ACquad.ipynb

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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "using LinearAlgebra\n",
    "using Jacobi\n",
    "using Test\n",
    "gr()\n",
    "Plots.GRBackend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# AC quad element with added more supplements basis "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetNormal (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function BilinearMap(coord_E, xhat, yhat)\n",
    "    \"\"\" \n",
    "    This function maps [xhat,yhat] in Ehat=[-1,1]^2 \n",
    "    to (x,y) in E.\n",
    "    Written based on eq. (3.3), (3.4) of AC paper 2016\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: coordinates of quadrilateral E .\n",
    "    coord_E is 4x2 array\n",
    "    coord_E = [[x1,y1],\n",
    "               [x2,y2],\n",
    "               [x3,y3],\n",
    "               [x4,y4]] with vertices numbering\n",
    "    3----4\n",
    "    |    |\n",
    "    1----2\n",
    "    [xhat, yhat] in Ehat\n",
    "    Output:\n",
    "    ------\n",
    "    [x;y]: mapped vector in E.\n",
    "    DF_E: Jacobian matrix\n",
    "    J_E: det(DF_E)\n",
    "    later we transform vector vhat to v by eq. (3.4)\n",
    "    v(x) = P_E(vhat)(x) = (DF_E/J_E)*vhat(xhat)\n",
    "    \"\"\"\n",
    "    m = length(xhat)\n",
    "    P = 0.25 * [(1 .- xhat).*(1 .- yhat) (1 .+ xhat).*(1 .- yhat) (1 .- xhat).*(1 .+ yhat) (1 .+ xhat).*(1 .+ yhat)]\n",
    "    P_E = P * coord_E\n",
    "    # x=F_E(xhat)\n",
    "    X = P_E\n",
    "    x = X[:, 1]\n",
    "    y = X[:, 2]\n",
    "    # gradient of P, 1st row = dP/dxhat, 2nd row=dP/dyhat\n",
    "    # GradP_(2m x 4)\n",
    "    GradP = 0.25 * [-(1 .- yhat) (1 .- yhat) -(1 .+ yhat) (1 .+ yhat);\n",
    "                    -(1 .- xhat) -(1 .+ xhat) (1 .- xhat) (1 .+ xhat)]\n",
    "\n",
    "    # JT= [[dx/dxhat, dy/dxhat],\n",
    "    #      [dx/dyhat, dy/dyhat]] (2m x 2)\n",
    "    JT = GradP * coord_E\n",
    "    dxdxhat = JT[1:m,1]\n",
    "    dydxhat = JT[1:m,2]\n",
    "    dxdyhat = JT[m+1:2*m,1]\n",
    "    dydyhat = JT[m+1:2*m,2]\n",
    "    J_E = dxdxhat .* dydyhat .- dydxhat .* dxdyhat\n",
    "    \n",
    "    DF_E = [dxdxhat dxdyhat;dydxhat dydyhat]\n",
    "    \n",
    "    return x, y, DF_E, J_E\n",
    "end\n",
    "\n",
    "\n",
    "function PrimeBasis(coord_E, xhat, yhat)\n",
    "\n",
    "    # original supplements in AC paper\n",
    "    sghat1 = [(1 .- xhat.^2) (2*xhat .* yhat)]\n",
    "    sghat2 = [(-2*xhat .* yhat) (yhat.^2 .- 1)]\n",
    "    # New supplements\n",
    "    sghat3 = [(2*yhat .* (1 .+ xhat .+ xhat.^3)) -((1 .+ 3*xhat.^2).* yhat.^2)]\n",
    "    sghat4 = [((1 .+ 3*yhat.^2).* xhat.^2) -(2*xhat .* (1 .+ yhat .+ yhat.^3))]\n",
    "    \n",
    "    # (x,y) are in E\n",
    "    x, y, DF_E, J_E = BilinearMap(coord_E, xhat, yhat)\n",
    "\n",
    "    m = length(xhat)\n",
    "    # we have 8 basis including sg1 and sg2\n",
    "    P = zeros(2*m,10)\n",
    "    P[1:m,1] = ones(m)\n",
    "    P[1:m,2] = x\n",
    "    P[1:m,3] = y\n",
    "    P[m+1:2*m,4] = ones(m)\n",
    "    P[m+1:2*m,5] = x\n",
    "    P[m+1:2*m,6] = y\n",
    "    for i=1:m\n",
    "    P[i,7] = dot(DF_E[i,:], sghat1[i,:]) / J_E[i]\n",
    "    P[i+m,7] = dot(DF_E[i+m,:], sghat1[i,:]) / J_E[i]\n",
    "    end\n",
    "    \n",
    "    for i=1:m\n",
    "    P[i,8] = dot(DF_E[i,:], sghat2[i,:]) / J_E[i]\n",
    "    P[i+m,8] = dot(DF_E[i+m,:], sghat2[i,:]) / J_E[i]\n",
    "    end\n",
    "    \n",
    "    for i=1:m\n",
    "    P[i,9] = dot(DF_E[i,:], sghat3[i,:]) / J_E[i]\n",
    "    P[i+m,9] = dot(DF_E[i+m,:], sghat3[i,:]) / J_E[i]\n",
    "    end\n",
    "    \n",
    "    for i=1:m\n",
    "    P[i,10] = dot(DF_E[i,:], sghat4[i,:]) / J_E[i]\n",
    "    P[i+m,10] = dot(DF_E[i+m,:], sghat4[i,:]) / J_E[i]\n",
    "    end\n",
    "    \n",
    "    return P\n",
    "end\n",
    "\n",
    "\n",
    "function GetNormal(coord_E, edge)\n",
    "\n",
    "    if edge == \"left\"\n",
    "        # left edge, (0,0,1)x(dxdyhat,dydyhat,0)\n",
    "        x, y, DF_E, J_E = BilinearMap(coord_E, [-1.], [0.])\n",
    "        dxdyhat = DF_E[1,2]\n",
    "        dydyhat = DF_E[2,2]\n",
    "        length2 = sqrt(dxdyhat*dxdyhat + dydyhat*dydyhat)\n",
    "        n = [-dydyhat; dxdyhat]/length2\n",
    "\n",
    "    elseif edge == \"right\"\n",
    "        # right edge, (0,0,-1)x(dxdyhat,dydyhat,0)\n",
    "        x, y, DF_E, J_E = BilinearMap(coord_E, [1.], [0.])\n",
    "        dxdyhat = DF_E[1,2]\n",
    "        dydyhat = DF_E[2,2]\n",
    "        length2 = sqrt(dxdyhat*dxdyhat + dydyhat*dydyhat)\n",
    "        n = [dydyhat; -dxdyhat]/length2\n",
    "\n",
    "    elseif edge == \"bottom\"\n",
    "        # bottom edge, (0,0,-1)x(dxdxhat,dydxhat,0)\n",
    "        x, y, DF_E, J_E = BilinearMap(coord_E, [0.], [-1.])\n",
    "        dxdxhat = DF_E[1,1]\n",
    "        dydxhat = DF_E[2,1]\n",
    "        length1 = sqrt(dxdxhat*dxdxhat + dydxhat*dydxhat)\n",
    "        n = [dydxhat; -dxdxhat]/length1\n",
    "\n",
    "    elseif edge == \"top\"\n",
    "        # top edge, (0,0,1)x(dxdxhat,dydxhat,0)\n",
    "        x, y, DF_E, J_E = BilinearMap(coord_E, [0.], [1.])\n",
    "        dxdxhat = DF_E[1,1]\n",
    "        dydxhat = DF_E[2,1]\n",
    "        length1 = sqrt(dxdxhat*dxdxhat + dydxhat*dydxhat)\n",
    "        n = [-dydxhat; dxdxhat]/length1\n",
    "\n",
    "    else\n",
    "        println(\"Error! Enter the Xhat=[xhat, yhat] on the edge of Ehat\")\n",
    "    \n",
    "    end\n",
    "\n",
    "    return n\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ConstraintMat (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ConstraintMat(coord_E, mode)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: is the coordinate of vertices of element.\n",
    "    Note\n",
    "    3---4\n",
    "    |   |\n",
    "    1---2\n",
    "    \n",
    "    mode: QR or SVD\n",
    "    Output:\n",
    "    ------\n",
    "    VM: the 8x8 vondermonde matrix\n",
    "    \"\"\"\n",
    "    nl = GetNormal(coord_E, \"left\")\n",
    "    nr = GetNormal(coord_E, \"right\")\n",
    "    nb = GetNormal(coord_E, \"bottom\")\n",
    "    nt = GetNormal(coord_E, \"top\")\n",
    "    \n",
    "    # Gauss Quadrature\n",
    "    Q = 3\n",
    "    alpha=0.0\n",
    "    beta = 0.0\n",
    "    q = zgj(Q, alpha, beta)\n",
    "    w = wgj(q, alpha, beta)\n",
    "    nm = -ones(Q)\n",
    "    np = ones(Q)\n",
    "    # these are (xhat,yhat) for bottom, right, top, left edges\n",
    "    nodes = [q nm np q q np nm q]\n",
    "    normals = [nb nr nt nl]\n",
    "    # Constraint matrix\n",
    "    L = zeros(4,10)\n",
    "    for i=1:4\n",
    "        for j=1:10\n",
    "            for k=1:Q\n",
    "                P = PrimeBasis(coord_E, [nodes[k,2*i-1]], [nodes[k,2*i]])\n",
    "                PdotN = dot(P[:,j],normals[:,i])\n",
    "                leg = legendre(q[k], 2)\n",
    "                L[i,j] += PdotN * leg * w[k]\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "\n",
    "    if mode == \"SVD\"\n",
    "        U, S, V = svd(L,full=true)\n",
    "        VL = V[:,3:10]\n",
    "    elseif mode == \"QR\"\n",
    "        Q, R =qr(L')\n",
    "        VL  = Q[:,3:10]\n",
    "    else\n",
    "        error(\"Define mode as QR or SVD\")\n",
    "    end\n",
    "    \n",
    "    return L, VL\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetACNodalBasis (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetACNodalBasis(coord_E, xhat, yhat, VL)\n",
    "    \"\"\"This function returns the AC Nodal basis at point Xhat=[xhat,yhat]\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: coordinate of element E as 4x2 array\n",
    "    (xhat,yhat): is the coordinate at reference element [-1,1]^2\n",
    "    VL: computed in ConstraintMat(coord_E) function\n",
    "    \n",
    "    Output:\n",
    "    -------\n",
    "    Nhat: the nodal basis computed at (xhat,yhat)\n",
    "    Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    local numbering is as follow \n",
    "     v7     v8\n",
    "    3---------4\n",
    "  v6|         |v4\n",
    "    |         |\n",
    "  v5|         |v3\n",
    "    1---------2\n",
    "     v1     v2\n",
    "    \"\"\"\n",
    "    nl = GetNormal(coord_E, \"left\")\n",
    "    nr = GetNormal(coord_E, \"right\")\n",
    "    nb = GetNormal(coord_E, \"bottom\")\n",
    "    nt = GetNormal(coord_E, \"top\")\n",
    "    normals = [nb nb nr nr nl nl nt nt]\n",
    "    node1 = [-1.;-1.]\n",
    "    node2 = [1.;-1.]\n",
    "    node3 = [-1.;1.]\n",
    "    node4 = [1.;1.]\n",
    "    nodes = [node1 node2 node2 node4 node1 node3 node3 node4]\n",
    "    \n",
    "    # vondermonde matrix, V_ij = phi_j(x_i).n_i\n",
    "    VM = zeros(8,8)\n",
    "    for i=1:8\n",
    "        for j=1:8\n",
    "            # original prime basis with dim=10\n",
    "            P = PrimeBasis(coord_E, [nodes[1,i]], [nodes[2,i]])\n",
    "            # reduced prime basis with dim=8\n",
    "            PP = P * VL\n",
    "            VM[i,j] = dot(PP[:,j],normals[:,i])\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    # create nodal basis from vondermonde (VM) and VL\n",
    "    P = PrimeBasis(coord_E, xhat, yhat)\n",
    "    invVM = inv(VM)\n",
    "    # nodal basis function\n",
    "    Nhat = P * VL * invVM\n",
    "\n",
    "    return Nhat\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coord_E = [0. 0.;1. 0.;0. 1.;1. 1.]\n",
    "L, VL = ConstraintMat(coord_E, \"QR\")\n",
    "\n",
    "meshgrid(x, y) = (repeat(x, outer=length(y)), repeat(y, inner=length(x)))\n",
    "n = 10\n",
    "xhat, yhat = meshgrid(LinRange(0,1,n), LinRange(0,1,n))\n",
    "N = GetACNodalBasis(coord_E, xhat, yhat, VL)\n",
    "# which N you want to plot\n",
    "base = 8\n",
    "u = N[1:n*n,base]\n",
    "v = N[n*n+1:2*n*n,base]\n",
    "scale = 0.2\n",
    "quiver(xhat, yhat, quiver=(scale*u, scale*v))"
   ]
  },
  {
   "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",
      "TestNodalBasisUniform | \u001b[32m  32  \u001b[39m\u001b[36m   32\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestNodalBasisUniform\", Any[], 32, false, false)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestNodalBasisUniform\" begin\n",
    "\n",
    "    coord_E = [0. 0.;.5 0.;0. .5;.5 .5]\n",
    "    L, VL = ConstraintMat(coord_E, \"QR\")\n",
    "    nl = GetNormal(coord_E, \"left\")\n",
    "    nr = GetNormal(coord_E, \"right\")\n",
    "    nb = GetNormal(coord_E, \"bottom\")\n",
    "    nt = GetNormal(coord_E, \"top\")\n",
    "    \n",
    "    err = 1e-14\n",
    "    \n",
    "    # check node 1\n",
    "    Nhat = GetACNodalBasis(coord_E,[-1], [-1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v1.nb=1 and v5.nl=1\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 0.;atol=err)\n",
    "    \n",
    "    # check node 2\n",
    "    Nhat = GetACNodalBasis(coord_E,[1], [-1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v2.nb=1 and v3.nr=1\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 0.;atol=err)\n",
    "    \n",
    "    # check node 3\n",
    "    Nhat = GetACNodalBasis(coord_E,[-1], [1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v6.nl=1 and v7.nt=1\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 0.;atol=err)\n",
    "    \n",
    "    # check node 4\n",
    "    Nhat = GetACNodalBasis(coord_E,[1], [1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v8.nt=1 and v4.nr=1\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 0.;atol=err)\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "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",
      "TestNodalBasisNonUniform | \u001b[32m  32  \u001b[39m\u001b[36m   32\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestNodalBasisNonUniform\", Any[], 32, false, false)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestNodalBasisNonUniform\" begin\n",
    "\n",
    "    coord_E = [0. 0.;1. 0.;.25 .5;.75 .75]\n",
    "    L, VL = ConstraintMat(coord_E, \"QR\")\n",
    "    nl = GetNormal(coord_E, \"left\")\n",
    "    nr = GetNormal(coord_E, \"right\")\n",
    "    nb = GetNormal(coord_E, \"bottom\")\n",
    "    nt = GetNormal(coord_E, \"top\")\n",
    "    \n",
    "    err = 1e-14\n",
    "    \n",
    "    # check node 1\n",
    "    Nhat = GetACNodalBasis(coord_E,[-1], [-1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v1.nb=1 and v5.nl=1\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 0.;atol=err)\n",
    "    \n",
    "    # check node 2\n",
    "    Nhat = GetACNodalBasis(coord_E,[1], [-1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v2.nb=1 and v3.nr=1\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 0.;atol=err)\n",
    "    \n",
    "    # check node 3\n",
    "    Nhat = GetACNodalBasis(coord_E,[-1], [1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v6.nl=1 and v7.nt=1\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 0.;atol=err)\n",
    "    \n",
    "    # check node 4\n",
    "    Nhat = GetACNodalBasis(coord_E,[1], [1], VL)\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v8.nt=1 and v4.nr=1\n",
    "    @test isapprox(dot(Nhat[:,4],nr), 1.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,8],nt), 1.;atol=err)\n",
    "    # check other nodes\n",
    "    @test isapprox(dot(Nhat[:,1],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,2],nb), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,3],nr), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,5],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,6],nl), 0.;atol=err)\n",
    "    @test isapprox(dot(Nhat[:,7],nt), 0.;atol=err)\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The following is based on AC paper we implmented previously"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetACNodalBasis2 (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function BilinearMap2(coord_E, xhat, yhat)\n",
    "    \"\"\" \n",
    "    This function maps [xhat,yhat] in Ehat=[-1,1]^2 \n",
    "    to (x,y) in E.\n",
    "    Written based on eq. (3.3), (3.4) of AC paper 2016\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: coordinates of quadrilateral E .\n",
    "    coord_E is 4x2 array\n",
    "    coord_E = [[x1,y1],\n",
    "               [x2,y2],\n",
    "               [x3,y3],\n",
    "               [x4,y4]] with vertices numbering\n",
    "    3----4\n",
    "    |    |\n",
    "    1----2\n",
    "    [xhat, yhat] in Ehat\n",
    "    Output:\n",
    "    ------\n",
    "    [x;y]: mapped vector in E.\n",
    "    DF_E: Jacobian matrix\n",
    "    J_E: det(DF_E)\n",
    "    later we transform vector vhat to v by eq. (3.4)\n",
    "    v(x) = P_E(vhat)(x) = (DF_E/J_E)*vhat(xhat)\n",
    "    \"\"\"\n",
    "    m = length(xhat)\n",
    "    P = 0.25 * [(1 .- xhat).*(1 .- yhat) (1 .+ xhat).*(1 .- yhat) (1 .- xhat).*(1 .+ yhat) (1 .+ xhat).*(1 .+ yhat)]\n",
    "    P_E = P * coord_E\n",
    "    # x=F_E(xhat)\n",
    "    X = P_E\n",
    "    x = X[:, 1]\n",
    "    y = X[:, 2]\n",
    "    # gradient of P, 1st row = dP/dxhat, 2nd row=dP/dyhat\n",
    "    # GradP_(2m x 4)\n",
    "    GradP = 0.25 * [-(1 .- yhat) (1 .- yhat) -(1 .+ yhat) (1 .+ yhat);\n",
    "                    -(1 .- xhat) -(1 .+ xhat) (1 .- xhat) (1 .+ xhat)]\n",
    "\n",
    "    # JT= [[dx/dxhat, dy/dxhat],\n",
    "    #      [dx/dyhat, dy/dyhat]] (2m x 2)\n",
    "    JT = GradP * coord_E\n",
    "    dxdxhat = JT[1:m,1]\n",
    "    dydxhat = JT[1:m,2]\n",
    "    dxdyhat = JT[m+1:2*m,1]\n",
    "    dydyhat = JT[m+1:2*m,2]\n",
    "    J_E = dxdxhat .* dydyhat .- dydxhat .* dxdyhat\n",
    "    \n",
    "    DF_E = [dxdxhat dxdyhat;dydxhat dydyhat]\n",
    "    \n",
    "    return x, y, DF_E, J_E\n",
    "end\n",
    "\n",
    "\n",
    "function PrimeBasis2(coord_E, xhat, yhat)\n",
    "\n",
    "    # original supplements in AC paper\n",
    "    # sigma_hat_1 = curl((1-xhat^2)*yhat)\n",
    "    # sigma_hat_2 = curl((1-yhat^2)*xhat)\n",
    "    sghat1 = [(1 .- xhat.^2) (2*xhat .* yhat)]\n",
    "    sghat2 = [(-2*xhat .* yhat) (yhat.^2 .- 1)]\n",
    "\n",
    "    # (x,y) are in E\n",
    "    x, y, DF_E, J_E = BilinearMap2(coord_E, xhat, yhat)\n",
    "\n",
    "    m = length(xhat)\n",
    "    # we have 8 basis including sg1 and sg2\n",
    "    P = zeros(2*m,8)\n",
    "    P[1:m,1] = ones(m)\n",
    "    P[1:m,2] = x\n",
    "    P[1:m,3] = y\n",
    "    P[m+1:2*m,4] = ones(m)\n",
    "    P[m+1:2*m,5] = x\n",
    "    P[m+1:2*m,6] = y\n",
    "    for i=1:m\n",
    "    P[i,7] = dot(DF_E[i,:], sghat1[i,:]) / J_E[i]\n",
    "    P[i+m,7] = dot(DF_E[i+m,:], sghat1[i,:]) / J_E[i]\n",
    "    end\n",
    "    \n",
    "    for i=1:m\n",
    "    P[i,8] = dot(DF_E[i,:], sghat2[i,:]) / J_E[i]\n",
    "    P[i+m,8] = dot(DF_E[i+m,:], sghat2[i,:]) / J_E[i]\n",
    "    end\n",
    "    \n",
    "    return P\n",
    "end\n",
    "\n",
    "\n",
    "function GetNormal2(coord_E, xhat, yhat)\n",
    "    \n",
    "\n",
    "    x, y, DF_E, J_E = BilinearMap2(coord_E, xhat, yhat)\n",
    "    \n",
    "    dxdxhat = DF_E[1,1]\n",
    "    dydxhat = DF_E[2,1]\n",
    "    length1 = sqrt(dxdxhat*dxdxhat + dydxhat*dydxhat)\n",
    "\n",
    "    dxdyhat = DF_E[1,2]\n",
    "    dydyhat = DF_E[2,2]\n",
    "    length2 = sqrt(dxdyhat*dxdyhat + dydyhat*dydyhat)\n",
    "\n",
    "    if (xhat[1] == -1. && -1. < yhat[1] < 1.)\n",
    "        # left edge, (0,0,1)x(dxdyhat,dydyhat,0)\n",
    "        n = [-dydyhat; dxdyhat]/length2\n",
    "\n",
    "    elseif (xhat[1] == 1. && -1. < yhat[1] < 1.)\n",
    "        # right edge, (0,0,-1)x(dxdyhat,dydyhat,0)\n",
    "        n = [dydyhat; -dxdyhat]/length2\n",
    "\n",
    "    elseif (yhat[1] == -1. && -1. < xhat[1] < 1.)\n",
    "        # bottom edge, (0,0,-1)x(dxdxhat,dydxhat,0)\n",
    "        n = [dydxhat; -dxdxhat]/length1\n",
    "\n",
    "    elseif (yhat[1] == 1. && -1. < xhat[1] < 1.)\n",
    "        # top edge, (0,0,1)x(dxdxhat,dydxhat,0)\n",
    "        n = [-dydxhat; dxdxhat]/length1\n",
    "\n",
    "    else\n",
    "        println(\"Error! Enter the Xhat=[xhat, yhat] on the edge of Ehat\")\n",
    "    \n",
    "    end\n",
    "\n",
    "    return n, x, y\n",
    "end\n",
    "\n",
    "\n",
    "function VondermondeMat2(coord_E)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: is the coordinate of vertices of element.\n",
    "    Note\n",
    "    3---4\n",
    "    |   |\n",
    "    1---2\n",
    "    Output:\n",
    "    ------\n",
    "    VM: the 8x8 vondermonde matrix\n",
    "    \"\"\"\n",
    "    nl, x, y = GetNormal2(coord_E, [-1.], [0.])\n",
    "    nr, x, y = GetNormal2(coord_E, [1.], [0.])\n",
    "    nb, x, y = GetNormal2(coord_E, [0.], [-1.])\n",
    "    nt, x, y = GetNormal2(coord_E, [0.], [1.])\n",
    "    normals = [nb nb nr nr nl nl nt nt]\n",
    "    node1 = [-1.;-1.]\n",
    "    node2 = [1.;-1.]\n",
    "    node3 = [-1.;1.]\n",
    "    node4 = [1.;1.]\n",
    "    nodes = [node1 node2 node2 node4 node1 node3 node3 node4]\n",
    "    # vondermonde matrix, V_ij = phi_j(x_i).n_i\n",
    "    VM = zeros(8,8)\n",
    "\n",
    "    for i=1:8\n",
    "        for j=1:8\n",
    "            P = PrimeBasis2(coord_E, [nodes[1,i]], [nodes[2,i]])\n",
    "            VM[i,j] = dot(P[:,j],normals[:,i])\n",
    "        end\n",
    "    end\n",
    "\n",
    "    return VM\n",
    "end\n",
    "\n",
    "\n",
    "function GetACNodalBasis2(coord_E, xhat, yhat)\n",
    "    \"\"\"This function returns the AC Nodal basis at point Xhat=[xhat,yhat]\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: coordinate of element E as 4x2 array\n",
    "    Xhat: is the coordinate at reference element [-1,1]^2\n",
    "\n",
    "    Output:\n",
    "    -------\n",
    "    Nhat: the nodal basis computed at Xhat=[xhat,yhat]\n",
    "    shape (2,8) as\n",
    "    Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    local numbering is as follow \n",
    "     v7     v8\n",
    "    3---------4\n",
    "  v6|         |v4\n",
    "    |         |\n",
    "  v5|         |v3\n",
    "    1---------2\n",
    "     v1     v2\n",
    "    \"\"\"\n",
    "    VM = VondermondeMat2(coord_E)\n",
    "    P = PrimeBasis2(coord_E, xhat, yhat)\n",
    "    invVM = inv(VM)\n",
    "    Nhat = P * invVM\n",
    "\n",
    "    return Nhat\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coord_E = [0. 0.;1. 0.;0. 1.;1. 1.]\n",
    "meshgrid(x, y) = (repeat(x, outer=length(y)), repeat(y, inner=length(x)))\n",
    "n = 10\n",
    "xhat, yhat = meshgrid(LinRange(0,1,n), LinRange(0,1,n))\n",
    "N2 = GetACNodalBasis2(coord_E, xhat, yhat)\n",
    "# which N you want to plot\n",
    "base = 8\n",
    "u = N2[1:n*n,base]\n",
    "v = N2[n*n+1:2*n*n,base]\n",
    "scale = 0.2\n",
    "quiver(xhat, yhat, quiver=(scale*u, scale*v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "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",
      "TestNodalBasisUniform | \u001b[32m  32  \u001b[39m\u001b[36m   32\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Test.DefaultTestSet(\"TestNodalBasisUniform\", Any[], 32, false)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@testset \"TestNodalBasisUniform\" begin\n",
    "\n",
    "    coord_E = [0. 0.;\n",
    "               .5 0.;\n",
    "               0. .5;\n",
    "               .5 .5]\n",
    "    \n",
    "    nl, x, y = GetNormal2(coord_E, [-1.], [0.])\n",
    "    nr, x, y = GetNormal2(coord_E, [1.], [0.])\n",
    "    nb, x, y = GetNormal2(coord_E, [0.], [-1.])\n",
    "    nt, x, y = GetNormal2(coord_E, [0.], [1.])\n",
    "    # check node 1\n",
    "    Nhat = GetACNodalBasis2(coord_E,[-1], [-1])\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v1.nb=1 and v5.nl=1\n",
    "    @test dot(Nhat[:,1],nb) == 1.\n",
    "    @test dot(Nhat[:,5],nl) == 1.\n",
    "    # check other nodes\n",
    "    @test dot(Nhat[:,2],nb) == 0.\n",
    "    @test dot(Nhat[:,3],nr) == 0.\n",
    "    @test dot(Nhat[:,4],nr) == 0.\n",
    "    @test dot(Nhat[:,6],nl) == 0.\n",
    "    @test dot(Nhat[:,7],nt) == 0.\n",
    "    @test dot(Nhat[:,8],nt) == 0.\n",
    "\n",
    "    # check node 2\n",
    "    Nhat = GetACNodalBasis2(coord_E,[1], [-1])\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v2.nb=1 and v3.nr=1\n",
    "    @test dot(Nhat[:,2],nb) == 1.\n",
    "    @test dot(Nhat[:,3],nr) == 1.\n",
    "    # check other nodes\n",
    "    @test dot(Nhat[:,1],nb) == 0.\n",
    "    @test dot(Nhat[:,4],nr) == 0.\n",
    "    @test dot(Nhat[:,5],nl) == 0.\n",
    "    @test dot(Nhat[:,6],nl) == 0.\n",
    "    @test dot(Nhat[:,7],nt) == 0.\n",
    "    @test dot(Nhat[:,8],nt) == 0.\n",
    "\n",
    "    # check node 3\n",
    "    Nhat = GetACNodalBasis2(coord_E,[-1], [1])\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v6.nl=1 and v7.nt=1\n",
    "    @test dot(Nhat[:,6],nl) == 1.\n",
    "    @test dot(Nhat[:,7],nt) == 1.\n",
    "    # check other nodes\n",
    "    @test dot(Nhat[:,1],nb) == 0.\n",
    "    @test dot(Nhat[:,2],nb) == 0.\n",
    "    @test dot(Nhat[:,3],nr) == 0.\n",
    "    @test dot(Nhat[:,4],nr) == 0.\n",
    "    @test dot(Nhat[:,5],nl) == 0.\n",
    "    @test dot(Nhat[:,8],nt) == 0.\n",
    "\n",
    "    # check node 4\n",
    "    Nhat = GetACNodalBasis2(coord_E,[1], [1])\n",
    "    # Note: Nhat = [v1,v2,v3,v4,v5,v6,v7,v8]\n",
    "    # check v8.nt=1 and v4.nr=1\n",
    "    @test dot(Nhat[:,4],nr) == 1.\n",
    "    @test dot(Nhat[:,8],nt) == 1.\n",
    "    # check other nodes\n",
    "    @test dot(Nhat[:,1],nb) == 0.\n",
    "    @test dot(Nhat[:,2],nb) == 0.\n",
    "    @test dot(Nhat[:,3],nr) == 0.\n",
    "    @test dot(Nhat[:,5],nl) == 0.\n",
    "    @test dot(Nhat[:,6],nl) == 0.\n",
    "    @test dot(Nhat[:,7],nt) == 0.\n",
    "\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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