PrismConstrainedWY.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LowRankApprox\n",
    "using Plots\n",
    "using LinearAlgebra\n",
    "using Jacobi\n",
    "using Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TrilinearMap (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function TrilinearMap(coord_E, xhat, yhat, zhat)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    coord_E: coordinate of element E as a 6x3 matrix\n",
    "    The prism is defined by cutting a cube [-1,1]^3 on diagonal y = -x\n",
    "    \n",
    "    Local numbering of Prism\n",
    "    \n",
    "        4--------6           z\n",
    "      / |                    |\n",
    "    5   |                    |\n",
    "    |   |                    /----y\n",
    "    |   |                   /\n",
    "    |   1 -------3         x\n",
    "    | /\n",
    "    2\n",
    "\n",
    "    \"\"\"\n",
    "    \n",
    "    m = length(xhat)\n",
    "    N1 = @. 0.25*(-xhat-yhat)*(1-zhat)\n",
    "    N2 = @. 0.25*(1+xhat)*(1-zhat)\n",
    "    N3 = @. 0.25*(1+yhat)*(1-zhat)\n",
    "    N4 = @. 0.25*(-xhat-yhat)*(1+zhat)\n",
    "    N5 = @. 0.25*(1+xhat)*(1+zhat)\n",
    "    N6 = @. 0.25*(1+yhat)*(1+zhat)\n",
    "    N = [N1 N2 N3 N4 N5 N6]\n",
    "    X = N * coord_E\n",
    "    # X(3 x m), 1st row x, 2nd row y, 3rd row z\n",
    "    X = X'\n",
    "    # gradient of N, [dN/dxhat; dN/dyhat; dN/dzhat]\n",
    "    GradN = zeros(3,m,6)\n",
    "    GradN[1,:,:] = @. 0.25*[-(1-zhat) (1-zhat) zeros(m) -(1+zhat) (1+zhat) zeros(m)]\n",
    "    GradN[2,:,:] = @. 0.25*[-(1-zhat) zeros(m) (1-zhat) -(1+zhat) zeros(m) (1+zhat)]\n",
    "    GradN[3,:,:] = @. 0.25*[-(-xhat-yhat) -(1+xhat) -(1+yhat) (-xhat-yhat) (1+xhat) (1+yhat)]\n",
    "\n",
    "    # JT = [[dx/dxhat, dy/dxhat, dz/dxhat],\n",
    "    #       [dx/dyhat, dy/dyhat, dz/dyhat],\n",
    "    #       [dx/dzhat, dy/dzhat, dz/dzhat]] (3m x 3)\n",
    "\n",
    "    JTxhat = GradN[1,:,:] * coord_E\n",
    "    dxdxhat = JTxhat[:,1]\n",
    "    dydxhat = JTxhat[:,2]\n",
    "    dzdxhat = JTxhat[:,3]\n",
    "    JTyhat = GradN[2,:,:] * coord_E\n",
    "    dxdyhat = JTyhat[:,1]\n",
    "    dydyhat = JTyhat[:,2]\n",
    "    dzdyhat = JTyhat[:,3]\n",
    "    JTzhat = GradN[3,:,:] * coord_E\n",
    "    dxdzhat = JTzhat[:,1]\n",
    "    dydzhat = JTzhat[:,2]\n",
    "    dzdzhat = JTzhat[:,3]\n",
    "    # compute det\n",
    "    j1 = @. (dxdxhat*dydyhat*dzdzhat + dxdyhat*dydzhat*dzdzhat + dxdzhat*dydxhat*dzdyhat)\n",
    "    j2 = @. (dxdxhat*dydzhat*dzdyhat + dxdyhat*dydxhat*dzdzhat + dxdzhat*dydyhat*dzdxhat)\n",
    "    detJ = @. j1 - j2\n",
    "    \n",
    "    J = zeros(3,m,3)\n",
    "    J[1,:,:] = [dxdxhat dxdyhat dxdzhat]\n",
    "    J[2,:,:] = [dydxhat dydyhat dydzhat]\n",
    "    J[3,:,:] = [dzdxhat dzdyhat dzdzhat]\n",
    "    \n",
    "    return X, J, detJ\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PrimeBasis (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function PrimeBasis(coord_E, xhat, yhat, zhat)\n",
    "\n",
    "    m = length(xhat)\n",
    "    # supplements from Wheeler, Xue, & Yotov, 2012\n",
    "    # I eliminate two vectors as Nate did\n",
    "    shat1 = [(xhat .* zhat) (-yhat .* zhat) (zeros(m))]\n",
    "    shat2 = [(zeros(m)) (-2*xhat .* zhat) (zeros(m))]  \n",
    "    shat3 = [(zeros(m)) (xhat .* yhat) (-xhat .* zhat)]\n",
    "    shat4 = [(zeros(m)) (2*yhat .* zhat) (-zhat.^2)]\n",
    "    shat5 = [(zeros(m)) (zeros(m)) (-2*xhat .* yhat)]\n",
    "    shat6 = [(zeros(m)) (xhat .* yhat.^2) (-2*xhat .* yhat .* zhat)]\n",
    "    shat7 = [(-xhat .* yhat) (zeros(m)) (yhat .* zhat)]\n",
    "    shat8 = [(-xhat.^2) (zeros(m)) (2*xhat .* zhat)]\n",
    "    shat9 = [(-2*yhat .* zhat) (zeros(m)) (zeros(m))]\n",
    "    shat10 = [(-2*xhat .* yhat .* zhat) (zeros(m)) (yhat .* zhat.^2)]\n",
    "    # X are in E\n",
    "    X, J, detJ = TrilinearMap(coord_E, xhat, yhat, zhat)\n",
    "    x = X[1,:]\n",
    "    y = X[2,:]\n",
    "    z = X[3,:]\n",
    "    # Prime basis in 3D space with dimension 22\n",
    "    P = zeros(3,m,22)\n",
    "    P[1,:,1] = ones(m)\n",
    "    P[1,:,2] = x\n",
    "    P[1,:,3] = y\n",
    "    P[1,:,4] = z\n",
    "    P[2,:,5] = ones(m)\n",
    "    P[2,:,6] = x\n",
    "    P[2,:,7] = y\n",
    "    P[2,:,8] = z\n",
    "    P[3,:,9] = ones(m)\n",
    "    P[3,:,10] = x\n",
    "    P[3,:,11] = y\n",
    "    P[3,:,12] = z\n",
    "    # supplement\n",
    "    P[1,:,13] = (J[1,:,1] .* shat1[:,1] + J[1,:,2] .* shat1[:,2] + J[1,:,3] .* shat1[:,3]) ./ detJ\n",
    "    P[2,:,13] = (J[2,:,1] .* shat1[:,1] + J[2,:,2] .* shat1[:,2] + J[2,:,3] .* shat1[:,3]) ./ detJ\n",
    "    P[3,:,13] = (J[3,:,1] .* shat1[:,1] + J[3,:,2] .* shat1[:,2] + J[3,:,3] .* shat1[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,14] = (J[1,:,1] .* shat2[:,1] + J[1,:,2] .* shat2[:,2] + J[1,:,3] .* shat2[:,3]) ./ detJ\n",
    "    P[2,:,14] = (J[2,:,1] .* shat2[:,1] + J[2,:,2] .* shat2[:,2] + J[2,:,3] .* shat2[:,3]) ./ detJ\n",
    "    P[3,:,14] = (J[3,:,1] .* shat2[:,1] + J[3,:,2] .* shat2[:,2] + J[3,:,3] .* shat2[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,15] = (J[1,:,1] .* shat3[:,1] + J[1,:,2] .* shat3[:,2] + J[1,:,3] .* shat3[:,3]) ./ detJ\n",
    "    P[2,:,15] = (J[2,:,1] .* shat3[:,1] + J[2,:,2] .* shat3[:,2] + J[2,:,3] .* shat3[:,3]) ./ detJ\n",
    "    P[3,:,15] = (J[3,:,1] .* shat3[:,1] + J[3,:,2] .* shat3[:,2] + J[3,:,3] .* shat3[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,16] = (J[1,:,1] .* shat4[:,1] + J[1,:,2] .* shat4[:,2] + J[1,:,3] .* shat4[:,3]) ./ detJ\n",
    "    P[2,:,16] = (J[2,:,1] .* shat4[:,1] + J[2,:,2] .* shat4[:,2] + J[2,:,3] .* shat4[:,3]) ./ detJ\n",
    "    P[3,:,16] = (J[3,:,1] .* shat4[:,1] + J[3,:,2] .* shat4[:,2] + J[3,:,3] .* shat4[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,17] = (J[1,:,1] .* shat5[:,1] + J[1,:,2] .* shat5[:,2] + J[1,:,3] .* shat5[:,3]) ./ detJ\n",
    "    P[2,:,17] = (J[2,:,1] .* shat5[:,1] + J[2,:,2] .* shat5[:,2] + J[2,:,3] .* shat5[:,3]) ./ detJ\n",
    "    P[3,:,17] = (J[3,:,1] .* shat5[:,1] + J[3,:,2] .* shat5[:,2] + J[3,:,3] .* shat5[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,18] = (J[1,:,1] .* shat6[:,1] + J[1,:,2] .* shat6[:,2] + J[1,:,3] .* shat6[:,3]) ./ detJ\n",
    "    P[2,:,18] = (J[2,:,1] .* shat6[:,1] + J[2,:,2] .* shat6[:,2] + J[2,:,3] .* shat6[:,3]) ./ detJ\n",
    "    P[3,:,18] = (J[3,:,1] .* shat6[:,1] + J[3,:,2] .* shat6[:,2] + J[3,:,3] .* shat6[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,19] = (J[1,:,1] .* shat7[:,1] + J[1,:,2] .* shat7[:,2] + J[1,:,3] .* shat7[:,3]) ./ detJ\n",
    "    P[2,:,19] = (J[2,:,1] .* shat7[:,1] + J[2,:,2] .* shat7[:,2] + J[2,:,3] .* shat7[:,3]) ./ detJ\n",
    "    P[3,:,19] = (J[3,:,1] .* shat7[:,1] + J[3,:,2] .* shat7[:,2] + J[3,:,3] .* shat7[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,20] = (J[1,:,1] .* shat8[:,1] + J[1,:,2] .* shat8[:,2] + J[1,:,3] .* shat8[:,3]) ./ detJ\n",
    "    P[2,:,20] = (J[2,:,1] .* shat8[:,1] + J[2,:,2] .* shat8[:,2] + J[2,:,3] .* shat8[:,3]) ./ detJ\n",
    "    P[3,:,20] = (J[3,:,1] .* shat8[:,1] + J[3,:,2] .* shat8[:,2] + J[3,:,3] .* shat8[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,21] = (J[1,:,1] .* shat9[:,1] + J[1,:,2] .* shat9[:,2] + J[1,:,3] .* shat9[:,3]) ./ detJ\n",
    "    P[2,:,21] = (J[2,:,1] .* shat9[:,1] + J[2,:,2] .* shat9[:,2] + J[2,:,3] .* shat9[:,3]) ./ detJ\n",
    "    P[3,:,21] = (J[3,:,1] .* shat9[:,1] + J[3,:,2] .* shat9[:,2] + J[3,:,3] .* shat9[:,3]) ./ detJ\n",
    "\n",
    "    P[1,:,22] = (J[1,:,1] .* shat10[:,1] + J[1,:,2] .* shat10[:,2] + J[1,:,3] .* shat10[:,3]) ./ detJ\n",
    "    P[2,:,22] = (J[2,:,1] .* shat10[:,1] + J[2,:,2] .* shat10[:,2] + J[2,:,3] .* shat10[:,3]) ./ detJ\n",
    "    P[3,:,22] = (J[3,:,1] .* shat10[:,1] + J[3,:,2] .* shat10[:,2] + J[3,:,3] .* shat10[:,3]) ./ detJ\n",
    "    \n",
    "    return P\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetNormal (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetNormal(coord_E, xhat, yhat, zhat, face)\n",
    "\n",
    "    X, J, detJ = TrilinearMap(coord_E, xhat, yhat, zhat)\n",
    "\n",
    "    dxdxhat = J[1,:,1]\n",
    "    dxdyhat = J[1,:,2]\n",
    "    dxdzhat = J[1,:,3]\n",
    "    \n",
    "    dydxhat = J[2,:,1]\n",
    "    dydyhat = J[2,:,2]\n",
    "    dydzhat = J[2,:,3]\n",
    "    \n",
    "    dzdxhat = J[3,:,1]\n",
    "    dzdyhat = J[3,:,2]\n",
    "    dzdzhat = J[3,:,3]\n",
    "\n",
    "    m = length(xhat)\n",
    "    \n",
    "    if face == \"left\"\n",
    "        \n",
    "        n1 = @. dydxhat*dzdzhat - dzdxhat*dydzhat\n",
    "        n2 = @. dzdxhat*dxdzhat - dxdxhat*dzdzhat\n",
    "        n3 = @. dxdxhat*dydzhat - dydxhat*dxdzhat\n",
    "        leng = @. sqrt(n1*n1 + n2*n2 + n3*n3)\n",
    "        n = zeros(3,m)\n",
    "        n[1,:] = n1 ./ leng\n",
    "        n[2,:] = n2 ./ leng\n",
    "        n[3,:] = n3 ./ leng\n",
    "      \n",
    "    elseif face == \"back\"\n",
    "        \n",
    "        n1 = @. dzdyhat*dydzhat - dydyhat*dzdzhat \n",
    "        n2 = @. dxdyhat*dzdzhat - dzdyhat*dxdzhat\n",
    "        n3 = @. dydyhat*dxdzhat - dxdyhat*dydzhat \n",
    "        leng = @. sqrt(n1*n1 + n2*n2 + n3*n3)\n",
    "        n = zeros(3,m)\n",
    "        n[1,:] = n1 ./ leng\n",
    "        n[2,:] = n2 ./ leng\n",
    "        n[3,:] = n3 ./ leng\n",
    "    \n",
    "    elseif face == \"diagonal\"\n",
    "        #dxdshat = -dxdxhat + dxdyhat, ...\n",
    "        dxdshat = J[1,:,2] - J[1,:,1]\n",
    "        dydshat = J[2,:,2] - J[2,:,1]\n",
    "        dzdshat = J[3,:,2] - J[3,:,1]\n",
    "        \n",
    "        n1 = @. dydshat*dzdzhat - dzdshat*dydzhat\n",
    "        n2 = @. dzdshat*dxdzhat - dxdshat*dzdzhat\n",
    "        n3 = @. dxdshat*dydzhat - dydshat*dxdzhat\n",
    "        leng = @. sqrt(n1*n1 + n2*n2 + n3*n3)\n",
    "        n = zeros(3,m)\n",
    "        n[1,:] = n1 ./ leng\n",
    "        n[2,:] = n2 ./ leng\n",
    "        n[3,:] = n3 ./ leng\n",
    "    \n",
    "    elseif face == \"top\"\n",
    "        \n",
    "        n1 = @. dydxhat*dzdyhat - dzdxhat*dydyhat \n",
    "        n2 = @. dzdxhat*dxdyhat - dxdxhat*dzdyhat \n",
    "        n3 = @. dxdxhat*dydyhat - dydxhat*dxdyhat \n",
    "        leng = @. sqrt(n1*n1 + n2*n2 + n3*n3)\n",
    "        n = zeros(3,m)\n",
    "        n[1,:] = n1 ./ leng\n",
    "        n[2,:] = n2 ./ leng\n",
    "        n[3,:] = n3 ./ leng\n",
    "    \n",
    "    elseif face == \"bottom\"\n",
    "        \n",
    "        n1 = @. dzdxhat*dydyhat - dydxhat*dzdyhat\n",
    "        n2 = @. dxdxhat*dzdyhat - dzdxhat*dxdyhat\n",
    "        n3 = @. dydxhat*dxdyhat - dxdxhat*dydyhat \n",
    "        leng = @. sqrt(n1*n1 + n2*n2 + n3*n3)\n",
    "        n = zeros(3,m)\n",
    "        n[1,:] = n1 ./ leng\n",
    "        n[2,:] = n2 ./ leng\n",
    "        n[3,:] = n3 ./ leng\n",
    "    else\n",
    "        error(\"face is not defined\")\n",
    "    end\n",
    "\n",
    "    return n, leng\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PlotElem (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function PlotElem(coord_E)\n",
    "    \n",
    "    x = coord_E[:,1]\n",
    "    y = coord_E[:,2]\n",
    "    z = coord_E[:,3]\n",
    "    nodes = [1 2 3 1 4 5 6 4]\n",
    "    n = length(nodes)\n",
    "    xx = zeros(n,1)\n",
    "    yy = zeros(n,1)\n",
    "    zz = zeros(n,1)\n",
    "    for j =1:n\n",
    "        xx[j] = x[nodes[j]] \n",
    "        yy[j] = y[nodes[j]] \n",
    "        zz[j] = z[nodes[j]]\n",
    "    end\n",
    "    plot(xx,yy,zz,color=\"black\",legend=false)\n",
    "    xx2 = zeros(2,1)\n",
    "    yy2 = zeros(2,1)\n",
    "    zz2 = zeros(2,1)\n",
    "    xx2[1] = x[2]; xx2[2] = x[5]\n",
    "    yy2[1] = y[2]; yy2[2] = y[5]\n",
    "    zz2[1] = z[2]; zz2[2] = z[5]\n",
    "    \n",
    "    xx3 = zeros(2,1)\n",
    "    yy3 = zeros(2,1)\n",
    "    zz3 = zeros(2,1)\n",
    "    xx3[1] = x[3]; xx3[2] = x[6]\n",
    "    yy3[1] = y[3]; yy3[2] = y[6]\n",
    "    zz3[1] = z[3]; zz3[2] = z[6]\n",
    "    \n",
    "    plot!(xx2,yy2,zz2,color=\"black\",legend=false)\n",
    "    p2 = plot!(xx3,yy3,zz3,color=\"black\",legend=false, title = \"3D view\")\n",
    "    \n",
    "    # to see from top \n",
    "    # bottom face\n",
    "    nodes = [1 2 3 1]\n",
    "    n = length(nodes)\n",
    "    xb = zeros(n,1)\n",
    "    yb = zeros(n,1)\n",
    "    for j =1:n\n",
    "        xb[j] = x[nodes[j]] \n",
    "        yb[j] = y[nodes[j]] \n",
    "    end\n",
    "    # top face\n",
    "    nodes = [4 5 6 4]\n",
    "    n = length(nodes)\n",
    "    xt = zeros(n,1)\n",
    "    yt = zeros(n,1)\n",
    "    for j =1:n\n",
    "        xt[j] = x[nodes[j]] \n",
    "        yt[j] = y[nodes[j]] \n",
    "    end\n",
    "    plot(xb,yb,color=\"black\",legend=false)\n",
    "    pp=plot!(xt,yt,color=\"black\",legend=false, title = \"Top View\")\n",
    "    \n",
    "    plot(p2,pp)\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "  1421.07,1447.87 2325.62,1447.87 1421.07,162.047 1421.07,1447.87 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip773)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# unstructured element\n",
    "#coord_E = [0. 0. 0.;1. 0. 0.;0. 1. 0.;1. 1. 1.;2. 1. 1.;1. 2. 1]\n",
    "# reference element\n",
    "coord_E = [-1. -1. -1.;1. -1. -1.;-1. 1. -1.;-1. -1. 1.;1. -1. 1.;-1. 1. 1.]\n",
    "PlotElem(coord_E)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test Normal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[37m\u001b[1mTest Summary: | \u001b[22m\u001b[39m\u001b[32m\u001b[1mPass  \u001b[22m\u001b[39m\u001b[36m\u001b[1mTotal\u001b[22m\u001b[39m\n",
      "TestNormal    | \u001b[32m   5  \u001b[39m\u001b[36m    5\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestNormal\", Any[], 5, false, false)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestNormal\" begin\n",
    "    \n",
    "    # unstructured element\n",
    "    # coord_E = [0. 0. 0.;1. 0. 0.;0. 1. 0.;1. 1. 1.;2. 1. 1.;1. 2. 1.]\n",
    "    # reference element\n",
    "    coord_E = [-1. -1. -1.;1. -1. -1.;-1. 1. -1.;-1. -1. 1.;1. -1. 1.;-1. 1. 1.]\n",
    "    \n",
    "    nl, le = GetNormal(coord_E, [0.], [-1.],[0.],\"left\")\n",
    "    nd, le = GetNormal(coord_E, [0.], [0.],[0.],\"diagonal\")\n",
    "    nbt, le = GetNormal(coord_E, [-0.5], [-0.5],[-1.],\"bottom\")\n",
    "    nt, le = GetNormal(coord_E, [-0.5], [-0.5],[1.],\"top\")\n",
    "    nbk, le = GetNormal(coord_E, [-1.], [0.],[0.],\"back\")\n",
    "\n",
    "    # check the left face normal\n",
    "    a = coord_E[2,:] - coord_E[1,:] \n",
    "    b = coord_E[4,:] - coord_E[1,:]\n",
    "    c = cross(a,b)\n",
    "    c = normalize(c)\n",
    "    @test nl[:] == c[:]\n",
    "\n",
    "    # check the back face normal \n",
    "    a = coord_E[4,:] - coord_E[1,:]\n",
    "    b = coord_E[3,:] - coord_E[1,:]\n",
    "    c = cross(a,b)\n",
    "    c = normalize(c)\n",
    "    @test nbk[:] == c[:]\n",
    "    \n",
    "    # check the diagonal face normal \n",
    "    a = coord_E[3,:] - coord_E[2,:]\n",
    "    b = coord_E[5,:] - coord_E[2,:]\n",
    "    c = cross(a,b)\n",
    "    c = normalize(c)\n",
    "    @test nd[:] == c[:]\n",
    "    \n",
    "    # check the top face normal \n",
    "    a = coord_E[5,:] - coord_E[4,:]\n",
    "    b = coord_E[6,:] - coord_E[4,:]\n",
    "    c = cross(a,b)\n",
    "    c = normalize(c)\n",
    "    @test nt[:] == c[:]\n",
    "    \n",
    "    # check the bottom face normal \n",
    "    a = coord_E[3,:] - coord_E[1,:]\n",
    "    b = coord_E[2,:] - coord_E[1,:]\n",
    "    c = cross(a,b)\n",
    "    c = normalize(c)\n",
    "    @test nbt[:] == c[:]\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dubiner (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "include(\"GetQuadratureTriangle.jl\") # returns w, q for integration over triangle (-1,-1), (1,-1), (-1,1)\n",
    "\n",
    "function GetQuadrature2D(Q)\n",
    "    \"\"\"\n",
    "    Gauss Quadrature over [-1,1]^2\n",
    "    Q: Number of Gauss points in 1D\n",
    "    \n",
    "    \"\"\"\n",
    "    # 1D Gauss\n",
    "    alpha = 0.0\n",
    "    beta = 0.0\n",
    "    q = zgj(Q, alpha, beta)\n",
    "    w = wgj(q, alpha, beta)\n",
    "    \n",
    "    w2 = zeros(Q*Q)\n",
    "    qx = zeros(Q*Q)\n",
    "    qy = zeros(Q*Q)\n",
    "    for i=1:Q\n",
    "        for j=1:Q\n",
    "            k = (i-1)*Q +j\n",
    "            qx[k] = q[j]\n",
    "            qy[k] = q[i]\n",
    "            w2[k] = w[j]*w[i]\n",
    "        end\n",
    "    end\n",
    "    return w2, qx, qy\n",
    "end\n",
    "\n",
    "function Legendre2D(x,y,i,j)\n",
    "    # Legendre2D define on [-1,1]^2\n",
    "    return legendre(x,i) .* legendre(y,j)\n",
    "end\n",
    "\n",
    "function Dubiner(x,y,i,j)\n",
    "    # Dubiner polynomial defined on triangle (-1,-1),(1,-1),(-1,1)\n",
    "    eta1 = 2*(1 .+ x) ./ clamp.(1 .- y,1e-12,10) .- 1\n",
    "    eta2 = y\n",
    "    D    = jacobi.(eta1, i, 0, 0)\n",
    "    D   *= (0.5*(1 .- eta2)) .^i\n",
    "    D   *= jacobi.(eta2, j, 2*i+1, 0)\n",
    "    return D\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[37m\u001b[1mTest Summary:   | \u001b[22m\u001b[39m\u001b[32m\u001b[1mPass  \u001b[22m\u001b[39m\u001b[36m\u001b[1mTotal\u001b[22m\u001b[39m\n",
      "TestIntegration | \u001b[32m   5  \u001b[39m\u001b[36m    5\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestIntegration\", Any[], 5, false, false)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestIntegration\" begin\n",
    "\n",
    "    # unstructured element\n",
    "     coord_E = [0. 0. 0.;1. 0. 0.;0. 1. 0.;1. 1. 1.;2. 1. 1.;1. 2. 1.]\n",
    "    # reference element\n",
    "    #coord_E = [-1. -1. -1.;1. -1. -1.;-1. 1. -1.;-1. -1. 1.;1. -1. 1.;-1. 1. 1.]\n",
    "   \n",
    "    # get quadrature on [-1,1]^2\n",
    "    Q1d_quad = 5 # No. pts in 1D\n",
    "    ww, q1, q2 = GetQuadrature2D(Q1d_quad)\n",
    "    m1 = -ones(Q1d_quad*Q1d_quad)\n",
    "    p1 = ones(Q1d_quad*Q1d_quad)\n",
    "    \n",
    "    # left face    \n",
    "    n, le = GetNormal(coord_E, q1, m1, q2, \"left\")\n",
    "    Integral = (ww .* le)' *(Legendre2D.(q1,q2,2,0) .* Legendre2D.(q1,q2,1,2))\n",
    "    @test isapprox(Integral, 0.;atol=1e-15)\n",
    "    \n",
    "    # back face\n",
    "    n, le = GetNormal(coord_E, m1, q1, q2, \"back\")\n",
    "    Integral = (ww .* le)' *(Legendre2D.(q1,q2,2,0) .* Legendre2D.(q1,q2,1,2))\n",
    "    @test isapprox(Integral, 0.;atol=1e-15)\n",
    "    \n",
    "    # diagonal face\n",
    "    n, le = GetNormal(coord_E, -q1, q1, q2, \"diagonal\")\n",
    "    Integral = (ww .* le)' *(Legendre2D.(q1,q2,2,0) .* Legendre2D.(q1,q2,1,2))\n",
    "    @test isapprox(Integral, 0.;atol=1e-15)\n",
    "    \n",
    "    # get quadrature on (-1,-1), (1,-1), (-1,1)\n",
    "    Q_tri = 16 # No. pts in 2D\n",
    "    ww, q = GetQuadratureTriangle(Q_tri)\n",
    "    q1 = q[:,1]\n",
    "    q2 = q[:,2]\n",
    "    m1 = -ones(Q_tri)\n",
    "    p1 = ones(Q_tri)\n",
    "    \n",
    "    # top face\n",
    "    n, le = GetNormal(coord_E, q1, q2, p1, \"top\")\n",
    "    Integral = (ww .* le)' *(Dubiner.(q1,q2,1,2) .* Dubiner.(q1,q2,2,2))\n",
    "    @test isapprox(Integral, 0.;atol=1e-15)\n",
    "    \n",
    "    # bottom face\n",
    "    n, le = GetNormal(coord_E, q1, q2, m1, \"bottom\")\n",
    "    Integral = (ww .* le)' *(Dubiner.(q1,q2,2,1) .* Dubiner.(q1,q2,2,2))\n",
    "    @test isapprox(Integral, 0.;atol=1e-15)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "IntegrateFace (generic function with 1 method)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function IntegrateFace(i,j,coord_E, face, Q1d_quad, Q_tri)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    i,j: degrees of Legendre polynomials L_i(s)L_j(t)\n",
    "    coord_E: physical coordinate of element E\n",
    "    face: face that you want to compute the normal component of the trace\n",
    "    Q1d_quad: No. pts in 1D for integration over quad faces\n",
    "    Q_tri: No. pts in 2D for integration over triangle faces\n",
    "    \n",
    "    Return:\n",
    "    this function returns int{P.n * L_i(s)L_j(t)}dA\n",
    "    where \"n\" is the normal on the physical face\n",
    "    \"L\" are Legendre polynomial\n",
    "\n",
    "    local numbering of Prism\n",
    "    \n",
    "        4--------6           z\n",
    "      / |                    |\n",
    "    5   |                    |\n",
    "    |   |                    /----y\n",
    "    |   |                   /\n",
    "    |   1 -------3         x\n",
    "    | /\n",
    "    2\n",
    "\n",
    "    \"\"\"\n",
    "    # get quadrature on [-1,1]^2\n",
    "    ww, q1, q2 = GetQuadrature2D(Q1d_quad)\n",
    "    m1 = -ones(Q1d_quad*Q1d_quad)\n",
    "    p1 = ones(Q1d_quad*Q1d_quad)\n",
    "    \n",
    "    # get quadrature on triangle (-1,-1), (1,-1), (-1,1)\n",
    "    wwt, qt = GetQuadratureTriangle(Q_tri)\n",
    "    q1t = qt[:,1]\n",
    "    q2t = qt[:,2]\n",
    "    m1t = -ones(Q_tri)\n",
    "    p1t = ones(Q_tri)\n",
    "    \n",
    "    # left face includes nodes 1,2,4,5 at yhat = -1\n",
    "    if face == \"left\"\n",
    "        n, le = GetNormal(coord_E, q1, m1, q2, face)\n",
    "        P = PrimeBasis(coord_E, q1, m1, q2)\n",
    "        PdotN = P[1,:,:] .* n[1,:] + P[2,:,:] .* n[2,:] + P[3,:,:] .* n[3,:]\n",
    "        trace = PdotN' *(ww .* le .* Legendre2D.(q1, q2,i,j))\n",
    "        \n",
    "    # back face includes nodes 1,3,4,6 at xhat = -1\n",
    "    elseif face == \"back\"\n",
    "        n, le = GetNormal(coord_E, m1, q1, q2, face)\n",
    "        P = PrimeBasis(coord_E, m1, q1, q2)\n",
    "        PdotN = P[1,:,:] .* n[1,:] + P[2,:,:] .* n[2,:] + P[3,:,:] .* n[3,:]\n",
    "        trace = PdotN' *(ww .* le .* Legendre2D.(q1, q2,i,j))\n",
    "        \n",
    "    # diagonal face includes nodes 2,3,5,6 at xhat = -yhat\n",
    "    elseif face == \"diagonal\"\n",
    "        n, le = GetNormal(coord_E, -q1, q1, q2, face)\n",
    "        P = PrimeBasis(coord_E, -q1, q1, q2)\n",
    "        PdotN = P[1,:,:] .* n[1,:] + P[2,:,:] .* n[2,:] + P[3,:,:] .* n[3,:]\n",
    "        trace = PdotN' *(ww .* le .* Legendre2D.(q1, q2,i,j))  \n",
    "        \n",
    "    # top face includes nodes 4,5,6 at zhat = 1\n",
    "    elseif face == \"top\"\n",
    "        n, le = GetNormal(coord_E, q1t, q2t, p1t, face)\n",
    "        P = PrimeBasis(coord_E, q1t, q2t, p1t);\n",
    "        PdotN = P[1,:,:] .* n[1,:] + P[2,:,:] .* n[2,:] + P[3,:,:] .* n[3,:]\n",
    "        trace = PdotN' *(wwt .* le .* Dubiner.(q1t, q2t,i,j))\n",
    "        \n",
    "    # bottom face includes nodes 1,2,3 at zhat = -1\n",
    "    elseif face == \"bottom\"\n",
    "        n, le = GetNormal(coord_E, q1t, q2t, m1t, face)\n",
    "        P = PrimeBasis(coord_E, q1t, q2t, m1t);\n",
    "        PdotN = P[1,:,:] .* n[1,:] + P[2,:,:] .* n[2,:] + P[3,:,:] .* n[3,:]\n",
    "        trace = PdotN' *(wwt .* le .* Dubiner.(q1t, q2t,i,j))\n",
    "    else\n",
    "        error(\"face is not defined\")\n",
    "    end\n",
    "    \n",
    "    return trace\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ConstraintMat (generic function with 1 method)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ConstraintMat(coord_E, Q1d_quad, Q_tri)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    coord_E: physical coordinate of element E\n",
    "    Q1d_quad: No. pts in 1D for integration over quad faces\n",
    "    Q_tri: No. pts in 2D for integration over triangle faces\n",
    "    \n",
    "    Return:\n",
    "    CM: Constraint Matrix\n",
    "    \"\"\"\n",
    "    \n",
    "    # On each quad face we compute trace_i,j\n",
    "    # for (i,j) = (2,0),(2,1),(2,2),(1,2),(0,2) ==> 15 rows for all quad\n",
    "    # On triangle face we have above (i,j)+(1,1) ==> 12 rows for all tri\n",
    "    n = [2 2 2 1 0]\n",
    "    quad = [\"left\",\"back\",\"diagonal\"]\n",
    "    # 15 constraint, 22 dim\n",
    "    L1 = zeros(15,22)\n",
    "    for i = 1:3 # loop over faces\n",
    "        for j = 1:length(n) # loop over legendre_i,j\n",
    "            trace = IntegrateFace(n[j], n[length(n)+1-j], coord_E, quad[i], Q1d_quad, Q_tri)\n",
    "            L1[j + length(n)*(i-1),:] = trace\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    L2 = zeros(6,22)\n",
    "    for j = 1:length(n) # loop over legendre_i,j\n",
    "        trace = IntegrateFace(n[j], n[length(n)+1-j], coord_E, \"top\", Q1d_quad, Q_tri)\n",
    "        L2[j,:] = trace\n",
    "    end\n",
    "    L2[6,:] = IntegrateFace(1, 1, coord_E, \"top\", Q1d_quad, Q_tri)\n",
    "\n",
    "    L3 = zeros(6,22)\n",
    "    for j = 1:length(n) # loop over legendre_i,j\n",
    "        trace = IntegrateFace(n[j], n[length(n)+1-j], coord_E, \"bottom\", Q1d_quad, Q_tri)\n",
    "        L2[j,:] = trace\n",
    "    end\n",
    "    L3[6,:] = IntegrateFace(1, 1, coord_E, \"bottom\", Q1d_quad, Q_tri)\n",
    "    \n",
    "    L = zeros(27,22)\n",
    "    L[1:15,:] = L1\n",
    "    L[16:21,:] = L2\n",
    "    L[22:27,:] = L3\n",
    "    return L\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×27 UpperTrapezoidal{Float64}:\n",
       " -0.92376  -2.33693e-18  -8.33284e-16  …  0.0  0.0  0.0  0.0  0.0\n",
       "  0.0       0.355556     -4.57967e-16     0.0  0.0  0.0  0.0  0.0\n",
       "  0.0       0.0          -0.377124        0.0  0.0  0.0  0.0  0.0\n",
       "  0.0       0.0           0.0             0.0  0.0  0.0  0.0  0.0\n",
       "  0.0       0.0           0.0             0.0  0.0  0.0  0.0  0.0\n",
       "  0.0       0.0           0.0          …  0.0  0.0  0.0  0.0  0.0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# unstructured element\n",
    "#coord_E = [0. 0. 0.;1. 0. 0.;0. 1. 0.;1. 1. 1.;2. 1. 1.;1. 2. 1.]\n",
    "# reference element\n",
    "coord_E = [-1. -1. -1.;1. -1. -1.;-1. 1. -1.;-1. -1. 1.;1. -1. 1.;-1. 1. 1.]\n",
    "Q1d_quad = 7\n",
    "Q_tri = 19\n",
    "L = ConstraintMat(coord_E, Q1d_quad, Q_tri)\n",
    "F=pqrfact(L')\n",
    "#Q = F[:Q]\n",
    "R1 = F[:R]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "clip (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function clip(x; threshold=1e-15)\n",
    "    y = copy(x)\n",
    "    y[abs.(y) .< threshold] .= 0\n",
    "    y\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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       "  1854.02,429.533 1878.02,429.533 1854.02,429.533 1854.02,92.2192 1878.02,92.2192 1854.02,92.2192 1854.02,47.2441 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L1 = clip(L)\n",
    "spy(L1, markersize = 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetNull (generic function with 2 methods)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetNull(L, mode)\n",
    "    \n",
    "    if mode == \"QR\"\n",
    "        F=pqrfact(L')\n",
    "        Qfull, _ = qr(F[:Q])\n",
    "        VL = Qfull[:, 5:end]\n",
    "        \n",
    "    elseif mode == \"SVD\"\n",
    "        \n",
    "        U, S, V = svd(L,full=true)\n",
    "        VL = V[:,5:end]\n",
    "    else\n",
    "        error(\"mode should be QR or SVD\")\n",
    "    end\n",
    "    return VL\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetNodalBasis (generic function with 1 method)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetNodalBasis(coord_E, xhat, yhat, zhat, VL)\n",
    "\n",
    "    nl, le = GetNormal(coord_E, [0.], [-1.],[0.],\"left\")\n",
    "    nd, le = GetNormal(coord_E, [0.], [0.],[0.],\"diagonal\")\n",
    "    nbt, le = GetNormal(coord_E, [-0.5], [-0.5],[-1.],\"bottom\")\n",
    "    nt, le = GetNormal(coord_E, [-0.5], [-0.5],[1.],\"top\")\n",
    "    nbk, le = GetNormal(coord_E, [-1.], [0.],[0.],\"back\")\n",
    "    normals = [nbk nl nbt nd nl nbt nbk nd nbt nbk nl nt nd nl nt nbk nd nt]\n",
    "    nd1 = [0.;0.;-1.]\n",
    "    nd2 = [1.;0.;-1.]\n",
    "    nd3 = [0.;1.;-1.]\n",
    "    nd4 = [0.;0.;1.]\n",
    "    nd5 = [1.;0.;1.]\n",
    "    nd6 = [0.;1.;1.]\n",
    "    nodes = [nd1 nd1 nd1 nd2 nd2 nd2 nd3 nd3 nd3 nd4 nd4 nd4 nd5 nd5 nd5 nd6 nd6 nd6]\n",
    "    # vondermonde matrix, V_ij = phi_j(x_i).n_i\n",
    "    VM = zeros(18,18)\n",
    "    for i=1:18\n",
    "        for j=1:18\n",
    "            # original prime basis with dim=22\n",
    "            P = PrimeBasis(coord_E, nodes[1,i], nodes[2,i], nodes[3,i])\n",
    "            # reduced prime basis with dim=18\n",
    "            PP = zeros(3,1,18)\n",
    "            PP[1,:,:] = P[1,:,:] * VL\n",
    "            PP[2,:,:] = P[2,:,:] * VL\n",
    "            PP[3,:,:] = P[3,:,:] * VL\n",
    "            VM[i,j] = PP[1,1,j] * normals[1,i] + PP[2,1,j] * normals[2,i] + PP[3,1,j] * normals[3,i]\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    # create nodal basis from vondermonde (VM) and VL\n",
    "    P = PrimeBasis(coord_E, xhat, yhat, zhat)\n",
    "    invVM = inv(VM)\n",
    "    # nodal basis\n",
    "    m = length(xhat)\n",
    "    Nhat = zeros(3,m,18)\n",
    "    Nhat[1,:,:] = P[1,:,:] * VL * invVM\n",
    "    Nhat[2,:,:] = P[2,:,:] * VL * invVM\n",
    "    Nhat[3,:,:] = P[3,:,:] * VL * invVM\n",
    "    return Nhat\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "IntegrateNormalTrace (generic function with 1 method)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function IntegrateNormalTrace(i,j,coord_E, face, Q1d_quad, Q_tri)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    i,j: degrees of Legendre polynomials L_i(s)L_j(t)\n",
    "    coord_E: physical coordinate of element E\n",
    "    face: face that you want to compute the normal component of the trace\n",
    "    Q1d_quad: No. pts in 1D for integration over quad faces\n",
    "    Q_tri: No. pts in 2D for integration over triangle faces\n",
    "    \n",
    "    Return:\n",
    "    this function returns int{N.n * L_i(s)L_j(t)}dA\n",
    "    where \"n\" is the normal on the physical face\n",
    "    \"L\" are Legendre polynomial, N is nodal basis\n",
    "\n",
    "    local numbering of Prism\n",
    "    \n",
    "        4--------6           z\n",
    "      / |                    |\n",
    "    5   |                    |\n",
    "    |   |                    /----y\n",
    "    |   |                   /\n",
    "    |   1 -------3         x\n",
    "    | /\n",
    "    2\n",
    "\n",
    "    \"\"\"\n",
    "    # get quadrature on [-1,1]^2\n",
    "    ww, q1, q2 = GetQuadrature2D(Q1d_quad)\n",
    "    m1 = -ones(Q1d_quad*Q1d_quad)\n",
    "    p1 = ones(Q1d_quad*Q1d_quad)\n",
    "    \n",
    "    # get quadrature on (-1,-1), (1,-1), (-1,1)\n",
    "    wwt, qt = GetQuadratureTriangle(Q_tri)\n",
    "    q1t = qt[:,1]\n",
    "    q2t = qt[:,2]\n",
    "    m1t = -ones(Q_tri)\n",
    "    p1t = ones(Q_tri)\n",
    "    \n",
    "    L = ConstraintMat(coord_E, Q1d_quad, Q_tri)\n",
    "    VL = GetNull(L, \"QR\")\n",
    "    \n",
    "    # left face includes nodes 1,2,4,5 at yhat = -1\n",
    "    if face == \"left\"\n",
    "        n, le = GetNormal(coord_E, q1, m1, q2, \"left\")\n",
    "        N = GetNodalBasis(coord_E, q1, m1, q2, VL)\n",
    "        Ndotn = N[1,:,:] .* n[1,:] + N[2,:,:] .* n[2,:] + N[3,:,:] .* n[3,:]\n",
    "        trace = Ndotn' *(ww .* le .* Legendre2D.(q1,q2,i,j))\n",
    "        \n",
    "    # back face includes nodes 1,3,4,6 at xhat = -1\n",
    "    elseif face == \"back\"\n",
    "        n, le = GetNormal(coord_E, m1, q1, q2, \"back\")\n",
    "        N = GetNodalBasis(coord_E, m1, q1, q2, VL)\n",
    "        Ndotn = N[1,:,:] .* n[1,:] + N[2,:,:] .* n[2,:] + N[3,:,:] .* n[3,:]\n",
    "        trace = Ndotn' *(ww .* le .* Legendre2D.(q1,q2,i,j))\n",
    "        \n",
    "    # diagonal face includes nodes 2,3,5,6 at xhat = -yhat\n",
    "    elseif face == \"diagonal\"\n",
    "        n, le = GetNormal(coord_E, -q1, q1, q2, \"diagonal\")\n",
    "        N = GetNodalBasis(coord_E, -q1, q1, q2, VL)\n",
    "        Ndotn = N[1,:,:] .* n[1,:] + N[2,:,:] .* n[2,:] + N[3,:,:] .* n[3,:]\n",
    "        trace = Ndotn' *(ww .* le .* Legendre2D.(q1,q2,i,j))  \n",
    "        \n",
    "    # top face includes nodes 4,5,6 at zhat = 1\n",
    "    elseif face == \"top\"\n",
    "        n, le = GetNormal(coord_E, q1t, q2t, p1t, \"top\")\n",
    "        N = GetNodalBasis(coord_E, q1t, q2t, p1t, VL)\n",
    "        Ndotn = N[1,:,:] .* n[1,:] + N[2,:,:] .* n[2,:] + N[3,:,:] .* n[3,:]\n",
    "        trace = Ndotn' *(wwt .* le .* Dubiner.(q1t,q2t,i,j))\n",
    "        \n",
    "    # bottom face includes nodes 1,2,3 at zhat = -1\n",
    "    elseif face == \"bottom\"\n",
    "        n, le = GetNormal(coord_E, q1t, q2t, m1t, \"bottom\")\n",
    "        N = GetNodalBasis(coord_E, q1t, q2t, m1t, VL)\n",
    "        Ndotn = N[1,:,:] .* n[1,:] + N[2,:,:] .* n[2,:] + N[3,:,:] .* n[3,:]\n",
    "        trace = Ndotn' *(wwt .* le .* Dubiner.(q1t,q2t,i,j))\n",
    "    else\n",
    "        error(\"face is not defined\")\n",
    "    end\n",
    "    \n",
    "    return trace\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NormalTrace (generic function with 1 method)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function NormalTrace(coord_E, face, Q1d_quad, Q_tri)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    coord_E: physical coordinate of element E\n",
    "    face: face that you want the normal trace\n",
    "    Q1d_quad: No. pts in 1D for integration over quad faces\n",
    "    Q_tri: No. pts in 2D for integration over triangle faces\n",
    "    \n",
    "    Return:\n",
    "    Computed normal trace on face\n",
    "    \"\"\"\n",
    "    \n",
    "    # On each quad face we compute trace_i,j\n",
    "    # for (i,j) = (2,0),(2,1),(2,2),(1,2),(0,2) \n",
    "    # On triangle face we have above (i,j)+(1,1)\n",
    "    \n",
    "    n = [2 2 2 1 0]\n",
    "    N = length(n)\n",
    "    \n",
    "    if face == \"top\" || face == \"bottom\"\n",
    "        # 6 constraint, 18 dim\n",
    "        L = zeros(N+1,18)\n",
    "        for j = 1:N # loop over legendre_i,j\n",
    "            trace = IntegrateNormalTrace(n[j], n[N+1-j], coord_E, face, Q1d_quad, Q_tri)\n",
    "            L[j,:] = trace\n",
    "        end\n",
    "        L[N+1,:] = IntegrateNormalTrace(1, 1, coord_E, face, Q1d_quad, Q_tri)\n",
    "    else\n",
    "        # 5 constraint, 18 dim\n",
    "        L = zeros(N,18)\n",
    "        for j = 1:N # loop over legendre_i,j\n",
    "            trace = IntegrateNormalTrace(n[j], n[N+1-j], coord_E, face, Q1d_quad, Q_tri)\n",
    "            L[j,:] = trace\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return L\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test Normal Trace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[37m\u001b[1mTest Summary:   | \u001b[22m\u001b[39m\u001b[32m\u001b[1mPass  \u001b[22m\u001b[39m\u001b[36m\u001b[1mTotal\u001b[22m\u001b[39m\n",
      "TestNormalTrace | \u001b[32m 486  \u001b[39m\u001b[36m  486\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestNormalTrace\", Any[], 486, false, false)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestNormalTrace\" begin\n",
    "    \n",
    "    # unstructured element\n",
    "    coord_E = [0. 0. 0.;1. 0. 0.;0. 1. 0.;1. 1. 1.;2. 1. 1.;1. 2. 1.]\n",
    "    # reference element\n",
    "    #coord_E = [-1. -1. -1.;1. -1. -1.;-1. 1. -1.;-1. -1. 1.;1. -1. 1.;-1. 1. 1.]\n",
    "    \n",
    "    err = 1e-12\n",
    "    Q1d_quad = 5\n",
    "    Q_tri = 19\n",
    "\n",
    "    trace = NormalTrace(coord_E, \"left\", Q1d_quad, Q_tri)\n",
    "    # check if they are zero\n",
    "    for i = 1:size(trace,1)\n",
    "        for j = 1:size(trace,2)\n",
    "            @test isapprox(trace[i,j], 0.;atol=err)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    trace = NormalTrace(coord_E, \"back\", Q1d_quad, Q_tri)\n",
    "    # check if they are zero\n",
    "    for i = 1:size(trace,1)\n",
    "        for j = 1:size(trace,2)\n",
    "            @test isapprox(trace[i,j], 0.;atol=err)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    trace = NormalTrace(coord_E, \"diagonal\", Q1d_quad, Q_tri)\n",
    "    # check if they are zero\n",
    "    for i = 1:size(trace,1)\n",
    "        for j = 1:size(trace,2)\n",
    "            @test isapprox(trace[i,j], 0.;atol=err)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    trace = NormalTrace(coord_E, \"top\", Q1d_quad, Q_tri)\n",
    "    # check if they are zero\n",
    "    for i = 1:size(trace,1)\n",
    "        for j = 1:size(trace,2)\n",
    "            @test isapprox(trace[i,j], 0.;atol=err)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    trace = NormalTrace(coord_E, \"bottom\", Q1d_quad, Q_tri)\n",
    "    # check if they are zero\n",
    "    for i = 1:size(trace,1)\n",
    "        for j = 1:size(trace,2)\n",
    "            @test isapprox(trace[i,j], 0.;atol=err)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test Nodal Value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[37m\u001b[1mTest Summary:  | \u001b[22m\u001b[39m\u001b[32m\u001b[1mPass  \u001b[22m\u001b[39m\u001b[36m\u001b[1mTotal\u001b[22m\u001b[39m\n",
      "TestNodalBasis | \u001b[32m 108  \u001b[39m\u001b[36m  108\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestNodalBasis\", Any[], 108, false, false)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestNodalBasis\" begin\n",
    "\n",
    "    # unstructured element\n",
    "    # coord_E = [0. 0. 0.;1. 0. 0.;0. 1. 0.;1. 1. 1.;2. 1. 1.;1. 2. 1.2]\n",
    "    # reference element\n",
    "    coord_E = [-1. -1. -1.;1. -1. -1.;-1. 1. -1.;-1. -1. 1.;1. -1. 1.;-1. 1. 1.]\n",
    "    \n",
    "    Q1d_quad = 5\n",
    "    Q_tri = 19\n",
    "    L = ConstraintMat(coord_E, Q1d_quad, Q_tri)\n",
    "    VL = GetNull(L, \"QR\")\n",
    "    \n",
    "    nl, le = GetNormal(coord_E, [0.], [-1.],[0.],\"left\")\n",
    "    nd, le = GetNormal(coord_E, [0.], [0.],[0.],\"diagonal\")\n",
    "    nbt, le = GetNormal(coord_E, [-0.5], [-0.5],[-1.],\"bottom\")\n",
    "    nt, le = GetNormal(coord_E, [-0.5], [-0.5],[1.],\"top\")\n",
    "    nbk, le = GetNormal(coord_E, [-1.], [0.],[0.],\"back\")\n",
    "    \n",
    "    err = 1e-12\n",
    "    #============= Node 1==============#\n",
    "    # check node 1\n",
    "    Nhat = GetNodalBasis(coord_E,[0.], [0.], [-1], VL)\n",
    "    @test isapprox(dot(Nhat[:,1,1],nbk), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,2],nl), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,3],nbt), 1.;atol=err)\n",
    "    # check node 2\n",
    "    @test isapprox(dot(Nhat[:,1,4],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,6],nbt), 0.;atol=err)\n",
    "    # check node 3\n",
    "    @test isapprox(dot(Nhat[:,1,7],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,8],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,9],nbt), 0.;atol=err)\n",
    "    # check node 4\n",
    "    @test isapprox(dot(Nhat[:,1,10],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,11],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,12],nt), 0.;atol=err)\n",
    "    # check node 5\n",
    "    @test isapprox(dot(Nhat[:,1,13],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,14],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,15],nt), 0.;atol=err)\n",
    "    # check node 6\n",
    "    @test isapprox(dot(Nhat[:,1,16],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,17],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,18],nt), 0.;atol=err)\n",
    "    \n",
    "    #============= Node 2==============#\n",
    "    # check node 1\n",
    "    Nhat = GetNodalBasis(coord_E,[1.], [0.], [-1], VL)\n",
    "    @test isapprox(dot(Nhat[:,1,1],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,2],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,3],nbt), 0.;atol=err)\n",
    "    # check node 2\n",
    "    @test isapprox(dot(Nhat[:,1,4],nd), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,5],nl), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,6],nbt), 1.;atol=err)\n",
    "    # check node 3\n",
    "    @test isapprox(dot(Nhat[:,1,7],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,8],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,9],nbt), 0.;atol=err)\n",
    "    # check node 4\n",
    "    @test isapprox(dot(Nhat[:,1,10],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,11],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,12],nt), 0.;atol=err)\n",
    "    # check node 5\n",
    "    @test isapprox(dot(Nhat[:,1,13],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,14],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,15],nt), 0.;atol=err)\n",
    "    # check node 6\n",
    "    @test isapprox(dot(Nhat[:,1,16],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,17],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,18],nt), 0.;atol=err)\n",
    "    #============= Node 3==============#\n",
    "    # check node 1\n",
    "    Nhat = GetNodalBasis(coord_E,[0.], [1.], [-1], VL)\n",
    "    @test isapprox(dot(Nhat[:,1,1],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,2],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,3],nbt), 0.;atol=err)\n",
    "    # check node 2\n",
    "    @test isapprox(dot(Nhat[:,1,4],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,6],nbt), 0.;atol=err)\n",
    "    # check node 3\n",
    "    @test isapprox(dot(Nhat[:,1,7],nbk), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,8],nd), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,9],nbt), 1.;atol=err)\n",
    "    # check node 4\n",
    "    @test isapprox(dot(Nhat[:,1,10],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,11],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,12],nt), 0.;atol=err)\n",
    "    # check node 5\n",
    "    @test isapprox(dot(Nhat[:,1,13],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,14],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,15],nt), 0.;atol=err)\n",
    "    # check node 6\n",
    "    @test isapprox(dot(Nhat[:,1,16],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,17],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,18],nt), 0.;atol=err)\n",
    "    #============= Node 4==============#\n",
    "    # check node 1\n",
    "    Nhat = GetNodalBasis(coord_E,[0.], [0.], [1], VL)\n",
    "    @test isapprox(dot(Nhat[:,1,1],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,2],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,3],nbt), 0.;atol=err)\n",
    "    # check node 2\n",
    "    @test isapprox(dot(Nhat[:,1,4],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,6],nbt), 0.;atol=err)\n",
    "    # check node 3\n",
    "    @test isapprox(dot(Nhat[:,1,7],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,8],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,9],nbt), 0.;atol=err)\n",
    "    # check node 4\n",
    "    @test isapprox(dot(Nhat[:,1,10],nbk), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,11],nl), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,12],nt), 1.;atol=err)\n",
    "    # check node 5\n",
    "    @test isapprox(dot(Nhat[:,1,13],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,14],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,15],nt), 0.;atol=err)\n",
    "    # check node 6\n",
    "    @test isapprox(dot(Nhat[:,1,16],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,17],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,18],nt), 0.;atol=err)\n",
    "    #============= Node 5==============#\n",
    "    # check node 1\n",
    "    Nhat = GetNodalBasis(coord_E,[1.], [0.], [1], VL)\n",
    "    @test isapprox(dot(Nhat[:,1,1],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,2],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,3],nbt), 0.;atol=err)\n",
    "    # check node 2\n",
    "    @test isapprox(dot(Nhat[:,1,4],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,6],nbt), 0.;atol=err)\n",
    "    # check node 3\n",
    "    @test isapprox(dot(Nhat[:,1,7],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,8],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,9],nbt), 0.;atol=err)\n",
    "    # check node 4\n",
    "    @test isapprox(dot(Nhat[:,1,10],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,11],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,12],nt), 0.;atol=err)\n",
    "    # check node 5\n",
    "    @test isapprox(dot(Nhat[:,1,13],nd), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,14],nl), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,15],nt), 1.;atol=err)\n",
    "    # check node 6\n",
    "    @test isapprox(dot(Nhat[:,1,16],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,17],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,18],nt), 0.;atol=err)\n",
    "    #============= Node 6==============#\n",
    "    # check node 1\n",
    "    Nhat = GetNodalBasis(coord_E,[0.], [1.], [1], VL)\n",
    "    @test isapprox(dot(Nhat[:,1,1],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,2],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,3],nbt), 0.;atol=err)\n",
    "    # check node 2\n",
    "    @test isapprox(dot(Nhat[:,1,4],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,6],nbt), 0.;atol=err)\n",
    "    # check node 3\n",
    "    @test isapprox(dot(Nhat[:,1,7],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,8],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,9],nbt), 0.;atol=err)\n",
    "    # check node 4\n",
    "    @test isapprox(dot(Nhat[:,1,10],nbk), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,11],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,12],nt), 0.;atol=err)\n",
    "    # check node 5\n",
    "    @test isapprox(dot(Nhat[:,1,13],nd), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,14],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,15],nt), 0.;atol=err)\n",
    "    # check node 6\n",
    "    @test isapprox(dot(Nhat[:,1,16],nbk), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,17],nd), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,1,18],nt), 1.;atol=err)\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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