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// MFEM Example 22
//
// Compile with: make ex22
//
// Sample runs: ex22 -m ../data/inline-segment.mesh -o 3
// ex22 -m ../data/inline-tri.mesh -o 3
// ex22 -m ../data/inline-quad.mesh -o 3
// ex22 -m ../data/inline-quad.mesh -o 3 -p 1
// ex22 -m ../data/inline-quad.mesh -o 3 -p 2
// ex22 -m ../data/inline-tet.mesh -o 2
// ex22 -m ../data/inline-hex.mesh -o 2
// ex22 -m ../data/inline-hex.mesh -o 2 -p 1
// ex22 -m ../data/inline-hex.mesh -o 2 -p 2
// ex22 -m ../data/star.mesh -r 1 -o 2 -sigma 10.0
//
// Description: This example code demonstrates the use of MFEM to define and
// solve simple complex-valued linear systems. It implements three
// variants of a damped harmonic oscillator:
//
// 1) A scalar H1 field
// -Div(a Grad u) - omega^2 b u + i omega c u = 0
//
// 2) A vector H(Curl) field
// Curl(a Curl u) - omega^2 b u + i omega c u = 0
//
// 3) A vector H(Div) field
// -Grad(a Div u) - omega^2 b u + i omega c u = 0
//
// In each case the field is driven by a forced oscillation, with
// angular frequency omega, imposed at the boundary or a portion
// of the boundary.
//
// In electromagnetics, the coefficients are typically named the
// permeability, mu = 1/a, permittivity, epsilon = b, and
// conductivity, sigma = c. The user can specify these constants
// using either set of names.
//
// The example also demonstrates how to display a time-varying
// solution as a sequence of fields sent to a single GLVis socket.
//
// We recommend viewing examples 1, 3 and 4 before viewing this
// example.
#include "mfem.hpp"
#include <fstream>
#include <iostream>
using namespace std;
using namespace mfem;
static double mu_ = 1.0;
static double epsilon_ = 1.0;
static double sigma_ = 20.0;
static double omega_ = 10.0;
double u0_real_exact(const Vector &);
double u0_imag_exact(const Vector &);
void u1_real_exact(const Vector &, Vector &);
void u1_imag_exact(const Vector &, Vector &);
void u2_real_exact(const Vector &, Vector &);
void u2_imag_exact(const Vector &, Vector &);
bool check_for_inline_mesh(const char * mesh_file);
int main(int argc, char *argv[])
{
// 1. Parse command-line options.
const char *mesh_file = "../data/inline-quad.mesh";
int ref_levels = 0;
int order = 1;
int prob = 0;
double freq = -1.0;
double a_coef = 0.0;
bool visualization = 1;
bool herm_conv = true;
bool exact_sol = true;
OptionsParser args(argc, argv);
args.AddOption(&mesh_file, "-m", "--mesh",
"Mesh file to use.");
args.AddOption(&ref_levels, "-r", "--refine",
"Number of times to refine the mesh uniformly.");
args.AddOption(&order, "-o", "--order",
"Finite element order (polynomial degree).");
args.AddOption(&prob, "-p", "--problem-type",
"Choose between 0: H_1, 1: H(Curl), or 2: H(Div) "
"damped harmonic oscillator.");
args.AddOption(&a_coef, "-a", "--stiffness-coef",
"Stiffness coefficient (spring constant or 1/mu).");
args.AddOption(&epsilon_, "-b", "--mass-coef",
"Mass coefficient (or epsilon).");
args.AddOption(&sigma_, "-c", "--damping-coef",
"Damping coefficient (or sigma).");
args.AddOption(&mu_, "-mu", "--permeability",
"Permeability of free space (or 1/(spring constant)).");
args.AddOption(&epsilon_, "-eps", "--permittivity",
"Permittivity of free space (or mass constant).");
args.AddOption(&sigma_, "-sigma", "--conductivity",
"Conductivity (or damping constant).");
args.AddOption(&freq, "-f", "--frequency",
"Frequency (in Hz).");
args.AddOption(&herm_conv, "-herm", "--hermitian", "-no-herm",
"--no-hermitian", "Use convention for Hermitian operators.");
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);
MFEM_VERIFY(prob >= 0 && prob <=2,
"Unrecognized problem type: " << prob);
if ( a_coef != 0.0 )
{
mu_ = 1.0 / a_coef;
}
if ( freq > 0.0 )
{
omega_ = 2.0 * M_PI * freq;
}
exact_sol = check_for_inline_mesh(mesh_file);
if (exact_sol)
{
cout << "Identified a mesh with known exact solution" << endl;
}
ComplexOperator::Convention conv =
herm_conv ? ComplexOperator::HERMITIAN : ComplexOperator::BLOCK_SYMMETRIC;
// 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 resolution. In this example we do
// 'ref_levels' of uniform refinement where the user specifies
// the number of levels with the '-r' option.
for (int l = 0; l < ref_levels; l++)
{
mesh->UniformRefinement();
}
// 4. Define a finite element space on the mesh. Here we use continuous
// Lagrange, Nedelec, or Raviart-Thomas finite elements of the specified
// order.
if (dim == 1 && prob != 0 )
{
cout << "Switching to problem type 0, H1 basis functions, "
<< "for 1 dimensional mesh." << endl;
prob = 0;
}
FiniteElementCollection *fec = NULL;
switch (prob)
{
case 0: fec = new H1_FECollection(order, dim); break;
case 1: fec = new ND_FECollection(order, dim); break;
case 2: fec = new RT_FECollection(order - 1, dim); break;
default: break; // This should be unreachable
}
FiniteElementSpace *fespace = new FiniteElementSpace(mesh, fec);
cout << "Number of finite element unknowns: " << fespace->GetTrueVSize()
<< endl;
// 5. Determine the list of true (i.e. conforming) essential boundary dofs.
// In this example, the boundary conditions are defined based on the type
// of mesh and the problem type.
Array<int> ess_tdof_list;
Array<int> ess_bdr;
if (mesh->bdr_attributes.Size())
{
ess_bdr.SetSize(mesh->bdr_attributes.Max());
ess_bdr = 1;
if (exact_sol)
{
switch (prob)
{
case 0: ess_bdr = 0; ess_bdr[0] = 1; break;
default: ess_bdr = 1; ess_bdr[2] = 0; break;
}
}
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.
ComplexLinearForm b(fespace, conv);
b.Vector::operator=(0.0);
// 7. Define the solution vector u as a complex finite element grid function
// corresponding to fespace. Initialize u with initial guess of 1+0i or
// the exact solution if it is known.
ComplexGridFunction u(fespace);
ComplexGridFunction * u_exact = NULL;
if (exact_sol) { u_exact = new ComplexGridFunction(fespace); }
FunctionCoefficient u0_r(u0_real_exact);
FunctionCoefficient u0_i(u0_imag_exact);
VectorFunctionCoefficient u1_r(dim, u1_real_exact);
VectorFunctionCoefficient u1_i(dim, u1_imag_exact);
VectorFunctionCoefficient u2_r(dim, u2_real_exact);
VectorFunctionCoefficient u2_i(dim, u2_imag_exact);
ConstantCoefficient zeroCoef(0.0);
ConstantCoefficient oneCoef(1.0);
Vector zeroVec(dim); zeroVec = 0.0;
Vector oneVec(dim); oneVec = 0.0; oneVec[(prob==2)?(dim-1):0] = 1.0;
VectorConstantCoefficient zeroVecCoef(zeroVec);
VectorConstantCoefficient oneVecCoef(oneVec);
u = 0.0;
switch (prob)
{
case 0:
u.ProjectBdrCoefficient(oneCoef, zeroCoef, ess_bdr);
if (exact_sol) { u_exact->ProjectCoefficient(u0_r, u0_i); }
break;
case 1:
u.ProjectBdrCoefficientTangent(oneVecCoef, zeroVecCoef, ess_bdr);
if (exact_sol) { u_exact->ProjectCoefficient(u1_r, u1_i); }
break;
case 2:
u.ProjectBdrCoefficientNormal(oneVecCoef, zeroVecCoef, ess_bdr);
if (exact_sol) { u_exact->ProjectCoefficient(u2_r, u2_i); }
break;
default: break; // This should be unreachable
}
if (visualization && exact_sol)
{
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock_r(vishost, visport);
socketstream sol_sock_i(vishost, visport);
sol_sock_r.precision(8);
sol_sock_i.precision(8);
sol_sock_r << "solution\n" << *mesh << u_exact->real()
<< "window_title 'Exact: Real Part'" << flush;
sol_sock_i << "solution\n" << *mesh << u_exact->imag()
<< "window_title 'Exact: Imaginary Part'" << flush;
}
// 8. Set up the sesquilinear form a(.,.) on the finite element space
// corresponding to the damped harmonic oscillator operator of the
// appropriate type:
//
// 0) A scalar H1 field
// -Div(a Grad) - omega^2 b + i omega c
//
// 1) A vector H(Curl) field
// Curl(a Curl) - omega^2 b + i omega c
//
// 2) A vector H(Div) field
// -Grad(a Div) - omega^2 b + i omega c
//
ConstantCoefficient stiffnessCoef(1.0/mu_);
ConstantCoefficient massCoef(-omega_ * omega_ * epsilon_);
ConstantCoefficient lossCoef(omega_ * sigma_);
ConstantCoefficient negMassCoef(omega_ * omega_ * epsilon_);
SesquilinearForm *a = new SesquilinearForm(fespace, conv);
switch (prob)
{
case 0:
a->AddDomainIntegrator(new DiffusionIntegrator(stiffnessCoef),
NULL);
a->AddDomainIntegrator(new MassIntegrator(massCoef),
new MassIntegrator(lossCoef));
break;
case 1:
a->AddDomainIntegrator(new CurlCurlIntegrator(stiffnessCoef),
NULL);
a->AddDomainIntegrator(new VectorFEMassIntegrator(massCoef),
new VectorFEMassIntegrator(lossCoef));
break;
case 2:
a->AddDomainIntegrator(new DivDivIntegrator(stiffnessCoef),
NULL);
a->AddDomainIntegrator(new VectorFEMassIntegrator(massCoef),
new VectorFEMassIntegrator(lossCoef));
break;
default: break; // This should be unreachable
}
// 8a. Set up the bilinear form for the preconditioner corresponding to the
// appropriate operator
//
// 0) A scalar H1 field
// -Div(a Grad) - omega^2 b + omega c
//
// 1) A vector H(Curl) field
// Curl(a Curl) + omega^2 b + omega c
//
// 2) A vector H(Div) field
// -Grad(a Div) - omega^2 b + omega c
//
BilinearForm *pcOp = new BilinearForm(fespace);
switch (prob)
{
case 0:
pcOp->AddDomainIntegrator(new DiffusionIntegrator(stiffnessCoef));
pcOp->AddDomainIntegrator(new MassIntegrator(massCoef));
pcOp->AddDomainIntegrator(new MassIntegrator(lossCoef));
break;
case 1:
pcOp->AddDomainIntegrator(new CurlCurlIntegrator(stiffnessCoef));
pcOp->AddDomainIntegrator(new VectorFEMassIntegrator(negMassCoef));
pcOp->AddDomainIntegrator(new VectorFEMassIntegrator(lossCoef));
break;
case 2:
pcOp->AddDomainIntegrator(new DivDivIntegrator(stiffnessCoef));
pcOp->AddDomainIntegrator(new VectorFEMassIntegrator(massCoef));
pcOp->AddDomainIntegrator(new VectorFEMassIntegrator(lossCoef));
break;
default: break; // This should be unreachable
}
// 9. Assemble the form and the corresponding linear system, applying any
// necessary transformations such as: assembly, eliminating boundary
// conditions, conforming constraints for non-conforming AMR, etc.
a->Assemble();
pcOp->Assemble();
OperatorHandle A;
Vector B, U;
a->FormLinearSystem(ess_tdof_list, u, b, A, U, B);
u = 0.0;
U = 0.0;
OperatorHandle PCOp;
pcOp->FormSystemMatrix(ess_tdof_list, PCOp);
{
ComplexSparseMatrix * Asp =
dynamic_cast<ComplexSparseMatrix*>(A.Ptr());
cout << "Size of linear system: "
<< 2 * Asp->real().Width() << endl << endl;
}
// 10. Define and apply a GMRES solver for AU=B with a block diagonal
// preconditioner based on the appropriate sparse smoother.
{
Array<int> blockOffsets;
blockOffsets.SetSize(3);
blockOffsets[0] = 0;
blockOffsets[1] = PCOp.Ptr()->Height();
blockOffsets[2] = PCOp.Ptr()->Height();
blockOffsets.PartialSum();
BlockDiagonalPreconditioner BDP(blockOffsets);
Operator * pc_r = NULL;
Operator * pc_i = NULL;
switch (prob)
{
case 0: // fallthrough to case 2
case 2:
pc_r = new DSmoother(*PCOp.As<SparseMatrix>());
break;
case 1:
pc_r = new GSSmoother(*PCOp.As<SparseMatrix>());
break;
default: break; // This should be unreachable
}
pc_i = new ScaledOperator(pc_r,
(conv == ComplexOperator::HERMITIAN) ?
1.0:-1.0);
BDP.SetDiagonalBlock(0, pc_r);
BDP.SetDiagonalBlock(1, pc_i);
BDP.owns_blocks = 1;
GMRESSolver gmres;
gmres.SetPreconditioner(BDP);
gmres.SetOperator(*A.Ptr());
gmres.SetRelTol(1e-12);
gmres.SetMaxIter(1000);
gmres.SetPrintLevel(1);
gmres.Mult(B, U);
}
// 11. Recover the solution as a finite element grid function and compute the
// errors if the exact solution is known.
a->RecoverFEMSolution(U, b, u);
if (exact_sol)
{
double err_r = -1.0;
double err_i = -1.0;
switch (prob)
{
case 0:
err_r = u.real().ComputeL2Error(u0_r);
err_i = u.imag().ComputeL2Error(u0_i);
break;
case 1:
err_r = u.real().ComputeL2Error(u1_r);
err_i = u.imag().ComputeL2Error(u1_i);
break;
case 2:
err_r = u.real().ComputeL2Error(u2_r);
err_i = u.imag().ComputeL2Error(u2_i);
break;
default: break; // This should be unreachable
}
cout << endl;
cout << "|| Re (u_h - u) ||_{L^2} = " << err_r << endl;
cout << "|| Im (u_h - u) ||_{L^2} = " << err_i << endl;
cout << endl;
}
// 12. Save the refined mesh and the solution. This output can be viewed
// later using GLVis: "glvis -m mesh -g sol".
{
ofstream mesh_ofs("refined.mesh");
mesh_ofs.precision(8);
mesh->Print(mesh_ofs);
ofstream sol_r_ofs("sol_r.gf");
ofstream sol_i_ofs("sol_i.gf");
sol_r_ofs.precision(8);
sol_i_ofs.precision(8);
u.real().Save(sol_r_ofs);
u.imag().Save(sol_i_ofs);
}
// 13. Send the solution by socket to a GLVis server.
if (visualization)
{
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock_r(vishost, visport);
socketstream sol_sock_i(vishost, visport);
sol_sock_r.precision(8);
sol_sock_i.precision(8);
sol_sock_r << "solution\n" << *mesh << u.real()
<< "window_title 'Solution: Real Part'" << flush;
sol_sock_i << "solution\n" << *mesh << u.imag()
<< "window_title 'Solution: Imaginary Part'" << flush;
}
if (visualization && exact_sol)
{
*u_exact -= u;
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock_r(vishost, visport);
socketstream sol_sock_i(vishost, visport);
sol_sock_r.precision(8);
sol_sock_i.precision(8);
sol_sock_r << "solution\n" << *mesh << u_exact->real()
<< "window_title 'Error: Real Part'" << flush;
sol_sock_i << "solution\n" << *mesh << u_exact->imag()
<< "window_title 'Error: Imaginary Part'" << flush;
}
if (visualization)
{
GridFunction u_t(fespace);
u_t = u.real();
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock(vishost, visport);
sol_sock.precision(8);
sol_sock << "solution\n" << *mesh << u_t
<< "window_title 'Harmonic Solution (t = 0.0 T)'"
<< "pause\n" << flush;
cout << "GLVis visualization paused."
<< " Press space (in the GLVis window) to resume it.\n";
int num_frames = 32;
int i = 0;
while (sol_sock)
{
double t = (double)(i % num_frames) / num_frames;
ostringstream oss;
oss << "Harmonic Solution (t = " << t << " T)";
add(cos( 2.0 * M_PI * t), u.real(),
sin(-2.0 * M_PI * t), u.imag(), u_t);
sol_sock << "solution\n" << *mesh << u_t
<< "window_title '" << oss.str() << "'" << flush;
i++;
}
}
// 14. Free the used memory.
delete a;
delete u_exact;
delete pcOp;
delete fespace;
delete fec;
delete mesh;
return 0;
}
bool check_for_inline_mesh(const char * mesh_file)
{
string file(mesh_file);
size_t p0 = file.find_last_of("/");
string s0 = file.substr((p0==string::npos)?0:(p0+1),7);
return s0 == "inline-";
}
complex<double> u0_exact(const Vector &x)
{
int dim = x.Size();
complex<double> i(0.0, 1.0);
complex<double> alpha = (epsilon_ * omega_ - i * sigma_);
complex<double> kappa = std::sqrt(mu_ * omega_* alpha);
return std::exp(-i * kappa * x[dim - 1]);
}
double u0_real_exact(const Vector &x)
{
return u0_exact(x).real();
}
double u0_imag_exact(const Vector &x)
{
return u0_exact(x).imag();
}
void u1_real_exact(const Vector &x, Vector &v)
{
int dim = x.Size();
v.SetSize(dim); v = 0.0; v[0] = u0_real_exact(x);
}
void u1_imag_exact(const Vector &x, Vector &v)
{
int dim = x.Size();
v.SetSize(dim); v = 0.0; v[0] = u0_imag_exact(x);
}
void u2_real_exact(const Vector &x, Vector &v)
{
int dim = x.Size();
v.SetSize(dim); v = 0.0; v[dim-1] = u0_real_exact(x);
}
void u2_imag_exact(const Vector &x, Vector &v)
{
int dim = x.Size();
v.SetSize(dim); v = 0.0; v[dim-1] = u0_imag_exact(x);
}
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