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material.cc
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	/**
	* @file material.cc
	*
	* @author Guillaume Anciaux <guillaume.anciaux@epfl.ch>
	* @author Aurelia Isabel Cuba Ramos <aurelia.cubaramos@epfl.ch>
	* @author Daniel Pino Muñoz <daniel.pinomunoz@epfl.ch>
	* @author Nicolas Richart <nicolas.richart@epfl.ch>
	* @author Marco Vocialta <marco.vocialta@epfl.ch>
	*
	* @date creation: Tue Jul 27 2010
	* @date last modification: Wed Feb 21 2018
	*
	* @brief Implementation of the common part of the material class
	*
	*
	* Copyright (©) 2010-2018 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 "material.hh"
	#include "mesh_iterators.hh"
	#include "solid_mechanics_model.hh"
	/* -------------------------------------------------------------------------- */

	namespace akantu {

	/* -------------------------------------------------------------------------- */
	Material::Material(SolidMechanicsModel & model, const ID & id)
	: Parsable(ParserType::_material, id), id(id), fem(model.getFEEngine()),
	model(model), spatial_dimension(this->model.getSpatialDimension()),
	element_filter("element_filter", id), stress("stress", *this),
	eigengradu("eigen_grad_u", this), gradu("grad_u", this),
	green_strain("green_strain", *this),
	piola_kirchhoff_2("piola_kirchhoff_2", *this),
	potential_energy("potential_energy", *this),
	interpolation_inverse_coordinates("interpolation inverse coordinates",
	*this),
	interpolation_points_matrices("interpolation points matrices", *this),
	eigen_grad_u(model.getSpatialDimension(), model.getSpatialDimension(),
	0.) {
	AKANTU_DEBUG_IN();

	this->registerParam("eigen_grad_u", eigen_grad_u, _pat_parsable,
	"EigenGradU");

	/// for each connectivity types allocate the element filer array of the
	/// material
	element_filter.initialize(model.getMesh(),
	_spatial_dimension = spatial_dimension,
	_element_kind = _ek_regular);
	this->initialize();

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	Material::Material(SolidMechanicsModel & model, UInt dim, const Mesh & mesh,
	FEEngine & fe_engine, const ID & id)
	: Parsable(ParserType::_material, id), id(id), fem(fe_engine), model(model),
	spatial_dimension(dim), element_filter("element_filter", id),
	stress("stress", *this, dim, fe_engine, this->element_filter),
	eigengradu("eigen_grad_u", *this, dim, fe_engine, this->element_filter),
	gradu("gradu", *this, dim, fe_engine, this->element_filter),
	green_strain("green_strain", *this, dim, fe_engine, this->element_filter),
	piola_kirchhoff_2("piola_kirchhoff_2", *this, dim, fe_engine,
	this->element_filter),
	potential_energy("potential_energy", *this, dim, fe_engine,
	this->element_filter),
	interpolation_inverse_coordinates("interpolation inverse_coordinates",
	*this, dim, fe_engine,
	this->element_filter),
	interpolation_points_matrices("interpolation points matrices", *this, dim,
	fe_engine, this->element_filter) {

	AKANTU_DEBUG_IN();

	element_filter.initialize(mesh, _spatial_dimension = spatial_dimension,
	_element_kind = _ek_regular);

	this->initialize();
	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	Material::~Material() = default;

	/* -------------------------------------------------------------------------- */
	void Material::initialize() {
	registerParam("rho", rho, Real(0.), _pat_parsable \| _pat_modifiable,
	"Density");
	registerParam("name", name, std::string(), _pat_parsable \| _pat_readable);
	registerParam("finite_deformation", finite_deformation, false,
	_pat_parsable \| _pat_readable, "Is finite deformation");
	registerParam("inelastic_deformation", inelastic_deformation, false,
	_pat_internal, "Is inelastic deformation");

	/// allocate gradu stress for local elements
	eigengradu.initialize(spatial_dimension * spatial_dimension);
	gradu.initialize(spatial_dimension * spatial_dimension);
	stress.initialize(spatial_dimension * spatial_dimension);

	potential_energy.initialize(1);

	this->model.registerEventHandler(*this);
	}

	/* -------------------------------------------------------------------------- */
	void Material::initMaterial() {
	AKANTU_DEBUG_IN();

	if (finite_deformation) {
	this->piola_kirchhoff_2.initialize(spatial_dimension * spatial_dimension);
	if (use_previous_stress) {
	this->piola_kirchhoff_2.initializeHistory();
	}
	this->green_strain.initialize(spatial_dimension * spatial_dimension);
	}

	if (use_previous_stress) {
	this->stress.initializeHistory();
	}
	if (use_previous_gradu) {
	this->gradu.initializeHistory();
	}

	this->resizeInternals();

	auto dim = model.getSpatialDimension();
	for (const auto & type :
	element_filter.elementTypes(_element_kind = _ek_regular)) {
	for (auto & eigen_gradu : make_view(eigengradu(type), dim, dim)) {
	eigen_gradu = eigen_grad_u;
	}
	}

	is_init = true;

	updateInternalParameters();

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::savePreviousState() {
	AKANTU_DEBUG_IN();

	for (auto pair : internal_vectors_real) {
	if (pair.second->hasHistory()) {
	pair.second->saveCurrentValues();
	}
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::restorePreviousState() {
	AKANTU_DEBUG_IN();

	for (auto pair : internal_vectors_real) {
	if (pair.second->hasHistory()) {
	pair.second->restorePreviousValues();
	}
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	/**
	* Compute the internal forces by assembling @f$\int_{e} \sigma_e \frac{\partial
	* \varphi}{\partial X} dX @f$
	*
	* @param[in] ghost_type compute the internal forces for _ghost or _not_ghost
	* element
	*/
	void Material::assembleInternalForces(GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	UInt spatial_dimension = model.getSpatialDimension();

	if (!finite_deformation) {

	auto & internal_force = const_cast<Array<Real> &>(model.getInternalForce());

	// Mesh & mesh = fem.getMesh();
	for (auto && type :
	element_filter.elementTypes(spatial_dimension, ghost_type)) {
	Array<UInt> & elem_filter = element_filter(type, ghost_type);
	UInt nb_element = elem_filter.size();

	if (nb_element == 0) {
	continue;
	}

	const Array<Real> & shapes_derivatives =
	fem.getShapesDerivatives(type, ghost_type);

	UInt size_of_shapes_derivatives = shapes_derivatives.getNbComponent();
	UInt nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);
	UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);

	/// compute @f$\sigma \frac{\partial \varphi}{\partial X}@f$ by
	/// @f$\mathbf{B}^t \mathbf{\sigma}_q@f$
	auto * sigma_dphi_dx =
	new Array<Real>(nb_element * nb_quadrature_points,
	size_of_shapes_derivatives, "sigma_x_dphi_/_dX");

	fem.computeBtD(stress(type, ghost_type), *sigma_dphi_dx, type, ghost_type,
	elem_filter);

	/**
	* compute @f$\int \sigma * \frac{\partial \varphi}{\partial X}dX@f$ by
	* @f$ \sum_q \mathbf{B}^t
	* \mathbf{\sigma}_q \overline w_q J_q@f$
	*/
	auto * int_sigma_dphi_dx =
	new Array<Real>(nb_element, nb_nodes_per_element * spatial_dimension,
	"int_sigma_x_dphi_/_dX");

	fem.integrate(sigma_dphi_dx, int_sigma_dphi_dx,
	size_of_shapes_derivatives, type, ghost_type, elem_filter);
	delete sigma_dphi_dx;

	/// assemble
	model.getDOFManager().assembleElementalArrayLocalArray(
	*int_sigma_dphi_dx, internal_force, type, ghost_type, -1,
	elem_filter);
	delete int_sigma_dphi_dx;
	}
	} else {
	switch (spatial_dimension) {
	case 1:
	this->assembleInternalForces<1>(ghost_type);
	break;
	case 2:
	this->assembleInternalForces<2>(ghost_type);
	break;
	case 3:
	this->assembleInternalForces<3>(ghost_type);
	break;
	}
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	/**
	* Compute the stress from the gradu
	*
	* @param[in] ghost_type compute the residual for _ghost or _not_ghost element
	*/
	void Material::computeAllStresses(GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	UInt spatial_dimension = model.getSpatialDimension();

	for (const auto & type :
	element_filter.elementTypes(spatial_dimension, ghost_type)) {
	Array<UInt> & elem_filter = element_filter(type, ghost_type);

	if (elem_filter.empty()) {
	continue;
	}
	Array<Real> & gradu_vect = gradu(type, ghost_type);

	/// compute @f$\nabla u@f$
	fem.gradientOnIntegrationPoints(model.getDisplacement(), gradu_vect,
	spatial_dimension, type, ghost_type,
	elem_filter);

	gradu_vect -= eigengradu(type, ghost_type);

	/// compute @f$\mathbf{\sigma}_q@f$ from @f$\nabla u@f$
	computeStress(type, ghost_type);
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::computeAllCauchyStresses(GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	AKANTU_DEBUG_ASSERT(finite_deformation, "The Cauchy stress can only be "
	"computed if you are working in "
	"finite deformation.");

	for (auto type : element_filter.elementTypes(spatial_dimension, ghost_type)) {
	switch (spatial_dimension) {
	case 1:
	this->StoCauchy<1>(type, ghost_type);
	break;
	case 2:
	this->StoCauchy<2>(type, ghost_type);
	break;
	case 3:
	this->StoCauchy<3>(type, ghost_type);
	break;
	}
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <UInt dim>
	void Material::StoCauchy(ElementType el_type, GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	auto gradu_it = this->gradu(el_type, ghost_type).begin(dim, dim);
	auto gradu_end = this->gradu(el_type, ghost_type).end(dim, dim);

	auto piola_it = this->piola_kirchhoff_2(el_type, ghost_type).begin(dim, dim);
	auto stress_it = this->stress(el_type, ghost_type).begin(dim, dim);

	for (; gradu_it != gradu_end; ++gradu_it, ++piola_it, ++stress_it) {
	Matrix<Real> & grad_u = *gradu_it;
	Matrix<Real> & piola = *piola_it;
	Matrix<Real> & sigma = *stress_it;

	auto F_tensor = gradUToF<dim>(grad_u);
	this->StoCauchy<dim>(F_tensor, piola, sigma);
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::setToSteadyState(GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	const Array<Real> & displacement = model.getDisplacement();

	// resizeInternalArray(gradu);

	UInt spatial_dimension = model.getSpatialDimension();

	for (auto type : element_filter.elementTypes(spatial_dimension, ghost_type)) {
	Array<UInt> & elem_filter = element_filter(type, ghost_type);
	Array<Real> & gradu_vect = gradu(type, ghost_type);

	/// compute @f$\nabla u@f$
	fem.gradientOnIntegrationPoints(displacement, gradu_vect, spatial_dimension,
	type, ghost_type, elem_filter);

	setToSteadyState(type, ghost_type);
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	/**
	* Compute the stiffness matrix by assembling @f$\int_{\omega} B^t \times D
	* \times B d\omega @f$
	*
	* @param[in] ghost_type compute the residual for _ghost or _not_ghost element
	*/
	void Material::assembleStiffnessMatrix(GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	UInt spatial_dimension = model.getSpatialDimension();

	for (auto type : element_filter.elementTypes(spatial_dimension, ghost_type)) {
	if (finite_deformation) {
	switch (spatial_dimension) {
	case 1: {
	assembleStiffnessMatrixNL<1>(type, ghost_type);
	assembleStiffnessMatrixL2<1>(type, ghost_type);
	break;
	}
	case 2: {
	assembleStiffnessMatrixNL<2>(type, ghost_type);
	assembleStiffnessMatrixL2<2>(type, ghost_type);
	break;
	}
	case 3: {
	assembleStiffnessMatrixNL<3>(type, ghost_type);
	assembleStiffnessMatrixL2<3>(type, ghost_type);
	break;
	}
	}
	} else {
	switch (spatial_dimension) {
	case 1: {
	assembleStiffnessMatrix<1>(type, ghost_type);
	break;
	}
	case 2: {
	assembleStiffnessMatrix<2>(type, ghost_type);
	break;
	}
	case 3: {
	assembleStiffnessMatrix<3>(type, ghost_type);
	break;
	}
	}
	}
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <UInt dim>
	void Material::assembleStiffnessMatrix(ElementType type, GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	Array<UInt> & elem_filter = element_filter(type, ghost_type);
	if (elem_filter.empty()) {
	AKANTU_DEBUG_OUT();
	return;
	}

	// const Array<Real> & shapes_derivatives =
	// fem.getShapesDerivatives(type, ghost_type);

	Array<Real> & gradu_vect = gradu(type, ghost_type);

	UInt nb_element = elem_filter.size();
	UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	UInt nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);

	gradu_vect.resize(nb_quadrature_points * nb_element);

	fem.gradientOnIntegrationPoints(model.getDisplacement(), gradu_vect, dim,
	type, ghost_type, elem_filter);

	UInt tangent_size = getTangentStiffnessVoigtSize(dim);

	auto * tangent_stiffness_matrix =
	new Array<Real>(nb_element * nb_quadrature_points,
	tangent_size * tangent_size, "tangent_stiffness_matrix");

	tangent_stiffness_matrix->zero();

	computeTangentModuli(type, *tangent_stiffness_matrix, ghost_type);

	/// compute @f$\mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	UInt bt_d_b_size = dim * nb_nodes_per_element;

	auto * bt_d_b = new Array<Real>(nb_element * nb_quadrature_points,
	bt_d_b_size * bt_d_b_size, "B^tDB");

	fem.computeBtDB(tangent_stiffness_matrix, bt_d_b, 4, type, ghost_type,
	elem_filter);

	delete tangent_stiffness_matrix;

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto * K_e = new Array<Real>(nb_element, bt_d_b_size * bt_d_b_size, "K_e");

	fem.integrate(bt_d_b, K_e, bt_d_b_size * bt_d_b_size, type, ghost_type,
	elem_filter);

	delete bt_d_b;

	model.getDOFManager().assembleElementalMatricesToMatrix(
	"K", "displacement", *K_e, type, ghost_type, _symmetric, elem_filter);
	delete K_e;

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <UInt dim>
	void Material::assembleStiffnessMatrixNL(ElementType type,
	GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	const Array<Real> & shapes_derivatives =
	fem.getShapesDerivatives(type, ghost_type);

	Array<UInt> & elem_filter = element_filter(type, ghost_type);
	// Array<Real> & gradu_vect = delta_gradu(type, ghost_type);

	UInt nb_element = elem_filter.size();
	UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	UInt nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);

	auto * shapes_derivatives_filtered = new Array<Real>(
	nb_element * nb_quadrature_points, dim * nb_nodes_per_element,
	"shapes derivatives filtered");

	FEEngine::filterElementalData(fem.getMesh(), shapes_derivatives,
	*shapes_derivatives_filtered, type, ghost_type,
	elem_filter);

	/// compute @f$\mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	UInt bt_s_b_size = dim * nb_nodes_per_element;

	auto * bt_s_b = new Array<Real>(nb_element * nb_quadrature_points,
	bt_s_b_size * bt_s_b_size, "B^tDB");

	UInt piola_matrix_size = getCauchyStressMatrixSize(dim);

	Matrix<Real> B(piola_matrix_size, bt_s_b_size);
	Matrix<Real> Bt_S(bt_s_b_size, piola_matrix_size);
	Matrix<Real> S(piola_matrix_size, piola_matrix_size);

	auto shapes_derivatives_filtered_it = shapes_derivatives_filtered->begin(
	spatial_dimension, nb_nodes_per_element);

	auto Bt_S_B_it = bt_s_b->begin(bt_s_b_size, bt_s_b_size);
	auto Bt_S_B_end = bt_s_b->end(bt_s_b_size, bt_s_b_size);
	auto piola_it = piola_kirchhoff_2(type, ghost_type).begin(dim, dim);

	for (; Bt_S_B_it != Bt_S_B_end;
	++Bt_S_B_it, ++shapes_derivatives_filtered_it, ++piola_it) {
	auto & Bt_S_B = *Bt_S_B_it;
	const auto & Piola_kirchhoff_matrix = *piola_it;

	setCauchyStressMatrix<dim>(Piola_kirchhoff_matrix, S);
	VoigtHelper<dim>::transferBMatrixToBNL(*shapes_derivatives_filtered_it, B,
	nb_nodes_per_element);
	Bt_S.template mul<true, false>(B, S);
	Bt_S_B.template mul<false, false>(Bt_S, B);
	}

	delete shapes_derivatives_filtered;

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto * K_e = new Array<Real>(nb_element, bt_s_b_size * bt_s_b_size, "K_e");

	fem.integrate(bt_s_b, K_e, bt_s_b_size * bt_s_b_size, type, ghost_type,
	elem_filter);

	delete bt_s_b;

	model.getDOFManager().assembleElementalMatricesToMatrix(
	"K", "displacement", *K_e, type, ghost_type, _symmetric, elem_filter);

	delete K_e;

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <UInt dim>
	void Material::assembleStiffnessMatrixL2(ElementType type,
	GhostType ghost_type) {
	AKANTU_DEBUG_IN();

	const Array<Real> & shapes_derivatives =
	fem.getShapesDerivatives(type, ghost_type);

	Array<UInt> & elem_filter = element_filter(type, ghost_type);
	Array<Real> & gradu_vect = gradu(type, ghost_type);

	UInt nb_element = elem_filter.size();
	UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	UInt nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);

	gradu_vect.resize(nb_quadrature_points * nb_element);

	fem.gradientOnIntegrationPoints(model.getDisplacement(), gradu_vect, dim,
	type, ghost_type, elem_filter);

	UInt tangent_size = getTangentStiffnessVoigtSize(dim);

	auto * tangent_stiffness_matrix =
	new Array<Real>(nb_element * nb_quadrature_points,
	tangent_size * tangent_size, "tangent_stiffness_matrix");

	tangent_stiffness_matrix->zero();

	computeTangentModuli(type, *tangent_stiffness_matrix, ghost_type);

	auto * shapes_derivatives_filtered = new Array<Real>(
	nb_element * nb_quadrature_points, dim * nb_nodes_per_element,
	"shapes derivatives filtered");
	FEEngine::filterElementalData(fem.getMesh(), shapes_derivatives,
	*shapes_derivatives_filtered, type, ghost_type,
	elem_filter);

	/// compute @f$\mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	UInt bt_d_b_size = dim * nb_nodes_per_element;

	auto * bt_d_b = new Array<Real>(nb_element * nb_quadrature_points,
	bt_d_b_size * bt_d_b_size, "B^tDB");

	Matrix<Real> B(tangent_size, dim * nb_nodes_per_element);
	Matrix<Real> B2(tangent_size, dim * nb_nodes_per_element);
	Matrix<Real> Bt_D(dim * nb_nodes_per_element, tangent_size);

	auto shapes_derivatives_filtered_it = shapes_derivatives_filtered->begin(
	spatial_dimension, nb_nodes_per_element);

	auto Bt_D_B_it = bt_d_b->begin(bt_d_b_size, bt_d_b_size);
	auto grad_u_it = gradu_vect.begin(dim, dim);
	auto D_it = tangent_stiffness_matrix->begin(tangent_size, tangent_size);
	auto D_end = tangent_stiffness_matrix->end(tangent_size, tangent_size);

	for (; D_it != D_end;
	++D_it, ++Bt_D_B_it, ++shapes_derivatives_filtered_it, ++grad_u_it) {
	const auto & grad_u = *grad_u_it;
	const auto & D = *D_it;
	auto & Bt_D_B = *Bt_D_B_it;

	// transferBMatrixToBL1<dim > (*shapes_derivatives_filtered_it, B,
	// nb_nodes_per_element);
	VoigtHelper<dim>::transferBMatrixToSymVoigtBMatrix(
	*shapes_derivatives_filtered_it, B, nb_nodes_per_element);
	VoigtHelper<dim>::transferBMatrixToBL2(*shapes_derivatives_filtered_it,
	grad_u, B2, nb_nodes_per_element);
	B += B2;
	Bt_D.template mul<true, false>(B, D);
	Bt_D_B.template mul<false, false>(Bt_D, B);
	}

	delete tangent_stiffness_matrix;
	delete shapes_derivatives_filtered;

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto * K_e = new Array<Real>(nb_element, bt_d_b_size * bt_d_b_size, "K_e");

	fem.integrate(bt_d_b, K_e, bt_d_b_size * bt_d_b_size, type, ghost_type,
	elem_filter);

	delete bt_d_b;

	model.getDOFManager().assembleElementalMatricesToMatrix(
	"K", "displacement", *K_e, type, ghost_type, _symmetric, elem_filter);
	delete K_e;

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <UInt dim>
	void Material::assembleInternalForces(GhostType ghost_type) {

	AKANTU_DEBUG_IN();

	Array<Real> & internal_force = model.getInternalForce();

	Mesh & mesh = fem.getMesh();
	for (auto type : element_filter.elementTypes(_ghost_type = ghost_type)) {
	const Array<Real> & shapes_derivatives =
	fem.getShapesDerivatives(type, ghost_type);

	Array<UInt> & elem_filter = element_filter(type, ghost_type);
	if (elem_filter.empty()) {
	continue;
	}
	UInt size_of_shapes_derivatives = shapes_derivatives.getNbComponent();
	UInt nb_element = elem_filter.size();
	UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	UInt nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);

	auto * shapesd_filtered = new Array<Real>(
	nb_element, size_of_shapes_derivatives, "filtered shapesd");

	FEEngine::filterElementalData(mesh, shapes_derivatives, *shapesd_filtered,
	type, ghost_type, elem_filter);

	Array<Real>::matrix_iterator shapes_derivatives_filtered_it =
	shapesd_filtered->begin(dim, nb_nodes_per_element);

	// Set stress vectors
	UInt stress_size = getTangentStiffnessVoigtSize(dim);

	// Set matrices B and BNL*
	UInt bt_s_size = dim * nb_nodes_per_element;

	auto * bt_s =
	new Array<Real>(nb_element * nb_quadrature_points, bt_s_size, "B^t*S");

	auto grad_u_it = this->gradu(type, ghost_type).begin(dim, dim);
	auto grad_u_end = this->gradu(type, ghost_type).end(dim, dim);
	auto stress_it = this->piola_kirchhoff_2(type, ghost_type).begin(dim, dim);
	shapes_derivatives_filtered_it =
	shapesd_filtered->begin(dim, nb_nodes_per_element);

	Array<Real>::matrix_iterator bt_s_it = bt_s->begin(bt_s_size, 1);

	Matrix<Real> B_tensor(stress_size, bt_s_size);
	Matrix<Real> B2_tensor(stress_size, bt_s_size);

	for (; grad_u_it != grad_u_end; ++grad_u_it, ++stress_it,
	++shapes_derivatives_filtered_it,
	++bt_s_it) {
	auto & grad_u = *grad_u_it;
	auto & r = *bt_s_it;
	auto & S = *stress_it;

	VoigtHelper<dim>::transferBMatrixToSymVoigtBMatrix(
	*shapes_derivatives_filtered_it, B_tensor, nb_nodes_per_element);

	VoigtHelper<dim>::transferBMatrixToBL2(*shapes_derivatives_filtered_it,
	grad_u, B2_tensor,
	nb_nodes_per_element);

	B_tensor += B2_tensor;

	auto S_vect = Material::stressToVoigt<dim>(S);
	Matrix<Real> S_voigt(S_vect.storage(), stress_size, 1);

	r.template mul<true, false>(B_tensor, S_voigt);
	}

	delete shapesd_filtered;

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto * r_e = new Array<Real>(nb_element, bt_s_size, "r_e");

	fem.integrate(bt_s, r_e, bt_s_size, type, ghost_type, elem_filter);

	delete bt_s;

	model.getDOFManager().assembleElementalArrayLocalArray(
	*r_e, internal_force, type, ghost_type, -1., elem_filter);
	delete r_e;
	}
	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::computePotentialEnergyByElements() {
	AKANTU_DEBUG_IN();

	for (auto type : element_filter.elementTypes(spatial_dimension, _not_ghost)) {
	computePotentialEnergy(type);
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::computePotentialEnergy(ElementType /unused/) {
	AKANTU_DEBUG_IN();
	AKANTU_TO_IMPLEMENT();
	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	Real Material::getPotentialEnergy() {
	AKANTU_DEBUG_IN();
	Real epot = 0.;

	computePotentialEnergyByElements();

	/// integrate the potential energy for each type of elements
	for (auto type : element_filter.elementTypes(spatial_dimension, _not_ghost)) {
	epot += fem.integrate(potential_energy(type, _not_ghost), type, _not_ghost,
	element_filter(type, _not_ghost));
	}

	AKANTU_DEBUG_OUT();
	return epot;
	}

	/* -------------------------------------------------------------------------- */
	Real Material::getPotentialEnergy(ElementType & type, UInt index) {
	AKANTU_DEBUG_IN();
	Real epot = 0.;

	Vector<Real> epot_on_quad_points(fem.getNbIntegrationPoints(type));

	computePotentialEnergyByElement(type, index, epot_on_quad_points);

	epot = fem.integrate(epot_on_quad_points, type, element_filter(type)(index));

	AKANTU_DEBUG_OUT();
	return epot;
	}

	/* -------------------------------------------------------------------------- */
	Real Material::getEnergy(const std::string & type) {
	AKANTU_DEBUG_IN();
	if (type == "potential") {
	return getPotentialEnergy();
	}
	AKANTU_DEBUG_OUT();
	return 0.;
	}

	/* -------------------------------------------------------------------------- */
	Real Material::getEnergy(const std::string & energy_id, ElementType type,
	UInt index) {
	AKANTU_DEBUG_IN();
	if (energy_id == "potential") {
	return getPotentialEnergy(type, index);
	}
	AKANTU_DEBUG_OUT();
	return 0.;
	}

	/* -------------------------------------------------------------------------- */
	void Material::initElementalFieldInterpolation(
	const ElementTypeMapArray<Real> & interpolation_points_coordinates) {
	AKANTU_DEBUG_IN();

	this->fem.initElementalFieldInterpolationFromIntegrationPoints(
	interpolation_points_coordinates, this->interpolation_points_matrices,
	this->interpolation_inverse_coordinates, &(this->element_filter));

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::interpolateStress(ElementTypeMapArray<Real> & result,
	const GhostType ghost_type) {

	this->fem.interpolateElementalFieldFromIntegrationPoints(
	this->stress, this->interpolation_points_matrices,
	this->interpolation_inverse_coordinates, result, ghost_type,
	&(this->element_filter));
	}

	/* -------------------------------------------------------------------------- */
	void Material::interpolateStressOnFacets(
	ElementTypeMapArray<Real> & result,
	ElementTypeMapArray<Real> & by_elem_result, const GhostType ghost_type) {

	interpolateStress(by_elem_result, ghost_type);

	UInt stress_size = this->stress.getNbComponent();

	const Mesh & mesh = this->model.getMesh();
	const Mesh & mesh_facets = mesh.getMeshFacets();

	for (auto type : element_filter.elementTypes(spatial_dimension, ghost_type)) {
	Array<UInt> & elem_fil = element_filter(type, ghost_type);
	Array<Real> & by_elem_res = by_elem_result(type, ghost_type);
	UInt nb_element = elem_fil.size();
	UInt nb_element_full = this->model.getMesh().getNbElement(type, ghost_type);
	UInt nb_interpolation_points_per_elem =
	by_elem_res.size() / nb_element_full;

	const Array<Element> & facet_to_element =
	mesh_facets.getSubelementToElement(type, ghost_type);
	ElementType type_facet = Mesh::getFacetType(type);
	UInt nb_facet_per_elem = facet_to_element.getNbComponent();
	UInt nb_quad_per_facet =
	nb_interpolation_points_per_elem / nb_facet_per_elem;
	Element element_for_comparison{type, 0, ghost_type};
	const Array<std::vector<Element>> * element_to_facet = nullptr;
	GhostType current_ghost_type = _casper;
	Array<Real> * result_vec = nullptr;

	Array<Real>::const_matrix_iterator result_it =
	by_elem_res.begin_reinterpret(
	stress_size, nb_interpolation_points_per_elem, nb_element_full);

	for (UInt el = 0; el < nb_element; ++el) {
	UInt global_el = elem_fil(el);
	element_for_comparison.element = global_el;
	for (UInt f = 0; f < nb_facet_per_elem; ++f) {

	Element facet_elem = facet_to_element(global_el, f);
	UInt global_facet = facet_elem.element;
	if (facet_elem.ghost_type != current_ghost_type) {
	current_ghost_type = facet_elem.ghost_type;
	element_to_facet = &mesh_facets.getElementToSubelement(
	type_facet, current_ghost_type);
	result_vec = &result(type_facet, current_ghost_type);
	}

	bool is_second_element =
	(*element_to_facet)(global_facet)[0] != element_for_comparison;

	for (UInt q = 0; q < nb_quad_per_facet; ++q) {
	Vector<Real> result_local(result_vec->storage() +
	(global_facet * nb_quad_per_facet + q) *
	result_vec->getNbComponent() +
	static_cast<UInt>(is_second_element) *
	stress_size,
	stress_size);

	const Matrix<Real> & result_tmp(result_it[global_el]);
	result_local = result_tmp(f * nb_quad_per_facet + q);
	}
	}
	}
	}
	}

	/* -------------------------------------------------------------------------- */
	template <typename T>
	const InternalField<T> &
	Material::getInternal([[gnu::unused]] const ID & int_id) const {
	AKANTU_TO_IMPLEMENT();
	return NULL;
	}

	/* -------------------------------------------------------------------------- */
	template <typename T>
	InternalField<T> & Material::getInternal([[gnu::unused]] const ID & int_id) {
	AKANTU_TO_IMPLEMENT();
	return NULL;
	}

	/* -------------------------------------------------------------------------- */
	template <>
	const InternalField<Real> & Material::getInternal(const ID & int_id) const {
	auto it = internal_vectors_real.find(getID() + ":" + int_id);
	if (it == internal_vectors_real.end()) {
	AKANTU_SILENT_EXCEPTION("The material " << name << "(" << getID()
	<< ") does not contain an internal "
	<< int_id << " ("
	<< (getID() + ":" + int_id) << ")");
	}
	return *it->second;
	}

	/* -------------------------------------------------------------------------- */
	template <> InternalField<Real> & Material::getInternal(const ID & int_id) {
	auto it = internal_vectors_real.find(getID() + ":" + int_id);
	if (it == internal_vectors_real.end()) {
	AKANTU_SILENT_EXCEPTION("The material " << name << "(" << getID()
	<< ") does not contain an internal "
	<< int_id << " ("
	<< (getID() + ":" + int_id) << ")");
	}
	return *it->second;
	}

	/* -------------------------------------------------------------------------- */
	template <>
	const InternalField<UInt> & Material::getInternal(const ID & int_id) const {
	auto it = internal_vectors_uint.find(getID() + ":" + int_id);
	if (it == internal_vectors_uint.end()) {
	AKANTU_SILENT_EXCEPTION("The material " << name << "(" << getID()
	<< ") does not contain an internal "
	<< int_id << " ("
	<< (getID() + ":" + int_id) << ")");
	}
	return *it->second;
	}

	/* -------------------------------------------------------------------------- */
	template <> InternalField<UInt> & Material::getInternal(const ID & int_id) {
	auto it = internal_vectors_uint.find(getID() + ":" + int_id);
	if (it == internal_vectors_uint.end()) {
	AKANTU_SILENT_EXCEPTION("The material " << name << "(" << getID()
	<< ") does not contain an internal "
	<< int_id << " ("
	<< (getID() + ":" + int_id) << ")");
	}
	return *it->second;
	}

	/* -------------------------------------------------------------------------- */
	template <typename T>
	const Array<T> & Material::getArray(const ID & vect_id, ElementType type,
	GhostType ghost_type) const {
	try {
	return this->template getInternal<T>(vect_id)(type, ghost_type);
	} catch (debug::Exception & e) {
	AKANTU_SILENT_EXCEPTION("The material " << name << "(" << getID()
	<< ") does not contain a vector "
	<< vect_id << " [" << e << "]");
	}
	}

	/* -------------------------------------------------------------------------- */
	template <typename T>
	Array<T> & Material::getArray(const ID & vect_id, ElementType type,
	GhostType ghost_type) {
	try {
	return this->template getInternal<T>(vect_id)(type, ghost_type);
	} catch (debug::Exception & e) {
	AKANTU_SILENT_EXCEPTION("The material " << name << "(" << getID()
	<< ") does not contain a vector "
	<< vect_id << " [" << e << "]");
	}
	}

	template const Array<Real> & Material::getArray(const ID & vect_id,
	ElementType type,
	GhostType ghost_type) const;
	/* -------------------------------------------------------------------------- */
	template Array<Real> & Material::getArray(const ID & vect_id, ElementType type,
	GhostType ghost_type);
	/* -------------------------------------------------------------------------- */
	template const Array<UInt> & Material::getArray(const ID & vect_id,
	ElementType type,
	GhostType ghost_type) const;
	/* -------------------------------------------------------------------------- */
	template Array<UInt> & Material::getArray(const ID & vect_id, ElementType type,
	GhostType ghost_type);

	/* -------------------------------------------------------------------------- */
	void Material::addElements(const Array<Element> & elements_to_add) {
	AKANTU_DEBUG_IN();

	UInt mat_id = model.getMaterialIndex(name);
	for (const auto & element : elements_to_add) {
	auto index = this->addElement(element);
	model.material_index(element) = mat_id;
	model.material_local_numbering(element) = index;
	}

	this->resizeInternals();

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::removeElements(const Array<Element> & elements_to_remove) {
	AKANTU_DEBUG_IN();

	auto el_begin = elements_to_remove.begin();
	auto el_end = elements_to_remove.end();

	if (elements_to_remove.empty()) {
	return;
	}

	auto & mesh = this->model.getMesh();

	ElementTypeMapArray<UInt> material_local_new_numbering(
	"remove mat filter elem", id);

	material_local_new_numbering.initialize(
	mesh, _element_filter = &element_filter, _element_kind = _ek_not_defined,
	_with_nb_element = true);

	ElementTypeMapArray<UInt> element_filter_tmp("element_filter_tmp", id);

	element_filter_tmp.initialize(mesh, _element_filter = &element_filter,
	_element_kind = _ek_not_defined);

	ElementTypeMap<UInt> new_ids, element_ids;

	for_each_element(
	mesh,
	[&](auto && el) {
	if (not new_ids(el.type, el.ghost_type)) {
	element_ids(el.type, el.ghost_type) = 0;
	}

	auto & element_id = element_ids(el.type, el.ghost_type);
	auto l_el = Element{el.type, element_id, el.ghost_type};
	if (std::find(el_begin, el_end, el) != el_end) {
	material_local_new_numbering(l_el) = UInt(-1);
	return;
	}

	element_filter_tmp(el.type, el.ghost_type).push_back(el.element);
	if (not new_ids(el.type, el.ghost_type)) {
	new_ids(el.type, el.ghost_type) = 0;
	}

	auto & new_id = new_ids(el.type, el.ghost_type);

	material_local_new_numbering(l_el) = new_id;
	model.material_local_numbering(el) = new_id;

	++new_id;
	++element_id;
	},
	_element_filter = &element_filter, _element_kind = _ek_not_defined);

	for (auto ghost_type : ghost_types) {
	for (const auto & type : element_filter.elementTypes(
	_ghost_type = ghost_type, _element_kind = _ek_not_defined)) {
	element_filter(type, ghost_type)
	.copy(element_filter_tmp(type, ghost_type));
	}
	}

	for (auto it = internal_vectors_real.begin();
	it != internal_vectors_real.end(); ++it) {
	it->second->removeIntegrationPoints(material_local_new_numbering);
	}

	for (auto it = internal_vectors_uint.begin();
	it != internal_vectors_uint.end(); ++it) {
	it->second->removeIntegrationPoints(material_local_new_numbering);
	}

	for (auto it = internal_vectors_bool.begin();
	it != internal_vectors_bool.end(); ++it) {
	it->second->removeIntegrationPoints(material_local_new_numbering);
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::resizeInternals() {
	AKANTU_DEBUG_IN();
	for (auto it = internal_vectors_real.begin();
	it != internal_vectors_real.end(); ++it) {
	it->second->resize();
	}

	for (auto it = internal_vectors_uint.begin();
	it != internal_vectors_uint.end(); ++it) {
	it->second->resize();
	}

	for (auto it = internal_vectors_bool.begin();
	it != internal_vectors_bool.end(); ++it) {
	it->second->resize();
	}
	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	void Material::onElementsAdded(const Array<Element> & /unused/,
	const NewElementsEvent & /unused/) {
	this->resizeInternals();
	}

	/* -------------------------------------------------------------------------- */
	void Material::onElementsRemoved(
	const Array<Element> & element_list,
	const ElementTypeMapArray<UInt> & new_numbering,
	[[gnu::unused]] const RemovedElementsEvent & event) {
	UInt my_num = model.getInternalIndexFromID(getID());

	ElementTypeMapArray<UInt> material_local_new_numbering(
	"remove mat filter elem", getID());

	auto el_begin = element_list.begin();
	auto el_end = element_list.end();

	for (auto && gt : ghost_types) {
	for (auto && type :
	new_numbering.elementTypes(_all_dimensions, gt, _ek_not_defined)) {

	if (not element_filter.exists(type, gt) \|\|
	element_filter(type, gt).empty()) {
	continue;
	}

	auto & elem_filter = element_filter(type, gt);
	auto & mat_indexes = this->model.material_index(type, gt);
	auto & mat_loc_num = this->model.material_local_numbering(type, gt);
	auto nb_element = this->model.getMesh().getNbElement(type, gt);

	// all materials will resize of the same size...
	mat_indexes.resize(nb_element);
	mat_loc_num.resize(nb_element);

	if (!material_local_new_numbering.exists(type, gt)) {
	material_local_new_numbering.alloc(elem_filter.size(), 1, type, gt);
	}

	auto & mat_renumbering = material_local_new_numbering(type, gt);
	const auto & renumbering = new_numbering(type, gt);
	Array<UInt> elem_filter_tmp;
	UInt ni = 0;
	Element el{type, 0, gt};

	for (UInt i = 0; i < elem_filter.size(); ++i) {
	el.element = elem_filter(i);
	if (std::find(el_begin, el_end, el) == el_end) {
	UInt new_el = renumbering(el.element);
	AKANTU_DEBUG_ASSERT(new_el != UInt(-1),
	"A not removed element as been badly renumbered");
	elem_filter_tmp.push_back(new_el);
	mat_renumbering(i) = ni;

	mat_indexes(new_el) = my_num;
	mat_loc_num(new_el) = ni;
	++ni;
	} else {
	mat_renumbering(i) = UInt(-1);
	}
	}

	elem_filter.resize(elem_filter_tmp.size());
	elem_filter.copy(elem_filter_tmp);
	}
	}

	for (auto it = internal_vectors_real.begin();
	it != internal_vectors_real.end(); ++it) {
	it->second->removeIntegrationPoints(material_local_new_numbering);
	}

	for (auto it = internal_vectors_uint.begin();
	it != internal_vectors_uint.end(); ++it) {
	it->second->removeIntegrationPoints(material_local_new_numbering);
	}

	for (auto it = internal_vectors_bool.begin();
	it != internal_vectors_bool.end(); ++it) {
	it->second->removeIntegrationPoints(material_local_new_numbering);
	}
	}

	/* -------------------------------------------------------------------------- */
	void Material::beforeSolveStep() { this->savePreviousState(); }

	/* -------------------------------------------------------------------------- */
	void Material::afterSolveStep(bool converged) {
	if (not converged) {
	this->restorePreviousState();
	return;
	}

	for (const auto & type : element_filter.elementTypes(
	_all_dimensions, _not_ghost, _ek_not_defined)) {
	this->updateEnergies(type);
	}
	}
	/* -------------------------------------------------------------------------- */
	void Material::onDamageIteration() { this->savePreviousState(); }

	/* -------------------------------------------------------------------------- */
	void Material::onDamageUpdate() {
	for (const auto & type : element_filter.elementTypes(
	_all_dimensions, _not_ghost, _ek_not_defined)) {
	this->updateEnergiesAfterDamage(type);
	}
	}

	/* -------------------------------------------------------------------------- */
	void Material::onDump() {
	if (this->isFiniteDeformation()) {
	this->computeAllCauchyStresses(_not_ghost);
	}
	}

	/* -------------------------------------------------------------------------- */
	void Material::printself(std::ostream & stream, int indent) const {
	std::string space(indent, AKANTU_INDENT);
	std::string type = getID().substr(getID().find_last_of(':') + 1);

	stream << space << "Material " << type << " [" << std::endl;
	Parsable::printself(stream, indent);
	stream << space << "]" << std::endl;
	}

	/* -------------------------------------------------------------------------- */
	/// extrapolate internal values
	void Material::extrapolateInternal(const ID & id, const Element & element,
	[[gnu::unused]] const Matrix<Real> & point,
	Matrix<Real> & extrapolated) {
	if (this->isInternal<Real>(id, element.kind())) {
	UInt nb_element =
	this->element_filter(element.type, element.ghost_type).size();
	const ID name = this->getID() + ":" + id;
	UInt nb_quads =
	this->internal_vectors_real[name]->getFEEngine().getNbIntegrationPoints(
	element.type, element.ghost_type);
	const Array<Real> & internal =
	this->getArray<Real>(id, element.type, element.ghost_type);
	UInt nb_component = internal.getNbComponent();
	Array<Real>::const_matrix_iterator internal_it =
	internal.begin_reinterpret(nb_component, nb_quads, nb_element);
	Element local_element = this->convertToLocalElement(element);

	/// instead of really extrapolating, here the value of the first GP
	/// is copied into the result vector. This works only for linear
	/// elements
	/// @todo extrapolate!!!!
	AKANTU_DEBUG_WARNING("This is a fix, values are not truly extrapolated");

	const Matrix<Real> & values = internal_it[local_element.element];
	UInt index = 0;
	Vector<Real> tmp(nb_component);
	for (UInt j = 0; j < values.cols(); ++j) {
	tmp = values(j);
	if (tmp.norm() > 0) {
	index = j;
	break;
	}
	}

	for (UInt i = 0; i < extrapolated.size(); ++i) {
	extrapolated(i) = values(index);
	}
	} else {
	Matrix<Real> default_values(extrapolated.rows(), extrapolated.cols(), 0.);
	extrapolated = default_values;
	}
	}

	/* -------------------------------------------------------------------------- */
	void Material::applyEigenGradU(const Matrix<Real> & prescribed_eigen_grad_u,
	const GhostType ghost_type) {

	for (auto && type : element_filter.elementTypes(_all_dimensions, _not_ghost,
	_ek_not_defined)) {
	if (element_filter(type, ghost_type).empty()) {
	continue;
	}

	for (auto & eigengradu : make_view(this->eigengradu(type, ghost_type),
	spatial_dimension, spatial_dimension)) {
	eigengradu = prescribed_eigen_grad_u;
	}
	}
	}

	/* -------------------------------------------------------------------------- */
	MaterialFactory & Material::getFactory() {
	return MaterialFactory::getInstance();
	}

	} // namespace akantu

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