""" This demo program solves a hyperelastic problem. It is implemented
in Python by Johan Hake following the C++ demo by Harish Narayanan"""
# Copyright (C) 2008-2010 Johan Hake and Garth N. Wells
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see .
#
# Modified by Harish Narayanan 2009
# Modified by Anders Logg 2011
#
# First added: 2009-10-11
# Last changed: 2012-11-12
# Begin demo
from dolfin import *
# Optimization options for the form compiler
parameters["form_compiler"]["cpp_optimize"] = True
ffc_options = {"optimize": True, \
"eliminate_zeros": True, \
"precompute_basis_const": True, \
"precompute_ip_const": True}
# Create mesh and define function space
mesh = UnitCubeMesh(24, 16, 16)
V = VectorFunctionSpace(mesh, "Lagrange", 1)
# Mark boundary subdomians
left = CompiledSubDomain("near(x[0], side) && on_boundary", side = 0.0)
right = CompiledSubDomain("near(x[0], side) && on_boundary", side = 1.0)
# Define Dirichlet boundary (x = 0 or x = 1)
c = Expression(("0.0", "0.0", "0.0"))
r = Expression(("scale*0.0",
"scale*(y0 + (x[1] - y0)*cos(theta) - (x[2] - z0)*sin(theta) - x[1])",
"scale*(z0 + (x[1] - y0)*sin(theta) + (x[2] - z0)*cos(theta) - x[2])"),
scale = 0.5, y0 = 0.5, z0 = 0.5, theta = pi/3)
bcl = DirichletBC(V, c, left)
bcr = DirichletBC(V, r, right)
bcs = [bcl, bcr]
# Define functions
du = TrialFunction(V) # Incremental displacement
v = TestFunction(V) # Test function
u = Function(V) # Displacement from previous iteration
B = Constant((0.0, -0.5, 0.0)) # Body force per unit volume
T = Constant((0.1, 0.0, 0.0)) # Traction force on the boundary
# Kinematics
d = u.geometric_dimension()
I = Identity(d) # Identity tensor
F = I + grad(u) # Deformation gradient
C = F.T*F # Right Cauchy-Green tensor
# Invariants of deformation tensors
Ic = tr(C)
J = det(F)
# Elasticity parameters
E, nu = 10.0, 0.3
mu, lmbda = Constant(E/(2*(1 + nu))), Constant(E*nu/((1 + nu)*(1 - 2*nu)))
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2
# Total potential energy
Pi = psi*dx - dot(B, u)*dx - dot(T, u)*ds
# Compute first variation of Pi (directional derivative about u in the direction of v)
F = derivative(Pi, u, v)
# Compute Jacobian of F
J = derivative(F, u, du)
# Solve variational problem
solve(F == 0, u, bcs, J=J,
form_compiler_parameters=ffc_options)
# Save solution in VTK format
file = File("displacement.pvd");
file << u;
# Plot and hold solution
plot(u, mode = "displacement", interactive = True)