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Lars Hubatsch
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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Standard Cahn-Hilliard Example that runs with fenics 2019.1\n",
"# Examples can be found at https://bitbucket.org/fenics-project/dolfin/src/master/python/demo/documented/cahn-hilliard/demo_cahn-hilliard.py.rst#rst-header-id1\n",
"# The resulting .pvd file can be opened using default settings in ParaView\n",
"\n",
"import random\n",
"from dolfin import *\n",
"# Class representing the intial conditions\n",
"class InitialConditions(UserExpression):\n",
" def __init__(self, **kwargs):\n",
" random.seed(2 + MPI.rank(MPI.comm_world))\n",
" super().__init__(**kwargs)\n",
" def eval(self, values, x):\n",
" values[0] = 0.63 + 0.02*(0.5 - random.random())\n",
" values[1] = 0.0\n",
" def value_shape(self):\n",
" return (2,)\n",
"# Class for interfacing with the Newton solver\n",
"class CahnHilliardEquation(NonlinearProblem):\n",
" def __init__(self, a, L):\n",
" NonlinearProblem.__init__(self)\n",
" self.L = L\n",
" self.a = a\n",
" def F(self, b, x):\n",
" assemble(self.L, tensor=b)\n",
" def J(self, A, x):\n",
" assemble(self.a, tensor=A)\n",
"# Model parameters\n",
"lmbda = 1.0e-02 # surface parameter\n",
"dt = 5.0e-14 # time step\n",
"theta = 0.5 # time stepping family, e.g. theta=1 -> backward Euler, theta=0.5 -> Crank-Nicolson\n",
"kBTG0 = 1\n",
"X = 2.5\n",
"kappa = 2\n",
"# Form compiler options\n",
"parameters[\"form_compiler\"][\"optimize\"] = True\n",
"parameters[\"form_compiler\"][\"cpp_optimize\"] = True\n",
"# Create mesh and build function space\n",
"mesh = UnitSquareMesh.create(96, 96, CellType.Type.quadrilateral)\n",
"P1 = FiniteElement(\"Lagrange\", mesh.ufl_cell(), 1)\n",
"ME = FunctionSpace(mesh, P1*P1)\n",
"# Define trial and test functions\n",
"du = TrialFunction(ME)\n",
"q, v = TestFunctions(ME)\n",
"# Define functions\n",
"u = Function(ME) # current solution\n",
"u0 = Function(ME) # solution from previous converged step\n",
"\n",
"# Split mixed functions\n",
"dc, dmu = split(du)\n",
"c, mu = split(u)\n",
"c0, mu0 = split(u0)\n",
"# Create intial conditions and interpolate\n",
"u_init = InitialConditions(degree=1)\n",
"u.interpolate(u_init)\n",
"u0.interpolate(u_init)\n",
"# Compute the chemical potential df/dc\n",
"c = variable(c)\n",
"f = 100*c**2*(1-c)**2\n",
"dfdc = diff(f, c)\n",
"# mu_(n+theta)\n",
"mu_mid = (1.0-theta)*mu0 + theta*mu\n",
"c_mid = (1.0-theta)*c0 + theta*c\n",
"# Weak statement of the equations\n",
"# L0 = c*q*dx - c0*q*dx + dt*dot(grad(mu_mid), grad(q))*dx\n",
"# L0 needs to include the next time step via theta, while L1 is indendent of \n",
"# time and therefore doesn't need discretization in time, check this is true with Chris!\n",
"# In particular the c_mid in L0!\n",
"L0 = (c*q*dx - c0*q*dx - dt*((1+2*X*c_mid)*mu_mid- 2*X*dot(grad(mu_mid), grad(mu_mid))\n",
" +2*X*(2*c_mid*dot(grad(c_mid), grad(c_mid)) + c_mid**2*mu_mid) \n",
" -kappa*(dot(grad(c_mid), grad(mu_mid)) - 2*c_mid*dot(grad(c_mid), grad(mu_mid))))*q*dx\n",
" -kappa*(c_mid**2*dot(grad(mu_mid), grad(q))+2*c_mid*q*dot(grad(mu_mid), grad(c_mid)))*dx)\n",
"# dt*((1+2*X*c_mid)*mu_mid - 2*X*dot(grad(mu_mid), grad(mu_mid)) + 2*X*(2*c_mid*dot(grad(c_mid), grad(c_mid)) + c_mid**2*mu_mid)\n",
"# -kappa*(dot(grad(c_mid), grad(mu_mid)) - 2*c_mid*dot(grad(c_mid), grad(mu_mid)))*q\n",
"# -kappa*(c_mid**2*dot(grad(mu_mid), grad(q))+2*c_mid*q*dot(grad(mu_mid), grad(c_mid))))*dx)\n",
"# L1 = mu*v*dx - dfdc*v*dx - lmbda*dot(grad(c), grad(v))*dx\n",
"L1 = mu*v*dx + dot(grad(c), grad(v))*dx\n",
"L = L0 + L1\n",
"# Compute directional derivative about u in the direction of du (Jacobian)\n",
"a = derivative(L, u, du)\n",
"# Create nonlinear problem and Newton solver\n",
"problem = CahnHilliardEquation(a, L)\n",
"solver = NewtonSolver()\n",
"solver.parameters[\"linear_solver\"] = \"lu\"\n",
"solver.parameters[\"convergence_criterion\"] = \"incremental\"\n",
"solver.parameters[\"relative_tolerance\"] = 1e-6\n",
"# Output file\n",
"file = File(\"output.pvd\", \"compressed\")\n",
"\n",
"# Step in time\n",
"t = 0.0\n",
"T = 5*dt\n",
"while (t < T):\n",
" t += dt\n",
" u0.vector()[:] = u.vector()\n",
" solver.solve(problem, u.vector())\n",
" file << (u.split()[0], t)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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