// This function assembles the system matrix and right-hand-side
// for the discrete form of our wave equation.
void assemble_wave(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, "Wave");
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
// Get a constant reference to the mesh object.
const MeshBase & mesh = es.get_mesh();
// Get a reference to the system we are solving.
LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Wave");
// 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.
const DofMap & dof_map = system.get_dof_map();
// The dimension that we are running.
const unsigned int dim = mesh.mesh_dimension();
// Copy the speed of sound to a local variable.
const Real speed = es.parameters.get<Real>("speed");
// 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);
// Build a Finite Element object of the specified type. Since the
// \p FEBase::build() member dynamically creates memory we will
// store the object as an \p UniquePtr<FEBase>. Check ex5 for details.
UniquePtr<FEBase> fe (FEBase::build(dim, fe_type));
// Do the same for an infinite element.
UniquePtr<FEBase> inf_fe (FEBase::build_InfFE(dim, fe_type));
// A 2nd order Gauss quadrature rule for numerical integration.
QGauss qrule (dim, SECOND);
// Tell the finite element object to use our quadrature rule.
fe->attach_quadrature_rule (&qrule);
// Due to its internal structure, the infinite element handles
// quadrature rules differently. It takes the quadrature
// rule which has been initialized for the FE object, but
// creates suitable quadrature rules by @e itself. The user
// need not worry about this.
inf_fe->attach_quadrature_rule (&qrule);
// Define data structures to contain the element matrix
// and right-hand-side vector contribution. Following
// basic finite element terminology we will denote these
// "Ke", "Ce", "Me", and "Fe" for the stiffness, damping
// and mass matrices, and the load vector. Note that in
// Acoustics, these descriptors though do @e not match the
// true physical meaning of the projectors. The final
// overall system, however, resembles the conventional
// notation again.
DenseMatrix<Number> Ke;
DenseMatrix<Number> Ce;
DenseMatrix<Number> Me;
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)
{
// 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);
// The mesh contains both finite and infinite elements. These
// elements are handled through different classes, namely
// \p FE and \p InfFE, respectively. However, since both
// are derived from \p FEBase, they share the same interface,
// and overall burden of coding is @e greatly reduced through
// using a pointer, which is adjusted appropriately to the
// current element type.
FEBase * cfe = libmesh_nullptr;
// This here is almost the only place where we need to
// distinguish between finite and infinite elements.
// For faster computation, however, different approaches
// may be feasible.
//
// Up to now, we do not know what kind of element we
// have. Aske the element of what type it is:
if (elem->infinite())
{
// We have an infinite element. Let \p cfe point
// to our \p InfFE object. This is handled through
// an UniquePtr. Through the \p UniquePtr::get() we "borrow"
// the pointer, while the \p UniquePtr \p inf_fe is
// still in charge of memory management.
cfe = inf_fe.get();
}
else
{
// This is a conventional finite element. Let \p fe handle it.
cfe = fe.get();
// Boundary conditions.
// Here we just zero the rhs-vector. For natural boundary
// conditions check e.g. previous examples.
{
// Zero the RHS for this element.
Fe.resize (dof_indices.size());
system.rhs->add_vector (Fe, dof_indices);
} // end boundary condition section
} // else if (elem->infinite())
// This is slightly different from the Poisson solver:
// Since the finite element object may change, we have to
// initialize the constant references to the data fields
// each time again, when a new element is processed.
//
// The element Jacobian * quadrature weight at each integration point.
const std::vector<Real> & JxW = cfe->get_JxW();
// The element shape functions evaluated at the quadrature points.
const std::vector<std::vector<Real> > & phi = cfe->get_phi();
// The element shape function gradients evaluated at the quadrature
// points.
const std::vector<std::vector<RealGradient> > & dphi = cfe->get_dphi();
// The infinite elements need more data fields than conventional FE.
// These are the gradients of the phase term \p dphase, an additional
// radial weight for the test functions \p Sobolev_weight, and its
// gradient.
//
// Note that these data fields are also initialized appropriately by
// the \p FE method, so that the weak form (below) is valid for @e both
// finite and infinite elements.
const std::vector<RealGradient> & dphase = cfe->get_dphase();
const std::vector<Real> & weight = cfe->get_Sobolev_weight();
const std::vector<RealGradient> & dweight = cfe->get_Sobolev_dweight();
// Now this is all independent of whether we use an \p FE
// or an \p InfFE. Nice, hm? ;-)
//
// Compute the element-specific data, as described
// in previous examples.
cfe->reinit (elem);
// Zero the element matrices. Boundary conditions were already
// processed in the \p FE-only section, see above.
Ke.resize (dof_indices.size(), dof_indices.size());
Ce.resize (dof_indices.size(), dof_indices.size());
Me.resize (dof_indices.size(), dof_indices.size());
// The total number of quadrature points for infinite elements
// @e has to be determined in a different way, compared to
// conventional finite elements. This type of access is also
// valid for finite elements, so this can safely be used
// anytime, instead of asking the quadrature rule, as
// seen in previous examples.
unsigned int max_qp = cfe->n_quadrature_points();
// Loop over the quadrature points.
for (unsigned int qp=0; qp<max_qp; qp++)
{
// Similar to the modified access to the number of quadrature
// points, the number of shape functions may also be obtained
// in a different manner. This offers the great advantage
// of being valid for both finite and infinite elements.
const unsigned int n_sf = cfe->n_shape_functions();
// Now we will build the element matrices. Since the infinite
// elements are based on a Petrov-Galerkin scheme, the
// resulting system matrices are non-symmetric. The additional
// weight, described before, is part of the trial space.
//
// For the finite elements, though, these matrices are symmetric
// just as we know them, since the additional fields \p dphase,
// \p weight, and \p dweight are initialized appropriately.
//
// test functions: weight[qp]*phi[i][qp]
// trial functions: phi[j][qp]
// phase term: phase[qp]
//
// derivatives are similar, but note that these are of type
// Point, not of type Real.
for (unsigned int i=0; i<n_sf; i++)
for (unsigned int j=0; j<n_sf; j++)
{
// (ndt*Ht + nHt*d) * nH
Ke(i,j) +=
(
(dweight[qp] * phi[i][qp] // Point * Real = Point
+ // +
dphi[i][qp] * weight[qp] // Point * Real = Point
) * dphi[j][qp]
) * JxW[qp];
// (d*Ht*nmut*nH - ndt*nmu*Ht*H - d*nHt*nmu*H)
Ce(i,j) +=
(
(dphase[qp] * dphi[j][qp]) // (Point * Point) = Real
* weight[qp] * phi[i][qp] // * Real * Real = Real
- // -
(dweight[qp] * dphase[qp]) // (Point * Point) = Real
* phi[i][qp] * phi[j][qp] // * Real * Real = Real
- // -
(dphi[i][qp] * dphase[qp]) // (Point * Point) = Real
* weight[qp] * phi[j][qp] // * Real * Real = Real
) * JxW[qp];
// (d*Ht*H * (1 - nmut*nmu))
Me(i,j) +=
(
(1. - (dphase[qp] * dphase[qp])) // (Real - (Point * Point)) = Real
* phi[i][qp] * phi[j][qp] * weight[qp] // * Real * Real * Real = Real
) * JxW[qp];
} // end of the matrix summation loop
} // end of quadrature point loop
// The element matrices are now built for this element.
// Collect them in Ke, and then add them to the global matrix.
// The \p SparseMatrix::add_matrix() member does this for us.
Ke.add(1./speed , Ce);
Ke.add(1./(speed*speed), Me);
// 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(Ke, dof_indices);
system.matrix->add_matrix (Ke, dof_indices);
} // end of element loop
// Note that we have not applied any boundary conditions so far.
// Here we apply a unit load at the node located at (0,0,0).
{
// Iterate over local nodes
MeshBase::const_node_iterator nd = mesh.local_nodes_begin();
const MeshBase::const_node_iterator nd_end = mesh.local_nodes_end();
for (; nd != nd_end; ++nd)
{
// Get a reference to the current node.
const Node & node = **nd;
// Check the location of the current node.
if (std::abs(node(0)) < TOLERANCE &&
std::abs(node(1)) < TOLERANCE &&
std::abs(node(2)) < TOLERANCE)
{
// The global number of the respective degree of freedom.
unsigned int dn = node.dof_number(0,0,0);
system.rhs->add (dn, 1.);
}
}
}
#else
// dummy assert
libmesh_assert_not_equal_to (es.get_mesh().mesh_dimension(), 1);
#endif //ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
}
// Stiffness and RHS assembly
// Equation references are from Samanta and Zabaras, 2005
void EnergySystem :: assemble(){
// GENERAL VARIABLES
// Get a constant reference to the mesh object.
const MeshBase& mesh = this->get_mesh();
// The dimension that we are running
const unsigned int dim = this->ndim();
// FEM THERMODYNAMIC RELATIONSHIPS (ThermoEq Class)
// Determine the FEM type (should be same for all ThermoEq variables)
FEType fe_type_thermo = thermo->variable_type(0);
// Build FE object; accessed via a pointer
AutoPtr<FEBase> fe_thermo(FEBase::build(dim, fe_type_thermo));
// Setup a quadrature rule
QGauss qrule_thermo(dim, fe_type_thermo.default_quadrature_order());
// Link FE and Quadrature
fe_thermo->attach_quadrature_rule(&qrule_thermo);
// References to shape functions and derivatives
const vector<std::vector<Real> >& N_thermo = fe_thermo->get_phi();
const vector<std::vector<RealGradient> >& B_thermo = fe_thermo->get_dphi();
// Setup a DOF map
const DofMap& dof_map_thermo = thermo->get_dof_map();
// FEM MOMENTUM EQUATION
// Determine the FEM type
FEType fe_type_momentum = momentum->variable_type(0);
// Build FE object; accessed via a pointer
AutoPtr<FEBase> fe_momentum(FEBase::build(dim, fe_type_momentum));
// Setup a quadrature rule
QGauss qrule_momentum(dim, fe_type_momentum.default_quadrature_order());
// Link FE and Quadrature
fe_momentum->attach_quadrature_rule(&qrule_momentum);
// References to shape functions and derivatives
const vector<std::vector<Real> >& N_momentum = fe_momentum->get_phi();
// Setup a DOF map
const DofMap& dof_map_momentum = momentum->get_dof_map();
// FEM ENERGY EQ. RELATIONSHIPS
// Get a constant reference to the Finite Element type
// for the first (and only) variable in the system.
FEType fe_type = this->variable_type(0);
// Build a Finite Element object of the specified type
AutoPtr<FEBase> fe (FEBase::build(dim, fe_type));
AutoPtr<FEBase> fe_face (FEBase::build(dim, fe_type));
// A Gauss quadrature rule for numerical integration.
// Let the \p FEType object decide what order rule is appropriate.
QGauss qrule (dim, fe_type.default_quadrature_order());
QGauss qface (dim-1, fe_type.default_quadrature_order());
// Tell the finite element object to use our quadrature rule.
fe->attach_quadrature_rule(&qrule);
fe_face->attach_quadrature_rule(&qface);
// Here we define some references to cell-specific data that
// will be used to assemble the linear system. We will start
// with the element Jacobian * quadrature weight at each integration point.
const vector<Real>& JxW = fe->get_JxW();
const vector<Real>& JxW_face = fe_face->get_JxW();
// The element shape functions evaluated at the quadrature points.
const vector<std::vector<Real> >& N = fe->get_phi();
const vector<std::vector<Real> >& N_face = fe_face->get_phi();
// Element shape function gradients evaluated at quadrature points
const vector<std::vector<RealGradient> >& B = fe->get_dphi();
// The XY locations of the quadrature points used for face integration
const vector<Point>& qface_points = fe_face->get_xyz();
// A reference to the \p DofMap objects
const DofMap& dof_map = this->get_dof_map(); // this system
// DEFINE VECTOR AND MATRIX VARIABLES
// Define data structures to contain the element matrix
// and right-hand-side vector contribution (Eq. 107)
DenseMatrix<Number> Me; // [\hat{M} + \hat{M}_{\delta}]
DenseMatrix<Number> Ne; // [\hat{N} + \hat{N}_{\delta}]
DenseMatrix<Number> Ke; // [\hat{K} + \hat{K}_{\delta}]
DenseVector<Number> Fe; // [\hat{F} + \hat{F}_{\delta}]
//DenseVector<Number> Fe_old; // element force vector (previous time)
DenseVector<Number> h; // element enthalpy vector (previous time)
DenseVector<Number> h_dot;
//DenseVector<Number> delta_h_dot;
DenseMatrix<Number> Mstar; // general time integration stiffness matrix (Eq. 125)
DenseVector<Number> R; // general time integration force vector (Eq. 126)
// Storage vectors for the degree of freedom indices
std::vector<unsigned int> dof_indices; // this system (h)
// std::vector<unsigned int> dof_indices_hdot;
// std::vector<unsigned int> dof_indices_deltahdot;
std::vector<unsigned int> dof_indices_velocity; // this system
std::vector<unsigned int> dof_indices_rho; // ThermoEq density
std::vector<unsigned int> dof_indices_tmp; // ThermoEq temperature
std::vector<unsigned int> dof_indices_f; // ThermoEq liquid fraction
std::vector<unsigned int> dof_indices_eps; // ThermoEq epsilon
// Define the necessary constants
const Number gamma = get_constant<Number>("gamma");
const Number dt = get_constant<Number>("dt"); // time step
Real time = this->time; // current time
const Number ks = thermo->get_constant<Number>("conductivity_solid");
const Number kf = thermo->get_constant<Number>("conductivity_fluid");
const Number cs = thermo->get_constant<Number>("specific_heat_solid");
const Number cf = thermo->get_constant<Number>("specific_heat_fluid");
const Number Te = thermo->get_constant<Number>("eutectic_temperature");
const Number hf = thermo->get_constant<Number>("latent_heat");
// Index of density variable in ThermoEq system
const unsigned int rho_idx = thermo->variable_number("density");
const unsigned int tmp_idx = thermo->variable_number("temperature");
const unsigned int f_idx = thermo->variable_number("liquid_mass_fraction");
const unsigned int eps_idx = thermo->variable_number("epsilon");
// Loop over all the elements in the mesh that are on local processor
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){
// Pointer to the element current element
const Elem* elem = *el;
// Get the degree of freedom indices for the current element
dof_map.dof_indices(elem, dof_indices, 0);
//dof_map.dof_indices(elem, dof_indices_hdot, 1);
//dof_map.dof_indices(elem, dof_indices_deltahdot, 2);
dof_map_momentum.dof_indices(elem, dof_indices_velocity);
dof_map_thermo.dof_indices(elem, dof_indices_rho, rho_idx);
dof_map_thermo.dof_indices(elem, dof_indices_tmp, tmp_idx);
dof_map_thermo.dof_indices(elem, dof_indices_f, f_idx);
dof_map_thermo.dof_indices(elem, dof_indices_eps, eps_idx);
// Compute the element-specific data for the current element
fe->reinit (elem);
fe_thermo->reinit(elem);
fe_momentum->reinit(elem);
// Zero the element matrices and vectors
Me.resize (dof_indices.size(), dof_indices.size()); // [\hat{M} + \hat{M}_{\delta}]
Ne.resize (dof_indices.size(), dof_indices.size()); //
Ke.resize (dof_indices.size(), dof_indices.size());
Fe.resize (dof_indices.size());
// Extract a vector of quadrature x,y,z coordinates
const vector<Point> qp_vec = fe->get_xyz();
// Compute the element length, h
Number elem_length = thermo->element_length(elem);
// Compute the RHS and mass and stiffness matrix for this element (Me)
for (unsigned int qp = 0; qp < qrule.n_points(); qp++){
// Get the velocity vector at this point (old value)
VectorValue<Number> v;
for (unsigned int i = 0; i < N_momentum.size(); i++){
for (unsigned int j = 0; j < dim; j++){
v(j) += N_momentum[i][qp] * momentum->old_solution(dof_indices_velocity[2*i+j]);
}
}
// Compute ThermoEq variables; must be mapped from node to quadrature points
Number T = 0;
Gradient grad_T;
Number f = 0;
Gradient grad_f;
Number rho = 0;
Number rho_old = 0;
Number eps = 0;
for (unsigned int i = 0; i < N_thermo.size(); i++){
T += N_thermo[i][qp] * thermo->current_solution(dof_indices_tmp[i]);
grad_T.add_scaled(B_thermo[i][qp], thermo->current_solution(dof_indices_tmp[i]));
f += N_thermo[i][qp] * thermo->current_solution(dof_indices_f[i]);
grad_f.add_scaled(B_thermo[i][qp], thermo->current_solution(dof_indices_f[i]));
rho += N_thermo[i][qp] * thermo->current_solution(dof_indices_rho[i]);
rho_old += N_thermo[i][qp] * thermo->old_solution(dof_indices_rho[i]);
eps += N_thermo[i][qp] * thermo->current_solution(dof_indices_eps[i]);
}
// Compute EnergySystem variables
Gradient grad_h;
for (unsigned int i = 0; i < B.size(); i++){
grad_h.add_scaled(B[i][qp], this->current_solution(dof_indices[i]));
}
// Compute T_{,k}^h v_k^h and f_{,k} v_k^h summation terms for F
Number Tv = 0;
Number fv = 0;
for (unsigned int i = 0; i < dim; i++){
Tv += grad_T(i) * v(i);
fv += grad_f(i) * v(i);
}
// Compute the time derivative of density
const Number drho_dt = (rho - rho_old)/dt;
// Compute alpha term of Eq. 69
const Number alpha = this->alpha(grad_T, grad_h, f);
// Extract tau_1 stabilization term
const Number tau_1 = thermo->tau_1(qp_vec[qp], elem_length);
// Loop through the components and construct matrices
for (unsigned int i = 0; i < N.size(); i++){
// Compute advective stabilization term (Eq. A, p. 1777)
const Number d = tau_1 * v * B[i][qp] / f - tau_1 * 1/rho * drho_dt * (1-f)/f * N[i][qp];
// Force vector, Eq. 77
Number F1 = JxW[qp] * (N[i][qp] + d) * rho * (1 - f) * (cf - cs) * Tv;
Number F2 = JxW[qp] * (N[i][qp] + d) * rho * fv * ((cf - cs) * (T - Te) + hf);
Number F3 = JxW[qp] * (N[i][qp] + d) * drho_dt * (1 - f) * ((cf - cs) * (T - Te) + hf);
Fe(i) += F1 + F2 + F3;
// Build the stiffness matrices
for (unsigned int j = 0; j < N.size(); j++){
// Mass matrix, Eq. 108
Me(i,j) += JxW[qp] * rho * ((N[i][qp] + d) * N[j][qp]);
// Stiffness matrix one, Ne, Eq. 109
Ne(i,j) += JxW[qp] * rho * ((N[i][qp] + d) * (v * B[j][qp]));
// Stiffness matrix two, Ke, Eq. 110
Ke(i,j) += JxW[qp]*((eps*kf + (1 - eps)*ks) * alpha * B[i][qp] * B[j][qp]);
}
}
}
printf("Me:\n");
Me.print(std::cout);
printf("\nNe:\n");
Ne.print(std::cout);
printf("\nKe:\n");
Ke.print(std::cout);
printf("\nFe:\n");
Fe.print(std::cout);
h.resize(dof_indices.size());
h_dot.resize(dof_indices.size());
// delta_h_dot.resize(dof_indices_deltahdot.size());
for (unsigned int i = 0; i < dof_indices.size(); i++){
h(i) = this->old_solution(dof_indices[i]);
h_dot(i) = this->get_vector("h_dot")(dof_indices[i]);
// delta_h_dot(i) = this->old_solution(dof_indices_deltahdot[i]);
}
this->get_matrix("M").add_matrix(Me, dof_indices);
this->get_matrix("N").add_matrix(Ne, dof_indices);
this->get_matrix("K").add_matrix(Ke, dof_indices);
this->get_vector("F").add_vector(Fe, dof_indices);
Mstar.resize(dof_indices.size(), dof_indices.size());
R.resize(dof_indices.size());
// Me + gamma*dt*(Ke + Ne);
Mstar.add(1,Me);
Mstar.add(gamma*dt,Ke);
Mstar.add(gamma*dt,Ne);
this->matrix->add_matrix(Mstar, dof_indices);
R.add(1,Fe);
DenseVector<Number> a(dof_indices.size());
Me.vector_mult(a, h_dot);
R.add(-1, a);
DenseMatrix<Number> B(dof_indices.size(), dof_indices.size());
DenseVector<Number> b(dof_indices.size());
B.add(1,Ne);
B.add(1,Ke);
B.vector_mult(b, h);
R.add(-1,b);
this->rhs->add_vector(R, dof_indices);
/*
// BOUNDARY CONDITIONS
// Loop through each side of the element for applying boundary conditions
for (unsigned int s = 0; s < elem->n_sides(); s++){
// Only consider the side if it does not have a neighbor
if (elem->neighbor(s) == NULL){
// Pointer to current element side
const AutoPtr<Elem> side = elem->side(s);
// Boundary ID of the current side
int boundary_id = (mesh.boundary_info)->boundary_id(elem, s);
// Get index of the boundary class with the same id
// this vector is empty if there is no match and only
// contains a single value if there is a match
std::vector<int> idx = get_boundary_index(boundary_id);
// Continue of there is a match
if(!idx.empty()){
// Compute the shape function values on the element face
fe_face->reinit(elem, s);
// Create a shared pointer to the boundary class
boost::shared_ptr<HeatEqBoundaryBase> ptr = bc_ptrs[idx[0]];
// Determine the type of boundary considered
std::string type = ptr->type;
// Loop through quadrature points
for (unsigned int qp = 0; qp < qface.n_points(); qp++){
// DIRICHLET (libMesh version; handled at initialization)
if(type.compare("dirichlet") == 0){
// The dirichlet conditions are handled at initlization
// but I don't want to throw an error if they are
// encountered, so just do nothing
// NEUMANN condition
} else if(type.compare("neumann") == 0){
// Current and past flux values
const Number q = ptr->q(qface_points[qp], time);
const Number q_old = ptr->q(qface_points[qp], time - dt);
// Add values to Fe
for (unsigned int i = 0; i < psi.size(); i++){
Fe(i) += JxW_face[qp] * q * psi[i][qp];
Fe_old(i) += JxW_face[qp] * q_old * psi[i][qp];
}
// CONVECTION boundary
} else if(type.compare("convection") == 0){
// Current and past h and T_inf
const Number h = ptr->h(qface_points[qp], time);
const Number h_old = ptr->h(qface_points[qp], time - dt);
const Number Tinf = ptr->Tinf(qface_points[qp], time);
const Number Tinf_old = ptr->Tinf(qface_points[qp], time - dt);
// Add values to Ke and Fe
for (unsigned int i = 0; i < psi.size(); i++){
Fe(i) += (1) * JxW_face[qp] * h * Tinf * psi[i][qp];
Fe_old(i) += (1) * JxW_face[qp] * h_old * Tinf_old * psi[i][qp];
for (unsigned int j = 0; j < psi.size(); j++){
Ke(i,j) += JxW_face[qp] * psi[i][qp] * h * psi[j][qp];
}
}
// Un-registerd type
} else {
printf("WARNING! The boundary type, %s, was not understood!\n", type.c_str());
} // (end) type.compare(...) statemenst
} //(end) for (int qp = 0; qp < qface.n_points(); qp++)
} // (end) if(!idx.empty)
} // (end) if (elem->neighbor(s) == NULL){
} // (end) for (int s = 0; s < elem->n_sides(); s++)
// Zero the pervious time-step temperature vector for this element
u_old.resize(dof_indices.size());
// Gather the temperatures at the nodes
for (unsigned int i = 0; i < psi.size(); i++){
u_old(i) = this->old_solution(dof_indices[i]);
}
// Build K_hat and F_hat (appends existing)
K_hat.resize(dof_indices.size(), dof_indices.size());
F_hat.resize(dof_indices.size());
build_stiffness_and_rhs(K_hat, F_hat, Me, Ke, Fe_old, Fe, u_old, dt, theta);
// Applies the dirichlet constraints to K_hat and F_hat
dof_map.heterogenously_constrain_element_matrix_and_vector(K_hat, F_hat, dof_indices);
// Apply the local components to the global K and F
this->matrix->add_matrix(K_hat, dof_indices);
this->rhs->add_vector(F_hat, dof_indices);
*/
} // (end) for ( ; el != end_el; ++el)
//update_rhs();
} // (end) assemble()