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125ff22 Jun 30, 2016
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@tzanio @jakubcerveny
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// MFEM Example 5
//
// Compile with: make ex5
//
// Sample runs: ex5 -m ../data/square-disc.mesh
// ex5 -m ../data/star.mesh
// ex5 -m ../data/beam-tet.mesh
// ex5 -m ../data/beam-hex.mesh
// ex5 -m ../data/escher.mesh
// ex5 -m ../data/fichera.mesh
//
// Description: This example code solves a simple 2D/3D mixed Darcy problem
// corresponding to the saddle point system
// k*u + grad p = f
// - div u = g
// with natural boundary condition -p = <given pressure>.
// Here, we use a given exact solution (u,p) and compute the
// corresponding r.h.s. (f,g). We discretize with Raviart-Thomas
// finite elements (velocity u) and piecewise discontinuous
// polynomials (pressure p).
//
// The example demonstrates the use of the BlockMatrix class, as
// well as the collective saving of several grid functions in a
// VisIt (visit.llnl.gov) visualization format.
//
// We recommend viewing examples 1-4 before viewing this example.
#include "mfem.hpp"
#include <fstream>
#include <iostream>
#include <algorithm>
using namespace std;
using namespace mfem;
// Define the analytical solution and forcing terms / boundary conditions
void uFun_ex(const Vector & x, Vector & u);
double pFun_ex(const Vector & x);
void fFun(const Vector & x, Vector & f);
double gFun(const Vector & x);
double f_natural(const Vector & x);
int main(int argc, char *argv[])
{
StopWatch chrono;
// 1. Parse command-line options.
const char *mesh_file = "../data/star.mesh";
int order = 1;
bool visualization = 1;
OptionsParser args(argc, argv);
args.AddOption(&mesh_file, "-m", "--mesh",
"Mesh file to use.");
args.AddOption(&order, "-o", "--order",
"Finite element order (polynomial degree).");
args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
"--no-visualization",
"Enable or disable GLVis visualization.");
args.Parse();
if (!args.Good())
{
args.PrintUsage(cout);
return 1;
}
args.PrintOptions(cout);
// 2. Read the mesh from the given mesh file. We can handle triangular,
// quadrilateral, tetrahedral, hexahedral, surface and volume meshes with
// the same code.
Mesh *mesh = new Mesh(mesh_file, 1, 1);
int 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 10,000
// elements.
{
int ref_levels =
(int)floor(log(10000./mesh->GetNE())/log(2.)/dim);
for (int l = 0; l < ref_levels; l++)
{
mesh->UniformRefinement();
}
}
// 4. Define a finite element space on the mesh. Here we use the
// Raviart-Thomas finite elements of the specified order.
FiniteElementCollection *hdiv_coll(new RT_FECollection(order, dim));
FiniteElementCollection *l2_coll(new L2_FECollection(order, dim));
FiniteElementSpace *R_space = new FiniteElementSpace(mesh, hdiv_coll);
FiniteElementSpace *W_space = new FiniteElementSpace(mesh, l2_coll);
// 5. Define the BlockStructure of the problem, i.e. define the array of
// offsets for each variable. The last component of the Array is the sum
// of the dimensions of each block.
Array<int> block_offsets(3); // number of variables + 1
block_offsets[0] = 0;
block_offsets[1] = R_space->GetVSize();
block_offsets[2] = W_space->GetVSize();
block_offsets.PartialSum();
std::cout << "***********************************************************\n";
std::cout << "dim(R) = " << block_offsets[1] - block_offsets[0] << "\n";
std::cout << "dim(W) = " << block_offsets[2] - block_offsets[1] << "\n";
std::cout << "dim(R+W) = " << block_offsets.Last() << "\n";
std::cout << "***********************************************************\n";
// 6. Define the coefficients, analytical solution, and rhs of the PDE.
ConstantCoefficient k(1.0);
VectorFunctionCoefficient fcoeff(dim, fFun);
FunctionCoefficient fnatcoeff(f_natural);
FunctionCoefficient gcoeff(gFun);
VectorFunctionCoefficient ucoeff(dim, uFun_ex);
FunctionCoefficient pcoeff(pFun_ex);
// 7. Allocate memory (x, rhs) for the analytical solution and the right hand
// side. Define the GridFunction u,p for the finite element solution and
// linear forms fform and gform for the right hand side. The data
// allocated by x and rhs are passed as a reference to the grid functions
// (u,p) and the linear forms (fform, gform).
BlockVector x(block_offsets), rhs(block_offsets);
LinearForm *fform(new LinearForm);
fform->Update(R_space, rhs.GetBlock(0), 0);
fform->AddDomainIntegrator(new VectorFEDomainLFIntegrator(fcoeff));
fform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(fnatcoeff));
fform->Assemble();
LinearForm *gform(new LinearForm);
gform->Update(W_space, rhs.GetBlock(1), 0);
gform->AddDomainIntegrator(new DomainLFIntegrator(gcoeff));
gform->Assemble();
// 8. Assemble the finite element matrices for the Darcy operator
//
// D = [ M B^T ]
// [ B 0 ]
// where:
//
// M = \int_\Omega k u_h \cdot v_h d\Omega u_h, v_h \in R_h
// B = -\int_\Omega \div u_h q_h d\Omega u_h \in R_h, q_h \in W_h
BilinearForm *mVarf(new BilinearForm(R_space));
MixedBilinearForm *bVarf(new MixedBilinearForm(R_space, W_space));
mVarf->AddDomainIntegrator(new VectorFEMassIntegrator(k));
mVarf->Assemble();
mVarf->Finalize();
SparseMatrix &M(mVarf->SpMat());
bVarf->AddDomainIntegrator(new VectorFEDivergenceIntegrator);
bVarf->Assemble();
bVarf->Finalize();
SparseMatrix & B(bVarf->SpMat());
B *= -1.;
SparseMatrix *BT = Transpose(B);
BlockMatrix darcyMatrix(block_offsets);
darcyMatrix.SetBlock(0,0, &M);
darcyMatrix.SetBlock(0,1, BT);
darcyMatrix.SetBlock(1,0, &B);
// 9. Construct the operators for preconditioner
//
// P = [ diag(M) 0 ]
// [ 0 B diag(M)^-1 B^T ]
//
// Here we use Symmetric Gauss-Seidel to approximate the inverse of the
// pressure Schur Complement
SparseMatrix *MinvBt = Transpose(B);
Vector Md(M.Height());
M.GetDiag(Md);
for (int i = 0; i < Md.Size(); i++)
{
MinvBt->ScaleRow(i, 1./Md(i));
}
SparseMatrix *S = Mult(B, *MinvBt);
Solver *invM, *invS;
invM = new DSmoother(M);
#ifndef MFEM_USE_SUITESPARSE
invS = new GSSmoother(*S);
#else
invS = new UMFPackSolver(*S);
#endif
invM->iterative_mode = false;
invS->iterative_mode = false;
BlockDiagonalPreconditioner darcyPrec(block_offsets);
darcyPrec.SetDiagonalBlock(0, invM);
darcyPrec.SetDiagonalBlock(1, invS);
// 10. Solve the linear system with MINRES.
// Check the norm of the unpreconditioned residual.
int maxIter(1000);
double rtol(1.e-6);
double atol(1.e-10);
chrono.Clear();
chrono.Start();
MINRESSolver solver;
solver.SetAbsTol(atol);
solver.SetRelTol(rtol);
solver.SetMaxIter(maxIter);
solver.SetOperator(darcyMatrix);
solver.SetPreconditioner(darcyPrec);
solver.SetPrintLevel(1);
x = 0.0;
solver.Mult(rhs, x);
chrono.Stop();
if (solver.GetConverged())
std::cout << "MINRES converged in " << solver.GetNumIterations()
<< " iterations with a residual norm of " << solver.GetFinalNorm() << ".\n";
else
std::cout << "MINRES did not converge in " << solver.GetNumIterations()
<< " iterations. Residual norm is " << solver.GetFinalNorm() << ".\n";
std::cout << "MINRES solver took " << chrono.RealTime() << "s. \n";
// 11. Create the grid functions u and p. Compute the L2 error norms.
GridFunction u, p;
u.MakeRef(R_space, x.GetBlock(0), 0);
p.MakeRef(W_space, x.GetBlock(1), 0);
int order_quad = max(2, 2*order+1);
const IntegrationRule *irs[Geometry::NumGeom];
for (int i=0; i < Geometry::NumGeom; ++i)
{
irs[i] = &(IntRules.Get(i, order_quad));
}
double err_u = u.ComputeL2Error(ucoeff, irs);
double norm_u = ComputeLpNorm(2., ucoeff, *mesh, irs);
double err_p = p.ComputeL2Error(pcoeff, irs);
double norm_p = ComputeLpNorm(2., pcoeff, *mesh, irs);
std::cout << "|| u_h - u_ex || / || u_ex || = " << err_u / norm_u << "\n";
std::cout << "|| p_h - p_ex || / || p_ex || = " << err_p / norm_p << "\n";
// 12. Save the mesh and the solution. This output can be viewed later using
// GLVis: "glvis -m ex5.mesh -g sol_u.gf" or "glvis -m ex5.mesh -g
// sol_p.gf".
{
ofstream mesh_ofs("ex5.mesh");
mesh_ofs.precision(8);
mesh->Print(mesh_ofs);
ofstream u_ofs("sol_u.gf");
u_ofs.precision(8);
u.Save(u_ofs);
ofstream p_ofs("sol_p.gf");
p_ofs.precision(8);
p.Save(p_ofs);
}
// 13. Save data in the VisIt format
VisItDataCollection visit_dc("Example5", mesh);
visit_dc.RegisterField("velocity", &u);
visit_dc.RegisterField("pressure", &p);
visit_dc.Save();
// 14. Send the solution by socket to a GLVis server.
if (visualization)
{
char vishost[] = "localhost";
int visport = 19916;
socketstream u_sock(vishost, visport);
u_sock.precision(8);
u_sock << "solution\n" << *mesh << u << "window_title 'Velocity'" << endl;
socketstream p_sock(vishost, visport);
p_sock.precision(8);
p_sock << "solution\n" << *mesh << p << "window_title 'Pressure'" << endl;
}
// 15. Free the used memory.
delete fform;
delete gform;
delete invM;
delete invS;
delete S;
delete MinvBt;
delete BT;
delete mVarf;
delete bVarf;
delete W_space;
delete R_space;
delete l2_coll;
delete hdiv_coll;
delete mesh;
return 0;
}
void uFun_ex(const Vector & x, Vector & u)
{
double xi(x(0));
double yi(x(1));
double zi(0.0);
if (x.Size() == 3)
{
zi = x(2);
}
u(0) = - exp(xi)*sin(yi)*cos(zi);
u(1) = - exp(xi)*cos(yi)*cos(zi);
if (x.Size() == 3)
{
u(2) = exp(xi)*sin(yi)*sin(zi);
}
}
// Change if needed
double pFun_ex(const Vector & x)
{
double xi(x(0));
double yi(x(1));
double zi(0.0);
if (x.Size() == 3)
{
zi = x(2);
}
return exp(xi)*sin(yi)*cos(zi);
}
void fFun(const Vector & x, Vector & f)
{
f = 0.0;
}
double gFun(const Vector & x)
{
if (x.Size() == 3)
{
return -pFun_ex(x);
}
else
{
return 0;
}
}
double f_natural(const Vector & x)
{
return (-pFun_ex(x));
}