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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 Nicolas Richart <nicolas.richart@epfl.ch>
			*
			* @date creation: Thu Feb 21 2013
			* @date last modification: Wed Nov 29 2017
			*
			* @brief Implementation of the inline templated function of the element class
			* descriptions
			*
			*
			* Copyright (©) 2014-2018 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 <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 auto
			ElementClass<element_type, element_kind>::getFacetTypes() {
			return VectorProxy<const ElementType>(
			element_class_extra_geom_property::facet_type.data(),
			geometrical_element::getNbFacetTypes());
			}

			/* -------------------------------------------------------------------------- */
			/* GeometricalElement */
			/* -------------------------------------------------------------------------- */
			template <GeometricalType geometrical_type, GeometricalShapeType shape>
			inline constexpr 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 MatrixProxy<const UInt>(
			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 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 UInt
			GeometricalElement<geometrical_type, shape>::getNbFacetsPerElement(UInt t) {
			return geometrical_property::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. + std::numeric_limits<Real>::epsilon())) &&
			(coords(i) <= (1. + std::numeric_limits<Real>::epsilon())));
			}
			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) >= -(Math::getTolerance())) &&
			(coords(i) <= (1. + Math::getTolerance())));
			sum += coords(i);
			}
			if (in) {
			return (in && (sum <= (1. + Math::getTolerance())));
			}
			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 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>
			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(),
			InterpolationProperty<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(
			InterpolationProperty<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(
			InterpolationProperty<interpolation_type>::natural_space_dimension,
			InterpolationProperty<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, 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.storage(), spatial_dimension, 1);

			// initial guess
			// Matrix<Real> natural_guess(natural_coords.storage(), dimension, 1);
			natural_coords.zero();

			// real space coordinates provided by initial guess
			Matrix<Real> physical_guess(spatial_dimension /*changed from
			dimension */
			,
			1);

			// objective function f = real_coords - physical_guess
			Matrix<Real> f(spatial_dimension /changed from 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(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.storage(), 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<L_2>();
			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
			G.mul<false, true>(J, J);

			// inverse G
			Ginv.inverse(G);

			// compute F
			F.mul<true, false>(J, Ginv);

			// compute increment
			dxit.mul<true, false>(f, F);

			dxi = dxit.transpose();

			// update our guess
			natural_coords += Vector<Real>(dxi(0));

			inverse_map_error = update_f();
			iterations++;
			}
			}

			/* -------------------------------------------------------------------------- */
			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, UInt max_iterations, 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, max_iterations, tolerance);
			}
			}

			} // namespace akantu

			#endif /* AKANTU_ELEMENT_CLASS_TMPL_HH_ */

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rAKA/src/fe_engine/element_class_tmpl.hh
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element_class_tmpl.hh
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