{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solid Mechanics Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# dependencies\n",
    "using LFAToolkit\n",
    "using LinearAlgebra\n",
    "using Pkg\n",
    "Pkg.activate(\"./\")\n",
    "Pkg.instantiate()\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "code_folding": [
     20,
     49
    ]
   },
   "outputs": [],
   "source": [
    "# setup\n",
    "mesh = Mesh3D(1.0, 1.0, 1.0)\n",
    "finep = 3\n",
    "coarsep = 2\n",
    "numbercomponents = 1\n",
    "dimension = 3\n",
    "finebasis = TensorH1LagrangeBasis(finep, finep, numbercomponents, dimension)\n",
    "coarsebasis = TensorH1LagrangeBasis(coarsep, finep, numbercomponents, dimension)\n",
    "ctofbasis = TensorH1LagrangeBasis(coarsep, finep, numbercomponents, dimension, lagrangequadrature = true)\n",
    "\n",
    "# constants\n",
    "e = 1E6                     # Young's modulus\n",
    "ν = 0.3                     # Poisson's ratio\n",
    "K = e/(3*(1 - 2*ν))         # bulk modulus\n",
    "λ = e*ν/((1 + ν)*(1 - 2*ν)) # Lamé parameters\n",
    "μ = e/(2*(1 + ν))\n",
    "\n",
    "# state\n",
    "gradu = [1; 2; 3]*ones(1, 3);\n",
    "\n",
    "function neohookeanweakform(deltadu::Array{Float64}, w::Array{Float64})\n",
    "    # dP = dF S + F dS\n",
    "\n",
    "    # deformation gradient\n",
    "    F = gradu + I\n",
    "    J = det(F)\n",
    "    # Green-Lagrange strain tensor\n",
    "    E = (gradu*gradu' + gradu'*gradu)/2\n",
    "    # right Cauchy-Green tensor\n",
    "    C = 2*E + I\n",
    "    C_inv = C^-1\n",
    "    # second Piola-Kirchhoff\n",
    "    S = λ*log(J)*C_inv + 2*μ*C_inv*E\n",
    "\n",
    "    # delta du\n",
    "    deltadu = deltadu'\n",
    "    # dF\n",
    "    dF = deltadu + I\n",
    "    # deltaE\n",
    "    deltaE = (deltadu*deltadu' + deltadu'*deltadu)/2\n",
    "    # dS\n",
    "    dS = λ*sum(C_inv.*deltaE)*C_inv + 2*(μ - λ*log(J))*C_inv*deltaE*C_inv\n",
    "    # dP\n",
    "    dP = dF*S + F*dS\n",
    "\n",
    "    return [dP']\n",
    "end\n",
    "\n",
    "# linearized Neo-Hookean operators\n",
    "function makeoperator(basis::TensorBasis)\n",
    "    inputs = [\n",
    "        OperatorField(basis, [EvaluationMode.gradient], \"gradent of deformation\"),\n",
    "        OperatorField(basis, [EvaluationMode.quadratureweights], \"quadrature weights\"),\n",
    "    ]\n",
    "    outputs = [\n",
    "        OperatorField(\n",
    "            basis,\n",
    "            [EvaluationMode.gradient],\n",
    "            \"test function gradient of deformation\",\n",
    "        ),\n",
    "    ]\n",
    "    return Operator(neohookeanweakform, mesh, inputs, outputs)\n",
    "end\n",
    "fineoperator = makeoperator(finebasis)\n",
    "coarseoperator = makeoperator(coarsebasis)\n",
    "\n",
    "# Chebyshev smoother\n",
    "chebyshev = Chebyshev(fineoperator)\n",
    "\n",
    "# p-multigrid preconditioner\n",
    "multigrid =\n",
    "    PMultigrid(fineoperator, coarseoperator, chebyshev, [ctofbasis, ctofbasis, ctofbasis])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "code_folding": [
     8,
     41
    ]
   },
   "outputs": [],
   "source": [
    "# full operator symbols\n",
    "numberruns = 250\n",
    "maxeigenvalue = 0\n",
    "θ_min = -π/2\n",
    "θ_max = 3π/2\n",
    "\n",
    "# compute and plot smoothing factor\n",
    "# -- 1D --\n",
    "if dimension == 1\n",
    "    # setup\n",
    "    ω = [0.636]\n",
    "    v = [1, 1]\n",
    "    maxeigenvalues = zeros(numberruns)\n",
    "\n",
    "    # compute\n",
    "    for i in 1:numberruns\n",
    "        θ = [θ_min + (θ_max - θ_min)*i/numberruns]\n",
    "        if abs(θ[1]  % 2π) >  π/128\n",
    "            A = computesymbols(multigrid, ω, v, θ)\n",
    "            eigenvalues = [abs(val) for val in eigvals(A)]\n",
    "            maxeigenvalues[i] = max(eigenvalues...)\n",
    "            maxeigenvalue = max(maxeigenvalue, maxeigenvalues[i])\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # plot\n",
    "    println(\"max eigenvalue: \", maxeigenvalue)\n",
    "    xrange = θ_min/π:(θ_max - θ_min)/π/(numberruns-1):θ_max/π\n",
    "    plot(\n",
    "        xrange,\n",
    "        xlabel=\"θ/π\",\n",
    "        xtickfont=font(12, \"Courier\"),\n",
    "        maxeigenvalues,\n",
    "        ytickfont=font(12, \"Courier\"),\n",
    "        ylabel=\"spectral radius\",\n",
    "        linewidth=3,\n",
    "        legend=:none,\n",
    "        title=\"P-Multigrid Error Symbol\"\n",
    "    )\n",
    "    ylims!(0.0, max(maxeigenvalues...) * 1.1)\n",
    "# -- 2D --\n",
    "elseif dimension == 2\n",
    "    # setup\n",
    "    ω = [0.636]\n",
    "    v = [1, 1]\n",
    "    maxeigenvalues = zeros(numberruns, numberruns)\n",
    "\n",
    "    # compute\n",
    "    for i in 1:numberruns, j in 1:numberruns\n",
    "        θ = [\n",
    "            θ_min + (θ_max - θ_min)*i/numberruns,\n",
    "            θ_min + (θ_max - θ_min)*j/numberruns\n",
    "        ]\n",
    "        if sqrt(abs(θ[1] % 2π)^2 + abs(θ[2] % 2π)^2) > π/128\n",
    "            A = computesymbols(multigrid, ω, v, θ)\n",
    "            eigenvalues = [abs(val) for val in eigvals(A)]\n",
    "            maxeigenvalues[i, j] = max(eigenvalues...)\n",
    "            maxeigenvalue = max(maxeigenvalue, maxeigenvalues[i, j])\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # plot\n",
    "    println(\"max eigenvalue: \", maxeigenvalue)\n",
    "    xrange = θ_min/π:(θ_max - θ_min)/π/(numberruns-1):θ_max/π\n",
    "    heatmap(\n",
    "        xrange,\n",
    "        xlabel=\"θ/π\",\n",
    "        xtickfont=font(12, \"Courier\"),\n",
    "        xrange,\n",
    "        ylabel=\"θ/π\",\n",
    "        ytickfont=font(12, \"Courier\"),\n",
    "        maxeigenvalues,\n",
    "        title=\"P-Multigrid Error Symbol\",\n",
    "        transpose=true,\n",
    "        aspect_ratio=:equal\n",
    "    )\n",
    "    xlims!(θ_min/π, θ_max/π)\n",
    "    ylims!(θ_min/π, θ_max/π)\n",
    "end"
   ]
  }
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