ex4_boundary_second_derivs.cc

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

// Basic include file needed for the mesh functionality.
#include "libmesh/libmesh.h"
#include "libmesh/mesh.h"
#include "libmesh/mesh_generation.h"
#include "libmesh/exodusII_io.h"
#include "libmesh/gnuplot_io.h"
#include "libmesh/linear_implicit_system.h"
#include "libmesh/equation_systems.h"

// Define the Finite Element object.
#include "libmesh/fe.h"

// Define Gauss quadrature rules.
#include "libmesh/quadrature_gauss.h"

// Define the DofMap, which handles degree of freedom
// indexing.
#include "libmesh/dof_map.h"

// Define useful datatypes for finite element
// matrix and vector components.
#include "libmesh/sparse_matrix.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/dense_matrix.h"
#include "libmesh/dense_vector.h"

// Define the PerfLog, a performance logging utility.
// It is useful for timing events in a code and giving
// you an idea where bottlenecks lie.
#include "libmesh/perf_log.h"

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

#include "libmesh/string_to_enum.h"
#include "libmesh/getpot.h"

// Additional includes needed for MatGetColoring stuff
#include "libmesh/petsc_matrix.h"

// To use constraint-based Dirichlet boundary conditions.
#include "libmesh/dirichlet_boundaries.h"
#include "libmesh/analytic_function.h"

using namespace libMesh;

// Function prototype.  This is the function that will assemble
// the linear system for our Poisson problem.  Note that the
// function will take the \p EquationSystems object and the
// name of the system we are assembling as input.  From the
// \p EquationSystems object we have acess to the \p Mesh and
// other objects we might need.
void assemble_poisson(EquationSystems& es,
                      const std::string& system_name);

// Exact solution function prototype.
Real exact_solution (const Real x,
                     const Real y,
                     const Real /*z*/)
{
  // static const Real pi = acos(-1.);
  // return cos(.5*pi*x)*sin(.5*pi*y)*cos(.5*pi*z);

  // Exact solution has Laplacian = 4
  return x*x + y*y;
}


// This wraps the user's existing exact_solution() function into an interface
// that works with DirichletBoundaries.
void exact_solution_wrapper (DenseVector<Number> & output,
                             const Point & p,
                             const Real)
{
  output(0) = exact_solution(p(0),
                             (LIBMESH_DIM>1)?p(1):0,
                             (LIBMESH_DIM>2)?p(2):0);
}


// Begin the main program.
int main (int argc, char** argv)
{
  // Initialize libMesh and any dependent libaries, like in example 2.
  LibMeshInit init (argc, argv);

  // Declare a performance log for the main program
  // PerfLog perf_main("Main Program");

  // Create a GetPot object to parse the command line
  GetPot command_line (argc, argv);

  // Check for proper calling arguments.
  if (argc < 3)
    {
      if (init.comm().rank() == 0)
        libmesh_error_msg("Usage:\n" << "\t " << argv[0] << " -d 2(3)" << " -n 15");
    }

  // Brief message to the user regarding the program name
  // and command line arguments.
  else
    {
      std::cout << "Running " << argv[0];

      for (int i=1; i<argc; i++)
        std::cout << " " << argv[i];

      std::cout << std::endl << std::endl;
    }


  // Read problem dimension from command line.  Use int
  // instead of unsigned since the GetPot overload is ambiguous
  // otherwise.
  int dim = 2;
  if ( command_line.search(1, "-d") )
    dim = command_line.next(dim);

  // Skip higher-dimensional examples on a lower-dimensional libMesh build
  libmesh_example_requires(dim <= LIBMESH_DIM, "2D/3D support");

  // Create a mesh with user-defined dimension.
  // Read number of elements from command line
  int ps = 15;
  if ( command_line.search(1, "-n") )
    ps = command_line.next(ps);

  // Read FE order from command line
  std::string order = "SECOND";
  if ( command_line.search(2, "-Order", "-o") )
    order = command_line.next(order);

  // Read FE Family from command line
  std::string family = "LAGRANGE";
  if ( command_line.search(2, "-FEFamily", "-f") )
    family = command_line.next(family);

  // Cannot use discontinuous basis.
  if ((family == "MONOMIAL") || (family == "XYZ"))
    {
      if (init.comm().rank() == 0)
        libmesh_error_msg("ex4 currently requires a C^0 (or higher) FE basis.");
    }

  Mesh mesh (init.comm(), dim);


  // Use the MeshTools::Generation mesh generator to create a uniform
  // grid on the square [-1,1]^D.  We instruct the mesh generator
  // to build a mesh of 8x8 \p Quad9 elements in 2D, or \p Hex27
  // elements in 3D.  Building these higher-order elements allows
  // us to use higher-order approximation, as in example 3.

  Real halfwidth = dim > 1 ? 1. : 0.;
  Real halfheight = dim > 2 ? 1. : 0.;

  if ((family == "LAGRANGE") && (order == "FIRST"))
    {
      // No reason to use high-order geometric elements if we are
      // solving with low-order finite elements.
      MeshTools::Generation::build_cube (mesh,
                                         ps,
                                         (dim>1) ? ps : 0,
                                         (dim>2) ? ps : 0,
                                         -1., 1.,
                                         -halfwidth, halfwidth,
                                         -halfheight, halfheight,
                                         (dim==1)    ? EDGE2 :
                                         ((dim == 2) ? QUAD4 : HEX8));
    }


  else // order == SECOND
    {
      MeshTools::Generation::build_cube (mesh,
                                         ps,
                                         (dim>1) ? ps : 0,
                                         (dim>2) ? ps : 0,
                                         -1., 1.,
                                         -halfwidth, halfwidth,
                                         -halfheight, halfheight,
                                         (dim==1)    ? EDGE3 :
                                         ((dim == 2) ? QUAD9 : HEX27));
    }


  // Print information about the mesh to the screen.
  mesh.print_info();


  // Create an equation systems object.
  EquationSystems equation_systems (mesh);

  // Declare the system and its variables.
  // Create a system named "Poisson"
  LinearImplicitSystem& system =
    equation_systems.add_system<LinearImplicitSystem> ("Poisson");


  // Add the variable "u" to "Poisson".  "u"
  // will be approximated using second-order approximation.
    unsigned int u_var = system.add_variable("u",
                                             Utility::string_to_enum<Order>   (order),
                                             Utility::string_to_enum<FEFamily>(family));

  // Give the system a pointer to the matrix assembly
  // function.
  system.attach_assemble_function (assemble_poisson);

  // Construct a Dirichlet boundary condition object

  // Indicate which boundary IDs we impose the BC on
  // We either build a line, a square or a cube, and
  // here we indicate the boundaries IDs in each case
  std::set<boundary_id_type> boundary_ids;
  // the dim==1 mesh has two boundaries with IDs 0 and 1
  boundary_ids.insert(0);
  boundary_ids.insert(1);
  // the dim==2 mesh has four boundaries with IDs 0, 1, 2 and 3
  if (dim>=2)
    {
      boundary_ids.insert(2);
      boundary_ids.insert(3);
    }
  // the dim==3 mesh has four boundaries with IDs 0, 1, 2, 3, 4 and 5
  if (dim==3)
    {
      boundary_ids.insert(4);
      boundary_ids.insert(5);
    }

  // Create a vector storing the variable numbers which the BC applies to
  std::vector<unsigned int> variables(1);
  variables[0] = u_var;

  // Create an AnalyticFunction object that we use to project the BC
  // This function just calls the function exact_solution via exact_solution_wrapper
  AnalyticFunction<Number> exact_solution_object(exact_solution_wrapper);

  DirichletBoundary dirichlet_bc(boundary_ids,
                                 variables,
                                 &exact_solution_object);

  // We must add the Dirichlet boundary condition _before_
  // we call equation_systems.init()
  system.get_dof_map().add_dirichlet_boundary(dirichlet_bc);

  // Initialize the data structures for the equation system.
  equation_systems.init();

  // Print information about the system to the screen.
  equation_systems.print_info();
  mesh.print_info();

  // Solve the system "Poisson", just like example 2.
  equation_systems.get_system("Poisson").solve();

  // Do an extra assembly with the solved system.  This should
  // hopefully show non-zero second derivatives in our print
  // statements...
  equation_systems.get_system("Poisson").assemble();
  system.matrix->close();

  // After solving the system write the solution
  // to a GMV-formatted plot file.
  if(dim == 1)
  {
    GnuPlotIO plot(mesh,"Example 4, 1D",GnuPlotIO::GRID_ON);
    plot.write_equation_systems("out_1",equation_systems);
  }
  else
  {
    ExodusII_IO (mesh).write_equation_systems ("ex4_boundary_second_derivs.e", equation_systems);
  }

  // All done.
  return 0;
}




// We now define the matrix assembly function for the
// Poisson system.  We need to first compute element
// matrices and right-hand sides, and then take into
// account the boundary conditions, which will be handled
// via a penalty method.
void assemble_poisson(EquationSystems& es,
                      const std::string& system_name)
{
  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert (system_name == "Poisson");


  // Declare a performance log.  Give it a descriptive
  // string to identify what part of the code we are
  // logging, since there may be many PerfLogs in an
  // application.
  PerfLog perf_log ("Matrix Assembly");

    // Get a constant reference to the mesh object.
  const MeshBase& mesh = es.get_mesh();

  // The dimension that we are running
  const unsigned int dim = mesh.mesh_dimension();

  // Get a reference to the LinearImplicitSystem we are solving
  LinearImplicitSystem& system = es.get_system<LinearImplicitSystem>("Poisson");

  // Get the variable number for "u".
  const unsigned int u_var = system.variable_number ("u");

  // A reference to the \p DofMap object for this system.  The \p DofMap
  // object handles the index translation from node and element numbers
  // to degree of freedom numbers.  We will talk more about the \p DofMap
  // in future examples.
  const DofMap& dof_map = system.get_dof_map();

  // Get a constant reference to the Finite Element type
  // for the first (and only) variable in the system.
  FEType fe_type = dof_map.variable_type(0);

  // Build a Finite Element object of the specified type.  Since the
  // \p FEBase::build() member dynamically creates memory we will
  // store the object as an \p UniquePtr<FEBase>.  This can be thought
  // of as a pointer that will clean up after itself.
  UniquePtr<FEBase> fe (FEBase::build(dim, fe_type));

  // Use 2x2 (2D) quadrature to reduce the amount of output and mimic the
  // MOOSE example.
  QGauss qrule (dim, THIRD);

  // Tell the finite element object to use our quadrature rule.
  fe->attach_quadrature_rule (&qrule);

  // Declare a special finite element object for
  // boundary integration.
  UniquePtr<FEBase> fe_face (FEBase::build(dim, fe_type));

  // Use 2-pt (1D) quadrature to reduce the amount of output and mimic the
  // MOOSE example.
  QGauss qface(dim-1, THIRD);

  // Tell the finte element object to use our
  // quadrature rule.
  fe_face->attach_quadrature_rule (&qface);

  // Here we define some references to cell-specific data that
  // will be used to assemble the linear system.
  // We begin with the element Jacobian * quadrature weight at each
  // integration point.
  const std::vector<Real>& JxW = fe->get_JxW();

  // The physical XY locations of the quadrature points on the element.
  // These might be useful for evaluating spatially varying material
  // properties at the quadrature points.
  const std::vector<Point>& q_point = fe->get_xyz();

  // The element shape functions evaluated at the quadrature points.
  const std::vector<std::vector<Real> >& phi = fe->get_phi();

  // The element shape function gradients evaluated at the quadrature
  // points.
  const std::vector<std::vector<RealGradient> >& dphi = fe->get_dphi();

  // Define data structures to contain the element matrix
  // and right-hand-side vector contribution.  Following
  // basic finite element terminology we will denote these
  // "Ke" and "Fe". More detail is in example 3.
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  // This vector will hold the degree of freedom indices for
  // the element.  These define where in the global system
  // the element degrees of freedom get mapped.
  std::vector<unsigned int> dof_indices;
  std::vector<dof_id_type> dof_indices_u;

  // Now we will loop over all the elements in the mesh.
  // We will compute the element matrix and right-hand-side
  // contribution.  See example 3 for a discussion of the
  // element iterators.
  MeshBase::const_element_iterator       el     = mesh.active_local_elements_begin();
  const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end();

  for ( ; el != end_el; ++el)
    {
      // Start logging the shape function initialization.
      // This is done through a simple function call with
      // the name of the event to log.
      perf_log.push("elem init");

      // Store a pointer to the element we are currently
      // working on.  This allows for nicer syntax later.
      const Elem* elem = *el;

      // Get the degree of freedom indices for the
      // current element.  These define where in the global
      // matrix and right-hand-side this element will
      // contribute to.
      dof_map.dof_indices (elem, dof_indices);

      // Compute the element-specific data for the current
      // element.  This involves computing the location of the
      // quadrature points (q_point) and the shape functions
      // (phi, dphi) for the current element.
      fe->reinit (elem);

      // Zero the element matrix and right-hand side before
      // summing them.  We use the resize member here because
      // the number of degrees of freedom might have changed from
      // the last element.  Note that this will be the case if the
      // element type is different (i.e. the last element was a
      // triangle, now we are on a quadrilateral).
      Ke.resize (dof_indices.size(),
                 dof_indices.size());

      Fe.resize (dof_indices.size());

      // Get dof indices for the u variable
      dof_map.dof_indices (elem, dof_indices_u, u_var);

      // Stop logging the shape function initialization.
      // If you forget to stop logging an event the PerfLog
      // object will probably catch the error and abort.
      perf_log.pop("elem init");

      // Now we will build the element matrix.  This involves
      // a double loop to integrate the test funcions (i) against
      // the trial functions (j).
      //
      // We have split the numeric integration into two loops
      // so that we can log the matrix and right-hand-side
      // computation seperately.
      //
      // Now start logging the element matrix computation
      perf_log.push ("Ke");

      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        for (unsigned int i=0; i<phi.size(); i++)
          for (unsigned int j=0; j<phi.size(); j++)
            Ke(i,j) += JxW[qp]*(dphi[i][qp]*dphi[j][qp]);


      // Stop logging the matrix computation
      perf_log.pop ("Ke");

      // Now we build the element right-hand-side contribution.
      // This involves a single loop in which we integrate the
      // "forcing function" in the PDE against the test functions.
      //
      // Start logging the right-hand-side computation
      perf_log.push ("Fe");

      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          // fxy is the forcing function for the Poisson equation.
          // In this case we set fxy to be a finite difference
          // Laplacian approximation to the (known) exact solution.
          //
          // We will use the second-order accurate FD Laplacian
          // approximation, which in 2D on a structured grid is
          //
          // u_xx + u_yy = (u(i-1,j) + u(i+1,j) +
          //                u(i,j-1) + u(i,j+1) +
          //                -4*u(i,j))/h^2
          //
          // Since the value of the forcing function depends only
          // on the location of the quadrature point (q_point[qp])
          // we will compute it here, outside of the i-loop
          const Real x = q_point[qp](0);
          const Real y = q_point[qp](1);
          const Real z = q_point[qp](2);
          const Real eps = 1.e-3;

          const Real uxx = (exact_solution(x-eps,y,z) +
                            exact_solution(x+eps,y,z) +
                            -2.*exact_solution(x,y,z))/eps/eps;

          const Real uyy = (exact_solution(x,y-eps,z) +
                            exact_solution(x,y+eps,z) +
                            -2.*exact_solution(x,y,z))/eps/eps;

          const Real uzz = (exact_solution(x,y,z-eps) +
                            exact_solution(x,y,z+eps) +
                            -2.*exact_solution(x,y,z))/eps/eps;

          Real fxy = - (uxx + uyy + ((dim==2) ? 0. : uzz));
          // libMesh::out << "fxy=" << fxy << std::endl;

          // Add the RHS contribution
          for (unsigned int i=0; i<phi.size(); i++)
            Fe(i) += JxW[qp]*fxy*phi[i][qp];
        }

      // Stop logging the right-hand-side computation
      perf_log.pop ("Fe");

      // At this point the interior element integration has
      // been completed.  However, we have not yet addressed
      // boundary conditions.  For this example we will only
      // consider simple Dirichlet boundary conditions imposed
      // via the penalty method. This is discussed at length in
      // example 3.
      {

        // Start logging the boundary condition computation
        perf_log.push ("BCs");

        // The following loops over the sides of the element.
        // If the element has no neighbor on a side then that
        // side MUST live on a boundary of the domain.
        for (unsigned int side=0; side<elem->n_sides(); side++)
          if (elem->neighbor(side) == NULL)
            {
              // BCs are now handled by the DirichletBoundaries object!
              // The penalty value.  \frac{1}{\epsilon}
              // in the discussion above.
              // const Real penalty = 1.e10;

              // The value of the shape functions at the quadrature
              // points.
              const std::vector<std::vector<Real> >&  phi_face = fe_face->get_phi();

              // The Jacobian * Quadrature Weight at the quadrature
              // points on the face.
              const std::vector<Real>& JxW_face = fe_face->get_JxW();

              // The XYZ locations (in physical space) of the
              // quadrature points on the face.  This is where
              // we will interpolate the boundary value function.
              const std::vector<Point >& qface_point = fe_face->get_xyz();

              // Request the second derivative values.
              const std::vector<std::vector<RealTensor> > & d2phi_face = fe_face->get_d2phi();

              // Compute the shape function values on the element
              // face.
              fe_face->reinit(elem, side);

              // Get the boundary id for the current side
              std::vector<boundary_id_type> ids;
              mesh.get_boundary_info().boundary_ids(elem, side, ids);

              // Set a flag if we're on side 1.
              bool on_side_1 = std::count(ids.begin(), ids.end(), 1);

              // Loop over the face quadrature points for integration.
              for (unsigned int qp=0; qp<qface.n_points(); qp++)
              {
                // Check values: if we're on the righ boundary (id==1), print the d2phi_face tensor at this qp.
                if (on_side_1)
                  {
                    // Print a blank line to separate results of different qp's.
                    libMesh::out << std::endl;

                    // Verify that the second derivatives sum to 0.
                    RealTensor sum_d2phi;

                    // Compute second_u by summing u_i * d2phi[i][qp]
                    RealTensor second_u;

                    // Compute Laplacian(u) by summing u_i * Laplacian (phi_i)
                    Real lap_u = 0.;

                    libMesh::out << "qface_point[" << qp << "]=" << qface_point[qp] << std::endl;
                    for (unsigned i=0; i<d2phi_face.size(); ++i)
                      {
                        // Debugging output: print d2phi_face and u at every qp.
                        // libMesh::out << "d2phi_face[" << i << "][" << qp << "]=\n" << d2phi_face[i][qp] << std::endl;
                        // libMesh::out << "u[" << i << "]=" << system.current_solution (dof_indices_u[i]) << std::endl;

                        sum_d2phi += d2phi_face[i][qp];

                        // D2(u) = sum u_i D2(phi_i)
                        second_u += d2phi_face[i][qp] * system.current_solution (dof_indices_u[i]);

                        lap_u += d2phi_face[i][qp].tr() * system.current_solution (dof_indices_u[i]);
                      }

                    // If the sum of the second derivatives is not zero, throw an error
                    if (sum_d2phi.size() > 1.e-12)
                      {
                        libMesh::out << "sum_d2phi[qp=" << qp << "]=\n" << sum_d2phi << std::endl;
                        libmesh_error_msg("Error! Second derivatives should sum to zero!");
                      }
                    libMesh::out << "second_u[qp=" << qp << "]=\n" << second_u;
                    libMesh::out << "lap_u=" << lap_u << std::endl;
                  }

                // BCs are now handled by the DirichletBoundaries object!

//                // The location on the boundary of the current
//                // face quadrature point.
//                const Real xf = qface_point[qp](0);
//                const Real yf = qface_point[qp](1);
//                const Real zf = qface_point[qp](2);
//
//                // The boundary value.
//                const Real value = exact_solution(xf, yf, zf);
//
//                // Verify the exact solution is 1 + y**2 on side 1.  Since the
//                // 2-pt quadrature rule points are at (1,sqrt(3)/3), we should
//                // get u = 1 + 3/9 = 1.33333333.
//                if (on_side_1)
//                  libMesh::out << "exact solution value=" << value << std::endl;
//
//                // Matrix contribution of the L2 projection.
//                for (unsigned int i=0; i<phi_face.size(); i++)
//                  for (unsigned int j=0; j<phi_face.size(); j++)
//                    Ke(i,j) += JxW_face[qp]*penalty*phi_face[i][qp]*phi_face[j][qp];
//
//                // Right-hand-side contribution of the L2
//                // projection.
//                for (unsigned int i=0; i<phi_face.size(); i++)
//                  Fe(i) += JxW_face[qp]*penalty*value*phi_face[i][qp];
              }
            }


        // Stop logging the boundary condition computation
        perf_log.pop ("BCs");
      }

      // If this assembly program were to be used on an adaptive mesh,
      // we would have to apply any hanging node constraint equations
      // dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

      // We call heterogenously_constrain_element_matrix_and_vector()
      // to impose a inhomogeneous Dirichlet boundary conditions.
      dof_map.heterogenously_constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

      // The element matrix and right-hand-side are now built
      // for this element.  Add them to the global matrix and
      // right-hand-side vector.  The \p SparseMatrix::add_matrix()
      // and \p NumericVector::add_vector() members do this for us.
      // Start logging the insertion of the local (element)
      // matrix and vector into the global matrix and vector
      perf_log.push ("matrix insertion");

      system.matrix->add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);

      // Start logging the insertion of the local (element)
      // matrix and vector into the global matrix and vector
      perf_log.pop ("matrix insertion");
    }

  // That's it.  We don't need to do anything else to the
  // PerfLog.  When it goes out of scope (at this function return)
  // it will print its log to the screen. Pretty easy, huh?
}