ex16.cpp

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//                                MFEM Example 16
//                             SUNDIALS Modification
//
// Compile with: make ex16
//
// Sample runs:  ex16
//               ex16 -m ../../data/inline-tri.mesh
//               ex16 -m ../../data/disc-nurbs.mesh -tf 2
//               ex16 -s 12 -a 0.0 -k 1.0
//               ex16 -s 8 -a 1.0 -k 0.0 -dt 1e-4 -tf 5e-2 -vs 25
//               ex16 -s 9 -a 0.5 -k 0.5 -o 4 -dt 1e-4 -tf 2e-2 -vs 25
//               ex16 -s 10 -dt 1.0e-4 -tf 4.0e-2 -vs 40
//               ex16 -m ../../data/fichera-q2.mesh
//               ex16 -m ../../data/escher.mesh
//               ex16 -m ../../data/beam-tet.mesh -tf 10 -dt 0.1
//               ex16 -m ../../data/amr-quad.mesh -o 4 -r 0
//               ex16 -m ../../data/amr-hex.mesh -o 2 -r 0
//
// Description:  This example solves a time dependent nonlinear heat equation
//               problem of the form du/dt = C(u), with a non-linear diffusion
//               operator C(u) = \nabla \cdot (\kappa + \alpha u) \nabla u.
//
//               The example demonstrates the use of nonlinear operators (the
//               class ConductionOperator defining C(u)), as well as their
//               implicit time integration. Note that implementing the method
//               ConductionOperator::ImplicitSolve is the only requirement for
//               high-order implicit (SDIRK) time integration. By default, this
//               example uses the SUNDIALS ODE solvers from CVODE and ARKODE.
//
//               We recommend viewing examples 2, 9 and 10 before viewing this
//               example.
 
#include "mfem.hpp"
#include <fstream>
#include <iostream>
 
using namespace std;
using namespace mfem;
 
/** After spatial discretization, the conduction model can be written as:
 *
 *     du/dt = M^{-1}(-Ku)
 *
 *  where u is the vector representing the temperature, M is the mass matrix,
 *  and K is the diffusion operator with diffusivity depending on u:
 *  (\kappa + \alpha u).
 *
 *  Class ConductionOperator represents the right-hand side of the above ODE.
 */
class ConductionOperator : public TimeDependentOperator
{
protected:
   FiniteElementSpace &fespace;
   Array<int> ess_tdof_list; // this list remains empty for pure Neumann b.c.
 
   BilinearForm *M;
   BilinearForm *K;
 
   SparseMatrix Mmat, Kmat;
   SparseMatrix *T; // T = M + dt K
 
   CGSolver M_solver; // Krylov solver for inverting the mass matrix M
   DSmoother M_prec;  // Preconditioner for the mass matrix M
 
   CGSolver T_solver; // Implicit solver for T = M + dt K
   DSmoother T_prec;  // Preconditioner for the implicit solver
 
   double alpha, kappa;
 
   mutable Vector z; // auxiliary vector
 
public:
   ConductionOperator(FiniteElementSpace &f, double alpha, double kappa,
                      const Vector &u);
 
   virtual void Mult(const Vector &u, Vector &du_dt) const;
 
   /** Solve the Backward-Euler equation: k = f(u + dt*k, t), for the unknown k.
       This is the only requirement for high-order SDIRK implicit integration.*/
   virtual void ImplicitSolve(const double dt, const Vector &u, Vector &k);
 
   /// Custom Jacobian system solver for the SUNDIALS time integrators.
   /** For the ODE system represented by ConductionOperator
 
           M du/dt = -K(u),
 
       this class facilitates the solution of linear systems of the form
 
           (M + γK) y = M b,
 
       for given b, u (not used), and γ = GetTimeStep(). */
 
   /** Setup the system (M + dt K) x = M b. This method is used by the implicit
       SUNDIALS solvers. */
   virtual int SUNImplicitSetup(const Vector &x, const Vector &fx,
                                int jok, int *jcur, double gamma);
 
   /** Solve the system (M + dt K) x = M b. This method is used by the implicit
       SUNDIALS solvers. */
   virtual int SUNImplicitSolve(const Vector &b, Vector &x, double tol);
 
   /// Update the diffusion BilinearForm K using the given true-dof vector `u`.
   void SetParameters(const Vector &u);
 
   virtual ~ConductionOperator();
};
 
double InitialTemperature(const Vector &x);
 
int main(int argc, char *argv[])
{
   // 0. Initialize SUNDIALS.
   Sundials::Init();
 
   // 1. Parse command-line options.
   const char *mesh_file = "../../data/star.mesh";
   int ref_levels = 2;
   int order = 2;
   int ode_solver_type = 9; // CVODE implicit BDF
   double t_final = 0.5;
   double dt = 1.0e-2;
   double alpha = 1.0e-2;
   double kappa = 0.5;
   bool visualization = true;
   bool visit = false;
   int vis_steps = 5;
 
   // Relative and absolute tolerances for CVODE and ARKODE.
   const double reltol = 1e-4, abstol = 1e-4;
 
   int precision = 8;
   cout.precision(precision);
 
   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",
                  "Order (degree) of the finite elements.");
   args.AddOption(&ode_solver_type, "-s", "--ode-solver",
                  "ODE solver:\n\t"
                  "1  - Forward Euler,\n\t"
                  "2  - RK2,\n\t"
                  "3  - RK3 SSP,\n\t"
                  "4  - RK4,\n\t"
                  "5  - Backward Euler,\n\t"
                  "6  - SDIRK 2,\n\t"
                  "7  - SDIRK 3,\n\t"
                  "8  - CVODE (implicit Adams),\n\t"
                  "9  - CVODE (implicit BDF),\n\t"
                  "10 - ARKODE (default explicit),\n\t"
                  "11 - ARKODE (explicit Fehlberg-6-4-5),\n\t"
                  "12 - ARKODE (default impicit).");
   args.AddOption(&t_final, "-tf", "--t-final",
                  "Final time; start time is 0.");
   args.AddOption(&dt, "-dt", "--time-step",
                  "Time step.");
   args.AddOption(&alpha, "-a", "--alpha",
                  "Alpha coefficient.");
   args.AddOption(&kappa, "-k", "--kappa",
                  "Kappa coefficient offset.");
   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.AddOption(&visit, "-visit", "--visit-datafiles", "-no-visit",
                  "--no-visit-datafiles",
                  "Save data files for VisIt (visit.llnl.gov) visualization.");
   args.AddOption(&vis_steps, "-vs", "--visualization-steps",
                  "Visualize every n-th timestep.");
   args.Parse();
   if (!args.Good())
   {
      args.PrintUsage(cout);
      return 1;
   }
   if (ode_solver_type < 1 || ode_solver_type > 12)
   {
      cout << "Unknown ODE solver type: " << ode_solver_type << '\n';
      return 3;
   }
   args.PrintOptions(cout);
 
   // 2. Read the mesh from the given mesh file. We can handle triangular,
   //    quadrilateral, tetrahedral and hexahedral 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, where 'ref_levels' is a
   //    command-line parameter.
   for (int lev = 0; lev < ref_levels; lev++)
   {
      mesh->UniformRefinement();
   }
 
   // 4. Define the vector finite element space representing the current and the
   //    initial temperature, u_ref.
   H1_FECollection fe_coll(order, dim);
   FiniteElementSpace fespace(mesh, &fe_coll);
 
   int fe_size = fespace.GetTrueVSize();
   cout << "Number of temperature unknowns: " << fe_size << endl;
 
   GridFunction u_gf(&fespace);
 
   // 5. Set the initial conditions for u. All boundaries are considered
   //    natural.
   FunctionCoefficient u_0(InitialTemperature);
   u_gf.ProjectCoefficient(u_0);
   Vector u;
   u_gf.GetTrueDofs(u);
 
   // 6. Initialize the conduction operator and the visualization.
   ConductionOperator oper(fespace, alpha, kappa, u);
 
   u_gf.SetFromTrueDofs(u);
   {
      ofstream omesh("ex16.mesh");
      omesh.precision(precision);
      mesh->Print(omesh);
      ofstream osol("ex16-init.gf");
      osol.precision(precision);
      u_gf.Save(osol);
   }
 
   VisItDataCollection visit_dc("Example16", mesh);
   visit_dc.RegisterField("temperature", &u_gf);
   if (visit)
   {
      visit_dc.SetCycle(0);
      visit_dc.SetTime(0.0);
      visit_dc.Save();
   }
 
   socketstream sout;
   if (visualization)
   {
      char vishost[] = "localhost";
      int  visport   = 19916;
      sout.open(vishost, visport);
      if (!sout)
      {
         cout << "Unable to connect to GLVis server at "
              << vishost << ':' << visport << endl;
         visualization = false;
         cout << "GLVis visualization disabled.\n";
      }
      else
      {
         sout.precision(precision);
         sout << "solution\n" << *mesh << u_gf;
         sout << "pause\n";
         sout << flush;
         cout << "GLVis visualization paused."
              << " Press space (in the GLVis window) to resume it.\n";
      }
   }
 
   // 7. Define the ODE solver used for time integration.
   double t = 0.0;
   ODESolver *ode_solver = NULL;
   CVODESolver *cvode = NULL;
   ARKStepSolver *arkode = NULL;
   switch (ode_solver_type)
   {
      // MFEM explicit methods
      case 1: ode_solver = new ForwardEulerSolver; break;
      case 2: ode_solver = new RK2Solver(0.5); break; // midpoint method
      case 3: ode_solver = new RK3SSPSolver; break;
      case 4: ode_solver = new RK4Solver; break;
      // MFEM implicit L-stable methods
      case 5: ode_solver = new BackwardEulerSolver; break;
      case 6: ode_solver = new SDIRK23Solver(2); break;
      case 7: ode_solver = new SDIRK33Solver; break;
      // CVODE
      case 8:
         cvode = new CVODESolver(CV_ADAMS);
         cvode->Init(oper);
         cvode->SetSStolerances(reltol, abstol);
         cvode->SetMaxStep(dt);
         ode_solver = cvode; break;
      case 9:
         cvode = new CVODESolver(CV_BDF);
         cvode->Init(oper);
         cvode->SetSStolerances(reltol, abstol);
         cvode->SetMaxStep(dt);
         ode_solver = cvode; break;
      // ARKODE
      case 10:
      case 11:
         arkode = new ARKStepSolver(ARKStepSolver::EXPLICIT);
         arkode->Init(oper);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
         if (ode_solver_type == 11)
         {
            arkode->SetERKTableNum(ARKODE_FEHLBERG_13_7_8);
         }
         ode_solver = arkode; break;
      case 12:
         arkode = new ARKStepSolver(ARKStepSolver::IMPLICIT);
         arkode->Init(oper);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
         ode_solver = arkode; break;
   }
 
   // Initialize MFEM integrators, SUNDIALS integrators are initialized above
   if (ode_solver_type < 8) { ode_solver->Init(oper); }
 
   // Since we want to update the diffusion coefficient after every time step,
   // we need to use the "one-step" mode of the SUNDIALS solvers.
   if (cvode) { cvode->SetStepMode(CV_ONE_STEP); }
   if (arkode) { arkode->SetStepMode(ARK_ONE_STEP); }
 
   // 8. Perform time-integration (looping over the time iterations, ti, with a
   //    time-step dt).
   cout << "Integrating the ODE ..." << endl;
   tic_toc.Clear();
   tic_toc.Start();
 
   bool last_step = false;
   for (int ti = 1; !last_step; ti++)
   {
      double dt_real = min(dt, t_final - t);
 
      // Note that since we are using the "one-step" mode of the SUNDIALS
      // solvers, they will, generally, step over the final time and will not
      // explicitly perform the interpolation to t_final as they do in the
      // "normal" step mode.
 
      ode_solver->Step(u, t, dt_real);
 
      last_step = (t >= t_final - 1e-8*dt);
 
      if (last_step || (ti % vis_steps) == 0)
      {
         cout << "step " << ti << ", t = " << t << endl;
         if (cvode) { cvode->PrintInfo(); }
         if (arkode) { arkode->PrintInfo(); }
 
         u_gf.SetFromTrueDofs(u);
         if (visualization)
         {
            sout << "solution\n" << *mesh << u_gf << flush;
         }
 
         if (visit)
         {
            visit_dc.SetCycle(ti);
            visit_dc.SetTime(t);
            visit_dc.Save();
         }
      }
      oper.SetParameters(u);
   }
   tic_toc.Stop();
   cout << "Done, " << tic_toc.RealTime() << "s." << endl;
 
   // 9. Save the final solution. This output can be viewed later using GLVis:
   //    "glvis -m ex16.mesh -g ex16-final.gf".
   {
      ofstream osol("ex16-final.gf");
      osol.precision(precision);
      u_gf.Save(osol);
   }
 
   // 10. Free the used memory.
   delete ode_solver;
   delete mesh;
 
   return 0;
}
 
ConductionOperator::ConductionOperator(FiniteElementSpace &f, double al,
                                       double kap, const Vector &u)
   : TimeDependentOperator(f.GetTrueVSize(), 0.0), fespace(f), M(NULL), K(NULL),
     T(NULL), z(height)
{
   const double rel_tol = 1e-8;
 
   M = new BilinearForm(&fespace);
   M->AddDomainIntegrator(new MassIntegrator());
   M->Assemble();
   M->FormSystemMatrix(ess_tdof_list, Mmat);
 
   M_solver.iterative_mode = false;
   M_solver.SetRelTol(rel_tol);
   M_solver.SetAbsTol(0.0);
   M_solver.SetMaxIter(50);
   M_solver.SetPrintLevel(0);
   M_solver.SetPreconditioner(M_prec);
   M_solver.SetOperator(Mmat);
 
   alpha = al;
   kappa = kap;
 
   T_solver.iterative_mode = false;
   T_solver.SetRelTol(rel_tol);
   T_solver.SetAbsTol(0.0);
   T_solver.SetMaxIter(100);
   T_solver.SetPrintLevel(0);
   T_solver.SetPreconditioner(T_prec);
 
   SetParameters(u);
}
 
void ConductionOperator::Mult(const Vector &u, Vector &du_dt) const
{
   // Compute:
   //    du_dt = M^{-1}*-K(u)
   // for du_dt
   Kmat.Mult(u, z);
   z.Neg(); // z = -z
   M_solver.Mult(z, du_dt);
}
 
void ConductionOperator::ImplicitSolve(const double dt,
                                       const Vector &u, Vector &du_dt)
{
   // Solve the equation:
   //    du_dt = M^{-1}*[-K(u + dt*du_dt)]
   // for du_dt
   if (T) { delete T; }
   T = Add(1.0, Mmat, dt, Kmat);
   T_solver.SetOperator(*T);
   Kmat.Mult(u, z);
   z.Neg();
   T_solver.Mult(z, du_dt);
}
 
void ConductionOperator::SetParameters(const Vector &u)
{
   GridFunction u_alpha_gf(&fespace);
   u_alpha_gf.SetFromTrueDofs(u);
   for (int i = 0; i < u_alpha_gf.Size(); i++)
   {
      u_alpha_gf(i) = kappa + alpha*u_alpha_gf(i);
   }
 
   delete K;
   K = new BilinearForm(&fespace);
 
   GridFunctionCoefficient u_coeff(&u_alpha_gf);
 
   K->AddDomainIntegrator(new DiffusionIntegrator(u_coeff));
   K->Assemble();
   K->FormSystemMatrix(ess_tdof_list, Kmat);
}
 
int ConductionOperator::SUNImplicitSetup(const Vector &x,
                                         const Vector &fx, int jok, int *jcur,
                                         double gamma)
{
   // Setup the ODE Jacobian T = M + gamma K.
   if (T) { delete T; }
   T = Add(1.0, Mmat, gamma, Kmat);
   T_solver.SetOperator(*T);
   *jcur = 1;
   return (0);
}
 
int ConductionOperator::SUNImplicitSolve(const Vector &b, Vector &x, double tol)
{
   // Solve the system A x = z => (M - gamma K) x = M b.
   Mmat.Mult(b, z);
   T_solver.Mult(z, x);
   return (0);
}
 
ConductionOperator::~ConductionOperator()
{
   delete T;
   delete M;
   delete K;
}
 
double InitialTemperature(const Vector &x)
{
   if (x.Norml2() < 0.5)
   {
      return 2.0;
   }
   else
   {
      return 1.0;
   }
}