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NumGy_LV07.qmd
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---
title: "Numerical Simulation Methods in Geophysics, Part 7: Finite Elements"
subtitle: "1. MGPY+MGIN"
author: "[email protected]"
title-slide-attributes:
data-background-image: pics/tubaf-logo.png
data-background-size: 20%
data-background-position: 10% 95%
format:
tubaf-revealjs:
html-math-method: mathjax
chalkboard: true
include-in-header:
- text: |
<script>
window.MathJax = {
loader: {
load: ['[tex]/physics']
},
tex: {
packages: {'[+]': ['physics']}
}
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</script>
slide-number: c/t
transition: slide
transition-speed: fast
menu:
side: left
# jupyter: python3
---
## Recap Finite Elements
* weak form of PDE: integral over product with test function $\vb w$
* approximate $u$ with shape functions $u=\sum u_i \vb v_i$
* Galerkin's method: same function space for $\vb w$ and $\vb v$
:::{.callout-tip icon="false" title="Difference of FE to FD"}
Solution $u$ is described on the whole space and approximates the solution, not the PDE!
Any source function $f(x)$ can be integrated on the whole space!
:::
## Recap (cont)
:::{.callout-tip icon="false" title="Generality of FE"}
Arbitrary base functions $v_i$ can be used to describe $u$
:::
* started with piece-wise linear (hat) functions
* system identical to FD for $\Delta x=$const and $a$=const
## Method of weighted residuals
PDE $\mathfrak{L}(u)=f$ $\Rightarrow$ approximated by $u_h$
residual $R=L_h(u)-f$ to be minimized, integrating over modelling domain
$$ \int_\Omega w R \dd\Omega = \int_\Omega w \mathfrak{L}(u_h)\dd\Omega - \int_\Omega w f \dd\Omega = 0 $$
with approximation $u_h(\vb r)=\sum_j^M u_j \vb v_j(\vb r)$
($\vb v$ basis / shape functions, $\vb w$ test / trial functions)
## Bilinear form for Poisson equation
Solve $\vb A \vb x = \vb b$ with $A_{ij}=(\grad v_i, \grad v_j)$ and $b_i=(\vb v_i, f)$, where
$$(\vb a, \vb b)=\int_\Omega \vb a \cdot \vb b\,\dd\Omega = \sum\limits_{c=i}^M \int_{\Omega_c} \vb a \cdot \vb b\,\dd\Omega_c$$
Solve the integrals either analytically or numerically
## The general solution
Solving any integral using (Gaussian) quadrature
$$\int g(x) \dd x \approx \sum_q g(x_q) w_q$$
$$f_i^c=\int_{\Omega_c} v_i f \dd x \approx \sum_q v(x_q^c)f(x_q^c)w_q^c$$
$$a_{ij}^c = \int_c a_c \grad v_i \cdot \grad v_j=\sum a_c \grad v_i(x_q^c)\cdot \grad v_j(x_q^c) w_q^c $$
## Gaussian quadrature
:::: {.columns}
::: {.column width="45%"}
{width="100%"}
:::
::: {.column width="55%"}
`quadratureRules(c.shape(), 5)`
* optimum quadrature on reference triangle for a given order (5)
:::
::::
## Gaussian quadrature
:::: {.columns}
::: {.column width="45%"}
{width="100%"}
:::
::: {.column width="55%"}
`quadratureRules(c, 2)`
* optimum quadrature on arbitrary triangle for order 2
:::
::::
## Coordinate transformation
1D: local coordinate $\xi=\frac{x-x_i}{x_{i+1}-x_i}$ (0..1)
$u(\xi)=c_1+c_2\xi$
$u_0=u(0)=c_1$, $u_1=u(1)=c_1+c_2$ $c_2=u_1-u_0$
$$\Rightarrow u(\xi)=u_0+\xi(u_1-u_0)=u_0(1-\xi)+u_1\xi=u_i v_i + u_1 v_1$$
## Quadratic elements
$u(\xi)=c_1+c_2\xi+c_3\xi^2$
nodes at $x_0$, $x_{1/2}$, $x_1$
$u_i=u(0)=c_1$, $u_1=c_1+c_2+c_3$
$u_{1/2}=c_1+c_2/2+c_3/4$
$$u(\xi)=u_0(1-3\xi+2\xi^2)+u_{1/2}(4\xi-4\xi^2)+u_1(-\xi+2\xi^2)$$
## Quadratic elements
$$u(\xi)=u_0(3\xi+2\xi^2)+u_{1/2}(4\xi-4\xi^2)+u_1(-\xi+2\xi^2)$$
```{python}
#| output-location: column
#| eval: true
#| echo: true
import numpy as np
import matplotlib.pyplot as plt
x=np.linspace(0, 1, 101)
# Plot velocity distribution.
plt.plot(x, 1-3*x+2*x**2)
plt.plot(x, 4*x-4*x**2)
plt.plot(x, -x+2*x**2)
plt.plot(0, 0, "o", color="C0", ms=8)
plt.plot(0.5, 0, "o", color="C1", ms=8)
plt.plot(1, 0, "o", color="C2", ms=8)
plt.grid()
```
## Cubic elements
<!-- $$u(\xi)=u_0(3\xi+2\xi^2)+u_{1/2}(4\xi-4\xi^2)+u_1(-\xi+2\xi^2)$$ -->
```{python}
#| output-location: column
#| eval: true
#| echo: true
import numpy as np
import matplotlib.pyplot as plt
x=np.linspace(0, 1, 101)
# Plot velocity distribution.
plt.plot(x, 1-3*x**2+2*x**3)
plt.plot(x, x-2*x**2+x**3)
plt.plot(x, -x**2+x**3)
plt.plot(x, 3*x**2-2*x**3)
for i in range(4):
plt.plot(i/(3), 0, "o",
color=f"C{i}", ms=8)
plt.grid()
```
## Triangles with linear shape functions
:::: {.columns}
::: {.column width="60%"}
$$x=x_1+(x_2-x_1)\xi+(x_3-x_1)\eta$$
$$y=y_1+(y_2-y_1)\xi+(y_3-y_1)\eta$$
:::
::: {.column width="40%"}
```{python}
#| eval: true
#| echo: false
plt.fill_betweenx([1, 0], [0, 1])
plt.text(0.05, 0.05, "P1")
plt.text(1, 0.05, "P2")
plt.text(0.05, 1, "P3")
plt.xlabel(r"$\xi$")
plt.ylabel(r"$\eta$");
```
:::
::::
## Triangle
$$u(\xi)=u_1(1-\xi-\eta)+u_2\xi+u_3\eta$$
```{python}
#| output-location: column
#| eval: true
#| echo: true
import pygimli as pg
import pygimli.meshtools as mt
shape = mt.createPolygon(
[[0, 0], [1, 0],[0, 1]],
isClosed=True)
mesh = mt.createMesh(shape, area=0.01)
mx = pg.x(mesh)
my = pg.y(mesh)
# Plot velocity distribution.
fig, ax = plt.subplots()
pg.show(mesh, 1-mx-my, ax=ax,
nLevs=11, label="u");
```
<!-- ## Triangle
$$u(\xi)=u_1(1-\xi-\eta)+u_2\xi+u_3\eta$$
```{python}
#| output-location: column
#| eval: true
#| echo: true
fig, ax = plt.subplots()
pg.show(mesh, mx, ax=ax, nLevs=11);
```
## Triangle
$$u(\xi)=u_1(1-\xi-\eta)+u_2\xi+u_3\eta$$
```{python}
#| output-location: column
#| eval: true
#| echo: true
fig, ax = plt.subplots()
pg.show(mesh, my, ax=ax, nLevs=11);
```
-->
## Triangle linear shape functions
$$u(\xi)=u_1(1-\xi-\eta)+u_2\xi+u_3\eta$$
```{python}
#| eval: true
#| echo: false
fig, ax = plt.subplots(figsize=(12, 6), ncols=3, sharex=True, sharey=True)
fig.tight_layout()
pg.show(mesh, 1-mx-my, ax=ax[0], nLevs=11);
pg.show(mesh, mx, ax=ax[1], nLevs=11);
pg.show(mesh, my, ax=ax[2], nLevs=11);
```
## Verification
1. Method of Manufactured Solutions (MMS)
* manufacture a smooth u
* generate $f$ matching approximation of $u$
2. Method of Exact Solutions (MES)
* find parameters for which an analytic solution exists
3. Perform convergence tests for increasingly smaller $h$
* approximation error $E(h)\lt C h^n$ test for some $h$
## Green's functions
The Green's function $G$ is the solution for a Dirac source $\delta$
$$ \mathfrak{L} G = \delta $$
The solution can then be obtained by convolution
$$u=G*f$$