Beispiel #1
0
void assemble_poisson(EquationSystems & es,
                      const std::string & system_name)
{
  libmesh_assert_equal_to (system_name, "Poisson");

  const MeshBase & mesh = es.get_mesh();
  const unsigned int dim = mesh.mesh_dimension();
  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Poisson");

  const DofMap & dof_map = system.get_dof_map();

  FEType fe_type = dof_map.variable_type(0);
  UniquePtr<FEBase> fe (FEBase::build(dim, fe_type));
  QGauss qrule (dim, FIFTH);
  fe->attach_quadrature_rule (&qrule);
  UniquePtr<FEBase> fe_face (FEBase::build(dim, fe_type));
  QGauss qface(dim-1, FIFTH);
  fe_face->attach_quadrature_rule (&qface);

  const std::vector<Real> & JxW = fe->get_JxW();
  const std::vector<std::vector<Real> > & phi = fe->get_phi();
  const std::vector<std::vector<RealGradient> > & dphi = fe->get_dphi();

  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  std::vector<dof_id_type> dof_indices;

  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)
    {
      const Elem * elem = *el;

      dof_map.dof_indices (elem, dof_indices);

      fe->reinit (elem);

      Ke.resize (dof_indices.size(),
                 dof_indices.size());

      Fe.resize (dof_indices.size());

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

      dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

      system.matrix->add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);
    }
}
Beispiel #2
0
void DenseMatrix<T>::vector_mult (DenseVector<T>& dest,
                                  const DenseVector<T>& arg) const
{
  // Make sure the input sizes are compatible
  libmesh_assert_equal_to (this->n(), arg.size());

  // Resize and clear dest.
  // Note: DenseVector::resize() also zeros the vector.
  dest.resize(this->m());

  // Short-circuit if the matrix is empty
  if(this->m() == 0 || this->n() == 0)
    return;

  if (this->use_blas_lapack)
    this->_matvec_blas(1., 0., dest, arg);
  else
    {
      const unsigned int n_rows = this->m();
      const unsigned int n_cols = this->n();

      for(unsigned int i=0; i<n_rows; i++)
	for(unsigned int j=0; j<n_cols; j++)
	  dest(i) += (*this)(i,j)*arg(j);
    }
}
///////////////////////////////////////////////////////////////////////////////
//* performs a matrix-vector multiply with optional transposes BLAS version
void MultMv(const Matrix<double> &A, const Vector<double> &v, DenseVector<double> &c,
            const bool At, double a, double b)
{
  static char t[2] = {'N','T'};
  char *ta=t+At;
  int sA[2] = {A.nRows(), A.nCols()};  // sizes of A
  int sV[2] = {v.size(), 1};           // sizes of v
  
  GCK(A, v, sA[!At]!=sV[0], "MultAB<double>: matrix-vector multiply");
  if (c.size() != sA[At])
  {
    c.resize(sA[At]); // set size of C to final size
    c.zero();
  }
  // get pointers to the matrix sizes needed by BLAS
  int *M = sA+At;  // # of rows in op[A]  (op[A] = A' if At='T' else A)
  int *N = sV+1;   // # of cols in op[B]
  int *K = sA+!At; // # of cols in op[A] or # of rows in op[B]

  double *pa=A.ptr(), *pv=v.ptr(), *pc=c.ptr();

#ifdef COL_STORAGE
  dgemm_(ta, t, M, N, K, &a, pa, sA, pv, sV, &b, pc, M); 
#else
  dgemm_(t, ta, N, M, K, &a, pv, sV+1, pa, sA+1, &b, pc, N); 
#endif
}
void DenseMatrix<T>::_svd_lapack (DenseVector<T>& sigma, DenseMatrix<T>& U, DenseMatrix<T>& VT)
{
  // The calling sequence for dgetrf is:
  // DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO )


  //  JOBU    (input) CHARACTER*1
  //          Specifies options for computing all or part of the matrix U:
  //          = 'A':  all M columns of U are returned in array U:
  //          = 'S':  the first min(m,n) columns of U (the left singular
  //                  vectors) are returned in the array U;
  //          = 'O':  the first min(m,n) columns of U (the left singular
  //                  vectors) are overwritten on the array A;
  //          = 'N':  no columns of U (no left singular vectors) are
  //                  computed.
  char JOBU = 'S';

  //  JOBVT   (input) CHARACTER*1
  //          Specifies options for computing all or part of the matrix
  //          V**T:
  //          = 'A':  all N rows of V**T are returned in the array VT;
  //          = 'S':  the first min(m,n) rows of V**T (the right singular
  //                  vectors) are returned in the array VT;
  //          = 'O':  the first min(m,n) rows of V**T (the right singular
  //                  vectors) are overwritten on the array A;
  //          = 'N':  no rows of V**T (no right singular vectors) are
  //                  computed.
  char JOBVT = 'S';

  std::vector<T> sigma_val;
  std::vector<T> U_val;
  std::vector<T> VT_val;

  _svd_helper(JOBU, JOBVT, sigma_val, U_val, VT_val);

  // Load the singular values into sigma, ignore U_val and VT_val
  sigma.resize(sigma_val.size());
  for(unsigned int i=0; i<sigma_val.size(); i++)
    sigma(i) = sigma_val[i];

  int M = this->n();
  int N = this->m();
  int min_MN = (M < N) ? M : N;
  U.resize(M,min_MN);
  for(unsigned int i=0; i<U.m(); i++)
    for(unsigned int j=0; j<U.n(); j++)
    {
      unsigned int index = i + j*U.n();  // Column major storage
      U(i,j) = U_val[index];
    }

  VT.resize(min_MN,N);
  for(unsigned int i=0; i<VT.m(); i++)
    for(unsigned int j=0; j<VT.n(); j++)
    {
      unsigned int index = i + j*U.n(); // Column major storage
      VT(i,j) = VT_val[index];
    }

}
void AssembleOptimization::assemble_A_and_F()
{
  A_matrix->zero();
  F_vector->zero();

  const MeshBase & mesh = _sys.get_mesh();

  const unsigned int dim = mesh.mesh_dimension();
  const unsigned int u_var = _sys.variable_number ("u");

  const DofMap & dof_map = _sys.get_dof_map();
  FEType fe_type = dof_map.variable_type(u_var);
  UniquePtr<FEBase> fe (FEBase::build(dim, fe_type));
  QGauss qrule (dim, fe_type.default_quadrature_order());
  fe->attach_quadrature_rule (&qrule);

  const std::vector<Real> & JxW = fe->get_JxW();
  const std::vector<std::vector<Real> > & phi = fe->get_phi();
  const std::vector<std::vector<RealGradient> > & dphi = fe->get_dphi();

  std::vector<dof_id_type> dof_indices;

  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  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)
    {
      const Elem * elem = *el;

      dof_map.dof_indices (elem, dof_indices);

      const unsigned int n_dofs = dof_indices.size();

      fe->reinit (elem);

      Ke.resize (n_dofs, n_dofs);
      Fe.resize (n_dofs);

      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          for (unsigned int dof_i=0; dof_i<n_dofs; dof_i++)
            {
              for (unsigned int dof_j=0; dof_j<n_dofs; dof_j++)
                {
                  Ke(dof_i, dof_j) += JxW[qp] * (dphi[dof_j][qp]* dphi[dof_i][qp]);
                }
              Fe(dof_i) += JxW[qp] * phi[dof_i][qp];
            }
        }

      A_matrix->add_matrix (Ke, dof_indices);
      F_vector->add_vector (Fe, dof_indices);
    }

  A_matrix->close();
  F_vector->close();
}
Beispiel #6
0
void DenseMatrix<T>::vector_mult_transpose (DenseVector<T>& dest,
                                            const DenseVector<T>& arg) const
{
  // Make sure the input sizes are compatible
  libmesh_assert_equal_to (this->m(), arg.size());

  // Resize and clear dest.
  // Note: DenseVector::resize() also zeros the vector.
  dest.resize(this->n());

  // Short-circuit if the matrix is empty
  if(this->m() == 0)
    return;

  if (this->use_blas_lapack)
    {
      this->_matvec_blas(1., 0., dest, arg, /*trans=*/true);
    }
  else
    {
      const unsigned int n_rows = this->m();
      const unsigned int n_cols = this->n();

      // WORKS
      // for(unsigned int j=0; j<n_cols; j++)
      //   for(unsigned int i=0; i<n_rows; i++)
      //     dest(j) += (*this)(i,j)*arg(i);

      // ALSO WORKS, (i,j) just swapped
      for(unsigned int i=0; i<n_cols; i++)
	for(unsigned int j=0; j<n_rows; j++)
	  dest(i) += (*this)(j,i)*arg(j);
    }
}
 void operator() (const Point & p,
                  const Real time,
                  DenseVector<Number> & output)
 {
   output.resize(1);
   output(0) = (*this)(p,time);
   return;
 }
Beispiel #8
0
		void value(DenseVector<Number>& output, const Point& p, const Real){
			
			// Gather the time from the system
			Real t = get_system().time;
			
			// Return the values of k and cp
			output.resize(2);
			output(0) = k(p,t);
			output(1) = cp(p,t);
		}		
Beispiel #9
0
 virtual void operator() (const Point & p,
                          const Real,
                          DenseVector<Number> & output)
 {
   output.resize(2);
   output.zero();
   const Real x=p(0);
   // Set the parabolic weighting
   output(_T_var) = -x * (1 - x);
 }
Beispiel #10
0
 virtual void operator() (const Point& p,
                          const Real,
                          DenseVector<Number>& output)
   {
     output.resize(2);
     output.zero();
     const Real y=p(1);
     // Set the parabolic inflow boundary conditions at stations 0 & 1
     output(_u_var) = (_sign)*((y-2) * (y-3));
     output(_v_var) = 0;      
   }
Beispiel #11
0
void initial_concentration(DenseVector<Number>& output, const Point& p, const Real){
	output.resize(1);
	if (p(0) == -1 && p(1) == -1){
		output(0) = 0.192; // x-direction
	} else if (p(0) == 1 && p(1) == -1){
		output(0) = 0.192;
	} else if (p(0) == 1 && p(1) == 1){
		output(0) = 0.192;
	} else if (p(0) == -1 && p(1) == 1){
		output(0) = 0.192;
	}
}
Beispiel #12
0
void
dataLoad(std::istream & stream, DenseVector<Real> & v, void * /*context*/)
{
  unsigned int n = 0;
  stream.read((char *) &n, sizeof(n));
  v.resize(n);
  for (unsigned int i = 0; i < n; i++)
  {
    Real r = 0;
    stream.read((char *) &r, sizeof(r));
    v(i) = r;
  }
}
Beispiel #13
0
void new_enthalpy(DenseVector<Number>& output, const Point& p, const Real){
	output.resize(1);
	double x = 0.9;
	if (p(0) == -1 && p(1) == -1){
		output(0) = x*3.8002; // x-direction
	} else if (p(0) == 1 && p(1) == -1){
		output(0) = x*2.8499;
	} else if (p(0) == 1 && p(1) == 1){
		output(0) = x*3.5154;
	} else if (p(0) == -1 && p(1) == 1){
		output(0) = x*4.2149;
	} 
}
Beispiel #14
0
// A function for space dependent conductivity; time is ignored by libmesh
void func1(DenseVector<Number>& output, const Point& p, const Real t){
	
	// Display the time, it does not get updated (only show at 0,0)
	if(p(0) == 0 && p(1) == 0){
		printf("\nTime = %f\n", t);
	}
	
	// Define pi
	const double pi = boost::math::constants::pi<double>(); 
	
	// Update the output vector
	output.resize(2); 
	output(0) = 1 + exp(-t) * sin(pi*p(0)) * sin(pi*p(1));
	output(1) = 10 + exp(-t) * sin(pi*p(0)) * sin(pi*p(1));
}
Beispiel #15
0
DenseVector<Number>
FDKernel::perturbedResidual(unsigned int varnum, unsigned int perturbationj, Real perturbation_scale, Real& perturbation)
{
    DenseVector<Number> re;
    re.resize(_var.dofIndices().size());
    re.zero();

    MooseVariable& var = _sys.getVariable(_tid,varnum);
    var.computePerturbedElemValues(perturbationj,perturbation_scale,perturbation);
    precalculateResidual();
    for (_i = 0; _i < _test.size(); _i++)
        for (_qp = 0; _qp < _qrule->n_points(); _qp++)
            re(_i) += _JxW[_qp] * _coord[_qp] * computeQpResidual();
    var.restoreUnperturbedElemValues();
    return re;
}
Beispiel #16
0
void initial_velocity(DenseVector<Number>& output, const Point& p, const Real){

 	output.resize(2);
	if (p(0) == -1 && p(1) == -1){
		output(0) = 1; // x-direction
		output(1) = 2; // y-direction
	} else if (p(0) == 1 && p(1) == -1){
		output(0) = 1;
		output(1) = 0;		
	} else if (p(0) == 1 && p(1) == 1){
		output(0) = 1;
		output(1) = 1;	
	} else if (p(0) == -1 && p(1) == 1){
		output(0) = 0;
		output(1) = 0;	
	}
}
Beispiel #17
0
void func2(DenseVector<Number>& output, const Point& p, const Real, const Parameters& parameters){
		
	// Extract the time	
	Real t = parameters.get<Real>("time");

	// Display the time, only show at 0,0
	if(p(0) == 0 && p(1) == 0){
		printf("\nTime = %f\n", t);
	}
	
	// Define pi
	const double pi = boost::math::constants::pi<double>(); 

	// Update the output vector
	output.resize(2); 
	output(0) = 1 + exp(-t) * sin(pi*p(0)) * sin(pi*p(1));
	output(1) = 10 + exp(-t) * sin(pi*p(0)) * sin(pi*p(1));
}
Beispiel #18
0
void initial_temperature(DenseVector<Number>& output, const Point& p, const Real){
	output.resize(3);
	if (p(0) == -1 && p(1) == -1){
		output(0) = 287; 
	} else if (p(0) == 1 && p(1) == -1){
		output(0) = 287;
	} else if (p(0) == 1 && p(1) == 1){
		output(0) = 287;
	} else if (p(0) == -1 && p(1) == 1){
		output(0) = 287;
	}
	
	// h_dot and delta_h_dot
	//! \todo this needs to be handled internally
	output(1) = 0;
	output(2) = 0;
	
}
Beispiel #19
0
bool SolveDeficit(DenseMatrix< VariableArray2<double> > &A,
		DenseVector<VariableArray1<double> > &x, DenseVector<VariableArray1<double> > &rhs, double deficitTolerance)
{
	DenseMatrix< VariableArray2<double> > A2=A;
	DenseVector<VariableArray1<double> > rhs2=rhs;

	UG_ASSERT(A.num_rows() == rhs.size(), "");
	UG_ASSERT(A.num_cols() == x.size(), "");

	size_t iNonNullRows;
	x.resize(A.num_cols());
	for(size_t i=0; i<x.size(); i++)
		x[i] = 0.0;
	std::vector<size_t> interchange;
	if(Decomp(A, rhs, iNonNullRows, interchange, deficitTolerance) == false) return false;

//	A.maple_print("Adecomp");
//	rhs.maple_print("rhs decomp");

	for(int i=iNonNullRows-1; i>=0; i--)
	{
		double d=A(i,i);
		double s=0;
		for(size_t k=i+1; k<A.num_cols(); k++)
			s += A(i,k)*x[interchange[k]];
		x[interchange[i]] = (rhs[i] - s)/d;
	}
	DenseVector<VariableArray1<double> > f;
	f = A2*x - rhs2;
	if(VecNormSquared(f) > 1e-2)
	{
		UG_LOGN("iNonNullRows = " << iNonNullRows);
		UG_LOG("solving was wrong:");
		UG_LOGN(CPPString(A2, "Aold"));
		rhs2.maple_print("rhs");
		UG_LOGN(CPPString(A, "Adecomp"));
		rhs.maple_print("rhsDecomp");
		x.maple_print("x");
		f.maple_print("f");

	}

	return true;
}
void DenseMatrix<T>::_svd_lapack (DenseVector<T>& sigma)
{
  // The calling sequence for dgetrf is:
  // DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO )


  //  JOBU    (input) CHARACTER*1
  //          Specifies options for computing all or part of the matrix U:
  //          = 'A':  all M columns of U are returned in array U:
  //          = 'S':  the first min(m,n) columns of U (the left singular
  //                  vectors) are returned in the array U;
  //          = 'O':  the first min(m,n) columns of U (the left singular
  //                  vectors) are overwritten on the array A;
  //          = 'N':  no columns of U (no left singular vectors) are
  //                  computed.
  char JOBU = 'N';

  //  JOBVT   (input) CHARACTER*1
  //          Specifies options for computing all or part of the matrix
  //          V**T:
  //          = 'A':  all N rows of V**T are returned in the array VT;
  //          = 'S':  the first min(m,n) rows of V**T (the right singular
  //                  vectors) are returned in the array VT;
  //          = 'O':  the first min(m,n) rows of V**T (the right singular
  //                  vectors) are overwritten on the array A;
  //          = 'N':  no rows of V**T (no right singular vectors) are
  //                  computed.
  char JOBVT = 'N';

  std::vector<T> sigma_val;
  std::vector<T> U_val;
  std::vector<T> VT_val;

  _svd_helper(JOBU, JOBVT, sigma_val, U_val, VT_val);

  // Load the singular values into sigma, ignore U_val and VT_val
  const unsigned int n_sigma_vals =
    libmesh_cast_int<unsigned int>(sigma_val.size());
  sigma.resize(n_sigma_vals);
  for(unsigned int i=0; i<n_sigma_vals; i++)
    sigma(i) = sigma_val[i];

}
Beispiel #21
0
void DenseMatrix<T>::_lu_back_substitute (const DenseVector<T>& b,
					  DenseVector<T>& x ) const
{
  const unsigned int
    n_cols = this->n();

  libmesh_assert_equal_to (this->m(), n_cols);
  libmesh_assert_equal_to (this->m(), b.size());

  x.resize (n_cols);

  // A convenient reference to *this
  const DenseMatrix<T>& A = *this;

  // Temporary vector storage.  We use this instead of
  // modifying the RHS.
  DenseVector<T> z = b;

  // Lower-triangular "top to bottom" solve step, taking into account pivots
  for (unsigned int i=0; i<n_cols; ++i)
    {
      // Swap
      if (_pivots[i] != static_cast<int>(i))
	std::swap( z(i), z(_pivots[i]) );

      x(i) = z(i);

      for (unsigned int j=0; j<i; ++j)
	x(i) -= A(i,j)*x(j);

      x(i) /= A(i,i);
    }

  // Upper-triangular "bottom to top" solve step
  const unsigned int last_row = n_cols-1;

  for (int i=last_row; i>=0; --i)
    {
      for (int j=i+1; j<static_cast<int>(n_cols); ++j)
	x(i) -= A(i,j)*x(j);
    }
}
Beispiel #22
0
void DenseMatrix<T>::vector_mult_add (DenseVector<T>& dest,
                                      const T factor,
                                      const DenseVector<T>& arg) const
{
  // Short-circuit if the matrix is empty
  if(this->m() == 0)
  {
    dest.resize(0);
    return;
  }

  if (this->use_blas_lapack)
    this->_matvec_blas(factor, 1., dest, arg);
  else
    {
      DenseVector<T> temp(arg.size());
      this->vector_mult(temp, arg);
      dest.add(factor, temp);
    }
}
Beispiel #23
0
void DenseMatrix<T>::_cholesky_back_substitute (const DenseVector<T2>& b,
						DenseVector<T2>& x) const
{
  // Shorthand notation for number of rows and columns.
  const unsigned int
    n_rows = this->m(),
    n_cols = this->n();

  // Just to be really sure...
  libmesh_assert_equal_to (n_rows, n_cols);

  // A convenient reference to *this
  const DenseMatrix<T>& A = *this;

  // Now compute the solution to Ax =b using the factorization.
  x.resize(n_rows);

  // Solve for Ly=b
  for (unsigned int i=0; i<n_cols; ++i)
    {
      T2 temp = b(i);

      for (unsigned int k=0; k<i; ++k)
	temp -= A(i,k)*x(k);

      x(i) = temp / A(i,i);
    }

  // Solve for L^T x = y
  for (unsigned int i=0; i<n_cols; ++i)
    {
      const unsigned int ib = (n_cols-1)-i;

      for (unsigned int k=(ib+1); k<n_cols; ++k)
	x(ib) -= A(k,ib) * x(k);

      x(ib) /= A(ib,ib);
    }
}
// The matrix assembly function to be called at each time step to
// prepare for the linear solve.
void assemble_stokes (EquationSystems & es,
                      const std::string & system_name)
{
  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert_equal_to (system_name, "Navier-Stokes");

  // 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 Stokes system object.
  TransientLinearImplicitSystem & navier_stokes_system =
    es.get_system<TransientLinearImplicitSystem> ("Navier-Stokes");

  // Numeric ids corresponding to each variable in the system
  const unsigned int u_var = navier_stokes_system.variable_number ("u");
  const unsigned int v_var = navier_stokes_system.variable_number ("v");
  const unsigned int p_var = navier_stokes_system.variable_number ("p");
  const unsigned int alpha_var = navier_stokes_system.variable_number ("alpha");

  // Get the Finite Element type for "u".  Note this will be
  // the same as the type for "v".
  FEType fe_vel_type = navier_stokes_system.variable_type(u_var);

  // Get the Finite Element type for "p".
  FEType fe_pres_type = navier_stokes_system.variable_type(p_var);

  // Build a Finite Element object of the specified type for
  // the velocity variables.
  UniquePtr<FEBase> fe_vel  (FEBase::build(dim, fe_vel_type));

  // Build a Finite Element object of the specified type for
  // the pressure variables.
  UniquePtr<FEBase> fe_pres (FEBase::build(dim, fe_pres_type));

  // A Gauss quadrature rule for numerical integration.
  // Let the FEType object decide what order rule is appropriate.
  QGauss qrule (dim, fe_vel_type.default_quadrature_order());

  // Tell the finite element objects to use our quadrature rule.
  fe_vel->attach_quadrature_rule (&qrule);
  fe_pres->attach_quadrature_rule (&qrule);

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

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

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

  // The element shape functions for the pressure variable
  // evaluated at the quadrature points.
  const std::vector<std::vector<Real> > & psi = fe_pres->get_phi();

  // The value of the linear shape function gradients at the quadrature points
  // const std::vector<std::vector<RealGradient> > & dpsi = fe_pres->get_dphi();

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

  // 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".
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  DenseSubMatrix<Number>
    Kuu(Ke), Kuv(Ke), Kup(Ke),
    Kvu(Ke), Kvv(Ke), Kvp(Ke),
    Kpu(Ke), Kpv(Ke), Kpp(Ke);
  DenseSubMatrix<Number> Kalpha_p(Ke), Kp_alpha(Ke);

  DenseSubVector<Number>
    Fu(Fe),
    Fv(Fe),
    Fp(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<dof_id_type> dof_indices;
  std::vector<dof_id_type> dof_indices_u;
  std::vector<dof_id_type> dof_indices_v;
  std::vector<dof_id_type> dof_indices_p;
  std::vector<dof_id_type> dof_indices_alpha;

  // Find out what the timestep size parameter is from the system, and
  // the value of theta for the theta method.  We use implicit Euler (theta=1)
  // for this simulation even though it is only first-order accurate in time.
  // The reason for this decision is that the second-order Crank-Nicolson
  // method is notoriously oscillatory for problems with discontinuous
  // initial data such as the lid-driven cavity.  Therefore,
  // we sacrifice accuracy in time for stability, but since the solution
  // reaches steady state relatively quickly we can afford to take small
  // timesteps.  If you monitor the initial nonlinear residual for this
  // simulation, you should see that it is monotonically decreasing in time.
  const Real dt    = es.parameters.get<Real>("dt");
  // const Real time  = es.parameters.get<Real>("time");
  const Real theta = 1.;

  // Now we will loop over all the elements in the mesh that
  // live on the local processor. We will compute the element
  // matrix and right-hand-side contribution.  Since the mesh
  // will be refined we want to only consider the ACTIVE elements,
  // hence we use a variant of the active_elem_iterator.
  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)
    {
      // 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);
      dof_map.dof_indices (elem, dof_indices_u, u_var);
      dof_map.dof_indices (elem, dof_indices_v, v_var);
      dof_map.dof_indices (elem, dof_indices_p, p_var);
      dof_map.dof_indices (elem, dof_indices_alpha, alpha_var);

      const unsigned int n_dofs   = dof_indices.size();
      const unsigned int n_u_dofs = dof_indices_u.size();
      const unsigned int n_v_dofs = dof_indices_v.size();
      const unsigned int n_p_dofs = dof_indices_p.size();

      // 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_vel->reinit  (elem);
      fe_pres->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 (n_dofs, n_dofs);
      Fe.resize (n_dofs);

      // Reposition the submatrices...  The idea is this:
      //
      //         -           -          -  -
      //        | Kuu Kuv Kup |        | Fu |
      //   Ke = | Kvu Kvv Kvp |;  Fe = | Fv |
      //        | Kpu Kpv Kpp |        | Fp |
      //         -           -          -  -
      //
      // The DenseSubMatrix.repostition () member takes the
      // (row_offset, column_offset, row_size, column_size).
      //
      // Similarly, the DenseSubVector.reposition () member
      // takes the (row_offset, row_size)
      Kuu.reposition (u_var*n_u_dofs, u_var*n_u_dofs, n_u_dofs, n_u_dofs);
      Kuv.reposition (u_var*n_u_dofs, v_var*n_u_dofs, n_u_dofs, n_v_dofs);
      Kup.reposition (u_var*n_u_dofs, p_var*n_u_dofs, n_u_dofs, n_p_dofs);

      Kvu.reposition (v_var*n_v_dofs, u_var*n_v_dofs, n_v_dofs, n_u_dofs);
      Kvv.reposition (v_var*n_v_dofs, v_var*n_v_dofs, n_v_dofs, n_v_dofs);
      Kvp.reposition (v_var*n_v_dofs, p_var*n_v_dofs, n_v_dofs, n_p_dofs);

      Kpu.reposition (p_var*n_u_dofs, u_var*n_u_dofs, n_p_dofs, n_u_dofs);
      Kpv.reposition (p_var*n_u_dofs, v_var*n_u_dofs, n_p_dofs, n_v_dofs);
      Kpp.reposition (p_var*n_u_dofs, p_var*n_u_dofs, n_p_dofs, n_p_dofs);

      // Also, add a row and a column to constrain the pressure
      Kp_alpha.reposition (p_var*n_u_dofs, p_var*n_u_dofs+n_p_dofs, n_p_dofs, 1);
      Kalpha_p.reposition (p_var*n_u_dofs+n_p_dofs, p_var*n_u_dofs, 1, n_p_dofs);


      Fu.reposition (u_var*n_u_dofs, n_u_dofs);
      Fv.reposition (v_var*n_u_dofs, n_v_dofs);
      Fp.reposition (p_var*n_u_dofs, n_p_dofs);

      // Now we will build the element matrix and right-hand-side.
      // Constructing the RHS requires the solution and its
      // gradient from the previous timestep.  This must be
      // calculated at each quadrature point by summing the
      // solution degree-of-freedom values by the appropriate
      // weight functions.
      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          // Values to hold the solution & its gradient at the previous timestep.
          Number u = 0., u_old = 0.;
          Number v = 0., v_old = 0.;
          Number p_old = 0.;
          Gradient grad_u, grad_u_old;
          Gradient grad_v, grad_v_old;

          // Compute the velocity & its gradient from the previous timestep
          // and the old Newton iterate.
          for (unsigned int l=0; l<n_u_dofs; l++)
            {
              // From the old timestep:
              u_old += phi[l][qp]*navier_stokes_system.old_solution (dof_indices_u[l]);
              v_old += phi[l][qp]*navier_stokes_system.old_solution (dof_indices_v[l]);
              grad_u_old.add_scaled (dphi[l][qp], navier_stokes_system.old_solution (dof_indices_u[l]));
              grad_v_old.add_scaled (dphi[l][qp], navier_stokes_system.old_solution (dof_indices_v[l]));

              // From the previous Newton iterate:
              u += phi[l][qp]*navier_stokes_system.current_solution (dof_indices_u[l]);
              v += phi[l][qp]*navier_stokes_system.current_solution (dof_indices_v[l]);
              grad_u.add_scaled (dphi[l][qp], navier_stokes_system.current_solution (dof_indices_u[l]));
              grad_v.add_scaled (dphi[l][qp], navier_stokes_system.current_solution (dof_indices_v[l]));
            }

          // Compute the old pressure value at this quadrature point.
          for (unsigned int l=0; l<n_p_dofs; l++)
            p_old += psi[l][qp]*navier_stokes_system.old_solution (dof_indices_p[l]);

          // Definitions for convenience.  It is sometimes simpler to do a
          // dot product if you have the full vector at your disposal.
          const NumberVectorValue U_old (u_old, v_old);
          const NumberVectorValue U     (u,     v);
          const Number u_x = grad_u(0);
          const Number u_y = grad_u(1);
          const Number v_x = grad_v(0);
          const Number v_y = grad_v(1);

          // First, an i-loop over the velocity degrees of freedom.
          // We know that n_u_dofs == n_v_dofs so we can compute contributions
          // for both at the same time.
          for (unsigned int i=0; i<n_u_dofs; i++)
            {
              Fu(i) += JxW[qp]*(u_old*phi[i][qp] -                            // mass-matrix term
                                (1.-theta)*dt*(U_old*grad_u_old)*phi[i][qp] + // convection term
                                (1.-theta)*dt*p_old*dphi[i][qp](0)  -         // pressure term on rhs
                                (1.-theta)*dt*(grad_u_old*dphi[i][qp]) +      // diffusion term on rhs
                                theta*dt*(U*grad_u)*phi[i][qp]);              // Newton term


              Fv(i) += JxW[qp]*(v_old*phi[i][qp] -                             // mass-matrix term
                                (1.-theta)*dt*(U_old*grad_v_old)*phi[i][qp] +  // convection term
                                (1.-theta)*dt*p_old*dphi[i][qp](1) -           // pressure term on rhs
                                (1.-theta)*dt*(grad_v_old*dphi[i][qp]) +       // diffusion term on rhs
                                theta*dt*(U*grad_v)*phi[i][qp]);               // Newton term


              // Note that the Fp block is identically zero unless we are using
              // some kind of artificial compressibility scheme...

              // Matrix contributions for the uu and vv couplings.
              for (unsigned int j=0; j<n_u_dofs; j++)
                {
                  Kuu(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp] +                // mass matrix term
                                       theta*dt*(dphi[i][qp]*dphi[j][qp]) +   // diffusion term
                                       theta*dt*(U*dphi[j][qp])*phi[i][qp] +  // convection term
                                       theta*dt*u_x*phi[i][qp]*phi[j][qp]);   // Newton term

                  Kuv(i,j) += JxW[qp]*theta*dt*u_y*phi[i][qp]*phi[j][qp];     // Newton term

                  Kvv(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp] +                // mass matrix term
                                       theta*dt*(dphi[i][qp]*dphi[j][qp]) +   // diffusion term
                                       theta*dt*(U*dphi[j][qp])*phi[i][qp] +  // convection term
                                       theta*dt*v_y*phi[i][qp]*phi[j][qp]);   // Newton term

                  Kvu(i,j) += JxW[qp]*theta*dt*v_x*phi[i][qp]*phi[j][qp];     // Newton term
                }

              // Matrix contributions for the up and vp couplings.
              for (unsigned int j=0; j<n_p_dofs; j++)
                {
                  Kup(i,j) += JxW[qp]*(-theta*dt*psi[j][qp]*dphi[i][qp](0));
                  Kvp(i,j) += JxW[qp]*(-theta*dt*psi[j][qp]*dphi[i][qp](1));
                }
            }

          // Now an i-loop over the pressure degrees of freedom.  This code computes
          // the matrix entries due to the continuity equation.  Note: To maintain a
          // symmetric matrix, we may (or may not) multiply the continuity equation by
          // negative one.  Here we do not.
          for (unsigned int i=0; i<n_p_dofs; i++)
            {
              Kp_alpha(i,0) += JxW[qp]*psi[i][qp];
              Kalpha_p(0,i) += JxW[qp]*psi[i][qp];
              for (unsigned int j=0; j<n_u_dofs; j++)
                {
                  Kpu(i,j) += JxW[qp]*psi[i][qp]*dphi[j][qp](0);
                  Kpv(i,j) += JxW[qp]*psi[i][qp]*dphi[j][qp](1);
                }
            }
        } // end of the quadrature point qp-loop


      // 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. The penalty method used here
      // is equivalent (for Lagrange basis functions) to lumping
      // the matrix resulting from the L2 projection penalty
      // approach introduced in example 3.
      {
        // The penalty value.  \f$ \frac{1}{\epsilon} \f$
        const Real penalty = 1.e10;

        // 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 s=0; s<elem->n_sides(); s++)
          if (elem->neighbor(s) == libmesh_nullptr)
            {
              UniquePtr<Elem> side (elem->build_side(s));

              // Loop over the nodes on the side.
              for (unsigned int ns=0; ns<side->n_nodes(); ns++)
                {
                  // Boundary ids are set internally by
                  // build_square().
                  // 0=bottom
                  // 1=right
                  // 2=top
                  // 3=left

                  // Set u = 1 on the top boundary, 0 everywhere else
                  const Real u_value =
                    (mesh.get_boundary_info().has_boundary_id(elem, s, 2))
                    ? 1. : 0.;

                  // Set v = 0 everywhere
                  const Real v_value = 0.;

                  // Find the node on the element matching this node on
                  // the side.  That defined where in the element matrix
                  // the boundary condition will be applied.
                  for (unsigned int n=0; n<elem->n_nodes(); n++)
                    if (elem->node_id(n) == side->node_id(ns))
                      {
                        // Matrix contribution.
                        Kuu(n,n) += penalty;
                        Kvv(n,n) += penalty;

                        // Right-hand-side contribution.
                        Fu(n) += penalty*u_value;
                        Fv(n) += penalty*v_value;
                      }
                } // end face node loop
            } // end if (elem->neighbor(side) == libmesh_nullptr)
      } // end boundary condition section

      // 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);

      // 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 SparseMatrix::add_matrix()
      // and NumericVector::add_vector() members do this for us.
      navier_stokes_system.matrix->add_matrix (Ke, dof_indices);
      navier_stokes_system.rhs->add_vector    (Fe, dof_indices);
    } // end of element loop

  // We can set the mean of the pressure by setting Falpha
  navier_stokes_system.rhs->add(navier_stokes_system.rhs->size()-1, 10.);
}
Beispiel #25
0
// Define the matrix assembly function for the 1D PDE we are solving
void assemble_1D(EquationSystems& es, const std::string& system_name)
{

#ifdef LIBMESH_ENABLE_AMR

  // It is a good idea to check we are solving the correct system
  libmesh_assert_equal_to (system_name, "1D");

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

  // The dimension we are using, i.e. dim==1
  const unsigned int dim = mesh.mesh_dimension();

  // Get a reference to the system we are solving
  LinearImplicitSystem& system = es.get_system<LinearImplicitSystem>("1D");

  // Get a reference to the DofMap object for this system. The DofMap object
  // handles the index translation from node and element numbers to degree of
  // freedom numbers. DofMap's are discussed in more detail 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. The build
  // function dynamically allocates memory so we use an UniquePtr in this case.
  // An UniquePtr is a pointer that cleans up after itself. See examples 3 and 4
  // for more details on UniquePtr.
  UniquePtr<FEBase> fe(FEBase::build(dim, fe_type));

  // Tell the finite element object to use fifth order Gaussian quadrature
  QGauss qrule(dim,FIFTH);
  fe->attach_quadrature_rule(&qrule);

  // Here we define some references to cell-specific data that will be used to
  // assemble the linear system.

  // The element Jacobian * quadrature weight at each integration point.
  const std::vector<Real>& JxW = fe->get_JxW();

  // 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();

  // Declare a dense matrix and dense vector to hold the element matrix
  // and right-hand-side contribution
  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<dof_id_type> dof_indices;

  // We now loop over all the active elements in the mesh in order to calculate
  // the matrix and right-hand-side contribution from each element. Use a
  // const_element_iterator to loop over the elements. We make
  // el_end const as it is used only for the stopping condition of the loop.
  MeshBase::const_element_iterator el     = mesh.active_local_elements_begin();
  const MeshBase::const_element_iterator el_end = mesh.active_local_elements_end();

  // Note that ++el is preferred to el++ when using loops with iterators
  for( ; el != el_end; ++el)
    {
      // It is convenient to store a pointer to the current element
      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);

      // Store the number of local degrees of freedom contained in this element
      const int n_dofs = dof_indices.size();

      // We resize and zero out Ke and Fe (resize() also clears the matrix and
      // vector). In this example, all elements in the mesh are EDGE3's, so
      // Ke will always be 3x3, and Fe will always be 3x1. If the mesh contained
      // different element types, then the size of Ke and Fe would change.
      Ke.resize(n_dofs, n_dofs);
      Fe.resize(n_dofs);


      // Now loop over quadrature points to handle numerical integration
      for(unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          // Now build the element matrix and right-hand-side using loops to
          // integrate the test functions (i) against the trial functions (j).
          for(unsigned int i=0; i<phi.size(); i++)
            {
              Fe(i) += JxW[qp]*phi[i][qp];

              for(unsigned int j=0; j<phi.size(); j++)
                {
                  Ke(i,j) += JxW[qp]*(1.e-3*dphi[i][qp]*dphi[j][qp] +
                                      phi[i][qp]*phi[j][qp]);
                }
            }
        }


      // At this point we have completed the matrix and RHS summation. The
      // final step is to apply boundary conditions, which in this case are
      // simple Dirichlet conditions with u(0) = u(1) = 0.

      // Define the penalty parameter used to enforce the BC's
      double penalty = 1.e10;

      // Loop over the sides of this element. For a 1D element, the "sides"
      // are defined as the nodes on each edge of the element, i.e. 1D elements
      // have 2 sides.
      for(unsigned int s=0; s<elem->n_sides(); s++)
        {
          // If this element has a NULL neighbor, then it is on the edge of the
          // mesh and we need to enforce a boundary condition using the penalty
          // method.
          if(elem->neighbor(s) == NULL)
            {
              Ke(s,s) += penalty;
              Fe(s)   += 0*penalty;
            }
        }

      // This is a function call that is necessary when using adaptive
      // mesh refinement. See Adaptivity Example 2 for more details.
      dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

      // Add Ke and Fe to the global matrix and right-hand-side.
      system.matrix->add_matrix(Ke, dof_indices);
      system.rhs->add_vector(Fe, dof_indices);
    }
#endif // #ifdef LIBMESH_ENABLE_AMR
}
Beispiel #26
0
void MeshFunction::operator() (const Point& p,
                               const Real,
                               DenseVector<Number>& output,
                               const std::set<subdomain_id_type>* subdomain_ids)
{
  libmesh_assert (this->initialized());

  const Elem* element = this->find_element(p,subdomain_ids);

  if (!element)
    {
      output = _out_of_mesh_value;
    }
  else
    {
      // resize the output vector to the number of output values
      // that the user told us
      output.resize (cast_int<unsigned int>
                     (this->_system_vars.size()));


      {
        const unsigned int dim = element->dim();


        /*
         * Get local coordinates to feed these into compute_data().
         * Note that the fe_type can safely be used from the 0-variable,
         * since the inverse mapping is the same for all FEFamilies
         */
        const Point mapped_point (FEInterface::inverse_map (dim,
                                                            this->_dof_map.variable_type(0),
                                                            element,
                                                            p));


        // loop over all vars
        for (unsigned int index=0; index < this->_system_vars.size(); index++)
          {
            /*
             * the data for this variable
             */
            const unsigned int var = _system_vars[index];
            const FEType& fe_type = this->_dof_map.variable_type(var);

            /**
             * Build an FEComputeData that contains both input and output data
             * for the specific compute_data method.
             */
            {
              FEComputeData data (this->_eqn_systems, mapped_point);

              FEInterface::compute_data (dim, fe_type, element, data);

              // where the solution values for the var-th variable are stored
              std::vector<dof_id_type> dof_indices;
              this->_dof_map.dof_indices (element, dof_indices, var);

              // interpolate the solution
              {
                Number value = 0.;

                for (unsigned int i=0; i<dof_indices.size(); i++)
                  value += this->_vector(dof_indices[i]) * data.shape[i];

                output(index) = value;
              }

            }

            // next variable
          }
      }
    }

  // all done
  return;
}
// Here we define the matrix assembly routine for
// the Helmholtz system.  This function will be
// called to form the stiffness matrix and right-hand side.
void assemble_helmholtz(EquationSystems & es,
                        const std::string & system_name)
{

  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert_equal_to (system_name, "Helmholtz");

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

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

  // Get a reference to our system, as before
  FrequencySystem & f_system =
    es.get_system<FrequencySystem> (system_name);

  // A const reference to the DofMap object for this system.  The DofMap
  // object handles the index translation from node and element numbers
  // to degree of freedom numbers.
  const DofMap & dof_map = f_system.get_dof_map();

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

  // For the admittance boundary condition,
  // get the fluid density
  const Real rho = es.parameters.get<Real>("rho");

  // In here, we will add the element matrices to the
  // <i>additional</i> matrices "stiffness_mass", "damping",
  // and the additional vector "rhs", not to the members
  // "matrix" and "rhs".  Therefore, get writable
  // references to them
  SparseMatrix<Number> & stiffness = f_system.get_matrix("stiffness");
  SparseMatrix<Number> & damping = f_system.get_matrix("damping");
  SparseMatrix<Number> & mass = f_system.get_matrix("mass");
  NumericVector<Number> & freq_indep_rhs = f_system.get_vector("rhs");

  // Some solver packages (PETSc) are especially picky about
  // allocating sparsity structure and truly assigning values
  // to this structure.  Namely, matrix additions, as performed
  // later, exhibit acceptable performance only for identical
  // sparsity structures.  Therefore, explicitly zero the
  // values in the collective matrix, so that matrix additions
  // encounter identical sparsity structures.
  SparseMatrix<Number> & matrix = *f_system.matrix;

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

  // A 5th order Gauss quadrature rule for numerical integration.
  QGauss qrule (dim, FIFTH);

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

  // The element Jacobian// quadrature weight at each integration point.
  const std::vector<Real> & JxW = fe->get_JxW();

  // 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();

  // Now this is slightly different from example 4.
  // We will not add directly to the global matrix,
  // but to the additional matrices "stiffness_mass" and "damping".
  // The same holds for the right-hand-side vector Fe, which we will
  // later on store in the additional vector "rhs".
  // The zero_matrix is used to explicitly induce the same sparsity
  // structure in the overall matrix.
  // see later on. (At least) the mass, and stiffness matrices, however,
  // are inherently real.  Therefore, store these as one complex
  // matrix.  This will definitely save memory.
  DenseMatrix<Number> Ke, Ce, Me, zero_matrix;
  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<dof_id_type> dof_indices;

  // Now we will loop over all the elements in the mesh.
  // We will compute the element matrix and right-hand-side
  // contribution.

  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 element initialization.
      START_LOG("elem init", "assemble_helmholtz");

      // 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 & resize the element matrix and right-hand side before
      // summing them, with different element types in the mesh this
      // is quite necessary.
      {
        const unsigned int n_dof_indices = dof_indices.size();

        Ke.resize          (n_dof_indices, n_dof_indices);
        Ce.resize          (n_dof_indices, n_dof_indices);
        Me.resize          (n_dof_indices, n_dof_indices);
        zero_matrix.resize (n_dof_indices, n_dof_indices);
        Fe.resize          (n_dof_indices);
      }

      // Stop logging the element initialization.
      STOP_LOG("elem init", "assemble_helmholtz");

      // Now loop over the quadrature points.  This handles
      // the numeric integration.
      START_LOG("stiffness & mass", "assemble_helmholtz");

      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          // Now we will build the element matrix.  This involves
          // a double loop to integrate the test funcions (i) against
          // the trial functions (j).  Note the braces on the rhs
          // of Ke(i,j): these are quite necessary to finally compute
          // Real*(Point*Point) = Real, and not something else...
          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]);
                Me(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp]);
              }
        }

      STOP_LOG("stiffness & mass", "assemble_helmholtz");

      // Now compute the contribution to the element matrix
      // (due to mixed boundary conditions) if the current
      // element lies on the boundary.
      //
      // 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) == libmesh_nullptr)
          {
            LOG_SCOPE("damping", "assemble_helmholtz");

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

            // Boundary integration requires one quadraure rule,
            // with dimensionality one less than the dimensionality
            // of the element.
            QGauss qface(dim-1, SECOND);

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

            // 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();

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

            // For the Robin BCs consider a normal admittance an=1
            // at some parts of the bounfdary
            const Real an_value = 1.;

            // Loop over the face quadrature points for integration.
            for (unsigned int qp=0; qp<qface.n_points(); qp++)
              {

                // Element matrix contributrion due to precribed
                // admittance boundary conditions.
                for (unsigned int i=0; i<phi_face.size(); i++)
                  for (unsigned int j=0; j<phi_face.size(); j++)
                    Ce(i,j) += rho*an_value*JxW_face[qp]*phi_face[i][qp]*phi_face[j][qp];
              }
          }

      // If this assembly program were to be used on an adaptive mesh,
      // we would have to apply any hanging node constraint equations
      // by uncommenting the following lines:
      // std::vector<unsigned int> dof_indicesC = dof_indices;
      // std::vector<unsigned int> dof_indicesM = dof_indices;
      // dof_map.constrain_element_matrix (Ke, dof_indices);
      // dof_map.constrain_element_matrix (Ce, dof_indicesC);
      // dof_map.constrain_element_matrix (Me, dof_indicesM);


      // Finally, simply add the contributions to the additional
      // matrices and vector.
      stiffness.add_matrix (Ke, dof_indices);
      damping.add_matrix (Ce, dof_indices);
      mass.add_matrix (Me, dof_indices);

      // For the overall matrix, explicitly zero the entries where
      // we added values in the other ones, so that we have
      // identical sparsity footprints.
      matrix.add_matrix(zero_matrix, dof_indices);

    } // loop el


  // It now remains to compute the rhs. Here, we simply
  // get the normal velocities values on the boundary from the
  // mesh data.
  {
    LOG_SCOPE("rhs", "assemble_helmholtz");

    // get a reference to the mesh data.
    const MeshData & mesh_data = es.get_mesh_data();

    // We will now loop over all nodes. In case nodal data
    // for a certain node is available in the MeshData, we simply
    // adopt the corresponding value for the rhs vector.
    // Note that normal data was given in the mesh data file,
    // i.e. one value per node
    libmesh_assert_equal_to (mesh_data.n_val_per_node(), 1);

    MeshBase::const_node_iterator       node_it  = mesh.nodes_begin();
    const MeshBase::const_node_iterator node_end = mesh.nodes_end();

    for ( ; node_it != node_end; ++node_it)
      {
        // the current node pointer
        const Node * node = *node_it;

        // check if the mesh data has data for the current node
        // and do for all components
        if (mesh_data.has_data(node))
          for (unsigned int comp=0; comp<node->n_comp(0, 0); comp++)
            {
              // the dof number
              unsigned int dn = node->dof_number(0, 0, comp);

              // set the nodal value
              freq_indep_rhs.set(dn, mesh_data.get_data(node)[0]);
            }
      }
  }

  // All done!
}
Beispiel #28
0
void assemble_elasticity(EquationSystems & es,
                         const std::string & libmesh_dbg_var(system_name))
{
  libmesh_assert_equal_to (system_name, "Elasticity");

  const MeshBase & mesh = es.get_mesh();

  const unsigned int dim = mesh.mesh_dimension();

  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Elasticity");

  const unsigned int u_var = system.variable_number ("u");
  const unsigned int v_var = system.variable_number ("v");

  const DofMap & dof_map = system.get_dof_map();
  FEType fe_type = dof_map.variable_type(0);
  UniquePtr<FEBase> fe (FEBase::build(dim, fe_type));
  QGauss qrule (dim, fe_type.default_quadrature_order());
  fe->attach_quadrature_rule (&qrule);

  UniquePtr<FEBase> fe_face (FEBase::build(dim, fe_type));
  QGauss qface(dim-1, fe_type.default_quadrature_order());
  fe_face->attach_quadrature_rule (&qface);

  const std::vector<Real> & JxW = fe->get_JxW();
  const std::vector<std::vector<RealGradient> > & dphi = fe->get_dphi();

  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  DenseSubMatrix<Number>
    Kuu(Ke), Kuv(Ke),
    Kvu(Ke), Kvv(Ke);

  DenseSubVector<Number>
    Fu(Fe),
    Fv(Fe);

  std::vector<dof_id_type> dof_indices;
  std::vector<dof_id_type> dof_indices_u;
  std::vector<dof_id_type> dof_indices_v;

  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)
    {
      const Elem * elem = *el;

      dof_map.dof_indices (elem, dof_indices);
      dof_map.dof_indices (elem, dof_indices_u, u_var);
      dof_map.dof_indices (elem, dof_indices_v, v_var);

      const unsigned int n_dofs   = dof_indices.size();
      const unsigned int n_u_dofs = dof_indices_u.size();
      const unsigned int n_v_dofs = dof_indices_v.size();

      fe->reinit (elem);

      Ke.resize (n_dofs, n_dofs);
      Fe.resize (n_dofs);

      Kuu.reposition (u_var*n_u_dofs, u_var*n_u_dofs, n_u_dofs, n_u_dofs);
      Kuv.reposition (u_var*n_u_dofs, v_var*n_u_dofs, n_u_dofs, n_v_dofs);

      Kvu.reposition (v_var*n_v_dofs, u_var*n_v_dofs, n_v_dofs, n_u_dofs);
      Kvv.reposition (v_var*n_v_dofs, v_var*n_v_dofs, n_v_dofs, n_v_dofs);

      Fu.reposition (u_var*n_u_dofs, n_u_dofs);
      Fv.reposition (v_var*n_u_dofs, n_v_dofs);

      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          for (unsigned int i=0; i<n_u_dofs; i++)
            for (unsigned int j=0; j<n_u_dofs; j++)
              {
                // Tensor indices
                unsigned int C_i, C_j, C_k, C_l;
                C_i=0, C_k=0;

                C_j=0, C_l=0;
                Kuu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=0;
                Kuu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=0, C_l=1;
                Kuu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=1;
                Kuu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));
              }

          for (unsigned int i=0; i<n_u_dofs; i++)
            for (unsigned int j=0; j<n_v_dofs; j++)
              {
                // Tensor indices
                unsigned int C_i, C_j, C_k, C_l;
                C_i=0, C_k=1;

                C_j=0, C_l=0;
                Kuv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=0;
                Kuv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=0, C_l=1;
                Kuv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=1;
                Kuv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));
              }

          for (unsigned int i=0; i<n_v_dofs; i++)
            for (unsigned int j=0; j<n_u_dofs; j++)
              {
                // Tensor indices
                unsigned int C_i, C_j, C_k, C_l;
                C_i=1, C_k=0;

                C_j=0, C_l=0;
                Kvu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=0;
                Kvu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=0, C_l=1;
                Kvu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=1;
                Kvu(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));
              }

          for (unsigned int i=0; i<n_v_dofs; i++)
            for (unsigned int j=0; j<n_v_dofs; j++)
              {
                // Tensor indices
                unsigned int C_i, C_j, C_k, C_l;
                C_i=1, C_k=1;

                C_j=0, C_l=0;
                Kvv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=0;
                Kvv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=0, C_l=1;
                Kvv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));

                C_j=1, C_l=1;
                Kvv(i,j) += JxW[qp]*(eval_elasticity_tensor(C_i, C_j, C_k, C_l) * dphi[i][qp](C_j)*dphi[j][qp](C_l));
              }
        }

      {
        for (unsigned int side=0; side<elem->n_sides(); side++)
          if (elem->neighbor_ptr(side) == libmesh_nullptr)
            {
              const std::vector<std::vector<Real> > & phi_face = fe_face->get_phi();
              const std::vector<Real> & JxW_face = fe_face->get_JxW();

              fe_face->reinit(elem, side);

              if (mesh.get_boundary_info().has_boundary_id (elem, side, 1)) // Apply a traction on the right side
                {
                  for (unsigned int qp=0; qp<qface.n_points(); qp++)
                    for (unsigned int i=0; i<n_v_dofs; i++)
                      Fv(i) += JxW_face[qp] * (-1.) * phi_face[i][qp];
                }
            }
      }

      dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

      system.matrix->add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);
    }
}
void assemble_stokes (EquationSystems& es,
                      const std::string& system_name)
{
  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert_equal_to (system_name, "Stokes");

  // 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 Convection-Diffusion system object.
  LinearImplicitSystem & system =
    es.get_system<LinearImplicitSystem> ("Stokes");

  // Numeric ids corresponding to each variable in the system
  const unsigned int u_var = system.variable_number ("u");
  const unsigned int v_var = system.variable_number ("v");
  const unsigned int p_var = system.variable_number ("p");

  // Get the Finite Element type for "u".  Note this will be
  // the same as the type for "v".
  FEType fe_vel_type = system.variable_type(u_var);

  // Get the Finite Element type for "p".
  FEType fe_pres_type = system.variable_type(p_var);

  // Build a Finite Element object of the specified type for
  // the velocity variables.
  UniquePtr<FEBase> fe_vel  (FEBase::build(dim, fe_vel_type));

  // Build a Finite Element object of the specified type for
  // the pressure variables.
  UniquePtr<FEBase> fe_pres (FEBase::build(dim, fe_pres_type));

  // A Gauss quadrature rule for numerical integration.
  // Let the \p FEType object decide what order rule is appropriate.
  QGauss qrule (dim, fe_vel_type.default_quadrature_order());

  // Tell the finite element objects to use our quadrature rule.
  fe_vel->attach_quadrature_rule (&qrule);
  fe_pres->attach_quadrature_rule (&qrule);

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

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

  // The element shape functions for the pressure variable
  // evaluated at the quadrature points.
  const std::vector<std::vector<Real> >& psi = fe_pres->get_phi();

  // 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();

  // 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".
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  DenseSubMatrix<Number>
    Kuu(Ke), Kuv(Ke), Kup(Ke),
    Kvu(Ke), Kvv(Ke), Kvp(Ke),
    Kpu(Ke), Kpv(Ke), Kpp(Ke);

  DenseSubVector<Number>
    Fu(Fe),
    Fv(Fe),
    Fp(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<dof_id_type> dof_indices;
  std::vector<dof_id_type> dof_indices_u;
  std::vector<dof_id_type> dof_indices_v;
  std::vector<dof_id_type> dof_indices_p;

  // Now we will loop over all the elements in the mesh that
  // live on the local processor. We will compute the element
  // matrix and right-hand-side contribution.  In case users later
  // modify this program to include refinement, we will be safe and
  // will only consider the active elements; hence we use a variant of
  // the \p active_elem_iterator.

  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)
    {
      // 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);
      dof_map.dof_indices (elem, dof_indices_u, u_var);
      dof_map.dof_indices (elem, dof_indices_v, v_var);
      dof_map.dof_indices (elem, dof_indices_p, p_var);

      const unsigned int n_dofs   = dof_indices.size();
      const unsigned int n_u_dofs = dof_indices_u.size();
      const unsigned int n_v_dofs = dof_indices_v.size();
      const unsigned int n_p_dofs = dof_indices_p.size();

      // 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_vel->reinit  (elem);
      fe_pres->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 (n_dofs, n_dofs);
      Fe.resize (n_dofs);

      // Reposition the submatrices...  The idea is this:
      //
      //         -           -          -  -
      //        | Kuu Kuv Kup |        | Fu |
      //   Ke = | Kvu Kvv Kvp |;  Fe = | Fv |
      //        | Kpu Kpv Kpp |        | Fp |
      //         -           -          -  -
      //
      // The \p DenseSubMatrix.repostition () member takes the
      // (row_offset, column_offset, row_size, column_size).
      //
      // Similarly, the \p DenseSubVector.reposition () member
      // takes the (row_offset, row_size)
      Kuu.reposition (u_var*n_u_dofs, u_var*n_u_dofs, n_u_dofs, n_u_dofs);
      Kuv.reposition (u_var*n_u_dofs, v_var*n_u_dofs, n_u_dofs, n_v_dofs);
      Kup.reposition (u_var*n_u_dofs, p_var*n_u_dofs, n_u_dofs, n_p_dofs);

      Kvu.reposition (v_var*n_v_dofs, u_var*n_v_dofs, n_v_dofs, n_u_dofs);
      Kvv.reposition (v_var*n_v_dofs, v_var*n_v_dofs, n_v_dofs, n_v_dofs);
      Kvp.reposition (v_var*n_v_dofs, p_var*n_v_dofs, n_v_dofs, n_p_dofs);

      Kpu.reposition (p_var*n_u_dofs, u_var*n_u_dofs, n_p_dofs, n_u_dofs);
      Kpv.reposition (p_var*n_u_dofs, v_var*n_u_dofs, n_p_dofs, n_v_dofs);
      Kpp.reposition (p_var*n_u_dofs, p_var*n_u_dofs, n_p_dofs, n_p_dofs);

      Fu.reposition (u_var*n_u_dofs, n_u_dofs);
      Fv.reposition (v_var*n_u_dofs, n_v_dofs);
      Fp.reposition (p_var*n_u_dofs, n_p_dofs);

      // Now we will build the element matrix.
      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          // Assemble the u-velocity row
          // uu coupling
          for (unsigned int i=0; i<n_u_dofs; i++)
            for (unsigned int j=0; j<n_u_dofs; j++)
              Kuu(i,j) += JxW[qp]*(dphi[i][qp]*dphi[j][qp]);

          // up coupling
          for (unsigned int i=0; i<n_u_dofs; i++)
            for (unsigned int j=0; j<n_p_dofs; j++)
              Kup(i,j) += -JxW[qp]*psi[j][qp]*dphi[i][qp](0);


          // Assemble the v-velocity row
          // vv coupling
          for (unsigned int i=0; i<n_v_dofs; i++)
            for (unsigned int j=0; j<n_v_dofs; j++)
              Kvv(i,j) += JxW[qp]*(dphi[i][qp]*dphi[j][qp]);

          // vp coupling
          for (unsigned int i=0; i<n_v_dofs; i++)
            for (unsigned int j=0; j<n_p_dofs; j++)
              Kvp(i,j) += -JxW[qp]*psi[j][qp]*dphi[i][qp](1);


          // Assemble the pressure row
          // pu coupling
          for (unsigned int i=0; i<n_p_dofs; i++)
            for (unsigned int j=0; j<n_u_dofs; j++)
              Kpu(i,j) += -JxW[qp]*psi[i][qp]*dphi[j][qp](0);

          // pv coupling
          for (unsigned int i=0; i<n_p_dofs; i++)
            for (unsigned int j=0; j<n_v_dofs; j++)
              Kpv(i,j) += -JxW[qp]*psi[i][qp]*dphi[j][qp](1);

        } // end of the quadrature point qp-loop

      // 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. The penalty method used here
      // is equivalent (for Lagrange basis functions) to lumping
      // the matrix resulting from the L2 projection penalty
      // approach introduced in example 3.
      {
        // 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 s=0; s<elem->n_sides(); s++)
          if (elem->neighbor(s) == NULL)
            {
              UniquePtr<Elem> side (elem->build_side(s));

              // Loop over the nodes on the side.
              for (unsigned int ns=0; ns<side->n_nodes(); ns++)
                {
                  // The location on the boundary of the current
                  // node.

                  // const Real xf = side->point(ns)(0);
                  const Real yf = side->point(ns)(1);

                  // The penalty value.  \f$ \frac{1}{\epsilon \f$
                  const Real penalty = 1.e10;

                  // The boundary values.

                  // Set u = 1 on the top boundary, 0 everywhere else
                  const Real u_value = (yf > .99) ? 1. : 0.;

                  // Set v = 0 everywhere
                  const Real v_value = 0.;

                  // Find the node on the element matching this node on
                  // the side.  That defined where in the element matrix
                  // the boundary condition will be applied.
                  for (unsigned int n=0; n<elem->n_nodes(); n++)
                    if (elem->node(n) == side->node(ns))
                      {
                        // Matrix contribution.
                        Kuu(n,n) += penalty;
                        Kvv(n,n) += penalty;

                        // Right-hand-side contribution.
                        Fu(n) += penalty*u_value;
                        Fv(n) += penalty*v_value;
                      }
                } // end face node loop
            } // end if (elem->neighbor(side) == NULL)
      } // end boundary condition section

      // 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);

      // 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 NumericMatrix::add_matrix()
      // and \p NumericVector::add_vector() members do this for us.
      system.matrix->add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);
    } // end of element loop

  // That's it.
  return;
}
void
MultiAppProjectionTransfer::assembleL2From(EquationSystems & es, const std::string & system_name)
{
  unsigned int n_apps = _multi_app->numGlobalApps();
  std::vector<NumericVector<Number> *> from_slns(n_apps, NULL);
  std::vector<MeshFunction *> from_fns(n_apps, NULL);
  std::vector<MeshTools::BoundingBox *> from_bbs(n_apps, NULL);

  // get bounding box, mesh function and solution for each subapp
  for (unsigned int i = 0; i < n_apps; i++)
  {
    if (!_multi_app->hasLocalApp(i))
      continue;

    MPI_Comm swapped = Moose::swapLibMeshComm(_multi_app->comm());

    FEProblem & from_problem = *_multi_app->appProblem(i);
    EquationSystems & from_es = from_problem.es();
    MeshBase & from_mesh = from_es.get_mesh();
    MeshTools::BoundingBox * app_box = new MeshTools::BoundingBox(MeshTools::processor_bounding_box(from_mesh, from_mesh.processor_id()));
    from_bbs[i] = app_box;

    MooseVariable & from_var = from_problem.getVariable(0, _from_var_name);
    System & from_sys = from_var.sys().system();
    unsigned int from_var_num = from_sys.variable_number(from_var.name());

    NumericVector<Number> * serialized_from_solution = NumericVector<Number>::build(from_sys.comm()).release();
    serialized_from_solution->init(from_sys.n_dofs(), false, SERIAL);
    // Need to pull down a full copy of this vector on every processor so we can get values in parallel
    from_sys.solution->localize(*serialized_from_solution);
    from_slns[i] = serialized_from_solution;

    MeshFunction * from_func = new MeshFunction(from_es, *serialized_from_solution, from_sys.get_dof_map(), from_var_num);
    from_func->init(Trees::ELEMENTS);
    from_func->enable_out_of_mesh_mode(NOTFOUND);
    from_fns[i] = from_func;

    Moose::swapLibMeshComm(swapped);
  }


  const MeshBase& mesh = es.get_mesh();
  const unsigned int dim = mesh.mesh_dimension();

  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>(system_name);

  FEType fe_type = system.variable_type(0);
  AutoPtr<FEBase> fe(FEBase::build(dim, fe_type));
  QGauss qrule(dim, fe_type.default_quadrature_order());
  fe->attach_quadrature_rule(&qrule);
  const std::vector<Real> & JxW = fe->get_JxW();
  const std::vector<std::vector<Real> > & phi = fe->get_phi();
  const std::vector<Point> & xyz = fe->get_xyz();

  const DofMap& dof_map = system.get_dof_map();
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;
  std::vector<dof_id_type> dof_indices;

  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)
  {
    const Elem* elem = *el;

    fe->reinit (elem);

    dof_map.dof_indices (elem, dof_indices);
    Ke.resize (dof_indices.size(), dof_indices.size());
    Fe.resize (dof_indices.size());

    for (unsigned int qp = 0; qp < qrule.n_points(); qp++)
    {
      Point qpt = xyz[qp];
      Real f = 0.;
      for (unsigned int app = 0; app < n_apps; app++)
      {
        Point pt = qpt - _multi_app->position(app);
        if (from_bbs[app] != NULL && from_bbs[app]->contains_point(pt))
        {
          MPI_Comm swapped = Moose::swapLibMeshComm(_multi_app->comm());
          f = (*from_fns[app])(pt);
          Moose::swapLibMeshComm(swapped);
          break;
        }
      }

      // Now compute the element matrix and RHS contributions.
      for (unsigned int i=0; i<phi.size(); i++)
      {
        // RHS
        Fe(i) += JxW[qp] * (f * phi[i][qp]);

        if (_compute_matrix)
          for (unsigned int j = 0; j < phi.size(); j++)
          {
            // The matrix contribution
            Ke(i,j) += JxW[qp] * (phi[i][qp] * phi[j][qp]);
          }
      }
      dof_map.constrain_element_matrix_and_vector(Ke, Fe, dof_indices);

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

  for (unsigned int i = 0; i < n_apps; i++)
  {
    delete from_fns[i];
    delete from_bbs[i];
    delete from_slns[i];
  }
}