// We now define the matrix assembly function for the
// Biharmonic system. We need to first compute element
// matrices and right-hand sides, and then take into
// account the boundary conditions, which will be handled
// via a penalty method.
void assemble_biharmonic(EquationSystems& es,
const std::string& system_name)
{
#ifdef LIBMESH_ENABLE_AMR
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
// It is a good idea to make sure we are assembling
// the proper system.
libmesh_assert_equal_to (system_name, "Biharmonic");
// Declare a performance log. Give it a descriptive
// string to identify what part of the code we are
// logging, since there may be many PerfLogs in an
// application.
PerfLog perf_log ("Matrix Assembly",false);
// Get a constant reference to the mesh object.
const MeshBase& mesh = es.get_mesh();
// The dimension that we are running
const unsigned int dim = mesh.mesh_dimension();
// Get a reference to the LinearImplicitSystem we are solving
LinearImplicitSystem& system = es.get_system<LinearImplicitSystem>("Biharmonic");
// A reference to the \p DofMap object for this system. The \p DofMap
// object handles the index translation from node and element numbers
// to degree of freedom numbers. We will talk more about the \p DofMap
// in future examples.
const DofMap& dof_map = system.get_dof_map();
// Get a constant reference to the Finite Element type
// for the first (and only) variable in the system.
FEType fe_type = dof_map.variable_type(0);
// Build a Finite Element object of the specified type. Since the
// \p FEBase::build() member dynamically creates memory we will
// store the object as an \p UniquePtr<FEBase>. This can be thought
// of as a pointer that will clean up after itself.
UniquePtr<FEBase> fe (FEBase::build(dim, fe_type));
// Quadrature rule for numerical integration.
// With 2D triangles, the Clough quadrature rule puts a Gaussian
// quadrature rule on each of the 3 subelements
UniquePtr<QBase> qrule(fe_type.default_quadrature_rule(dim));
// Tell the finite element object to use our quadrature rule.
fe->attach_quadrature_rule (qrule.get());
// Declare a special finite element object for
// boundary integration.
UniquePtr<FEBase> fe_face (FEBase::build(dim, fe_type));
// Boundary integration requires another quadraure rule,
// with dimensionality one less than the dimensionality
// of the element.
// In 1D, the Clough and Gauss quadrature rules are identical.
UniquePtr<QBase> qface(fe_type.default_quadrature_rule(dim-1));
// Tell the finte element object to use our
// quadrature rule.
fe_face->attach_quadrature_rule (qface.get());
// Here we define some references to cell-specific data that
// will be used to assemble the linear system.
// We begin with the element Jacobian * quadrature weight at each
// integration point.
const std::vector<Real>& JxW = fe->get_JxW();
// The physical XY locations of the quadrature points on the element.
// These might be useful for evaluating spatially varying material
// properties at the quadrature points.
const std::vector<Point>& q_point = fe->get_xyz();
// The element shape functions evaluated at the quadrature points.
const std::vector<std::vector<Real> >& phi = fe->get_phi();
// The element shape function second derivatives evaluated at the
// quadrature points. Note that for the simple biharmonic, shape
// function first derivatives are unnecessary.
const std::vector<std::vector<RealTensor> >& d2phi = fe->get_d2phi();
// For efficiency we will compute shape function laplacians n times,
// not n^2
std::vector<Real> shape_laplacian;
// Define data structures to contain the element matrix
// and right-hand-side vector contribution. Following
// basic finite element terminology we will denote these
// "Ke" and "Fe". More detail is in example 3.
DenseMatrix<Number> Ke;
DenseVector<Number> Fe;
// This vector will hold the degree of freedom indices for
// the element. These define where in the global system
// the element degrees of freedom get mapped.
std::vector<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. See
// example 3 for a discussion of the element iterators.
MeshBase::const_element_iterator el = mesh.active_local_elements_begin();
const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end();
for ( ; el != end_el; ++el)
{
// Start logging the shape function initialization.
// This is done through a simple function call with
// the name of the event to log.
perf_log.push("elem init");
// Store a pointer to the element we are currently
// working on. This allows for nicer syntax later.
const Elem* elem = *el;
// Get the degree of freedom indices for the
// current element. These define where in the global
// matrix and right-hand-side this element will
// contribute to.
dof_map.dof_indices (elem, dof_indices);
// Compute the element-specific data for the current
// element. This involves computing the location of the
// quadrature points (q_point) and the shape functions
// (phi, dphi) for the current element.
fe->reinit (elem);
// Zero the element matrix and right-hand side before
// summing them.
Ke.resize (dof_indices.size(),
dof_indices.size());
Fe.resize (dof_indices.size());
// Make sure there is enough room in this cache
shape_laplacian.resize(dof_indices.size());
// Stop logging the shape function initialization.
// If you forget to stop logging an event the PerfLog
// object will probably catch the error and abort.
perf_log.pop("elem init");
// Now we will build the element matrix. This involves
// a double loop to integrate laplacians of the test funcions
// (i) against laplacians of the trial functions (j).
//
// This step is why we need the Clough-Tocher elements -
// these C1 differentiable elements have square-integrable
// second derivatives.
//
// Now start logging the element matrix computation
perf_log.push ("Ke");
for (unsigned int qp=0; qp<qrule->n_points(); qp++)
{
for (unsigned int i=0; i<phi.size(); i++)
{
shape_laplacian[i] = d2phi[i][qp](0,0)+d2phi[i][qp](1,1);
if (dim == 3)
shape_laplacian[i] += d2phi[i][qp](2,2);
}
for (unsigned int i=0; i<phi.size(); i++)
for (unsigned int j=0; j<phi.size(); j++)
Ke(i,j) += JxW[qp]*
shape_laplacian[i]*shape_laplacian[j];
}
// Stop logging the matrix computation
perf_log.pop ("Ke");
// 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. Note that this is a fourth-order
// problem: Dirichlet boundary conditions include *both*
// boundary values and boundary normal fluxes.
{
// Start logging the boundary condition computation
perf_log.push ("BCs");
// The penalty values, for solution boundary trace and flux.
const Real penalty = 1e10;
const Real penalty2 = 1e10;
// 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)
{
// The value of the shape functions at the quadrature
// points.
const std::vector<std::vector<Real> >& phi_face =
fe_face->get_phi();
// The value of the shape function derivatives at the
// quadrature points.
const std::vector<std::vector<RealGradient> >& dphi_face =
fe_face->get_dphi();
// The Jacobian * Quadrature Weight at the quadrature
// points on the face.
const std::vector<Real>& JxW_face = fe_face->get_JxW();
// The XYZ locations (in physical space) of the
// quadrature points on the face. This is where
// we will interpolate the boundary value function.
const std::vector<Point>& qface_point = fe_face->get_xyz();
const std::vector<Point>& face_normals =
fe_face->get_normals();
// Compute the shape function values on the element
// face.
fe_face->reinit(elem, s);
// Loop over the face quagrature points for integration.
for (unsigned int qp=0; qp<qface->n_points(); qp++)
{
// The boundary value.
Number value = exact_solution(qface_point[qp],
es.parameters, "null",
"void");
Gradient flux = exact_2D_derivative(qface_point[qp],
es.parameters,
"null", "void");
// Matrix contribution of the L2 projection.
// Note that the basis function values are
// integrated against test function values while
// basis fluxes are integrated against test function
// fluxes.
for (unsigned int i=0; i<phi_face.size(); i++)
for (unsigned int j=0; j<phi_face.size(); j++)
Ke(i,j) += JxW_face[qp] *
(penalty * phi_face[i][qp] *
phi_face[j][qp] + penalty2
* (dphi_face[i][qp] *
face_normals[qp]) *
(dphi_face[j][qp] *
face_normals[qp]));
// Right-hand-side contribution of the L2
// projection.
for (unsigned int i=0; i<phi_face.size(); i++)
Fe(i) += JxW_face[qp] *
(penalty * value * phi_face[i][qp]
+ penalty2 *
(flux * face_normals[qp])
* (dphi_face[i][qp]
* face_normals[qp]));
}
}
// Stop logging the boundary condition computation
perf_log.pop ("BCs");
}
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]*forcing_function(q_point[qp]);
// The element matrix and right-hand-side are now built
// for this element. Add them to the global matrix and
// right-hand-side vector. The \p SparseMatrix::add_matrix()
// and \p NumericVector::add_vector() members do this for us.
// Start logging the insertion of the local (element)
// matrix and vector into the global matrix and vector
perf_log.push ("matrix insertion");
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);
// Stop logging the insertion of the local (element)
// matrix and vector into the global matrix and vector
perf_log.pop ("matrix insertion");
}
// That's it. We don't need to do anything else to the
// PerfLog. When it goes out of scope (at this function return)
// it will print its log to the screen. Pretty easy, huh?
#else
#endif // #ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
#endif // #ifdef LIBMESH_ENABLE_AMR
}
void Biharmonic::JR::residual_and_jacobian(const NumericVector<Number> & u,
NumericVector<Number> * R,
SparseMatrix<Number> * J,
NonlinearImplicitSystem &)
{
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
if (!R && !J)
return;
// Declare a performance log. Give it a descriptive
// string to identify what part of the code we are
// logging, since there may be many PerfLogs in an
// application.
PerfLog perf_log ("Biharmonic Residual and Jacobian", false);
// 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 = get_dof_map();
// Get a constant reference to the Finite Element type
// for the first (and only) variable in the system.
FEType fe_type = dof_map.variable_type(0);
// Build a Finite Element object of the specified type. Since the
// 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(_biharmonic._dim, fe_type));
// Quadrature rule for numerical integration.
// With 2D triangles, the Clough quadrature rule puts a Gaussian
// quadrature rule on each of the 3 subelements
UniquePtr<QBase> qrule(fe_type.default_quadrature_rule(_biharmonic._dim));
// Tell the finite element object to use our quadrature rule.
fe->attach_quadrature_rule (qrule.get());
// Here we define some references to element-specific data that
// will be used to assemble the linear system.
// We begin with the element Jacobian * quadrature weight at each
// integration point.
const std::vector<Real> & JxW = fe->get_JxW();
// The element shape functions evaluated at the quadrature points.
const std::vector<std::vector<Real> > & phi = fe->get_phi();
// The element shape functions' derivatives evaluated at the quadrature points.
const std::vector<std::vector<RealGradient> > & dphi = fe->get_dphi();
// The element shape functions' second derivatives evaluated at the quadrature points.
const std::vector<std::vector<RealTensor> > & d2phi = fe->get_d2phi();
// For efficiency we will compute shape function laplacians n times,
// not n^2
std::vector<Real> Laplacian_phi_qp;
// Define data structures to contain the element matrix
// and right-hand-side vector contribution. Following
// basic finite element terminology we will denote these
// "Je" and "Re". More detail is in example 3.
DenseMatrix<Number> Je;
DenseVector<Number> Re;
// 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;
// Old solution
const NumericVector<Number> & u_old = *old_local_solution;
// Now we will loop over all the elements in the mesh. We will
// compute the element matrix and right-hand-side contribution. See
// example 3 for a discussion of the element iterators.
MeshBase::const_element_iterator el = _biharmonic._mesh.active_local_elements_begin();
const MeshBase::const_element_iterator end_el = _biharmonic._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);
// Compute the element-specific data for the current
// element. This involves computing the location of the
// quadrature points (q_point) and the shape function
// values/derivatives (phi, dphi, d2phi) for the current element.
fe->reinit (elem);
// Zero the element matrix, the right-hand side and the Laplacian matrix
// before summing them.
if (J)
Je.resize(dof_indices.size(), dof_indices.size());
if (R)
Re.resize(dof_indices.size());
Laplacian_phi_qp.resize(dof_indices.size());
for (unsigned int qp=0; qp<qrule->n_points(); qp++)
{
// AUXILIARY QUANTITIES:
// Residual and Jacobian share a few calculations:
// at the very least, in the case of interfacial energy only with a constant mobility,
// both calculations use Laplacian_phi_qp; more is shared the case of a concentration-dependent
// mobility and bulk potentials.
Number
u_qp = 0.0,
u_old_qp = 0.0,
Laplacian_u_qp = 0.0,
Laplacian_u_old_qp = 0.0;
Gradient
grad_u_qp(0.0, 0.0, 0.0),
grad_u_old_qp(0.0, 0.0, 0.0);
Number
M_qp = 1.0,
M_old_qp = 1.0,
M_prime_qp = 0.0,
M_prime_old_qp = 0.0;
for (unsigned int i=0; i<phi.size(); i++)
{
Laplacian_phi_qp[i] = d2phi[i][qp](0, 0);
grad_u_qp(0) += u(dof_indices[i])*dphi[i][qp](0);
grad_u_old_qp(0) += u_old(dof_indices[i])*dphi[i][qp](0);
if (_biharmonic._dim > 1)
{
Laplacian_phi_qp[i] += d2phi[i][qp](1, 1);
grad_u_qp(1) += u(dof_indices[i])*dphi[i][qp](1);
grad_u_old_qp(1) += u_old(dof_indices[i])*dphi[i][qp](1);
}
if (_biharmonic._dim > 2)
{
Laplacian_phi_qp[i] += d2phi[i][qp](2, 2);
grad_u_qp(2) += u(dof_indices[i])*dphi[i][qp](2);
grad_u_old_qp(2) += u_old(dof_indices[i])*dphi[i][qp](2);
}
u_qp += phi[i][qp]*u(dof_indices[i]);
u_old_qp += phi[i][qp]*u_old(dof_indices[i]);
Laplacian_u_qp += Laplacian_phi_qp[i]*u(dof_indices[i]);
Laplacian_u_old_qp += Laplacian_phi_qp[i]*u_old(dof_indices[i]);
} // for i
if (_biharmonic._degenerate)
{
M_qp = 1.0 - u_qp*u_qp;
M_old_qp = 1.0 - u_old_qp*u_old_qp;
M_prime_qp = -2.0*u_qp;
M_prime_old_qp = -2.0*u_old_qp;
}
// ELEMENT RESIDUAL AND JACOBIAN
for (unsigned int i=0; i<phi.size(); i++)
{
// RESIDUAL
if (R)
{
Number ri = 0.0, ri_old = 0.0;
ri -= Laplacian_phi_qp[i]*M_qp*_biharmonic._kappa*Laplacian_u_qp;
ri_old -= Laplacian_phi_qp[i]*M_old_qp*_biharmonic._kappa*Laplacian_u_old_qp;
if (_biharmonic._degenerate)
{
ri -= (dphi[i][qp]*grad_u_qp)*M_prime_qp*(_biharmonic._kappa*Laplacian_u_qp);
ri_old -= (dphi[i][qp]*grad_u_old_qp)*M_prime_old_qp*(_biharmonic._kappa*Laplacian_u_old_qp);
}
if (_biharmonic._cahn_hillard)
{
if (_biharmonic._energy == DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_WELL)
{
ri += Laplacian_phi_qp[i]*M_qp*_biharmonic._theta_c*(u_qp*u_qp - 1.0)*u_qp;
ri_old += Laplacian_phi_qp[i]*M_old_qp*_biharmonic._theta_c*(u_old_qp*u_old_qp - 1.0)*u_old_qp;
if (_biharmonic._degenerate)
{
ri += (dphi[i][qp]*grad_u_qp)*M_prime_qp*_biharmonic._theta_c*(u_qp*u_qp - 1.0)*u_qp;
ri_old += (dphi[i][qp]*grad_u_old_qp)*M_prime_old_qp*_biharmonic._theta_c*(u_old_qp*u_old_qp - 1.0)*u_old_qp;
}
}// if (_biharmonic._energy == DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_WELL)
if (_biharmonic._energy == DOUBLE_OBSTACLE || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
{
ri -= Laplacian_phi_qp[i]*M_qp*_biharmonic._theta_c*u_qp;
ri_old -= Laplacian_phi_qp[i]*M_old_qp*_biharmonic._theta_c*u_old_qp;
if (_biharmonic._degenerate)
{
ri -= (dphi[i][qp]*grad_u_qp)*M_prime_qp*_biharmonic._theta_c*u_qp;
ri_old -= (dphi[i][qp]*grad_u_old_qp)*M_prime_old_qp*_biharmonic._theta_c*u_old_qp;
}
} // if (_biharmonic._energy == DOUBLE_OBSTACLE || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
if (_biharmonic._energy == LOG_DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
{
switch(_biharmonic._log_truncation)
{
case 2:
break;
case 3:
break;
default:
break;
}// switch(_biharmonic._log_truncation)
}// if (_biharmonic._energy == LOG_DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
}// if (_biharmonic._cahn_hillard)
Re(i) += JxW[qp]*((u_qp-u_old_qp)*phi[i][qp]-_biharmonic._dt*0.5*((2.0-_biharmonic._cnWeight)*ri + _biharmonic._cnWeight*ri_old));
} // if (R)
// JACOBIAN
if (J)
{
Number M_prime_prime_qp = 0.0;
if (_biharmonic._degenerate) M_prime_prime_qp = -2.0;
for (unsigned int j=0; j<phi.size(); j++)
{
Number ri_j = 0.0;
ri_j -= Laplacian_phi_qp[i]*M_qp*_biharmonic._kappa*Laplacian_phi_qp[j];
if (_biharmonic._degenerate)
{
ri_j -=
Laplacian_phi_qp[i]*M_prime_qp*phi[j][qp]*_biharmonic._kappa*Laplacian_u_qp +
(dphi[i][qp]*dphi[j][qp])*M_prime_qp*(_biharmonic._kappa*Laplacian_u_qp) +
(dphi[i][qp]*grad_u_qp)*(M_prime_prime_qp*phi[j][qp])*(_biharmonic._kappa*Laplacian_u_qp) +
(dphi[i][qp]*grad_u_qp)*(M_prime_qp)*(_biharmonic._kappa*Laplacian_phi_qp[j]);
}
if (_biharmonic._cahn_hillard)
{
if (_biharmonic._energy == DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_WELL)
{
ri_j +=
Laplacian_phi_qp[i]*M_prime_qp*phi[j][qp]*_biharmonic._theta_c*(u_qp*u_qp - 1.0)*u_qp +
Laplacian_phi_qp[i]*M_qp*_biharmonic._theta_c*(3.0*u_qp*u_qp - 1.0)*phi[j][qp] +
(dphi[i][qp]*dphi[j][qp])*M_prime_qp*_biharmonic._theta_c*(u_qp*u_qp - 1.0)*u_qp +
(dphi[i][qp]*grad_u_qp)*M_prime_prime_qp*_biharmonic._theta_c*(u_qp*u_qp - 1.0)*u_qp +
(dphi[i][qp]*grad_u_qp)*M_prime_qp*_biharmonic._theta_c*(3.0*u_qp*u_qp - 1.0)*phi[j][qp];
}// if (_biharmonic._energy == DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_WELL)
if (_biharmonic._energy == DOUBLE_OBSTACLE || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
{
ri_j -=
Laplacian_phi_qp[i]*M_prime_qp*phi[j][qp]*_biharmonic._theta_c*u_qp +
Laplacian_phi_qp[i]*M_qp*_biharmonic._theta_c*phi[j][qp] +
(dphi[i][qp]*dphi[j][qp])*M_prime_qp*_biharmonic._theta_c*u_qp +
(dphi[i][qp]*grad_u_qp)*M_prime_prime_qp*_biharmonic._theta_c*u_qp +
(dphi[i][qp]*grad_u_qp)*M_prime_qp*_biharmonic._theta_c*phi[j][qp];
} // if (_biharmonic._energy == DOUBLE_OBSTACLE || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
if (_biharmonic._energy == LOG_DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
{
switch(_biharmonic._log_truncation)
{
case 2:
break;
case 3:
break;
default:
break;
}// switch(_biharmonic._log_truncation)
}// if (_biharmonic._energy == LOG_DOUBLE_WELL || _biharmonic._energy == LOG_DOUBLE_OBSTACLE)
}// if (_biharmonic._cahn_hillard)
Je(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp] - 0.5*_biharmonic._dt*(2.0-_biharmonic._cnWeight)*ri_j);
} // for j
} // if (J)
} // for i
} // for qp
// 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.
// Start logging the insertion of the local (element)
// matrix and vector into the global matrix and vector
if (R)
{
// If the mesh has hanging nodes (e.g., as a result of refinement), those need to be constrained.
dof_map.constrain_element_vector(Re, dof_indices);
R->add_vector(Re, dof_indices);
}
if (J)
{
// If the mesh has hanging nodes (e.g., as a result of refinement), those need to be constrained.
dof_map.constrain_element_matrix(Je, dof_indices);
J->add_matrix(Je, dof_indices);
}
} // for el
#endif // LIBMESH_ENABLE_SECOND_DERIVATIVES
}
void ExactErrorEstimator::estimate_error (const System & system,
ErrorVector & error_per_cell,
const NumericVector<Number> * solution_vector,
bool estimate_parent_error)
{
// Ignore the fact that this variable is unused when !LIBMESH_ENABLE_AMR
libmesh_ignore(estimate_parent_error);
// The current mesh
const MeshBase & mesh = system.get_mesh();
// The dimensionality of the mesh
const unsigned int dim = mesh.mesh_dimension();
// The number of variables in the system
const unsigned int n_vars = system.n_vars();
// The DofMap for this system
const DofMap & dof_map = system.get_dof_map();
// Resize the error_per_cell vector to be
// the number of elements, initialize it to 0.
error_per_cell.resize (mesh.max_elem_id());
std::fill (error_per_cell.begin(), error_per_cell.end(), 0.);
// Prepare current_local_solution to localize a non-standard
// solution vector if necessary
if (solution_vector && solution_vector != system.solution.get())
{
NumericVector<Number> * newsol =
const_cast<NumericVector<Number> *>(solution_vector);
System & sys = const_cast<System &>(system);
newsol->swap(*sys.solution);
sys.update();
}
// Loop over all the variables in the system
for (unsigned int var=0; var<n_vars; var++)
{
// Possibly skip this variable
if (error_norm.weight(var) == 0.0) continue;
// The (string) name of this variable
const std::string & var_name = system.variable_name(var);
// The type of finite element to use for this variable
const FEType & fe_type = dof_map.variable_type (var);
UniquePtr<FEBase> fe (FEBase::build (dim, fe_type));
// Build an appropriate Gaussian quadrature rule
UniquePtr<QBase> qrule =
fe_type.default_quadrature_rule (dim,
_extra_order);
fe->attach_quadrature_rule (qrule.get());
// Prepare a global solution and a MeshFunction of the fine system if we need one
UniquePtr<MeshFunction> fine_values;
UniquePtr<NumericVector<Number> > fine_soln = NumericVector<Number>::build(system.comm());
if (_equation_systems_fine)
{
const System & fine_system = _equation_systems_fine->get_system(system.name());
std::vector<Number> global_soln;
// FIXME - we're assuming that the fine system solution gets
// used even when a different vector is used for the coarse
// system
fine_system.update_global_solution(global_soln);
fine_soln->init
(cast_int<numeric_index_type>(global_soln.size()), true,
SERIAL);
(*fine_soln) = global_soln;
fine_values = UniquePtr<MeshFunction>
(new MeshFunction(*_equation_systems_fine,
*fine_soln,
fine_system.get_dof_map(),
fine_system.variable_number(var_name)));
fine_values->init();
} else {
// Initialize functors if we're using them
for (unsigned int i=0; i != _exact_values.size(); ++i)
if (_exact_values[i])
_exact_values[i]->init();
for (unsigned int i=0; i != _exact_derivs.size(); ++i)
if (_exact_derivs[i])
_exact_derivs[i]->init();
for (unsigned int i=0; i != _exact_hessians.size(); ++i)
if (_exact_hessians[i])
_exact_hessians[i]->init();
}
// Request the data we'll need to compute with
fe->get_JxW();
fe->get_phi();
fe->get_dphi();
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
fe->get_d2phi();
#endif
fe->get_xyz();
#ifdef LIBMESH_ENABLE_AMR
// If we compute on parent elements, we'll want to do so only
// once on each, so we need to keep track of which we've done.
std::vector<bool> computed_var_on_parent;
if (estimate_parent_error)
computed_var_on_parent.resize(error_per_cell.size(), false);
#endif
// TODO: this ought to be threaded (and using subordinate
// MeshFunction objects in each thread rather than a single
// master)
// Iterate over all the active elements in the mesh
// that live on this processor.
MeshBase::const_element_iterator
elem_it = mesh.active_local_elements_begin();
const MeshBase::const_element_iterator
elem_end = mesh.active_local_elements_end();
for (;elem_it != elem_end; ++elem_it)
{
// e is necessarily an active element on the local processor
const Elem * elem = *elem_it;
const dof_id_type e_id = elem->id();
#ifdef LIBMESH_ENABLE_AMR
// See if the parent of element e has been examined yet;
// if not, we may want to compute the estimator on it
const Elem * parent = elem->parent();
// We only can compute and only need to compute on
// parents with all active children
bool compute_on_parent = true;
if (!parent || !estimate_parent_error)
compute_on_parent = false;
else
for (unsigned int c=0; c != parent->n_children(); ++c)
if (!parent->child_ptr(c)->active())
compute_on_parent = false;
if (compute_on_parent &&
!computed_var_on_parent[parent->id()])
{
computed_var_on_parent[parent->id()] = true;
// Compute a projection onto the parent
DenseVector<Number> Uparent;
FEBase::coarsened_dof_values(*(system.current_local_solution),
dof_map, parent, Uparent,
var, false);
error_per_cell[parent->id()] +=
static_cast<ErrorVectorReal>
(find_squared_element_error(system, var_name,
parent, Uparent,
fe.get(),
fine_values.get()));
}
#endif
// Get the local to global degree of freedom maps
std::vector<dof_id_type> dof_indices;
dof_map.dof_indices (elem, dof_indices, var);
const unsigned int n_dofs =
cast_int<unsigned int>(dof_indices.size());
DenseVector<Number> Uelem(n_dofs);
for (unsigned int i=0; i != n_dofs; ++i)
Uelem(i) = system.current_solution(dof_indices[i]);
error_per_cell[e_id] +=
static_cast<ErrorVectorReal>
(find_squared_element_error(system, var_name, elem,
Uelem, fe.get(),
fine_values.get()));
} // End loop over active local elements
} // End loop over variables
// Each processor has now computed the error contribuions
// for its local elements. We need to sum the vector
// and then take the square-root of each component. Note
// that we only need to sum if we are running on multiple
// processors, and we only need to take the square-root
// if the value is nonzero. There will in general be many
// zeros for the inactive elements.
// First sum the vector of estimated error values
this->reduce_error(error_per_cell, system.comm());
// Compute the square-root of each component.
{
LOG_SCOPE("std::sqrt()", "ExactErrorEstimator");
for (dof_id_type i=0; i<error_per_cell.size(); i++)
if (error_per_cell[i] != 0.)
{
libmesh_assert_greater (error_per_cell[i], 0.);
error_per_cell[i] = std::sqrt(error_per_cell[i]);
}
}
// If we used a non-standard solution before, now is the time to fix
// the current_local_solution
if (solution_vector && solution_vector != system.solution.get())
{
NumericVector<Number> * newsol =
const_cast<NumericVector<Number> *>(solution_vector);
System & sys = const_cast<System &>(system);
newsol->swap(*sys.solution);
sys.update();
}
}
// We now define the matrix and rhs vector assembly function
// for the shell system. This function implements the
// linear Kirchhoff-Love theory for thin shells. At the
// end we also take into account the boundary conditions
// here, using the penalty method.
void assemble_shell (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, "Shell");
// Get a constant reference to the mesh object.
const MeshBase& mesh = es.get_mesh();
// Get a reference to the shell system object.
LinearImplicitSystem & system = es.get_system<LinearImplicitSystem> ("Shell");
// Get the shell parameters that we need during assembly.
const Real h = es.parameters.get<Real> ("thickness");
const Real E = es.parameters.get<Real> ("young's modulus");
const Real nu = es.parameters.get<Real> ("poisson ratio");
const Real q = es.parameters.get<Real> ("uniform load");
// Compute the membrane stiffness \p K and the bending
// rigidity \p D from these parameters.
const Real K = E * h / (1-nu*nu);
const Real D = E * h*h*h / (12*(1-nu*nu));
// 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 w_var = system.variable_number ("w");
// Get the Finite Element type for "u". Note this will be
// the same as the type for "v" and "w".
FEType fe_type = system.variable_type (u_var);
// Build a Finite Element object of the specified type.
UniquePtr<FEBase> fe (FEBase::build(2, fe_type));
// A Gauss quadrature rule for numerical integration.
// For subdivision shell elements, a single Gauss point per
// element is sufficient, hence we use extraorder = 0.
const int extraorder = 0;
UniquePtr<QBase> qrule (fe_type.default_quadrature_rule (2, extraorder));
// Tell the finite element object to use our quadrature rule.
fe->attach_quadrature_rule (qrule.get());
// The element Jacobian * quadrature weight at each integration point.
const std::vector<Real>& JxW = fe->get_JxW();
// The surface tangents in both directions at the quadrature points.
const std::vector<RealGradient>& dxyzdxi = fe->get_dxyzdxi();
const std::vector<RealGradient>& dxyzdeta = fe->get_dxyzdeta();
// The second partial derivatives at the quadrature points.
const std::vector<RealGradient>& d2xyzdxi2 = fe->get_d2xyzdxi2();
const std::vector<RealGradient>& d2xyzdeta2 = fe->get_d2xyzdeta2();
const std::vector<RealGradient>& d2xyzdxideta = fe->get_d2xyzdxideta();
// The element shape function and its derivatives evaluated at the
// quadrature points.
const std::vector<std::vector<Real> >& phi = fe->get_phi();
const std::vector<std::vector<RealGradient> >& dphi = fe->get_dphi();
const std::vector<std::vector<RealTensor> >& d2phi = fe->get_d2phi();
// 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();
// Define data structures to contain the element stiffness 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), Kuw(Ke),
Kvu(Ke), Kvv(Ke), Kvw(Ke),
Kwu(Ke), Kwv(Ke), Kww(Ke);
DenseSubVector<Number>
Fu(Fe),
Fv(Fe),
Fw(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_w;
// 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;
// The ghost elements at the boundaries need to be excluded
// here, as they don't belong to the physical shell,
// but serve for a proper boundary treatment only.
libmesh_assert_equal_to (elem->type(), TRI3SUBDIVISION);
const Tri3Subdivision* sd_elem = static_cast<const Tri3Subdivision*> (elem);
if (sd_elem->is_ghost())
continue;
// 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_w, w_var);
const std::size_t n_dofs = dof_indices.size();
const std::size_t n_u_dofs = dof_indices_u.size();
const std::size_t n_v_dofs = dof_indices_v.size();
const std::size_t n_w_dofs = dof_indices_w.size();
// Compute the element-specific data for the current
// element. This involves computing the location of the
// quadrature points and the shape functions
// (phi, dphi, d2phi) for the current element.
fe->reinit (elem);
// Zero the element matrix and right-hand side before
// summing them. We use the resize member here because
// the number of degrees of freedom might have changed from
// the last element.
Ke.resize (n_dofs, n_dofs);
Fe.resize (n_dofs);
// Reposition the submatrices... The idea is this:
//
// - - - -
// | Kuu Kuv Kuw | | Fu |
// Ke = | Kvu Kvv Kvw |; Fe = | Fv |
// | Kwu Kwv Kww | | Fw |
// - - - -
//
// 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);
Kuw.reposition (u_var*n_u_dofs, w_var*n_u_dofs, n_u_dofs, n_w_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);
Kvw.reposition (v_var*n_v_dofs, w_var*n_v_dofs, n_v_dofs, n_w_dofs);
Kwu.reposition (w_var*n_w_dofs, u_var*n_w_dofs, n_w_dofs, n_u_dofs);
Kwv.reposition (w_var*n_w_dofs, v_var*n_w_dofs, n_w_dofs, n_v_dofs);
Kww.reposition (w_var*n_w_dofs, w_var*n_w_dofs, n_w_dofs, n_w_dofs);
Fu.reposition (u_var*n_u_dofs, n_u_dofs);
Fv.reposition (v_var*n_u_dofs, n_v_dofs);
Fw.reposition (w_var*n_u_dofs, n_w_dofs);
// Now we will build the element matrix and right-hand-side.
for (unsigned int qp=0; qp<qrule->n_points(); ++qp)
{
// First, we compute the external force resulting
// from a load q distributed uniformly across the plate.
// Since the load is supposed to be transverse to the plate,
// it affects the z-direction, i.e. the "w" variable.
for (unsigned int i=0; i<n_u_dofs; ++i)
Fw(i) += JxW[qp] * phi[i][qp] * q;
// Next, we assemble the stiffness matrix. This is only valid
// for the linear theory, i.e., for small deformations, where
// reference and deformed surface metrics are indistinguishable.
// Get the three surface basis vectors.
const RealVectorValue & a1 = dxyzdxi[qp];
const RealVectorValue & a2 = dxyzdeta[qp];
RealVectorValue a3 = a1.cross(a2);
const Real jac = a3.size(); // the surface Jacobian
libmesh_assert_greater (jac, 0);
a3 /= jac; // the shell director a3 is normalized to unit length
// Get the derivatives of the surface tangents.
const RealVectorValue & a11 = d2xyzdxi2[qp];
const RealVectorValue & a22 = d2xyzdeta2[qp];
const RealVectorValue & a12 = d2xyzdxideta[qp];
// Compute the three covariant components of the first
// fundamental form of the surface.
const RealVectorValue a(a1*a1, a2*a2, a1*a2);
// The elastic H matrix in Voigt's notation, computed from the
// covariant components of the first fundamental form rather
// than the contravariant components, exploiting that the
// contravariant first fundamental form is the inverse of the
// covatiant first fundamental form (hence the determinant etc.).
RealTensorValue H;
H(0,0) = a(1) * a(1);
H(0,1) = H(1,0) = nu * a(1) * a(0) + (1-nu) * a(2) * a(2);
H(0,2) = H(2,0) = -a(1) * a(2);
H(1,1) = a(0) * a(0);
H(1,2) = H(2,1) = -a(0) * a(2);
H(2,2) = 0.5 * ((1-nu) * a(1) * a(0) + (1+nu) * a(2) * a(2));
const Real det = a(0) * a(1) - a(2) * a(2);
libmesh_assert_not_equal_to (det * det, 0);
H /= det * det;
// Precompute come cross products for the bending part below.
const RealVectorValue a11xa2 = a11.cross(a2);
const RealVectorValue a22xa2 = a22.cross(a2);
const RealVectorValue a12xa2 = a12.cross(a2);
const RealVectorValue a1xa11 = a1.cross(a11);
const RealVectorValue a1xa22 = a1.cross(a22);
const RealVectorValue a1xa12 = a1.cross(a12);
const RealVectorValue a2xa3 = a2.cross(a3);
const RealVectorValue a3xa1 = a3.cross(a1);
// Loop over all pairs of nodes I,J.
for (unsigned int i=0; i<n_u_dofs; ++i)
{
for (unsigned int j=0; j<n_u_dofs; ++j)
{
// The membrane strain matrices in Voigt's notation.
RealTensorValue MI, MJ;
for (unsigned int k=0; k<3; ++k)
{
MI(0,k) = dphi[i][qp](0) * a1(k);
MI(1,k) = dphi[i][qp](1) * a2(k);
MI(2,k) = dphi[i][qp](1) * a1(k)
+ dphi[i][qp](0) * a2(k);
MJ(0,k) = dphi[j][qp](0) * a1(k);
MJ(1,k) = dphi[j][qp](1) * a2(k);
MJ(2,k) = dphi[j][qp](1) * a1(k)
+ dphi[j][qp](0) * a2(k);
}
// The bending strain matrices in Voigt's notation.
RealTensorValue BI, BJ;
for (unsigned int k=0; k<3; ++k)
{
const Real term_ik = dphi[i][qp](0) * a2xa3(k)
+ dphi[i][qp](1) * a3xa1(k);
BI(0,k) = -d2phi[i][qp](0,0) * a3(k)
+(dphi[i][qp](0) * a11xa2(k)
+ dphi[i][qp](1) * a1xa11(k)
+ (a3*a11) * term_ik) / jac;
BI(1,k) = -d2phi[i][qp](1,1) * a3(k)
+(dphi[i][qp](0) * a22xa2(k)
+ dphi[i][qp](1) * a1xa22(k)
+ (a3*a22) * term_ik) / jac;
BI(2,k) = 2 * (-d2phi[i][qp](0,1) * a3(k)
+(dphi[i][qp](0) * a12xa2(k)
+ dphi[i][qp](1) * a1xa12(k)
+ (a3*a12) * term_ik) / jac);
const Real term_jk = dphi[j][qp](0) * a2xa3(k)
+ dphi[j][qp](1) * a3xa1(k);
BJ(0,k) = -d2phi[j][qp](0,0) * a3(k)
+(dphi[j][qp](0) * a11xa2(k)
+ dphi[j][qp](1) * a1xa11(k)
+ (a3*a11) * term_jk) / jac;
BJ(1,k) = -d2phi[j][qp](1,1) * a3(k)
+(dphi[j][qp](0) * a22xa2(k)
+ dphi[j][qp](1) * a1xa22(k)
+ (a3*a22) * term_jk) / jac;
BJ(2,k) = 2 * (-d2phi[j][qp](0,1) * a3(k)
+(dphi[j][qp](0) * a12xa2(k)
+ dphi[j][qp](1) * a1xa12(k)
+ (a3*a12) * term_jk) / jac);
}
// The total stiffness matrix coupling the nodes
// I and J is a sum of membrane and bending
// contributions according to the following formula.
const RealTensorValue KIJ = JxW[qp] * K * MI.transpose() * H * MJ
+ JxW[qp] * D * BI.transpose() * H * BJ;
// Insert the components of the coupling stiffness
// matrix \p KIJ into the corresponding directional
// submatrices.
Kuu(i,j) += KIJ(0,0);
Kuv(i,j) += KIJ(0,1);
Kuw(i,j) += KIJ(0,2);
Kvu(i,j) += KIJ(1,0);
Kvv(i,j) += KIJ(1,1);
Kvw(i,j) += KIJ(1,2);
Kwu(i,j) += KIJ(2,0);
Kwv(i,j) += KIJ(2,1);
Kww(i,j) += KIJ(2,2);
}
}
} // end of the quadrature point qp-loop
// 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 non-ghost element loop
// Next, we apply the boundary conditions. In this case,
// all boundaries are clamped by the penalty method, using
// the special "ghost" nodes along the boundaries. Note
// that there are better ways to implement boundary conditions
// for subdivision shells. We use the simplest way here,
// which is known to be overly restrictive and will lead to
// a slightly too small deformation of the plate.
el = mesh.active_local_elements_begin();
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;
// For the boundary conditions, we only need to loop over
// the ghost elements.
libmesh_assert_equal_to (elem->type(), TRI3SUBDIVISION);
const Tri3Subdivision* gh_elem = static_cast<const Tri3Subdivision*> (elem);
if (!gh_elem->is_ghost())
continue;
// Find the side which is part of the physical plate boundary,
// that is, the boundary of the original mesh without ghosts.
for (unsigned int s=0; s<elem->n_sides(); ++s)
{
const Tri3Subdivision* nb_elem = static_cast<const Tri3Subdivision*> (elem->neighbor(s));
if (nb_elem == NULL || nb_elem->is_ghost())
continue;
/*
* Determine the four nodes involved in the boundary
* condition treatment of this side. The \p MeshTools::Subdiv
* namespace provides lookup tables \p next and \p prev
* for an efficient determination of the next and previous
* nodes of an element, respectively.
*
* n4
* / \
* / gh \
* n2 ---- n3
* \ nb /
* \ /
* n1
*/
Node* nodes [4]; // n1, n2, n3, n4
nodes[1] = gh_elem->get_node(s); // n2
nodes[2] = gh_elem->get_node(MeshTools::Subdivision::next[s]); // n3
nodes[3] = gh_elem->get_node(MeshTools::Subdivision::prev[s]); // n4
// The node in the interior of the domain, \p n1, is the
// hardest to find. Walk along the edges of element \p nb until
// we have identified it.
unsigned int n_int = 0;
nodes[0] = nb_elem->get_node(0);
while (nodes[0]->id() == nodes[1]->id() || nodes[0]->id() == nodes[2]->id())
nodes[0] = nb_elem->get_node(++n_int);
// The penalty value. \f$ \frac{1}{\epsilon} \f$
const Real penalty = 1.e10;
// With this simple method, clamped boundary conditions are
// obtained by penalizing the displacements of all four nodes.
// This ensures that the displacement field vanishes on the
// boundary side \p s.
for (unsigned int n=0; n<4; ++n)
{
const dof_id_type u_dof = nodes[n]->dof_number (system.number(), u_var, 0);
const dof_id_type v_dof = nodes[n]->dof_number (system.number(), v_var, 0);
const dof_id_type w_dof = nodes[n]->dof_number (system.number(), w_var, 0);
system.matrix->add (u_dof, u_dof, penalty);
system.matrix->add (v_dof, v_dof, penalty);
system.matrix->add (w_dof, w_dof, penalty);
}
}
} // end of ghost element loop
}