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	/**
	* Copyright (©) 2010-2023 EPFL (Ecole Polytechnique Fédérale de Lausanne)
	* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
	*
	* This file is part of Akantu
	*
	* 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()) {
	eigen_grad_u.fill(0.);

	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();
	}

	/* -------------------------------------------------------------------------- */
	Material::Material(SolidMechanicsModel & model, Int 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),
	eigen_grad_u(dim, dim) {
	eigen_grad_u.fill(0.);

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

	this->initialize();
	}

	/* -------------------------------------------------------------------------- */
	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);
	this->piola_kirchhoff_2.initializeHistory();
	this->green_strain.initialize(spatial_dimension * spatial_dimension);
	}

	this->stress.initializeHistory();
	this->gradu.initializeHistory();

	this->resizeInternals();

	auto dim = spatial_dimension;
	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();

	Int spatial_dimension = model.getSpatialDimension();

	tuple_dispatch<AllSpatialDimensions>(
	[&](auto && _) {
	constexpr auto dim = aka::decay_v<decltype(_)>;

	for (auto && type :
	element_filter.elementTypes(spatial_dimension, ghost_type)) {
	if (not finite_deformation) {
	this->assembleInternalForces<dim>(type, ghost_type);
	} else {
	this->assembleInternalForcesFiniteDeformation<dim>(type,
	ghost_type);
	}
	}
	},
	spatial_dimension);

	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();

	auto spatial_dimension = model.getSpatialDimension();

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

	if (elem_filter.empty()) {
	continue;
	}
	auto & 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)) {
	tuple_dispatch<AllSpatialDimensions>(
	[&](auto && _) {
	constexpr auto dim = aka::decay_v<decltype(_)>;
	this->StoCauchy<dim>(type, ghost_type);
	},
	spatial_dimension);
	}

	AKANTU_DEBUG_OUT();
	}

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

	for (auto && data :
	zip(make_view<dim, dim>(this->gradu(el_type, ghost_type)),
	make_view<dim, dim>(this->piola_kirchhoff_2(el_type, ghost_type)),
	make_view<dim, dim>(this->stress(el_type, ghost_type)))) {
	auto && grad_u = std::get<0>(data);
	auto && piola = std::get<1>(data);
	auto && sigma = std::get<2>(data);

	Matrix<Real, dim, dim> F = gradUToF<dim>(grad_u);
	this->StoCauchy<dim>(F, piola, sigma);
	}

	AKANTU_DEBUG_OUT();
	}

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

	const auto & displacement = model.getDisplacement();

	// resizeInternalArray(gradu);

	auto spatial_dimension = model.getSpatialDimension();

	for (auto type : element_filter.elementTypes(spatial_dimension, ghost_type)) {
	auto & elem_filter = element_filter(type, ghost_type);
	auto & 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();

	auto spatial_dimension = model.getSpatialDimension();

	tuple_dispatch<AllSpatialDimensions>(
	[&](auto && _) {
	constexpr auto dim = aka::decay_v<decltype(_)>;

	for (auto type :
	element_filter.elementTypes(spatial_dimension, ghost_type)) {
	if (finite_deformation) {
	this->assembleStiffnessMatrixFiniteDeformation<dim>(type,
	ghost_type);
	} else {
	this->assembleStiffnessMatrix<dim>(type, ghost_type);
	}
	}
	},
	spatial_dimension);

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <Int dim>
	void Material::assembleStiffnessMatrix(ElementType type, GhostType ghost_type) {
	tuple_dispatch<AllElementTypes>(
	[&](auto && enum_type) {
	constexpr auto type = aka::decay_v<decltype(enum_type)>;
	this->assembleStiffnessMatrix<dim, type>(ghost_type);
	},
	type);
	}

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

	const auto & elem_filter = element_filter(type, ghost_type);
	if (elem_filter.empty()) {
	AKANTU_DEBUG_OUT();
	return;
	}

	auto & gradu_vect = gradu(type, ghost_type);

	auto nb_element = elem_filter.size();
	auto nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);
	constexpr auto nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	constexpr auto tangent_size = getTangentStiffnessVoigtSize(dim);

	gradu_vect.resize(nb_quadrature_points * nb_element);

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

	auto tangent_stiffness_matrix = std::make_unique<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$\mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto bt_d_b_size = dim * nb_nodes_per_element;

	auto bt_d_b = std::make_unique<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);

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto K_e = std::make_unique<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);

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

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <Int dim>
	void Material::assembleStiffnessMatrixFiniteDeformation(ElementType type,
	GhostType ghost_type) {
	tuple_dispatch<AllElementTypes>(
	[&](auto && enum_type) {
	constexpr auto type = aka::decay_v<decltype(enum_type)>;
	this->assembleStiffnessMatrixNL<dim, type>(ghost_type);
	this->assembleStiffnessMatrixL2<dim, type>(ghost_type);
	},
	type);
	}

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

	const auto & shapes_derivatives = fem.getShapesDerivatives(type, ghost_type);
	const auto & elem_filter = element_filter(type, ghost_type);

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

	auto shapes_derivatives_filtered = std::make_unique<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$\mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	constexpr auto bt_s_b_size = dim * nb_nodes_per_element;
	constexpr auto piola_matrix_size = getCauchyStressMatrixSize(dim);

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

	Matrix<Real, piola_matrix_size, bt_s_b_size> B;
	Matrix<Real, piola_matrix_size, piola_matrix_size> S;

	for (auto && data :
	zip(make_view<bt_s_b_size, bt_s_b_size>(*bt_s_b),
	make_view<dim, dim>(piola_kirchhoff_2(type, ghost_type)),
	make_view<dim, nb_nodes_per_element>(
	*shapes_derivatives_filtered))) {
	auto && Bt_S_B = std::get<0>(data);
	auto && Piola_kirchhoff_matrix = std::get<1>(data);
	auto && shapes_derivatives = std::get<2>(data);

	setCauchyStressMatrix<dim>(Piola_kirchhoff_matrix, S);
	VoigtHelper<dim>::transferBMatrixToBNL(shapes_derivatives, B,
	nb_nodes_per_element);
	Bt_S_B = B.transpose() * S * B;
	}

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto K_e = std::make_unique<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);

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

	AKANTU_DEBUG_OUT();
	}

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

	const auto & shapes_derivatives = fem.getShapesDerivatives(type, ghost_type);

	auto & elem_filter = element_filter(type, ghost_type);
	auto & gradu_vect = gradu(type, ghost_type);

	auto nb_element = elem_filter.size();
	constexpr auto nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	auto 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);

	constexpr auto tangent_size = getTangentStiffnessVoigtSize(dim);

	auto tangent_stiffness_matrix = std::make_unique<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 = std::make_unique<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$\mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	constexpr auto bt_d_b_size = dim * nb_nodes_per_element;
	auto bt_d_b = std::make_unique<Array<Real>>(
	nb_element * nb_quadrature_points, bt_d_b_size * bt_d_b_size, "B^tDB");

	Matrix<Real, tangent_size, bt_d_b_size> B;
	Matrix<Real, tangent_size, bt_d_b_size> B2;

	for (auto && data :
	zip(make_view<bt_d_b_size, bt_d_b_size>(*bt_d_b),
	make_view<dim, dim>(gradu_vect),
	make_view<tangent_size, tangent_size>(*tangent_stiffness_matrix),
	make_view<dim, nb_nodes_per_element>(
	*shapes_derivatives_filtered))) {
	auto && Bt_D_B = std::get<0>(data);
	auto && grad_u = std::get<1>(data);
	auto && D = std::get<2>(data);
	auto && shapes_derivative = std::get<3>(data);

	// transferBMatrixToBL1<dim > (*shapes_derivatives_filtered_it, B,
	// nb_nodes_per_element);
	VoigtHelper<dim>::transferBMatrixToSymVoigtBMatrix(shapes_derivative, B,
	nb_nodes_per_element);
	VoigtHelper<dim>::transferBMatrixToBL2(shapes_derivative, grad_u, B2,
	nb_nodes_per_element);
	B += B2;
	Bt_D_B = B.transpose() * D * B;
	}

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto K_e = std::make_unique<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);

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

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	template <Int dim>
	void Material::assembleInternalForces(ElementType type, GhostType ghost_type) {
	tuple_dispatch<AllElementTypes>(
	[&](auto && enum_type) {
	constexpr auto type = aka::decay_v<decltype(enum_type)>;
	this->assembleInternalForces<dim, type>(ghost_type);
	},
	type);
	}

	/* -------------------------------------------------------------------------- */
	template <Int dim, ElementType type>
	void Material::assembleInternalForces(GhostType ghost_type) {
	auto & internal_force = model.getInternalForce();

	// Mesh & mesh = fem.getMesh();
	auto && elem_filter = element_filter(type, ghost_type);
	auto nb_element = elem_filter.size();

	if (nb_element == 0) {
	return;
	}

	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;
	}
	/* -------------------------------------------------------------------------- */
	template <Int dim>
	void Material::assembleInternalForcesFiniteDeformation(ElementType type,
	GhostType ghost_type) {
	tuple_dispatch<AllElementTypes>(
	[&](auto && enum_type) {
	constexpr auto type = aka::decay_v<decltype(enum_type)>;
	this->assembleInternalForcesFiniteDeformation<dim, type>(ghost_type);
	},
	type);
	}

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

	auto & internal_force = model.getInternalForce();
	auto & mesh = fem.getMesh();
	const auto & shapes_derivatives = fem.getShapesDerivatives(type, ghost_type);

	auto & elem_filter = element_filter(type, ghost_type);
	if (elem_filter.size() == 0)
	return;

	auto size_of_shapes_derivatives = shapes_derivatives.getNbComponent();
	auto nb_element = elem_filter.size();

	constexpr auto nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
	constexpr auto stress_size = getTangentStiffnessVoigtSize(dim);
	constexpr auto bt_s_size = dim * nb_nodes_per_element;

	auto nb_quadrature_points = fem.getNbIntegrationPoints(type, ghost_type);

	auto shapesd_filtered = std::make_unique<Array<Real>>(
	nb_element, size_of_shapes_derivatives, "filtered shapesd");

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

	// Set stress vectors
	auto bt_s = std::make_unique<Array<Real>>(nb_element * nb_quadrature_points,
	bt_s_size, "B^t*S");

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

	for (auto && data :
	zip(make_view<dim, dim>(this->gradu(type, ghost_type)),
	make_view<dim, dim>(this->piola_kirchhoff_2(type, ghost_type)),
	make_view<bt_s_size>(*bt_s),
	make_view<dim, nb_nodes_per_element>(*shapesd_filtered))) {
	auto && grad_u = std::get<0>(data);
	auto && S = std::get<1>(data);
	auto && r = std::get<2>(data);
	auto && shapes_derivative = std::get<3>(data);

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

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

	B_tensor += B2_tensor;
	auto && S_voight = Material::stressToVoigt<dim>(S);
	r = B_tensor.transpose() * S_voight;
	}

	/// compute @f$ k_e = \int_e \mathbf{B}^t * \mathbf{D} * \mathbf{B}@f$
	auto r_e = std::make_unique<Array<Real>>(nb_element, bt_s_size, "r_e");
	fem.integrate(bt_s, r_e, bt_s_size, type, ghost_type, elem_filter);

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

	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) { AKANTU_TO_IMPLEMENT(); }

	/* -------------------------------------------------------------------------- */
	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, Int index) {
	return getPotentialEnergy({type, index, _not_ghost});
	}

	/* -------------------------------------------------------------------------- */
	Real Material::getPotentialEnergy(const Element & element) {
	AKANTU_DEBUG_IN();

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

	auto epot = fem.integrate(epot_on_quad_points,
	{element.type,
	element_filter(element.type)(element.element),
	_not_ghost});

	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,
	const Element & element) {
	AKANTU_DEBUG_IN();
	if (energy_id == "potential") {
	return getPotentialEnergy(element);
	}
	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);

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

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

	for (auto type : element_filter.elementTypes(spatial_dimension, ghost_type)) {
	auto nb_element_full = mesh.getNbElement(type, ghost_type);

	auto nb_interpolation_points_per_elem =
	by_elem_result(type, ghost_type).size() / nb_element_full;

	const auto & facet_to_element =
	mesh_facets.getSubelementToElement(type, ghost_type);

	auto nb_facet_per_elem = facet_to_element.getNbComponent();
	auto nb_quad_per_facet =
	nb_interpolation_points_per_elem / nb_facet_per_elem;
	Element element{type, 0, ghost_type};

	auto && by_elem_res =
	make_view(by_elem_result(type, ghost_type), stress_size,
	nb_quad_per_facet, nb_facet_per_elem)
	.begin();

	for (auto global_el : element_filter(type, ghost_type)) {
	element.element = global_el;

	auto && result_per_elem = by_elem_res[global_el];

	for (Int f = 0; f < nb_facet_per_elem; ++f) {
	auto facet_elem = facet_to_element(global_el, f);
	Int is_second_element =
	mesh_facets.getElementToSubelement(facet_elem)[0] != element;

	auto && result_local =
	result.get(facet_elem, stress_size, 2, nb_quad_per_facet);

	for (auto && data : zip(result_local, result_per_elem(f))) {
	std::get<0>(data)(is_second_element) = std::get<1>(data);
	}
	}
	}
	}
	}

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

	if (elements_to_add.empty()) {
	return; // noops if no changes
	}

	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) {

	if (elements_to_remove.empty()) {
	return; // noops if no changes
	}

	AKANTU_DEBUG_IN();

	auto & mesh = this->model.getMesh();
	auto el_begin = elements_to_remove.begin();
	auto el_end = elements_to_remove.end();

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

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

	ElementTypeMap<Idx> 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_int.begin(); it != internal_vectors_int.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_int.begin(); it != internal_vectors_int.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<Idx> & new_numbering,
	[[gnu::unused]] const RemovedElementsEvent & event) {
	auto my_num = model.getInternalIndexFromID(getID());

	ElementTypeMapArray<Idx> 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<Idx> elem_filter_tmp;
	UInt ni = 0;
	Element el{type, 0, gt};

	for (auto && data : enumerate(elem_filter)) {
	el.element = std::get<1>(data);
	if (std::find(el_begin, el_end, el) == el_end) {
	auto new_el = renumbering(el.element);
	AKANTU_DEBUG_ASSERT(new_el != -1,
	"A not removed element as been badly renumbered");
	elem_filter_tmp.push_back(new_el);
	mat_renumbering(std::get<0>(data)) = ni;

	mat_indexes(new_el) = my_num;
	mat_loc_num(new_el) = ni;
	++ni;
	} else {
	mat_renumbering(std::get<0>(data)) = -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_int.begin(); it != internal_vectors_int.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,
	const Matrix<Real> & /point/,
	Matrix<Real> & extrapolated) {
	if (this->isInternal<Real>(id, element.kind())) {
	auto name = this->getID() + ":" + id;
	auto nb_quads =
	this->internal_vectors_real[name]->getFEEngine().getNbIntegrationPoints(
	element.type, element.ghost_type);
	const auto & internal =
	this->getArray<Real>(id, element.type, element.ghost_type);
	auto nb_component = internal.getNbComponent();
	auto internal_it = make_view(internal, nb_component, nb_quads).begin();
	auto 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");

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

	for (Int i = 0; i < extrapolated.size(); ++i) {
	extrapolated(i) = values(index);
	}
	} else {
	extrapolated.fill(0.);
	}
	}

	/* -------------------------------------------------------------------------- */
	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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