def SetParameters(self, u):
u_alpha_gf = mfem.GridFunction(self.fespace)
u_alpha_gf.SetFromTrueDofs(u)
for i in range(u_alpha_gf.Size()):
u_alpha_gf[i] = self.kappa + self.alpha * u_alpha_gf[i]
self.K = mfem.BilinearForm(self.fespace)
u_coeff = mfem.GridFunctionCoefficient(u_alpha_gf)
self.K.AddDomainIntegrator(mfem.DiffusionIntegrator(u_coeff))
self.K.Assemble()
self.K.FormSystemMatrix(self.ess_tdof_list, self.Kmat)
self.T = None
# 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.SparseMatrix()
B = mfem.Vector()
X = mfem.Vector()
a.FormLinearSystem(ess_tdof_list, x, b, A, X, B)
print("Size of linear system: " + str(A.Size()))
# 10. Solve
M = mfem.GSSmoother(A)
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)
# 4. Define a finite element space on the mesh. The polynomial order is # one (linear) by default, but this can be changed on the command line. fec = mfem.H1_FECollection(order, dim) fespace = mfem.FiniteElementSpace(mesh, fec) # 5. As in Example 1, 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 main loop. a = mfem.BilinearForm(fespace) b = mfem.LinearForm(fespace) one = mfem.ConstantCoefficient(1.0) zero = mfem.ConstantCoefficient(0.0) integ = mfem.DiffusionIntegrator(one) a.AddDomainIntegrator(integ) b.AddDomainIntegrator(mfem.DomainLFIntegrator(one)) # 6. The solution vector x and the associated finite element grid function # will be maintained over the AMR iterations. We initialize it to zero. x = mfem.GridFunction(fespace) x.Assign(0.0) # 7. All boundary attributes will be used for essential (Dirichlet) BC. ess_bdr = intArray(mesh.bdr_attributes.Max()) ess_bdr.Assign(1) # 9. Set up an error estimator. Here we use the Zienkiewicz-Zhu estimator # that uses the ComputeElementFlux method of the DiffusionIntegrator to # recover a smoothed flux (gradient) that is subtracted from the element
# create finite element function space
fec = mfem.H1_FECollection(1, mesh.Dimension()) # H1 order=1
fespace = mfem.FiniteElementSpace(mesh, fec)
#
ess_tdof_list = mfem.intArray()
ess_bdr = mfem.intArray([1] * mesh.bdr_attributes.Size())
fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list)
# constant coefficient
one = mfem.ConstantCoefficient(1.0)
# define Bilinear and Linear operator
a = mfem.BilinearForm(fespace)
a.AddDomainIntegrator(mfem.DiffusionIntegrator(sigma(2)))
a.Assemble()
b = mfem.LinearForm(fespace)
b.AddDomainIntegrator(mfem.DomainLFIntegrator(one))
b.Assemble()
# create gridfunction, which is where the solution vector is stored
x = mfem.GridFunction(fespace)
x.Assign(0.0)
# form linear equation (AX=B)
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()))
# the FEM linear system, which in this case is (f,phi_i) where f=1.0 and # phi_i are the basis functions in the test finite element fespace. one = mfem.ConstantCoefficient(1.0) F = mfem.LinearForm(test_space); F.AddDomainIntegrator(mfem.DomainLFIntegrator(one)) F.Assemble(); # 7. Set up the mixed bilinear form for the primal trial unknowns, B0, # the mixed bilinear form for the interfacial unknowns, Bhat, # the inverse stiffness matrix on the discontinuous test space, Sinv, # and the stiffness matrix on the continuous trial space, S0. ess_bdr = mfem.intArray(mesh.bdr_attributes.Max()) ess_bdr.Assign(1) B0 = mfem.MixedBilinearForm(x0_space,test_space); B0.AddDomainIntegrator(mfem.DiffusionIntegrator(one)) B0.Assemble() B0.EliminateTrialDofs(ess_bdr, x.GetBlock(x0_var), F) B0.Finalize() Bhat = mfem.MixedBilinearForm(xhat_space,test_space) Bhat.AddTraceFaceIntegrator(mfem.TraceJumpIntegrator()) Bhat.Assemble() Bhat.Finalize() Sinv = mfem.BilinearForm(test_space) Sum = mfem.SumIntegrator() Sum.AddIntegrator(mfem.DiffusionIntegrator(one)) Sum.AddIntegrator(mfem.MassIntegrator(one)) Sinv.AddDomainIntegrator(mfem.InverseIntegrator(Sum)) Sinv.Assemble()