# 5. Determine the list of true (i.e. conforming) essential boundary dofs.
# In this example, the boundary conditions are defined by marking all
# the boundary attributes from the mesh as essential (Dirichlet) and
# converting them to a list of true dofs.
ess_tdof_list = mfem.intArray()
if mesh.bdr_attributes.Size() > 0:
ess_bdr = mfem.intArray([1] * mesh.bdr_attributes.Max())
ess_bdr = mfem.intArray(mesh.bdr_attributes.Max())
ess_bdr.Assign(1)
fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list)
#6. Set up the linear form b(.) which corresponds to the right-hand side of
# the FEM linear system, which in this case is (1,phi_i) where phi_i are
# the basis functions in the finite element fespace.
b = mfem.LinearForm(fespace)
one = mfem.ConstantCoefficient(1.0)
b.AddDomainIntegrator(mfem.DomainLFIntegrator(one))
b.Assemble()
#7. Define the solution vector x as a finite element grid function
# corresponding to fespace. Initialize x with initial guess of zero,
# which satisfies the boundary conditions.
x = mfem.GridFunction(fespace)
x.Assign(0.0)
#8. Set up the bilinear form a(.,.) on the finite element space
# corresponding to the Laplacian operator -Delta, by adding the Diffusion
# domain integrator.
a = mfem.BilinearForm(fespace)
a.AddDomainIntegrator(mfem.DiffusionIntegrator(one))
#9. Assemble the bilinear form and the corresponding linear system,
# applying any necessary transformations such as: eliminating boundary
# conditions, applying conforming constraints for non-conforming AMR,
# static condensation, etc.
gcoeff = gFunc() ucoeff = uFunc_ex(dim) pcoeff = pFunc_ex() x = mfem.BlockVector(block_offsets) rhs = mfem.BlockVector(block_offsets) fform = mfem.LinearForm() fform.Update(R_space, rhs.GetBlock(0), 0) fform.AddDomainIntegrator(mfem.VectorFEDomainLFIntegrator(fcoeff)) fform.AddBoundaryIntegrator(mfem.VectorFEBoundaryFluxLFIntegrator(fnatcoeff)) fform.Assemble() gform = mfem.LinearForm() gform.Update(W_space, rhs.GetBlock(1), 0) gform.AddDomainIntegrator(mfem.DomainLFIntegrator(gcoeff)) gform.Assemble() mVarf = mfem.BilinearForm(R_space) bVarf = mfem.MixedBilinearForm(R_space, W_space) mVarf.AddDomainIntegrator(mfem.VectorFEMassIntegrator(k)) mVarf.Assemble() mVarf.Finalize() M = mVarf.SpMat() bVarf.AddDomainIntegrator(mfem.VectorFEDivergenceIntegrator()) bVarf.Assemble() bVarf.Finalize() B = bVarf.SpMat() B *= -1
fec = mfem.H1_FECollection(order, dim)
fespace = mfem.FiniteElementSpace(mesh, fec)
# 6. As in Example 1p, we set up bilinear and linear forms corresponding to
# the Laplace problem -\Delta u = 1. We don't assemble the discrete
# problem yet, this will be done in the inner loop.
a = mfem.BilinearForm(fespace)
b = mfem.LinearForm(fespace)
one = mfem.ConstantCoefficient(1.0)
bdr = BdrCoefficient()
rhs = RhsCoefficient()
integ = mfem.DiffusionIntegrator(one)
a.AddDomainIntegrator(integ)
b.AddDomainIntegrator(mfem.DomainLFIntegrator(rhs))
# 7. The solution vector x and the associated finite element grid function
# will be maintained over the AMR iterations.
x = mfem.GridFunction(fespace)
# 8. Connect to GLVis.
if visualization:
sout = mfem.socketstream("localhost", 19916)
sout.precision(8)
# 9. As in Example 6, we set up a Zienkiewicz-Zhu estimator that will be
# used to obtain element error indicators. The integrator needs to
# provide the method ComputeElementFlux. The smoothed flux space is a
# vector valued H1 space here.
flux_fespace = mfem.FiniteElementSpace(mesh, fec, sdim)
def run(order=1,
static_cond=False,
meshfile=def_meshfile,
visualization=False):
mesh = mfem.Mesh(meshfile, 1, 1)
dim = mesh.Dimension()
# 3. Refine the mesh to increase the resolution. In this example we do
# 'ref_levels' of uniform refinement. We choose 'ref_levels' to be the
# largest number that gives a final mesh with no more than 50,000
# elements.
ref_levels = int(np.floor(
np.log(50000. / mesh.GetNE()) / np.log(2.) / dim))
for x in range(ref_levels):
mesh.UniformRefinement()
#5. Define a finite element space on the mesh. Here we use vector finite
# elements, i.e. dim copies of a scalar finite element space. The vector
# dimension is specified by the last argument of the FiniteElementSpace
# constructor. For NURBS meshes, we use the (degree elevated) NURBS space
# associated with the mesh nodes.
if order > 0:
fec = mfem.H1_FECollection(order, dim)
elif mesh.GetNodes():
fec = mesh.GetNodes().OwnFEC()
prinr("Using isoparametric FEs: " + str(fec.Name()))
else:
order = 1
fec = mfem.H1_FECollection(order, dim)
fespace = mfem.FiniteElementSpace(mesh, fec)
print('Number of finite element unknowns: ' + str(fespace.GetTrueVSize()))
# 5. Determine the list of true (i.e. conforming) essential boundary dofs.
# In this example, the boundary conditions are defined by marking all
# the boundary attributes from the mesh as essential (Dirichlet) and
# converting them to a list of true dofs.
ess_tdof_list = mfem.intArray()
if mesh.bdr_attributes.Size() > 0:
ess_bdr = mfem.intArray([1] * mesh.bdr_attributes.Max())
ess_bdr = mfem.intArray(mesh.bdr_attributes.Max())
ess_bdr.Assign(1)
fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list)
#6. Set up the linear form b(.) which corresponds to the right-hand side of
# the FEM linear system, which in this case is (1,phi_i) where phi_i are
# the basis functions in the finite element fespace.
b = mfem.LinearForm(fespace)
one = mfem.ConstantCoefficient(1.0)
b.AddDomainIntegrator(mfem.DomainLFIntegrator(one))
b.Assemble()
#7. Define the solution vector x as a finite element grid function
# corresponding to fespace. Initialize x with initial guess of zero,
# which satisfies the boundary conditions.
x = mfem.GridFunction(fespace)
x.Assign(0.0)
#8. Set up the bilinear form a(.,.) on the finite element space
# corresponding to the Laplacian operator -Delta, by adding the Diffusion
# domain integrator.
a = mfem.BilinearForm(fespace)
a.AddDomainIntegrator(mfem.DiffusionIntegrator(one))
#9. Assemble the bilinear form and the corresponding linear system,
# applying any necessary transformations such as: eliminating boundary
# conditions, applying conforming constraints for non-conforming AMR,
# static condensation, etc.
if static_cond: a.EnableStaticCondensation()
a.Assemble()
A = mfem.OperatorPtr()
B = mfem.Vector()
X = mfem.Vector()
a.FormLinearSystem(ess_tdof_list, x, b, A, X, B)
print("Size of linear system: " + str(A.Height()))
# 10. Solve
AA = mfem.OperatorHandle2SparseMatrix(A)
M = mfem.GSSmoother(AA)
mfem.PCG(AA, M, B, X, 1, 200, 1e-12, 0.0)
# 11. Recover the solution as a finite element grid function.
a.RecoverFEMSolution(X, b, x)
# 12. Save the refined mesh and the solution. This output can be viewed later
# using GLVis: "glvis -m refined.mesh -g sol.gf".
mesh.Print('refined.mesh', 8)
x.Save('sol.gf', 8)
#13. Send the solution by socket to a GLVis server.
if (visualization):
sol_sock = mfem.socketstream("localhost", 19916)
sol_sock.precision(8)
sol_sock.send_solution(mesh, x)
def EvalValue(self, x):
l2 = np.sum(x**2)
return 7 * x[0] * x[1] / l2
class analytic_solution(mfem.PyCoefficient):
def EvalValue(self, x):
l2 = np.sum(x**2)
return x[0] * x[1] / l2
b = mfem.LinearForm(fespace)
one = mfem.ConstantCoefficient(1.0)
rhs_coef = analytic_rhs()
sol_coef = analytic_solution()
b.AddDomainIntegrator(mfem.DomainLFIntegrator(rhs_coef))
b.Assemble()
# 6. Define the solution vector x as a finite element grid function
# corresponding to fespace. Initialize x with initial guess of zero.
x = mfem.GridFunction(fespace)
x.Assign(0.0)
# 7. Set up the bilinear form a(.,.) on the finite element space
# corresponding to the Laplacian operator -Delta, by adding the Diffusion
# and Mass domain integrators.
a = mfem.BilinearForm(fespace)
a.AddDomainIntegrator(mfem.DiffusionIntegrator(one))
a.AddDomainIntegrator(mfem.MassIntegrator(one))
# 8. Assemble the linear system, apply conforming constraints, etc.
