fenics_mpi_example.py

# Extended Standard Cahn-Hilliard Example to Binary Flory Huggins 
# as discussed on 28/11/2019
# Example 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
# Runs with fenics 2019.01
# The resulting .pvd file can be opened using default settings in ParaView

import mshr as ms
import numpy as np
import time
import random
from dolfin import *
# Class representing the intial conditions
class InitialConditions(UserExpression):
    def __init__(self, **kwargs):
        random.seed(2 + MPI.rank(MPI.comm_world))
        super().__init__(**kwargs)
    def eval(self, values, x):
        if (x[0] > -0.1) & (x[0] < 0.1) & (x[1] > -0.1) & (x[1] < 0.1) & (x[2] > -0.1) & (x[2] < 0.1):
#             values[0] = 0.63 + 0.02*(0.5 - random.random())
            values[0] = 0.82
        else:
            values[0] = 0.3                
        values[1] = 0.0
#         values[0] = 0.63 + 0.02*(0.5 - random.random())
        values[1] = 0.0
    def value_shape(self):
        return (2,)
# Class for interfacing with the Newton solver
class CahnHilliardEquation(NonlinearProblem):
    def __init__(self, a, L):
        NonlinearProblem.__init__(self)
        self.L = L
        self.a = a
    def F(self, b, x):
        assemble(self.L, tensor=b)
    def J(self, A, x):
        assemble(self.a, tensor=A)
# Model parameters
kappa  = 0.25*10e-04  # surface parameter
dt     = 10.0e-02  # time step
X = 2.2 # Chi Flory Huggins parameter
# time stepping family, e.g.: 
# theta=1 -> backward Euler, theta=0.5 -> Crank-Nicolson
theta  = 0.5      
# Form compiler optionsmai
parameters["form_compiler"]["optimize"]     = True
parameters["form_compiler"]["cpp_optimize"] = True
# Create mesh and build function space
x_mesh = 10*96
y_mesh = 1
# mesh = UnitSquareMesh.create(x_mesh, y_mesh, CellType.Type.quadrilateral)
# mesh = RectangleMesh(Point(0, 0), Point(1, 0.02), x_mesh, y_mesh)
domain = ms.Sphere(Point(0, 0, 0), 1.0)
mesh = ms.generate_mesh(domain, 50)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
ME = FunctionSpace(mesh, P1*P1)
# Define trial and test functions
du    = TrialFunction(ME)
q, v  = TestFunctions(ME)
# Define functions
u   = Function(ME)  # current solution
u0  = Function(ME)  # solution from previous converged step

# Split mixed functions
dc, dmu = split(du)
c,  mu  = split(u)
c0, mu0 = split(u0)
# Create intial conditions and interpolate
u_init = InitialConditions(degree=1)
u.interpolate(u_init)
u0.interpolate(u_init)
# Compute the chemical potential df/dc
c = variable(c)
dfdc = ln(c/(1-c))+(1-2*X*c)
# mu_(n+theta)
mu_mid = (1.0-theta)*mu0 + theta*mu
# Weak statement of the equations
L0 = c*q*dx - c0*q*dx + dt*c*(1-c)*dot(grad(mu_mid), grad(q))*dx
L1 = mu*v*dx - (ln(c/(1-c))+X*(1-2*c))*v*dx - kappa*dot(grad(c), grad(v))*dx 
L = L0 + L1
# Compute directional derivative about u in the direction of du (Jacobian)
a = derivative(L, u, du)
# Create nonlinear problem and Newton solver
problem = CahnHilliardEquation(a, L)
solver = NewtonSolver()
solver.parameters["linear_solver"] = "gmres"
#solver.parameters["preconditioner"] = "hypre_euclid"
solver.parameters["preconditioner"] = "petsc_amg"
#solver.parameters["linear_solver"] = "mumps"
#solver.parameters["preconditioner"] = "ilu"
solver.parameters["convergence_criterion"] = "incremental"
solver.parameters["relative_tolerance"] = 1e-6
# Output file
file = File("output.pvd", "compressed")

# Step in time
t = 0.0
T = 2*dt
ti = time.time()
while (t < T):
    file << (u.split()[0], t)
    t += dt
    u0.vector()[:] = u.vector()
    solver.solve(problem, u.vector())
print(time.time()-ti)