ex9.c

一个用来实现偏微分方程中网格的计算库

/* $Id: ex9.C 2837 2008-05-08 17:23:37Z roystgnr $ */

/* The Next Great Finite Element Library. */
/* Copyright (C) 2003  Benjamin S. Kirk */

/* This library is free software; you can redistribute it and/or */
/* modify it under the terms of the GNU Lesser General Public */
/* License as published by the Free Software Foundation; either */
/* version 2.1 of the License, or (at your option) any later version. */

/* This library is distributed in the hope that it will be useful, */
/* but WITHOUT ANY WARRANTY; without even the implied warranty of */
/* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU */
/* Lesser General Public License for more details. */

/* You should have received a copy of the GNU Lesser General Public */
/* License along with this library; if not, write to the Free Software */
/* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA */



 // <h1>Example 9 - Solving a Transient Linear System in Parallel</h1>
 //
 // This example shows how a simple, linear transient
 // system can be solved in parallel.  The system is simple
 // scalar convection-diffusion with a specified external
 // velocity.  The initial condition is given, and the
 // solution is advanced in time with a standard Crank-Nicolson
 // time-stepping strategy.

// C++ include files that we need
#include <iostream>
#include <algorithm>
#include <math.h>

// Basic include file needed for the mesh functionality.
#include "libmesh.h"
#include "mesh.h"
#include "mesh_refinement.h"
#include "gmv_io.h"
#include "equation_systems.h"
#include "fe.h"
#include "quadrature_gauss.h"
#include "dof_map.h"
#include "sparse_matrix.h"
#include "numeric_vector.h"
#include "dense_matrix.h"
#include "dense_vector.h"

// Some (older) compilers do not offer full stream 
// functionality, OStringStream works around this.
#include "o_string_stream.h"

// This example will solve a linear transient system,
// so we need to include the \p TransientLinearImplicitSystem definition.
#include "linear_implicit_system.h"
#include "transient_system.h"
#include "vector_value.h"

// The definition of a geometric element
#include "elem.h"

// Function prototype.  This function will assemble the system
// matrix and right-hand-side at each time step.  Note that
// since the system is linear we technically do not need to
// assmeble the matrix at each time step, but we will anyway.
// In subsequent examples we will employ adaptive mesh refinement,
// and with a changing mesh it will be necessary to rebuild the
// system matrix.
void assemble_cd (EquationSystems& es,
                  const std::string& system_name);

// Function prototype.  This function will initialize the system.
// Initialization functions are optional for systems.  They allow
// you to specify the initial values of the solution.  If an
// initialization function is not provided then the default (0)
// solution is provided.
void init_cd (EquationSystems& es,
              const std::string& system_name);

// Exact solution function prototype.  This gives the exact
// solution as a function of space and time.  In this case the
// initial condition will be taken as the exact solution at time 0,
// as will the Dirichlet boundary conditions at time t.
Real exact_solution (const Real x,
                     const Real y,
                     const Real t);

Number exact_value (const Point& p,
                    const Parameters& parameters,
                    const std::string&,
                    const std::string&)
{
  return exact_solution(p(0), p(1), parameters.get<Real> ("time"));
}



// We can now begin the main program.  Note that this
// example will fail if you are using complex numbers
// since it was designed to be run only with real numbers.
int main (int argc, char** argv)
{
  // Initialize libMesh.
  LibMeshInit init (argc, argv);

#ifndef ENABLE_AMR
  if (libMesh::processor_id() == 0)
    std::cerr << "ERROR: This example requires libMesh to be\n"
              << "compiled with AMR support!"
              << std::endl;
  return 0;
#else

  // Create a two-dimensional mesh.
  Mesh mesh (2);
      
  // Read the mesh from file.  This is the coarse mesh that will be used
  // in example 10 to demonstrate adaptive mesh refinement.  Here we will
  // simply read it in and uniformly refine it 5 times before we compute
  // with it.
  mesh.read ("../ex10/mesh.xda");
  
  // Create a MeshRefinement object to handle refinement of our mesh.
  // This class handles all the details of mesh refinement and coarsening.
  MeshRefinement mesh_refinement (mesh);
  
  // Uniformly refine the mesh 5 times.  This is the
  // first time we use the mesh refinement capabilities
  // of the library.
  mesh_refinement.uniformly_refine (5);
  
  // Print information about the mesh to the screen.
  mesh.print_info();
  
  // Create an equation systems object.
  EquationSystems equation_systems (mesh);
  
  // Add a transient system to the EquationSystems
  // object named "Convection-Diffusion".
  TransientLinearImplicitSystem & system = 
    equation_systems.add_system<TransientLinearImplicitSystem> ("Convection-Diffusion");
    
  // Adds the variable "u" to "Convection-Diffusion".  "u"
  // will be approximated using first-order approximation.
  system.add_variable ("u", FIRST);
    
  // Give the system a pointer to the matrix assembly
  // and initialization functions.
  system.attach_assemble_function (assemble_cd);
  system.attach_init_function (init_cd);
    
  // Initialize the data structures for the equation system.
  equation_systems.init ();
    
  // Prints information about the system to the screen.
  equation_systems.print_info();
    
  // Write out the initial conditions.
  GMVIO(mesh).write_equation_systems ("out_000.gmv",
                                      equation_systems);
  
  // The Convection-Diffusion system requires that we specify
  // the flow velocity.  We will specify it as a RealVectorValue
  // data type and then use the Parameters object to pass it to
  // the assemble function.
  equation_systems.parameters.set<RealVectorValue>("velocity") = 
    RealVectorValue (0.8, 0.8);
  
  // Solve the system "Convection-Diffusion".  This will be done by
  // looping over the specified time interval and calling the
  // solve() member at each time step.  This will assemble the
  // system and call the linear solver.
  const Real dt = 0.025;
  Real time     = 0.;
  
  for (unsigned int t_step = 0; t_step < 50; t_step++)
    {
      // Incremenet the time counter, set the time and the
      // time step size as parameters in the EquationSystem.
      time += dt;

      equation_systems.parameters.set<Real> ("time") = time;
      equation_systems.parameters.set<Real> ("dt")   = dt;

      // A pretty update message
      std::cout << " Solving time step ";
      
      // Since some compilers fail to offer full stream
      // functionality, libMesh offers a string stream
      // to work around this.  Note that for other compilers,
      // this is just a set of preprocessor macros and therefore
      // should cost nothing (compared to a hand-coded string stream).
      // We use additional curly braces here simply to enforce data
      // locality.
      {
        OStringStream out;

        OSSInt(out,2,t_step);
        out << ", time=";
        OSSRealzeroleft(out,6,3,time);
        out <<  "...";
        std::cout << out.str() << std::endl;
      }
      
      // At this point we need to update the old
      // solution vector.  The old solution vector
      // will be the current solution vector from the
      // previous time step.  We will do this by extracting the
      // system from the \p EquationSystems object and using
      // vector assignment.  Since only \p TransientSystems
      // (and systems derived from them) contain old solutions
      // we need to specify the system type when we ask for it.
      TransientLinearImplicitSystem&  system =
        equation_systems.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");

      *system.old_local_solution = *system.current_local_solution;
      
      // Assemble & solve the linear system
      equation_systems.get_system("Convection-Diffusion").solve();
      
      // Output evey 10 timesteps to file.
      if ( (t_step+1)%10 == 0)
        {
          OStringStream file_name;

          file_name << "out_";
          OSSRealzeroright(file_name,3,0,t_step+1);
          file_name << ".gmv";

          GMVIO(mesh).write_equation_systems (file_name.str(),
                                              equation_systems);
        }
    }
#endif // #ifdef ENABLE_AMR

  // All done.  
  return 0;
}

// We now define the function which provides the
// initialization routines for the "Convection-Diffusion"
// system.  This handles things like setting initial
// conditions and boundary conditions.
void init_cd (EquationSystems& es,
              const std::string& system_name)
{
  // It is a good idea to make sure we are initializing