A FEniCS Tutorial - FEniCS Project

"""
FEniCS program for the deflection w(x,y) of a membrane:
-Laplace(w) = p = Gaussian function, in a unit circle,
with w = 0 on the boundary.
"""
from dolfin import *
import numpy
# Set pressure function:
T = 10.0 # tension
A = 1.0 # pressure amplitude
R = 0.3 # radius of domain
theta = 0.2
x0 = 0.6*R*cos(theta)
y0 = 0.6*R*sin(theta)
sigma = 0.025
sigma = 50 # large value for verification
n = 40 # approx no of elements in radial direction
mesh = UnitCircle(n)
V = FunctionSpace(mesh, ’Lagrange’, 1)
# Define boundary condition w=0
def boundary(x, on_boundary):
return on_boundary
bc = DirichletBC(V, Constant(0.0), boundary)
# Define variational problem
w = TrialFunction(V)
v = TestFunction(V)
a = inner(nabla_grad(w), nabla_grad(v))*dx
f = Expression(’4*exp(-0.5*(pow((R*x[0] - x0)/sigma, 2)) ’
’ -0.5*(pow((R*x[1] - y0)/sigma, 2)))’,
R=R, x0=x0, y0=y0, sigma=sigma)
L = f*v*dx
# Compute solution
w = Function(V)
problem = LinearVariationalProblem(a, L, w, bc)
solver = LinearVariationalSolver(problem)
solver.parameters[’linear_solver’] = ’cg’
solver.parameters[’preconditioner’] = ’ilu’
solver.solve()
# Plot scaled solution, mesh and pressure
plot(mesh, title=’Mesh over scaled domain’)
plot(w, title=’Scaled deflection’)
f = interpolate(f, V)
plot(f, title=’Scaled pressure’)
# Find maximum real deflection
max_w = w.vector().array().max()
max_D = A*max_w/(8*pi*sigma*T)
print ’Maximum real deflection is’, max_D
# Verification for "flat" pressure (large sigma)
if sigma >= 50:
w_e = Expression("1 - x[0]*x[0] - x[1]*x[1]")
w_e = interpolate(w_e, V)
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