMpyModel +from FEMpy.Utils import AssemblyUtils + + +class AssemUnitTest(unittest.TestCase): + """Unit tests for the FEMpy model class""" + + def testConvertBCDictToLists(self): + """Test that the mesh file was read in correctly""" + BCDict = { + "BC1Name": {"DOF": [0, 1, 2], "Value": [0, 0, 0]}, + "BC2Name": {"DOF": [13, 46, 1385], "Value": [1.0, 1.0, -1.0]}, + "BC3Name": {"DOF": [837, 25], "Value": [1.0, 1.0]}, + } + bcDOF, bcValues = AssemblyUtils.convertBCDictToLists(BCDict) + bcDOF_out = [0, 1, 2, 13, 46, 1385, 837, 25] + bcValues_out = [0, 0, 0, 1.0, 1.0, -1.0, 1.0, 1.0] + self.assertEqual(bcDOF, bcDOF_out) + self.assertEqual(bcValues, bcValues_out) + + def testConvertLoadsDictToVector(self): + """Test that the mesh file was read in correctly""" + loadsDict = { + "Load1Name": {"DOF": [0, 1, 2], "Value": [0, 0, -1]}, + "Load2Name": {"DOF": [4, 5, 8], "Value": [1.0, 1.0, -1.0]}, + "Load3Name": {"DOF": [2, 10], "Value": [3.0, 4.0]}, + } + loads = AssemblyUtils.convertLoadsDictToVector(loadsDict, 12) + loads_out = np.array([0.0, 0.0, 2.0, 0.0, 1.0, 1.0, 0.0, 0.0, -1.0, 0.0, 4.0, 0.0]) + np.testing.assert_equal(loads, loads_out) + + def testLocalMatricesToCOOArrays(self): + """Test that the mesh file was read in correctly""" + + localMats = np.array( + [ + np.array( + [ + [1, 0, 0, 0], + [0, 1, 0, 0], + [0, 1, 1, 0], + [0, 0, 0, 1], + ] + ), + np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 2], [0, 0, 0, 2]]), + np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 2], [0, 0, 0, 2]]), + np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 2], [0, 0, 0, 2]]), + ] + ) + COO_expected_rows = np.array([0, 1, 2, 2, 3, 2, 3, 4, 4, 5, 4, 5, 6, 6, 7, 6, 7, 0, 0, 1]) + COO_expected_cols = np.array([0, 1, 1, 2, 3, 2, 3, 4, 5, 5, 4, 5, 6, 7, 7, 6, 7, 0, 1, 1]) + COO_expected_vals = np.array([1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2]) + localDOF = np.array([[0, 1, 2, 3], [2, 3, 4, 5], [4, 5, 6, 7], [6, 7, 0, 1]]) + + COO_comp_rows, COO_comp_cols, COO_comp_vals = AssemblyUtils.localMatricesToCOOArrays(localMats, localDOF) + np.testing.assert_equal(COO_comp_rows, COO_expected_rows) + np.testing.assert_equal(COO_comp_cols, COO_expected_cols) + np.testing.assert_equal(COO_comp_vals, COO_expected_vals) + + +if __name__ == "__main__": + unittest.main() diff --git a/tests/testConstitutive.py b/tests/testConstitutive.py new file mode 100644 index 0000000..37a5d1d --- /dev/null +++ b/tests/testConstitutive.py @@ -0,0 +1,110 @@ +""" +============================================================================== +Constitutive class unit tests +============================================================================== +@File : testConstitutive.py +@Date : 2022/12/08 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +from parameterized import parameterized_class +import unittest + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy.Constitutive import IsoPlaneStrain, IsoPlaneStress, Iso3D + + +testParams = [] + +for cm in [IsoPlaneStrain, IsoPlaneStress, Iso3D]: + for linear in [True, False]: + if cm in [IsoPlaneStrain, IsoPlaneStress]: + constitutiveModel = cm(E=70e9, nu=0.3, rho=2700, t=1.0, linear=linear) + else: + constitutiveModel = cm(E=70e9, nu=0.3, rho=2700, linear=linear) + testParams.append({"ConstitutiveModel": constitutiveModel}) + + +@parameterized_class(testParams) +class ConstitutiveUnitTest(unittest.TestCase): + def setUp(self) -> None: + + self.tol = 1e-10 + np.random.seed(0) + + self.numPoints = 5 + + self.numPerts = 3 + self.stepSize = 1e-200 + self.relCSTol = 1e-10 + self.absCSTol = 1e-10 + + self.numStates = self.ConstitutiveModel.numStates + self.numDim = self.ConstitutiveModel.numDim + self.numStrains = self.ConstitutiveModel.numStrains + self.u = np.random.rand(self.numPoints, self.numStates) + self.coords = np.random.rand(self.numPoints, self.numDim) + self.dudx = np.random.rand(self.numPoints, self.numStates, self.numDim) + + self.dvs = {} + for dvName in self.ConstitutiveModel.designVariables: + self.dvs[dvName] = ( + np.random.rand(self.numPoints) * 2 * self.ConstitutiveModel.designVariables[dvName]["defaultValue"] + ) + + def testStrainSens(self): + """Validate the strain sensitivity calculation against complex-step""" + strainSens = self.ConstitutiveModel.computeStrainStateGradSens(self.u, self.dudx, self.coords, self.dvs) + + for ii in range(self.numStates): + for jj in range(self.numDim): + dudx = self.dudx.astype("complex") + dudx[:, ii, jj] += self.stepSize * 1j + strainPert = self.ConstitutiveModel.computeStrains(self.u, dudx, self.coords, self.dvs) + + csSens = np.imag(strainPert) / self.stepSize + np.testing.assert_allclose(csSens, strainSens[:, :, ii, jj], rtol=self.relCSTol, atol=self.absCSTol) + + def testStressStrainSens(self): + """Validate the stress-strain sensitivity calculation against complex-step""" + strain = self.ConstitutiveModel.computeStrains(self.u, self.dudx, self.coords, self.dvs) + stressSens = self.ConstitutiveModel.computeStressStrainSens(strain, self.dvs) + + for ii in range(self.numStrains): + strainPert = strain.astype("complex") + strainPert[:, ii] += self.stepSize * 1j + stressPert = self.ConstitutiveModel.computeStresses(strainPert, self.dvs) + + csSens = np.imag(stressPert) / self.stepSize + np.testing.assert_allclose(csSens, stressSens[:, :, ii], rtol=self.relCSTol, atol=self.absCSTol) + + def testGetFunction(self): + """Check that, for each function the constitutive model says it has, it provides a function that has the correct signature""" + states = self.u + stateGrad = self.dudx + dvs = self.dvs + coords = self.coords + for funcName in self.ConstitutiveModel.functionNames: + f = self.ConstitutiveModel.getFunction(funcName) + values = f(states, stateGrad, coords, dvs) + np.testing.assert_equal(values.shape, (self.numPoints,)) + + def testGetWrongFunction(self): + """Check that asking for a function that doesn't exist raises an error""" + with self.assertRaises(ValueError): + self.ConstitutiveModel.getFunction("wrongFunction") + + +if __name__ == "__main__": + unittest.main() diff --git a/tests/testElements.py b/tests/testElements.py index a711906..3b5fae8 100644 --- a/tests/testElements.py +++ b/tests/testElements.py @@ -13,6 +13,7 @@ # ============================================================================== import unittest from parameterized import parameterized_class +import time # ============================================================================== # External Python modules @@ -22,44 +23,148 @@ # ============================================================================== # Extension modules # ============================================================================== -from FEMpy import QuadElement, Lagrange1dElement, serendipityQuadElement +from FEMpy.Elements import QuadElement2D, TriElement2D, HexElement3D +from FEMpy.Constitutive import IsoPlaneStrain, Iso3D # --- Elements to test: --- -# QuadElement: 1st to 4th order -# SerendipityQuad -# Lagrange1DElement: 1st to 4th order +# QuadElement2D: 1st to 3rd order +# TruElement2D: 1st to 3rd order +# HexElement3D: 1st and 2rd order -test_params = [] +knownQuadStiffnessMat = 1.0e10 * np.array( + [ + [4.4786, 2.3299, -1.5533, 0.7249, -3.1583, -2.0710, 0.2330, -0.9837], + [2.3299, 4.6080, 0.0518, 1.5533, -2.0710, -2.8994, -0.3107, -3.2618], + [-1.5533, 0.0518, 3.4430, -0.7766, -1.0873, -0.5695, -0.8025, 1.2944], + [0.7249, 1.5533, -0.7766, 5.6435, -1.2426, -4.9704, 1.2944, -2.2263], + [-3.1583, -2.0710, -1.0873, -1.2426, 5.7988, 2.5888, -1.5533, 0.7249], + [-2.0710, -2.8994, -0.5695, -4.9704, 2.5888, 6.3166, 0.0518, 1.5533], + [0.2330, -0.3107, -0.8025, 1.2944, -1.5533, 0.0518, 2.1228, -1.0355], + [-0.9837, -3.2618, 1.2944, -2.2263, 0.7249, 1.5533, -1.0355, 3.9349], + ] +) -for el in [Lagrange1dElement, QuadElement, serendipityQuadElement]: - if el in [QuadElement, Lagrange1dElement]: - for order in range(1, 5): +testParams = [] + +cm2D = IsoPlaneStrain(E=70e9, nu=0.3, rho=2700, t=1.0) +cm3D = Iso3D(E=70e9, nu=0.3, rho=2700) + +for el in [QuadElement2D, TriElement2D, HexElement3D]: + if el in [QuadElement2D, TriElement2D]: + for order in range(1, 4): + element = el(order=order) + testParams.append({"element": element, "name": element.name, "knownJac": False}) + if el == QuadElement2D: + testParams[0]["knownJac"] = True + testParams[0]["knownJacCoords"] = np.array([[0.0, 0.0], [2.0, 0.0], [2.0, 1.0], [0.0, 2.0]]) + testParams[0]["knownJacDVs"] = {"Thickness": 1.0} + testParams[0]["knownJacMat"] = knownQuadStiffnessMat + elif el in [HexElement3D]: + for order in range(1, 3): element = el(order=order) - test_params.append({"element": element, "name": element.name}) + testParams.append({"element": element, "name": element.name, "knownJac": False}) else: element = el() - test_params.append({"element": element, "name": element.name}) + testParams.append({"element": element, "name": element.name}) + for ii in range(len(testParams)): + element = testParams[ii]["element"] + if element.numDim == 2: + testParams[ii]["ConstitutiveModel"] = cm2D + elif element.numDim == 3: + testParams[ii]["ConstitutiveModel"] = cm3D + else: + raise ValueError("Invalid number of dimensions") -@parameterized_class(test_params) + +@parameterized_class(testParams) class ElementUnitTest(unittest.TestCase): def setUp(self) -> None: self.tol = 1e-10 self.numTestPoints = 4 + self.rng = np.random.default_rng(1) np.random.seed(1) - def testGetParamCoord(self): - error = self.element.testGetParamCoord(self.numTestPoints, maxIter=400, tol=self.tol * 1e-3) - np.testing.assert_allclose(error, 0, atol=self.tol, rtol=self.tol) + self.numElements = 10 + + self.numPerts = 3 + self.stepSize = 1e-6 + self.relFDTol = 1e-5 + self.absFDTol = 1e-5 def testShapeFunctionDerivatives(self): - error = self.element.testShapeFunctionDerivatives(self.numTestPoints) + error = self.element.testShapeFunctionDerivatives(self.numTestPoints, rng=self.rng) np.testing.assert_allclose(error, 0, atol=self.tol, rtol=self.tol) def testShapeFunctionSum(self): - shapeFuncSum = self.element.testShapeFunctionSum(self.numTestPoints) + shapeFuncSum = self.element.testShapeFunctionSum(self.numTestPoints, rng=self.rng) np.testing.assert_allclose(shapeFuncSum, 1, atol=self.tol, rtol=self.tol) + def testIdentityJacobian(self): + JacDiff = self.element.testIdentityJacobian(self.numTestPoints, rng=self.rng) + np.testing.assert_allclose(JacDiff, 0, atol=self.tol, rtol=self.tol) + + def testInterpolation(self): + error = self.element.testInterpolation(self.numTestPoints, rng=self.rng) + np.testing.assert_allclose(error, 0, atol=self.tol, rtol=self.tol) + + def testStateGradient(self): + stateGradDiff = self.element.testStateGradient(self.numTestPoints, rng=self.rng) + np.testing.assert_allclose(stateGradDiff, 0, atol=self.tol, rtol=self.tol) + + def testGetClosestPoints(self): + """Test that the closest points are found correctly + + This test works by chossing some random parametric coordinates, computing their real coordinates and then + checking that we get back the same parametric coordinates when we ask for the closest points to the real coordinates. + """ + # self.skipTest("Not working robustly yet") + error = self.element.testGetClosestPoints(self.numTestPoints, tol=self.tol * 1e-3, rng=self.rng) + np.testing.assert_allclose(error, 0, atol=self.tol * 1e7, rtol=self.tol * 1e7) + + def testZeroResidual(self): + """Validate that the residual is zero if all states are zero""" + nodeCoordinates = np.zeros((self.numElements, self.element.numNodes, self.element.numDim)) + for ii in range(self.numElements): + nodeCoordinates[ii] = self.element.getRandomElementCoordinates(rng=self.rng) + nodeStates = np.zeros_like(nodeCoordinates) + dvs = {"Thickness": np.ones(self.numElements)} + res = self.element.computeResiduals(nodeStates, nodeCoordinates, dvs, self.ConstitutiveModel) + self.assertEqual(res.shape, (self.numElements, self.element.numNodes, self.element.numStates)) + np.testing.assert_allclose(res, 0, atol=self.tol, rtol=self.tol) + + def testResidualJacobian(self): + """Test that the residual Jacobian is consistent with the residual using finite differences""" + nodeCoordinates = np.zeros((self.numElements, self.element.numNodes, self.element.numDim)) + for ii in range(self.numElements): + nodeCoordinates[ii] = self.element.getRandomElementCoordinates(rng=self.rng) + + nodeStates = self.rng.random(nodeCoordinates.shape) + dvs = {"Thickness": self.rng.random(self.numElements)} + + # This is a special case where we know what the stiffness matrix should be + if self.knownJac: + nodeCoordinates[0] = self.knownJacCoords + for dv, value in self.knownJacDVs.items(): + dvs[dv][0] = value + + # --- Compute the residual Jacobians --- + Jac = self.element.computeResidualJacobians(nodeStates, nodeCoordinates, dvs, self.ConstitutiveModel) + + if self.knownJac: + np.testing.assert_allclose(Jac[0], self.knownJacMat, rtol=1e-3) + + # --- Test some random finite difference perturbations of the residual against mat-vec products with the Jacobians --- + res0 = self.element.computeResiduals(nodeStates, nodeCoordinates, dvs, self.ConstitutiveModel) + for _ in range(self.numPerts): + pert = self.rng.random(nodeStates.shape) * self.stepSize - self.stepSize * 0.5 + resPert = self.element.computeResiduals(nodeStates + pert, nodeCoordinates, dvs, self.ConstitutiveModel) + resChange = resPert - res0 + + for ii in range(self.numElements): + jacVecProd = Jac[ii].dot(pert[ii].flatten()) + np.testing.assert_allclose(resChange[ii].flatten(), jacVecProd, atol=self.absFDTol, rtol=self.relFDTol) + if __name__ == "__main__": unittest.main() diff --git a/tests/testGaussQuad.py b/tests/testGaussQuad.py deleted file mode 100644 index 025e9bc..0000000 --- a/tests/testGaussQuad.py +++ /dev/null @@ -1,99 +0,0 @@ -""" -============================================================================== -Gauss Quadrature unit tests -============================================================================== -@File : testGaussQuad.py -@Date : 2021/07/29 -@Author : Alasdair Christison Gray -@Description : Unit tests for FEMpy's gauss quadrature integration scheme -""" - -# ============================================================================== -# Standard Python modules -# ============================================================================== -import unittest - -# ============================================================================== -# External Python modules -# ============================================================================== -import numpy as np -import scipy.integrate as integrate - -# ============================================================================== -# Extension modules -# ============================================================================== -from FEMpy.GaussQuad import gaussQuad1d, gaussQuad2d, gaussQuad3d - - -def TestFunc(x): - f = 1.0 - for i in range(1, 10): - f += x**i - return f - - -def TestFunc2d(x1, x2): - f = 1.0 - for i in range(1, 10): - f += x1**i - 3.0 * x2**i - return f - - -def TestFunc3d(x1, x2, x3): - f = 1.0 - for i in range(1, 10): - f += x1**i - 4.0 * x2**i + 3.0 * x3**i - return f - - -def TestMatFunc3d(x1, x2, x3): - A = np.zeros((len(x1), 3, 3)) - for i in range(len(x1)): - A[i] = np.array([[x1[i], 2.0, 3.0], [1.0, x2[i], 3.0], [1.0, 2.0, x3[i]]]) - return A - - -class GaussQuadUnitTest(unittest.TestCase): - """Test FEMpy's Gauss quadrature integration against scipy's integration methods""" - - def setUp(self) -> None: - self.precision = 8 - - def test_1d_gauss_quad(self): - gaussInt = gaussQuad1d(TestFunc, 6) - scipyInt = integrate.quad(TestFunc, -1.0, 1.0)[0] - self.assertAlmostEqual(gaussInt, scipyInt, places=self.precision) - - def test_1d_gauss_quad_nonStandard_limits(self): - gaussInt = gaussQuad1d(TestFunc, 6, a=-2.6, b=1.9) - scipyInt = integrate.quad(TestFunc, -2.6, 1.9)[0] - self.assertAlmostEqual(gaussInt, scipyInt, places=self.precision) - - def test_2d_gauss_quad(self): - gaussInt = gaussQuad2d(TestFunc2d, 6) - scipyInt = integrate.dblquad(TestFunc2d, -1.0, 1.0, -1.0, 1.0)[0] - self.assertAlmostEqual(gaussInt, scipyInt, places=self.precision) - - def test_2d_gauss_quad_nonStandard_limits(self): - gaussInt = gaussQuad2d(TestFunc2d, [6, 6], a=[-0.3, -2.3], b=[0.7, 1.6]) - scipyInt = integrate.dblquad(TestFunc2d, -2.3, 1.6, -0.3, 0.7)[0] - self.assertAlmostEqual(gaussInt, scipyInt, places=self.precision) - - def test_3d_gauss_quad(self): - gaussInt = gaussQuad3d(TestFunc3d, 6) - scipyInt = integrate.tplquad(TestFunc3d, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0)[0] - self.assertAlmostEqual(gaussInt, scipyInt, places=self.precision) - - def test_3d_gauss_quad_nonStandard_limits(self): - gaussInt = gaussQuad3d(TestFunc3d, [6, 6, 6], a=[-4.0, 2.0, 0.0], b=[1.0, 3.0, 4.0]) - scipyInt = integrate.tplquad(TestFunc3d, 0.0, 4.0, 2.0, 3.0, -4.0, 1.0)[0] - self.assertAlmostEqual(gaussInt, scipyInt, places=self.precision) - - def test_3d_mat_gauss_quad(self): - gaussInt = gaussQuad3d(TestMatFunc3d, 6, a=[-4.0, 2.0, 0.0], b=[1.0, 3.0, 4.0]) - trueInt = np.array([[-30.0, 40.0, 60.0], [20.0, 50.0, 60.0], [20.0, 40.0, 40.0]]) - np.testing.assert_allclose(gaussInt, trueInt, atol=10**-self.precision, rtol=10**-self.precision) - - -if __name__ == "__main__": - unittest.main() diff --git a/tests/testISOstresses.py b/tests/testISOstresses.py new file mode 100644 index 0000000..26d12bb --- /dev/null +++ b/tests/testISOstresses.py @@ -0,0 +1,97 @@ +""" +============================================================================== +FEMpy Model class unit tests +============================================================================== +@File : testModel.py +@Date : 2022/11/14 +@Author : M.A. Saja A.Kaiyoom and Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import unittest +import os +import numpy as np + +# ============================================================================== +# External Python modules +# ============================================================================== + +# ============================================================================== +# Extension modules +# ============================================================================== +# from FEMpy import FEMpyModel +from FEMpy.Constitutive.StressModels import IsoStress + + +class StressCalculationTest(unittest.TestCase): + """Unit tests for the FEMpy stress computation""" + + def setUp(self): + self.n = 10 + np.random.seed(1) + self.tol = 1e-12 + + self.E = 1000 + self.nu = 0.3 + + def testPlanestress(self): + """Test that the mesh file was read in correctly""" + + strain = np.random.rand(self.n, 3) + E = self.E + nu = self.nu + mat = E / (1 - nu**2) * np.array([[1, nu, 0], [nu, 1, 0], [0, 0, (1 - nu) / 2]]) + strain_exptected = np.zeros_like(strain) + for i in range(self.n): + strain_exptected[i] = mat @ strain[i] + strain_computed = IsoStress.isoPlaneStressStress(strain, E, nu) + + np.testing.assert_allclose(strain_computed, strain_exptected, atol=self.tol, rtol=self.tol) + + def testPlanestrain(self): + """Test that the mesh file was read in correctly""" + + strain = np.random.rand(self.n, 3) + E = self.E + nu = self.nu + mat = E / ((1 + nu) * (1 - nu * 2)) * np.array([[1 - nu, nu, 0], [nu, 1 - nu, 0], [0, 0, (1 - 2 * nu) / 2]]) + strain_exptected = np.zeros_like(strain) + for i in range(self.n): + strain_exptected[i] = mat @ strain[i] + strain_computed = IsoStress.isoPlaneStrainStress(strain, E, nu) + + np.testing.assert_allclose(strain_computed, strain_exptected, atol=self.tol, rtol=self.tol) + + def test3Dstress(self): + """Test that the mesh file was read in correctly""" + + strain = np.random.rand(self.n, 6) + E = self.E + nu = self.nu + mat = ( + E + / ((1 + nu) * (1 - nu * 2)) + * np.array( + [ + [1 - nu, nu, nu, 0, 0, 0], + [nu, 1 - nu, nu, 0, 0, 0], + [nu, nu, 1 - nu, 0, 0, 0], + [0, 0, 0, (1 - 2 * nu) / 2, 0, 0], + [0, 0, 0, 0, (1 - 2 * nu) / 2, 0], + [0, 0, 0, 0, 0, (1 - 2 * nu) / 2], + ] + ) + ) + strain_exptected = np.zeros_like(strain) + for i in range(self.n): + strain_exptected[i] = mat @ strain[i] + strain_computed = IsoStress.iso3DStress(strain, E, nu) + + np.testing.assert_allclose(strain_computed, strain_exptected, atol=self.tol, rtol=self.tol) + + +if __name__ == "__main__": + unittest.main() diff --git a/tests/testLinAlg.py b/tests/testLinAlg.py index cfb810e..bf5910b 100644 --- a/tests/testLinAlg.py +++ b/tests/testLinAlg.py @@ -21,7 +21,7 @@ # ============================================================================== # Extension modules # ============================================================================== -from FEMpy import det1, det2, det3, inv1, inv2, inv3 +from FEMpy.LinAlg import det1, det2, det3, inv1, inv2, inv3 class LinAlgUnitTest(unittest.TestCase): diff --git a/tests/testModel.py b/tests/testModel.py new file mode 100644 index 0000000..1ff2577 --- /dev/null +++ b/tests/testModel.py @@ -0,0 +1,97 @@ +""" +============================================================================== +FEMpy Model class unit tests +============================================================================== +@File : testModel.py +@Date : 2022/11/14 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import unittest +from parameterized import parameterized_class +import os +import numpy as np + +# ============================================================================== +# External Python modules +# ============================================================================== + +# ============================================================================== +# Extension modules +# ============================================================================== +from FEMpy import FEMpyModel +import FEMpy as fp + + +test_params = [] + +# Create a unit test for each mesh file in the Examples mesh folder +currentDir = os.path.dirname(os.path.realpath(__file__)) +meshDir = "../Examples/Meshes/" +meshDir = os.path.join(currentDir, meshDir) +# TODO: Fix the reference values for these tests +# test_params.append( +# {"meshFileName": os.path.join(meshDir, "WingboxL3.bdf"), "numPoints": 23158, "numElements": 24128, "numDim": 3} +# ) +test_params.append( + {"meshFileName": os.path.join(meshDir, "Plate.bdf"), "numPoints": 4225, "numElements": 4096, "numDim": 2} +) +# test_params.append( +# {"meshFileName": os.path.join(meshDir, "GMSHTest.msh"), "numPoints": 3622, "numElements": 798, "numDim": 2} +# ) +# test_params.append( +# {"meshFileName": os.path.join(meshDir, "LBracket.msh"), "numPoints": 9, "numElements": 798, "numDim": 2} +# ) + + +# --- Create constitutive model, 7000 series Aluminium --- +E = 71.7e9 +nu = 0.33 +rho = 2 + +# This thickness value is a design variable, by default all elements will use this value, but we can change it later if +# we want +t = 5e-3 +constitutiveModel = fp.Constitutive.IsoPlaneStress(E, nu, rho, t) + + +@parameterized_class(test_params) +class MeshReadTest(unittest.TestCase): + """Unit tests for the FEMpy model class""" + + def setUp(self) -> None: + self.model = FEMpyModel(constitutiveModel=constitutiveModel, meshFileName=self.meshFileName) + + def testMeshRead(self): + """Test that the mesh file was read in correctly""" + self.assertEqual(self.model.numNodes, self.numPoints) + self.assertEqual(self.model.numDim, self.numDim) + + +class ModelTest(unittest.TestCase): + def setUp(self) -> None: + self.model = FEMpyModel(constitutiveModel=constitutiveModel, meshFileName=os.path.join(meshDir, "Plate.bdf")) + + def testaddFixedBCToNodes(self): + """Test that the BC are added correctly""" + self.model.addFixedBCToNodes("myBC", [2, 3, 4], 0, 1) + DOF_computed = [4, 6, 8] + VAL_computed = [1.0, 1.0, 1.0] + + self.assertEqual(self.model.BCs["myBC"]["DOF"], DOF_computed) + self.assertEqual(self.model.BCs["myBC"]["Value"], VAL_computed) + + def testgetDOFfromNodeInds(self): + """Test that the BC are added correctly""" + DOF_expected = self.model.getDOFfromNodeInds(np.array([3, 8, 11, 23])) + DOF_computed = np.array([6, 7, 16, 17, 22, 23, 46, 47]) + + np.testing.assert_equal(DOF_expected, DOF_computed) + + +if __name__ == "__main__": + unittest.main() diff --git a/tests/testProblem.py b/tests/testProblem.py new file mode 100644 index 0000000..7ec1d2e --- /dev/null +++ b/tests/testProblem.py @@ -0,0 +1,147 @@ +""" +============================================================================== +Unit test for FEMpy matrix assembly +============================================================================== +@File : testAssembly.py +@Date : 2022/12/03 +@Author : Alasdair Christison Gray +@Description : +""" + +# ============================================================================== +# Standard Python modules +# ============================================================================== +import unittest + +# ============================================================================== +# External Python modules +# ============================================================================== +import numpy as np + +# ============================================================================== +# Extension modules +# ============================================================================== +import FEMpy as fp + + +class testProblem(unittest.TestCase): + def setUp(self): + np.random.rand(1) + # --- Create constitutive model, 7000 series Aluminium --- + E = 71.7e9 + nu = 0.33 + rho = 2780 + + # This thickness value is a design variable, by default all elements will use this value, but we can change it later if + # we want + t = 5e-3 + self.constitutiveModel = fp.Constitutive.IsoPlaneStress(E, nu, rho, t) + + # --- Define node coordinates and element connectivity arrays --- + # Mesh looks like this: + + # 0 --------------- 2 --------------- 4 + # | | | + # | | | + # | | | + # | | | + # | | | + # 1 --------------- 3 --------------- 5 + + self.nodeCoordinates = np.array([[0.0, 1.0], [0.0, 0.0], [1.2, 1.0], [0.9, 0.0], [2.0, 1.0], [2.0, 0.0]]) + # Add a random perturbation to the coordinates so that there is no cancellation of terms from the local + # stiffness matrices + self.nodeCoordinates += 1e-14 * np.random.rand(*self.nodeCoordinates.shape) + self.connectivity = {"quad": np.array([[0, 1, 3, 2], [2, 3, 5, 4]])} + + self.model = fp.FEMpyModel( + self.constitutiveModel, nodeCoords=self.nodeCoordinates, connectivity=self.connectivity + ) + self.problem = self.model.addProblem("Test") + self.problem.addFixedBCToNodes(name="YFixed", nodeInds=[1, 5], dof=1, value=1.0) + self.problem.addFixedBCToNodes(name="XFixed", nodeInds=5, dof=0, value=0.1) + self.problem.addLoadToNodes(name="YLoad", nodeInds=2, dof=1, value=-10.0) + + self.numDOF = self.nodeCoordinates.shape[0] * 2 + + expectedSparsity = np.zeros((self.numDOF, self.numDOF)) + expectedSparsity[:4, :8] = 1 + expectedSparsity[4:8, :] = 1 + expectedSparsity[8:, 4:] = 1 + self.expectedSparsity = expectedSparsity + + def testSparsityNoBC(self): + """Check the matrix sparsity pattern for a simple mesh with no boundary conditions""" + self.problem.updateJacobian(applyBCs=False) + self.matSparsityTest() + + def testSolveWithBC(self): + """Check that the sparsity is as expected when some BCs are applied + + The row associated with each constrained DOF should be zero except for the diagonal entry + """ + + expectedRHS = np.zeros(self.numDOF) + expectedRHS[3] = -1.0 + expectedRHS[5] = 10.0 + expectedRHS[11] = -1.0 + expectedRHS[10] = -0.1 + + self.problem.updateResidual() + np.testing.assert_equal(self.problem.Res, expectedRHS, err_msg="RHS is not as expected") + + self.expectedSparsity[[3, 10, 11], :] = 0 + self.expectedSparsity[[3, 10, 11], [3, 10, 11]] = 1.0 + + # Test that the BCs are applied correctly (i.e. the solution has the correct values) + self.problem.solve() + self.problem.updateResidual() + np.testing.assert_allclose( + self.problem.Res, np.zeros(self.numDOF), rtol=1e-6, atol=1e-6, err_msg="RHS is not zero after solving" + ) + u = self.problem.states.flatten() + np.testing.assert_approx_equal(u[3], 1.0, significant=10, err_msg="BC not applied correctly") + np.testing.assert_approx_equal(u[11], 1.0, significant=10, err_msg="BC not applied correctly") + np.testing.assert_approx_equal(u[10], 0.1, significant=10, err_msg="BC not applied correctly") + + self.matSparsityTest() + + def matSparsityTest(self): + expectedNonZeros = np.nonzero(self.expectedSparsity) + K = self.problem.Jacobian + KNonZeros = K.nonzero() + + comparisons = { + "Shape": [K.shape, self.expectedSparsity.shape], + "Row Sparsity": [KNonZeros[0], expectedNonZeros[0]], + "Column Sparsity": [KNonZeros[1], expectedNonZeros[1]], + } + for name, values in comparisons.items(): + with self.subTest(values=values): + np.testing.assert_equal( + values[0], + values[1], + err_msg=f"{name} of the assembled matrix is not as expected", + ) + + def testComputeFunction(self): + # test computation of total mass + expectedTotalMass = self.constitutiveModel.rho * 2 + np.testing.assert_allclose( + expectedTotalMass, + self.problem.computeFunction("Mass", elementReductionType="integrate", globalReductionType="sum"), + ) + + # test element reduction of mass + expectedValue = {"quad": np.array([self.constitutiveModel.rho, self.constitutiveModel.rho])} + np.testing.assert_equal(expectedValue, self.problem.computeFunction("Mass", elementReductionType="max")) + + # test element global reduction + np.testing.assert_equal( + self.constitutiveModel.rho * 2, + self.problem.computeFunction("Mass", elementReductionType="min", globalReductionType="sum"), + ) + + +if __name__ == "__main__": + unittest.main()