"""This demo program solves the Biharmonic equation,
nabla^4 u(x, y) = f(x, y)
on the unit square with source f given by
f(x, y) = 4 pi^4 sin(pi*x)*sin(pi*y)
and boundary conditions given by
u(x, y) = 0
nabla^2 u(x, y) = 0
using a discontinuous Galerkin formulation (interior penalty method).
"""
# Copyright (C) 2009 Kristian B. Oelgaard
#
# 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 Anders Logg, 2011
#
# First added: 2009-06-26
# Last changed: 2012-11-12
# Begin demo
from dolfin import *
# Optimization options for the form compiler
parameters["form_compiler"]["cpp_optimize"] = True
parameters["form_compiler"]["optimize"] = True
# Create mesh and define function space
mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "CG", 2)
# Define Dirichlet boundary
class DirichletBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary
class Source(Expression):
def eval(self, values, x):
values[0] = 4.0*pi**4*sin(pi*x[0])*sin(pi*x[1])
# Define boundary condition
u0 = Constant(0.0)
bc = DirichletBC(V, u0, DirichletBoundary())
# Define trial and test functions
u = TrialFunction(V)
v = TestFunction(V)
# Define normal component, mesh size and right-hand side
h = CellSize(mesh)
h_avg = (h('+') + h('-'))/2.0
n = FacetNormal(mesh)
f = Source()
# Penalty parameter
alpha = Constant(8.0)
# Define bilinear form
a = inner(div(grad(u)), div(grad(v)))*dx \
- inner(avg(div(grad(u))), jump(grad(v), n))*dS \
- inner(jump(grad(u), n), avg(div(grad(v))))*dS \
+ alpha('+')/h_avg*inner(jump(grad(u),n), jump(grad(v),n))*dS
# Define linear form
L = f*v*dx
# Solve variational problem
u = Function(V)
solve(a == L, u, bc)
# Save solution to file
file = File("biharmonic.pvd")
file << u
# Plot solution
plot(u, interactive=True)