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element_class_tmpl.hh
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/**
* @file element_class_tmpl.hh
*
* @author Aurelia Isabel Cuba Ramos <aurelia.cubaramos@epfl.ch>
* @author Thomas Menouillard <tmenouillard@stucky.ch>
* @author Mohit Pundir <mohit.pundir@epfl.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Thu Feb 21 2013
* @date last modification: Fri Dec 11 2020
*
* @brief Implementation of the inline templated function of the element class
* descriptions
*
*
* @section LICENSE
*
* Copyright (©) 2014-2021 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/>.
*
*/
/* -------------------------------------------------------------------------- */
//#include "element_class.hh"
#include "gauss_integration_tmpl.hh"
#include "aka_iterators.hh"
/* -------------------------------------------------------------------------- */
#include <type_traits>
/* -------------------------------------------------------------------------- */
#ifndef AKANTU_ELEMENT_CLASS_TMPL_HH_
#define AKANTU_ELEMENT_CLASS_TMPL_HH_
namespace akantu {
template <ElementType element_type, ElementKind element_kind>
inline constexpr decltype(auto)
ElementClass<element_type, element_kind>::getFacetTypes() {
return Eigen::Map<const Eigen::Matrix<
ElementType, geometrical_element::getNbFacetTypes(), 1>>(
element_class_extra_geom_property::facet_type.data());
}
/* -------------------------------------------------------------------------- */
/* GeometricalElement */
/* -------------------------------------------------------------------------- */
template <GeometricalType geometrical_type, GeometricalShapeType shape>
inline constexpr decltype(auto)
GeometricalElement<geometrical_type,
shape>::getFacetLocalConnectivityPerElement(UInt t) {
int pos = 0;
for (UInt i = 0; i < t; ++i) {
pos += geometrical_property::nb_facets[i] *
geometrical_property::nb_nodes_per_facet[i];
}
return Eigen::Map<const Eigen::Matrix<UInt, Eigen::Dynamic, Eigen::Dynamic>>(
geometrical_property::facet_connectivity_vect.data() + pos,
geometrical_property::nb_facets[t],
geometrical_property::nb_nodes_per_facet[t]);
}
/* -------------------------------------------------------------------------- */
template <GeometricalType geometrical_type, GeometricalShapeType shape>
inline constexpr UInt
GeometricalElement<geometrical_type, shape>::getNbFacetsPerElement() {
UInt total_nb_facets = 0;
for (UInt n = 0; n < geometrical_property::nb_facet_types; ++n) {
total_nb_facets += geometrical_property::nb_facets[n];
}
return total_nb_facets;
}
/* -------------------------------------------------------------------------- */
template <GeometricalType geometrical_type, GeometricalShapeType shape>
inline constexpr UInt
GeometricalElement<geometrical_type, shape>::getNbFacetsPerElement(UInt t) {
return geometrical_property::nb_facets[t];
}
/* -------------------------------------------------------------------------- */
template <GeometricalType geometrical_type, GeometricalShapeType shape>
template <class D>
inline bool GeometricalElement<geometrical_type, shape>::contains(
const Eigen::MatrixBase<D> & coords) {
return GeometricalShapeContains<shape>::contains(coords);
}
/* -------------------------------------------------------------------------- */
template <>
template <class D>
inline bool GeometricalShapeContains<_gst_point>::contains(
const Eigen::MatrixBase<D> & coords) {
return (coords(0) < std::numeric_limits<Real>::epsilon());
}
/* -------------------------------------------------------------------------- */
template <>
template <class D>
inline bool GeometricalShapeContains<_gst_square>::contains(
const Eigen::MatrixBase<D> & coords) {
bool in = true;
for (UInt i = 0; i < coords.size() && in; ++i) {
in &= ((coords(i) >= -(1. + std::numeric_limits<Real>::epsilon())) &&
(coords(i) <= (1. + std::numeric_limits<Real>::epsilon())));
}
return in;
}
/* -------------------------------------------------------------------------- */
template <>
template <class D>
inline bool GeometricalShapeContains<_gst_triangle>::contains(
const Eigen::MatrixBase<D> & coords) {
bool in = true;
Real sum = 0;
for (Int i = 0; (i < coords.size()) && in; ++i) {
in &= ((coords(i) >= -(Math::getTolerance())) &&
(coords(i) <= (1. + Math::getTolerance())));
sum += coords(i);
}
if (in) {
return (in && (sum <= (1. + Math::getTolerance())));
}
return in;
}
/* -------------------------------------------------------------------------- */
template <>
template <class D>
inline bool GeometricalShapeContains<_gst_prism>::contains(
const Eigen::MatrixBase<D> & coords) {
bool in = ((coords(0) >= -1.) && (coords(0) <= 1.)); // x in segment [-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>
template <typename D1, typename D2,
aka::enable_if_t<aka::are_matrices<D1, D2>::value> *>
inline void InterpolationElement<interpolation_type, kind>::computeShapes(
const Eigen::MatrixBase<D1> & Xs, const Eigen::MatrixBase<D2> & N_) {
Eigen::MatrixBase<D2> & N = const_cast<Eigen::MatrixBase<D2> &>(
N_); // as advised by the Eigen developers
for (auto && data : zip(Xs, N)) {
computeShapes(std::get<0>(data), std::get<1>(data));
}
}
/* -------------------------------------------------------------------------- */
template <InterpolationType interpolation_type, InterpolationKind kind>
template <class D>
inline void InterpolationElement<interpolation_type, kind>::computeDNDS(
const Eigen::MatrixBase<D> & Xs, Tensor3Base<Real> & dNdS) {
for (auto && data : zip(Xs, dNdS)) {
computeDNDS(std::get<0>(data), std::get<1>(data));
}
}
// /* --------------------------------------------------------------------------
// */
// /**
// * 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>
// template <typename Derived1, typename Derived2, typename Derived3,
// aka::enable_if_t<aka::conjunction<
// aka::is_matrice<Derived1>, aka::is_vector<Derived2>>::value> *
// = nullptr>
// inline auto
// InterpolationElement<interpolation_type, kind>::interpolate(
// const Eigen::MatrixBase<Derived1> & nodal_values,
// const Eigen::MatrixBase<Derived2> & shapes) {
// return nodal_values * shapes;
// }
/* -------------------------------------------------------------------------- */
/**
* 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>
template <typename Derived1, typename Derived2, typename Derived3,
aka::enable_if_matrices_t<Derived1, Derived2, Derived3> *>
inline void InterpolationElement<interpolation_type, kind>::interpolate(
const Eigen::MatrixBase<Derived1> & nodal_values,
const Eigen::MatrixBase<Derived2> & Ns,
const Eigen::MatrixBase<Derived3> & interpolated_) {
auto && interpolated = const_cast<Eigen::MatrixBase<Derived3> &>(
interpolated_); // as advised by the Eigen developers
UInt nb_points = Ns.cols();
for (UInt p = 0; p < nb_points; ++p) {
interpolated(p) = interpolate(nodal_values, Ns(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 decltype(auto)
// InterpolationElement<interpolation_type,
// kind>::interpolateOnNaturalCoordinates(
// const Ref<const VectorXr> & natural_coords,
// const Ref<const MatrixXr> & nodal_values, Ref<VectorXr> interpolated) {
// using interpolation = InterpolationProperty<interpolation_type>;
// Eigen::Matrix<Real, interpolation::nb_nodes_per_element, 1> shapes;
// computeShapes(natural_coords, shapes);
// return interpolate(nodal_values, shapes);
// }
/* -------------------------------------------------------------------------- */
/// @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>
template <typename Derived1, typename Derived2>
inline decltype(auto)
InterpolationElement<interpolation_type, kind>::gradientOnNaturalCoordinates(
const Eigen::MatrixBase<Derived1> & natural_coords,
const Eigen::MatrixBase<Derived2> & f) {
Eigen::Matrix<
Real, InterpolationProperty<interpolation_type>::natural_space_dimension,
InterpolationProperty<interpolation_type>::nb_nodes_per_element>
dnds;
computeDNDS(natural_coords, dnds);
return f * dnds.transpose();
}
/* -------------------------------------------------------------------------- */
/* ElementClass */
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void
ElementClass<type, kind>::computeJMat(const Tensor3<Real> & dnds,
const Ref<const MatrixXr> & node_coords,
Tensor3<Real> & J) {
UInt nb_points = dnds.size(2);
for (UInt p = 0; p < nb_points; ++p) {
computeJMat(dnds(p), node_coords, J(p));
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void
ElementClass<type, kind>::computeJMat(const Ref<const MatrixXr> & dnds,
const Ref<const MatrixXr> & node_coords,
Ref<MatrixXr> J) {
/// @f$ J = dxds = dnds * x @f$
J = dnds * node_coords.transpose();
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeJacobian(
const Ref<const MatrixXr> & natural_coords,
const Ref<const MatrixXr> & node_coords, Ref<VectorXr> jacobians) {
UInt nb_points = natural_coords.cols();
Matrix<Real, interpolation_property::natural_space_dimension,
interpolation_property::nb_nodes_per_element>
dnds;
Ref<MatrixXr> J(natural_coords.rows(), node_coords.rows());
for (UInt p = 0; p < nb_points; ++p) {
interpolation_element::computeDNDS(natural_coords(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,
Ref<VectorXr> 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 Ref<const MatrixXr> & J,
Real & jacobians) {
if (J.rows() == J.cols()) {
jacobians = Math::det<element_property::spatial_dimension>(J.data());
} 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) {
computeShapeDerivatives(J(p), dnds(p), shape_deriv(p));
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeShapeDerivatives(
const Ref<const MatrixXr> & J, const Ref<const MatrixXr> & dnds,
Ref<MatrixXr> shape_deriv) {
shape_deriv = J.inverse() * dnds;
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
inline void ElementClass<type, kind>::computeNormalsOnNaturalCoordinates(
const Ref<const MatrixXr> & coord, const Ref<const MatrixXr> & f,
Ref<MatrixXr> 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) {
J = interpolation_element::gradientOnNaturalCoordinates(coord.col(p), f);
if (dimension == 2) {
Math::normal2(J.data(), normals.col(p).data());
}
if (dimension == 3) {
Math::normal3(J.col(0).data(), J.col(1).data(), normals.col(p).data());
}
}
}
/* ------------------------------------------------------------------------- */
/**
* 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 Ref<const VectorXr> & real_coords,
const Ref<const MatrixXr> & node_coords,
Ref<VectorXr> natural_coords,
UInt max_iterations,
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.data(), spatial_dimension, 1);
// initial guess
natural_coords.zero();
// real space coordinates provided by initial guess
Matrix<Real> physical_guess(spatial_dimension, 1);
// objective function f = real_coords - physical_guess
Matrix<Real> f(spatial_dimension, 1);
// J Jacobian matrix computed on the natural_guess
Matrix<Real> J(dimension, spatial_dimension);
// J^t
Matrix<Real> Jt(spatial_dimension, dimension);
// G = J^t * J
Matrix<Real> G(dimension, dimension);
// Ginv = G^{-1}
Matrix<Real> Ginv(dimension, dimension);
// J = Ginv * J^t
Matrix<Real> F(spatial_dimension, dimension);
// dxi = \xi_{k+1} - \xi in the iterative process
Matrix<Real> dxi(dimension, 1);
Matrix<Real> dxit(1, dimension);
/* --------------------------- */
/* init before iteration loop */
/* --------------------------- */
// do interpolation
auto update_f = [&f, &physical_guess, &natural_coords, &node_coords,
&mreal_coords, spatial_dimension]() {
Vector<Real> physical_guess_v(physical_guess.data(), spatial_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
auto error = f.norm();
return error;
};
auto inverse_map_error = update_f();
/* --------------------------- */
/* iteration loop */
/* --------------------------- */
UInt iterations{0};
while (tolerance < inverse_map_error and iterations < max_iterations) {
// compute J^t
interpolation_element::gradientOnNaturalCoordinates(natural_coords,
node_coords, Jt);
J = Jt.transpose();
// compute G
auto && G = J.transpose() * J;
// compute F
auto && F = G.inverse() * J.transpose();
// compute increment
auto && dxi = F * f;
// update our guess
natural_coords += dxi;
inverse_map_error = update_f();
iterations++;
}
if(iterations >= max_iterations) {
AKANTU_EXCEPTION("The solver in inverse map did not converge");
}
}
/* -------------------------------------------------------------------------- */
template <ElementType type, ElementKind kind>
template <typename Derived1, typename Derived2, typename Derived3,
aka::enable_if_matrices_t<Derived1, Derived2, Derived3> *>
inline void ElementClass<type, kind>::inverseMap(
const Eigen::MatrixBase<Derived1> & real_coords,
const Eigen::MatrixBase<Derived2> & node_coords,
const Eigen::MatrixBase<Derived3> & natural_coords_, UInt max_iterations,
Real tolerance) {
Eigen::MatrixBase<Derived2> & natural_coords =
const_cast<Eigen::MatrixBase<Derived2> &>(
natural_coords_); // as advised by the Eigen developers
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, max_iterations, tolerance);
}
}
} // namespace akantu
#endif /* AKANTU_ELEMENT_CLASS_TMPL_HH_ */

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