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);
}
}
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();
}
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;
}
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);
}
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);
}
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;
}
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;
}
}
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;
}
}
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;
}
}
// 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));
}
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;
}
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;
}
}
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));
}
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;
}
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];
}
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);
}
}
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);
}
}
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.);
}
// 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
}
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!
}
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];
}
}