A FEniCS Tutorial - FEniCS Project

alpha = 3; beta = 1.2
u0 = Expression(’1 + x[0]*x[0] + alpha*x[1]*x[1] + beta*t’,
{’alpha’: alpha, ’beta’: beta})
u0.t = 0
Thisfunctionexpressionhasthecomponentsofxasindependentvariables, while
alpha, beta, and t are parameters. The parameters can either be set through a
dictionaryatconstructiontime,asdemonstratedforalphaandbeta,oranytime
through attributes in the function object, as shown for the t parameter.
The essential boundary conditions, along the whole boundary in this case,
are set in the usual way,
def boundary(x, on_boundary): # define the Dirichlet boundary
return on_boundary
bc = DirichletBC(V, u0, boundary)
We shall use u for the unknown u at the new time level and u_1 for u at the
previous time level. The initial value of u_1, implied by the initial condition on
u, can be computed by either projecting or interpolating I. The I(x,y) function
is available in the program through u0, as long as u0.t is zero. We can then
do
u_1 = interpolate(u0, V)
# or
u_1 = project(u0, V)
Note that we could, as an equivalent alternative to using project, define a 0 and
L 0 as we did in Section 1.9 and form the associated variational problem. To
actually recover the exact solution (76) to machine precision, it is important not
to compute the discrete initial condition by projecting I, but by interpolating I
so that the nodal values are exact at t = 0 (projection results in approximative
values at the nodes).
The definition of a and L goes as follows:
dt = 0.3
# time step
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(beta - 2 - 2*alpha)
a = u*v*dx + dt*inner(nabla_grad(u), nabla_grad(v))*dx
L = (u_1 + dt*f)*v*dx
A = assemble(a)
# assemble only once, before the time stepping
Finally, we perform the time stepping in a loop:
u = Function(V)
T = 2
t = dt
# the unknown at a new time level
# total simulation time
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