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element_class_tmpl.hh
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/**
* @file element_class_tmpl.hh
*
* @author Thomas Menouillard <tmenouillard@stucky.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Wed Jan 16 2013
* @date last modification: Fri Jun 13 2014
*
* @brief Implementation of the inline templated function of the element class
* descriptions
*
* @section LICENSE
*
* Copyright (©) 2014 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu is free software: you can redistribute it and/or modify it under the
* terms of the GNU Lesser General Public License as published by the Free
* Software Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
/* -------------------------------------------------------------------------- */
/* GaussIntegrationElement */
/* -------------------------------------------------------------------------- */
template<GaussIntergrationType type, UInt dimension, UInt order>
struct GaussIntergrationTypeDataHelper {
typedef GaussIntegrationTypeData<type, order> git_data;
static UInt getNbQuadraturePoints() {
return git_data::nb_quadrature_points;
}
static const Matrix<Real> getQuadraturePoints() {
return Matrix<Real>(git_data::quad_positions,
dimension,
git_data::nb_quadrature_points);
}
static const Vector<Real> getWeights() {
return Vector<Real>(git_data::quad_weights,
git_data::nb_quadrature_points);
}
};
/* -------------------------------------------------------------------------- */
template<UInt dimension, UInt order>
struct GaussIntergrationTypeDataHelper<_git_segment, dimension, order> {
typedef GaussIntegrationTypeData<_git_segment, order> git_data;
static UInt getNbQuadraturePoints() {
return Math::pow<dimension>(git_data::nb_quadrature_points);
}
static const Matrix<Real> getQuadraturePoints() {
UInt tot_nquad = getNbQuadraturePoints();
UInt nquad = git_data::nb_quadrature_points;
Matrix<Real> quads(dimension,
tot_nquad);
Vector<Real> pos(git_data::quad_positions,
nquad);
UInt offset = 1;
for (UInt d = 0; d < dimension; ++d) {
for (UInt n = 0, q = 0; n < tot_nquad; ++n, q += offset) {
UInt rq = q % tot_nquad + q / tot_nquad;
quads(d, rq) = pos(n % nquad);
}
offset *= nquad;
}
return quads;
}
static const Vector<Real> getWeights() {
UInt tot_nquad = getNbQuadraturePoints();
UInt nquad = git_data::nb_quadrature_points;
Vector<Real> quads_weights(tot_nquad, 1.);
Vector<Real> weights(git_data::quad_weights,
nquad);
UInt offset = 1;
for (UInt d = 0; d < dimension; ++d) {
for (UInt n = 0, q = 0; n < tot_nquad; ++n, q += offset) {
UInt rq = q % tot_nquad + q / tot_nquad;
quads_weights(rq) *= weights(n % nquad);
}
offset *= nquad;
}
return quads_weights;
}
};
template<ElementType element_type, UInt order>
const Matrix<Real> GaussIntegrationElement<element_type, order>::getQuadraturePoints() {
const InterpolationType itp_type = ElementClassProperty<element_type>::interpolation_type;
typedef InterpolationPorperty<itp_type> interpolation_property;
typedef GaussIntergrationTypeDataHelper<ElementClassProperty<element_type>::gauss_integration_type,
interpolation_property::natural_space_dimension,
order> data_helper;
Matrix<Real> tmp(data_helper::getQuadraturePoints());
return tmp;
}
/* -------------------------------------------------------------------------- */
template<ElementType element_type, UInt order>
const Vector<Real> GaussIntegrationElement<element_type, order>::getWeights() {
const InterpolationType itp_type = ElementClassProperty<element_type>::interpolation_type;
typedef InterpolationPorperty<itp_type> interpolation_property;
typedef GaussIntergrationTypeDataHelper<ElementClassProperty<element_type>::gauss_integration_type,
interpolation_property::natural_space_dimension,
order> data_helper;
Vector<Real> tmp(data_helper::getWeights());
return tmp;
}
/* -------------------------------------------------------------------------- */
template<ElementType element_type, UInt order>
UInt GaussIntegrationElement<element_type, order>::getNbQuadraturePoints() {
const InterpolationType itp_type = ElementClassProperty<element_type>::interpolation_type;
typedef InterpolationPorperty<itp_type> interpolation_property;
typedef GaussIntergrationTypeDataHelper<ElementClassProperty<element_type>::gauss_integration_type,
interpolation_property::natural_space_dimension,
order> data_helper;
return data_helper::getNbQuadraturePoints();
}
/* -------------------------------------------------------------------------- */
/* GeometricalElement */
/* -------------------------------------------------------------------------- */
template<GeometricalType geometrical_type, GeometricalShapeType shape>
inline const MatrixProxy<UInt>
GeometricalElement<geometrical_type, shape>::getFacetLocalConnectivityPerElement(UInt t) {
return MatrixProxy<UInt>(facet_connectivity[t], nb_facets[t], nb_nodes_per_facet[t]);
}
/* -------------------------------------------------------------------------- */
template<GeometricalType geometrical_type, GeometricalShapeType shape>
inline UInt
GeometricalElement<geometrical_type, shape>::getNbFacetsPerElement(UInt t) {
return nb_facets[t];
}
/* -------------------------------------------------------------------------- */
template<GeometricalType geometrical_type, GeometricalShapeType shape>
template <class vector_type>
inline bool
GeometricalElement<geometrical_type, shape>::contains(const vector_type & coords) {
return GeometricalShapeContains<shape>::contains(coords);
}
/* -------------------------------------------------------------------------- */
template<>
template <class vector_type>
inline bool
GeometricalShapeContains<_gst_point>::contains(const vector_type & coords) {
return (coords(0) < std::numeric_limits<Real>::epsilon());
}
/* -------------------------------------------------------------------------- */
template<>
template <class vector_type>
inline bool
GeometricalShapeContains<_gst_square>::contains(const vector_type & coords) {
bool in = true;
for (UInt i = 0; i < coords.size() && in; ++i)
in &= ((coords(i) >= -1.) && (coords(i) <= 1.));
return in;
}
/* -------------------------------------------------------------------------- */
template<>
template <class vector_type>
inline bool
GeometricalShapeContains<_gst_triangle>::contains(const vector_type & coords) {
bool in = true;
Real sum = 0;
for (UInt i = 0; (i < coords.size()) && in; ++i) {
in &= ((coords(i) >= 0) && (coords(i) <= 1.));
sum += coords(i);
}
if(in) return (in && (sum <= 1));
return in;
}
/* -------------------------------------------------------------------------- */
template<>
template <class vector_type>
inline bool
GeometricalShapeContains<_gst_prism>::contains(const vector_type & coords) {
bool in = ((coords(0) >= -1.) && (coords(0) <= 1.)); // x in segement [-1, 1]
// y and z in triangle
in &= ((coords(1) >= 0) && (coords(1) <= 1.));
in &= ((coords(2) >= 0) && (coords(2) <= 1.));
Real sum = coords(1) + coords(2);
return (in && (sum <= 1));
}
/* -------------------------------------------------------------------------- */
/* InterpolationElement */
/* -------------------------------------------------------------------------- */
template<InterpolationType interpolation_type, InterpolationKind kind>
inline void
InterpolationElement<interpolation_type, kind>::computeShapes(const Matrix<Real> & natural_coord,
Matrix<Real> & N) {
UInt nb_points = natural_coord.cols();
for (UInt p = 0; p < nb_points; ++p) {
Vector<Real> Np(N(p));
Vector<Real> ncoord_p(natural_coord(p));
computeShapes(ncoord_p, Np);
}
}
/* -------------------------------------------------------------------------- */
template<InterpolationType interpolation_type, InterpolationKind kind>
inline void
InterpolationElement<interpolation_type, kind>::computeDNDS(const Matrix<Real> & natural_coord,
Tensor3<Real> & dnds) {
UInt nb_points = natural_coord.cols();
for (UInt p = 0; p < nb_points; ++p) {
Matrix<Real> dnds_p(dnds(p));
Vector<Real> ncoord_p(natural_coord(p));
computeDNDS(ncoord_p, dnds_p);
}
}
/* -------------------------------------------------------------------------- */
/**
* interpolate on a point a field for which values are given on the
* node of the element using the shape functions at this interpolation point
*
* @param nodal_values values of the function per node @f$ f_{ij} = f_{n_i j} @f$ so it should be a matrix of size nb_nodes_per_element @f$\times@f$ nb_degree_of_freedom
* @param shapes value of shape functions at the interpolation point
* @param interpolated interpolated value of f @f$ f_j(\xi) = \sum_i f_{n_i j} N_i @f$
*/
template<InterpolationType interpolation_type, InterpolationKind kind>
inline void
InterpolationElement<interpolation_type, kind>::interpolate(const Matrix<Real> & nodal_values,
const Vector<Real> & shapes,
Vector<Real> & interpolated) {
Matrix<Real> interpm(interpolated.storage(), nodal_values.rows(), 1);
Matrix<Real> shapesm(shapes.storage(), InterpolationPorperty<interpolation_type>::nb_nodes_per_element, 1);
interpm.mul<false, false>(nodal_values, shapesm);
}
/* -------------------------------------------------------------------------- */
/**
* interpolate on several points a field for which values are given on the
* node of the element using the shape functions at the interpolation point
*
* @param nodal_values values of the function per node @f$ f_{ij} = f_{n_i j} @f$ so it should be a matrix of size nb_nodes_per_element @f$\times@f$ nb_degree_of_freedom
* @param shapes value of shape functions at the interpolation point
* @param interpolated interpolated values of f @f$ f_j(\xi) = \sum_i f_{n_i j} N_i @f$
*/
template<InterpolationType interpolation_type, InterpolationKind kind>
inline void
InterpolationElement<interpolation_type, kind>::interpolate(const Matrix<Real> & nodal_values,
const Matrix<Real> & shapes,
Matrix<Real> & interpolated) {
UInt nb_points = shapes.cols();
for (UInt p = 0; p < nb_points; ++p) {
Vector<Real> Np(shapes(p));
Vector<Real> interpolated_p(interpolated(p));
interpolate(nodal_values, Np, interpolated_p);
}
}
/* -------------------------------------------------------------------------- */
/**
* interpolate the field on a point given in natural coordinates the field which
* values are given on the node of the element
*
* @param natural_coords natural coordinates of point where to interpolate \xi
* @param nodal_values values of the function per node @f$ f_{ij} = f_{n_i j} @f$ so it should be a matrix of size nb_nodes_per_element @f$\times@f$ nb_degree_of_freedom
* @param interpolated interpolated value of f @f$ f_j(\xi) = \sum_i f_{n_i j} N_i @f$
*/
template<InterpolationType interpolation_type, InterpolationKind kind>
inline void
InterpolationElement<interpolation_type, kind>::interpolateOnNaturalCoordinates(const Vector<Real> & natural_coords,
const Matrix<Real> & nodal_values,
Vector<Real> & interpolated) {
Vector<Real> shapes(InterpolationPorperty<interpolation_type>::nb_nodes_per_element);
computeShapes(natural_coords, shapes);
interpolate(nodal_values, shapes, interpolated);
}
/* -------------------------------------------------------------------------- */
/// @f$ gradient_{ij} = \frac{\partial f_j}{\partial s_i} = \sum_k \frac{\partial N_k}{\partial s_i}f_{j n_k} @f$
template<InterpolationType interpolation_type, InterpolationKind kind>
inline void
InterpolationElement<interpolation_type, kind>::gradientOnNaturalCoordinates(const Vector<Real> & natural_coords,
const Matrix<Real> & f,
Matrix<Real> & gradient) {
Matrix<Real> dnds(InterpolationPorperty<interpolation_type>::natural_space_dimension,
InterpolationPorperty<interpolation_type>::nb_nodes_per_element);
computeDNDS(natural_coords, dnds);
gradient.mul<false, true>(f, dnds);
}
/* -------------------------------------------------------------------------- */
/* ElementClass */
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeJMat(const Tensor3<Real> & dnds,
const Matrix<Real> & node_coords,
Tensor3<Real> & J) {
UInt nb_points = dnds.size(2);
for (UInt p = 0; p < nb_points; ++p) {
Matrix<Real> J_p(J(p));
Matrix<Real> dnds_p(dnds(p));
computeJMat(dnds_p, node_coords, J_p);
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeJMat(const Matrix<Real> & dnds,
const Matrix<Real> & node_coords,
Matrix<Real> & J) {
/// @f$ J = dxds = dnds * x @f$
J.mul<false, true>(dnds, node_coords);
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeJacobian(const Matrix<Real> & natural_coords,
const Matrix<Real> & node_coords,
Vector<Real> & jacobians) {
UInt nb_points = natural_coords.cols();
Matrix<Real> dnds(interpolation_property::natural_space_dimension,
interpolation_property::nb_nodes_per_element);
Matrix<Real> J(natural_coords.rows(),
node_coords.rows());
for (UInt p = 0; p < nb_points; ++p) {
Vector<Real> ncoord_p(natural_coords(p));
interpolation_element::computeDNDS(ncoord_p, dnds);
computeJMat(dnds, node_coords, J);
computeJacobian(J, jacobians(p));
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeJacobian(const Tensor3<Real> & J,
Vector<Real> & jacobians) {
UInt nb_points = J.size(2);
for (UInt p = 0; p < nb_points; ++p) {
computeJacobian(J(p), jacobians(p));
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeJacobian(const Matrix<Real> & J,
Real & jacobians) {
if(J.rows() == J.cols()) {
jacobians = Math::det<element_property::spatial_dimension>(J.storage());
} else {
interpolation_element::computeSpecialJacobian(J, jacobians);
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void
ElementClass<type, kind>::computeShapeDerivatives(const Tensor3<Real> & J,
const Tensor3<Real> & dnds,
Tensor3<Real> & shape_deriv) {
UInt nb_points = J.size(2);
for (UInt p = 0; p < nb_points; ++p) {
Matrix<Real> shape_deriv_p(shape_deriv(p));
computeShapeDerivatives(J(p), dnds(p), shape_deriv_p);
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void
ElementClass<type, kind>::computeShapeDerivatives(const Matrix<Real> & J,
const Matrix<Real> & dnds,
Matrix<Real> & shape_deriv) {
Matrix<Real> inv_J(J.rows(), J.cols());
Math::inv<element_property::spatial_dimension>(J.storage(), inv_J.storage());
shape_deriv.mul<false, false>(inv_J, dnds);
}
/* -------------------------------------------------------------------------- */
template<ElementType type, ElementKind kind>
inline void
ElementClass<type, kind>::computeNormalsOnNaturalCoordinates(const Matrix<Real> & coord,
Matrix<Real> & f,
Matrix<Real> & normals) {
UInt dimension = normals.rows();
UInt nb_points = coord.cols();
AKANTU_DEBUG_ASSERT((dimension - 1) == interpolation_property::natural_space_dimension,
"cannot extract a normal because of dimension mismatch "
<< dimension - 1 << " " << interpolation_property::natural_space_dimension);
Matrix<Real> J(dimension, interpolation_property::natural_space_dimension);
for (UInt p = 0; p < nb_points; ++p) {
interpolation_element::gradientOnNaturalCoordinates(coord(p), f, J);
if (dimension == 2) {
Math::normal2(J.storage(), normals(p).storage());
}
if (dimension == 3){
Math::normal3(J(0).storage(), J(1).storage(), normals(p).storage());
}
}
}
/* ------------------------------------------------------------------------- */
/**
* In the non linear cases we need to iterate to find the natural coordinates @f$\xi@f$
* provided real coordinates @f$x@f$.
*
* We want to solve: @f$ x- \phi(\xi) = 0@f$ with @f$\phi(\xi) = \sum_I N_I(\xi) x_I@f$
* the mapping function which uses the nodal coordinates @f$x_I@f$.
*
* To that end we use the Newton method and the following series:
*
* @f$ \frac{\partial \phi(x_k)}{\partial \xi} \left( \xi_{k+1} - \xi_k \right) = x - \phi(x_k)@f$
*
* When we consider elements embedded in a dimension higher than them (2D triangle in a 3D space for example)
* @f$ J = \frac{\partial \phi(\xi_k)}{\partial \xi}@f$ is of dimension @f$dim_{space} \times dim_{elem}@f$ which
* is not invertible in most cases. Rather we can solve the problem:
*
* @f$ J^T J \left( \xi_{k+1} - \xi_k \right) = J^T \left( x - \phi(\xi_k) \right) @f$
*
* So that
*
* @f$ d\xi = \xi_{k+1} - \xi_k = (J^T J)^{-1} J^T \left( x - \phi(\xi_k) \right) @f$
*
* So that if the series converges we have:
*
* @f$ 0 = J^T \left( \phi(\xi_\infty) - x \right) @f$
*
* And we see that this is ill-posed only if @f$ J^T x = 0@f$ which means that the vector provided
* is normal to any tangent which means it is outside of the element itself.
*
* @param real_coords: the real coordinates the natural coordinates are sought for
* @param node_coords: the coordinates of the nodes forming the element
* @param natural_coords: output->the sought natural coordinates
* @param spatial_dimension: spatial dimension of the problem
*
**/
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::inverseMap(const Vector<Real> & real_coords,
const Matrix<Real> & node_coords,
Vector<Real> & natural_coords,
Real tolerance) {
UInt spatial_dimension = real_coords.size();
UInt dimension = natural_coords.size();
//matrix copy of the real_coords
Matrix<Real> mreal_coords(real_coords.storage(), spatial_dimension, 1);
//initial guess
// Matrix<Real> natural_guess(natural_coords.storage(), dimension, 1);
natural_coords.clear();
// real space coordinates provided by initial guess
Matrix<Real> physical_guess(dimension, 1);
// objective function f = real_coords - physical_guess
Matrix<Real> f(dimension, 1);
// dnds computed on the natural_guess
// Matrix<Real> dnds(interpolation_element::nb_nodes_per_element, spatial_dimension);
// J Jacobian matrix computed on the natural_guess
Matrix<Real> J(spatial_dimension, dimension);
// G = J^t * J
Matrix<Real> G(spatial_dimension, spatial_dimension);
// Ginv = G^{-1}
Matrix<Real> Ginv(spatial_dimension, spatial_dimension);
// J = Ginv * J^t
Matrix<Real> F(spatial_dimension, dimension);
// dxi = \xi_{k+1} - \xi in the iterative process
Matrix<Real> dxi(spatial_dimension, 1);
/* --------------------------- */
/* init before iteration loop */
/* --------------------------- */
// do interpolation
Vector<Real> physical_guess_v(physical_guess.storage(), dimension);
interpolation_element::interpolateOnNaturalCoordinates(natural_coords,
node_coords,
physical_guess_v);
// compute initial objective function value f = real_coords - physical_guess
f = mreal_coords;
f -= physical_guess;
// compute initial error
Real inverse_map_error = f.norm<L_2>();
/* --------------------------- */
/* iteration loop */
/* --------------------------- */
while(tolerance < inverse_map_error) {
//compute J^t
interpolation_element::gradientOnNaturalCoordinates(natural_coords, node_coords, J);
//compute G
G.mul<true, false>(J, J);
// inverse G
Ginv.inverse(G);
//compute F
F.mul<false, true>(Ginv, J);
//compute increment
dxi.mul<false,false>(F, f);
//update our guess
natural_coords += Vector<Real>(dxi(0));
//interpolate
interpolation_element::interpolateOnNaturalCoordinates(natural_coords,
node_coords,
physical_guess_v);
// compute error
f = mreal_coords;
f -= physical_guess;
inverse_map_error = f.norm<L_2>();
}
// memcpy(natural_coords.storage(), natural_guess.storage(), sizeof(Real) * natural_coords.size());
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::inverseMap(const Matrix<Real> & real_coords,
const Matrix<Real> & node_coords,
Matrix<Real> & natural_coords,
Real tolerance) {
UInt nb_points = real_coords.cols();
for (UInt p = 0; p < nb_points; ++p) {
Vector<Real> X(real_coords(p));
Vector<Real> ncoord_p(natural_coords(p));
inverseMap(X, node_coords, ncoord_p, tolerance);
}
}
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