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material_reinforcement_tmpl.hh
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			/**
			* @file material_reinforcement_tmpl.hh
			*
			* @author Lucas Frerot <lucas.frerot@epfl.ch>
			*
			* @date creation: Wed Mar 25 2015
			* @date last modification: Tue Feb 20 2018
			*
			* @brief Reinforcement material
			*
			*
			* Copyright (©) 2015-2018 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 "aka_common.hh"
			#include "aka_voigthelper.hh"
			#include "material_reinforcement.hh"

			namespace akantu {

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			MaterialReinforcement<Mat, dim>::MaterialReinforcement(
			EmbeddedInterfaceModel & model, const ID & id)
			: Mat(model, 1, model.getInterfaceMesh(),
			model.getFEEngine("EmbeddedInterfaceFEEngine"), id),
			emodel(model),
			gradu_embedded("gradu_embedded", *this, 1,
			model.getFEEngine("EmbeddedInterfaceFEEngine"),
			this->element_filter),
			directing_cosines("directing_cosines", *this, 1,
			model.getFEEngine("EmbeddedInterfaceFEEngine"),
			this->element_filter),
			pre_stress("pre_stress", *this, 1,
			model.getFEEngine("EmbeddedInterfaceFEEngine"),
			this->element_filter),
			area(1.0), shape_derivatives() {
			AKANTU_DEBUG_IN();
			this->initialize();
			AKANTU_DEBUG_OUT();
			}

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::initialize() {
			AKANTU_DEBUG_IN();

			this->registerParam("area", area, _pat_parsable \| _pat_modifiable,
			"Reinforcement cross-sectional area");
			this->registerParam("pre_stress", pre_stress, _pat_parsable \| _pat_modifiable,
			"Uniform pre-stress");

			// this->unregisterInternal(this->stress);

			// Fool the AvgHomogenizingFunctor
			// stress.initialize(dim * dim);

			// Reallocate the element filter
			this->element_filter.initialize(this->emodel.getInterfaceMesh(),
			_spatial_dimension = 1);
			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			MaterialReinforcement<Mat, dim>::~MaterialReinforcement() {
			AKANTU_DEBUG_IN();

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::initMaterial() {
			Mat::initMaterial();

			gradu_embedded.initialize(dim * dim);
			pre_stress.initialize(1);

			/// We initialise the stuff that is not going to change during the simulation
			this->initFilters();
			this->allocBackgroundShapeDerivatives();
			this->initBackgroundShapeDerivatives();
			this->initDirectingCosines();
			}

			/* -------------------------------------------------------------------------- */
			/// Initialize the filter for background elements
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::initFilters() {
			for (auto gt : ghost_types) {
			for (auto && type : emodel.getInterfaceMesh().elementTypes(1, gt)) {
			std::string shaped_id = "filter";
			if (gt == _ghost)
			shaped_id += ":ghost";

			auto & background =
			background_filter(std::make_unique<ElementTypeMapArray<UInt>>(
			"bg_" + shaped_id, this->name),
			type, gt);
			auto & foreground = foreground_filter(
			std::make_unique<ElementTypeMapArray<UInt>>(shaped_id, this->name),
			type, gt);
			foreground->initialize(emodel.getMesh(), _nb_component = 1,
			_ghost_type = gt);
			background->initialize(emodel.getMesh(), _nb_component = 1,
			_ghost_type = gt);

			// Computing filters
			for (auto && bg_type : background->elementTypes(dim, gt)) {
			filterInterfaceBackgroundElements(
			(foreground)(bg_type), (background)(bg_type), bg_type, type, gt);
			}
			}
			}
			}

			/* -------------------------------------------------------------------------- */
			/// Construct a filter for a (interface_type, background_type) pair
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::filterInterfaceBackgroundElements(
			Array<UInt> & foreground, Array<UInt> & background,
			const ElementType & type, const ElementType & interface_type,
			GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			foreground.resize(0);
			background.resize(0);

			Array<Element> & elements =
			emodel.getInterfaceAssociatedElements(interface_type, ghost_type);
			Array<UInt> & elem_filter = this->element_filter(interface_type, ghost_type);

			for (auto & elem_id : elem_filter) {
			Element & elem = elements(elem_id);
			if (elem.type == type) {
			background.push_back(elem.element);
			foreground.push_back(elem_id);
			}
			}

			AKANTU_DEBUG_OUT();
			}

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

			namespace detail {
			class BackgroundShapeDInitializer : public ElementTypeMapArrayInitializer {
			public:
			BackgroundShapeDInitializer(UInt spatial_dimension, FEEngine & engine,
			const ElementType & foreground_type,
			const ElementTypeMapArray<UInt> & filter,
			const GhostType & ghost_type)
			: ElementTypeMapArrayInitializer(
			[](const ElementType & bgtype, const GhostType &) {
			return ShapeFunctions::getShapeDerivativesSize(bgtype);
			},
			spatial_dimension, ghost_type, _ek_regular) {
			auto nb_quad = engine.getNbIntegrationPoints(foreground_type);
			// Counting how many background elements are affected by elements of
			// interface_type
			for (auto type : filter.elementTypes(this->spatial_dimension)) {
			// Inserting size
			array_size_per_bg_type(filter(type).size() * nb_quad, type,
			this->ghost_type);
			}
			}

			auto elementTypes() const -> decltype(auto) {
			return array_size_per_bg_type.elementTypes();
			}

			UInt size(const ElementType & bgtype) const {
			return array_size_per_bg_type(bgtype, this->ghost_type);
			}

			protected:
			ElementTypeMap<UInt> array_size_per_bg_type;
			};
			} // namespace detail

			/**
			* Background shape derivatives need to be stored per background element
			* types but also per embedded element type, which is why they are stored
			* in an ElementTypeMap<ElementTypeMapArray<Real> *>. The outer ElementTypeMap
			* refers to the embedded types, and the inner refers to the background types.
			*/
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::allocBackgroundShapeDerivatives() {
			AKANTU_DEBUG_IN();

			for (auto gt : ghost_types) {
			for (auto && type : emodel.getInterfaceMesh().elementTypes(1, gt)) {
			std::string shaped_id = "embedded_shape_derivatives";
			if (gt == _ghost)
			shaped_id += ":ghost";

			auto & shaped_etma = shape_derivatives(
			std::make_unique<ElementTypeMapArray<Real>>(shaped_id, this->name),
			type, gt);
			shaped_etma->initialize(
			detail::BackgroundShapeDInitializer(
			emodel.getSpatialDimension(),
			emodel.getFEEngine("EmbeddedInterfaceFEEngine"), type,
			*background_filter(type, gt), gt),
			0, true);
			}
			}

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::initBackgroundShapeDerivatives() {
			AKANTU_DEBUG_IN();

			for (auto interface_type :
			this->element_filter.elementTypes(this->spatial_dimension)) {
			for (auto type : background_filter(interface_type)->elementTypes(dim)) {
			computeBackgroundShapeDerivatives(interface_type, type, _not_ghost,
			this->element_filter(interface_type));
			}
			}

			AKANTU_DEBUG_OUT();
			}

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::computeBackgroundShapeDerivatives(
			const ElementType & interface_type, const ElementType & bg_type,
			GhostType ghost_type, const Array<UInt> & filter) {
			auto & interface_engine = emodel.getFEEngine("EmbeddedInterfaceFEEngine");
			auto & engine = emodel.getFEEngine();
			auto & interface_mesh = emodel.getInterfaceMesh();

			const auto nb_nodes_elem_bg = Mesh::getNbNodesPerElement(bg_type);
			// const auto nb_strss = VoigtHelper<dim>::size;
			const auto nb_quads_per_elem =
			interface_engine.getNbIntegrationPoints(interface_type);

			Array<Real> quad_pos(0, dim, "interface_quad_pos");
			interface_engine.interpolateOnIntegrationPoints(interface_mesh.getNodes(),
			quad_pos, dim, interface_type,
			ghost_type, filter);
			auto & background_shapesd =
			(*shape_derivatives(interface_type, ghost_type))(bg_type, ghost_type);
			auto & background_elements =
			(*background_filter(interface_type, ghost_type))(bg_type, ghost_type);
			auto & foreground_elements =
			(*foreground_filter(interface_type, ghost_type))(bg_type, ghost_type);

			auto shapesd_begin =
			background_shapesd.begin(dim, nb_nodes_elem_bg, nb_quads_per_elem);
			auto quad_begin = quad_pos.begin(dim, nb_quads_per_elem);

			for (auto && tuple : zip(background_elements, foreground_elements)) {
			UInt bg = std::get<0>(tuple), fg = std::get<1>(tuple);
			for (UInt i = 0; i < nb_quads_per_elem; ++i) {
			Matrix<Real> shapesd = Tensor3<Real>(shapesd_begin[fg])(i);
			Vector<Real> quads = Matrix<Real>(quad_begin[fg])(i);

			engine.computeShapeDerivatives(quads, bg, bg_type, shapesd, ghost_type);
			}
			}
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::initDirectingCosines() {
			AKANTU_DEBUG_IN();

			Mesh & mesh = emodel.getInterfaceMesh();

			const UInt voigt_size = VoigtHelper<dim>::size;
			directing_cosines.initialize(voigt_size);

			for (auto && type : mesh.elementTypes(1, _not_ghost)) {
			computeDirectingCosines(type, _not_ghost);
			// computeDirectingCosines(*type_it, _ghost);
			}

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::assembleStiffnessMatrix(
			GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			Mesh & interface_mesh = emodel.getInterfaceMesh();

			for (auto && type : interface_mesh.elementTypes(1, _not_ghost)) {
			assembleStiffnessMatrix(type, ghost_type);
			}

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::assembleInternalForces(
			GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			Mesh & interface_mesh = emodel.getInterfaceMesh();

			for (auto && type : interface_mesh.elementTypes(1, _not_ghost)) {
			this->assembleInternalForces(type, ghost_type);
			}

			AKANTU_DEBUG_OUT();
			}

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::computeAllStresses(GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			Mesh & interface_mesh = emodel.getInterfaceMesh();
			for (auto && type : interface_mesh.elementTypes(_ghost_type = ghost_type)) {
			computeGradU(type, ghost_type);
			this->computeStress(type, ghost_type);
			addPrestress(type, ghost_type);
			}

			AKANTU_DEBUG_OUT();
			}

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::addPrestress(const ElementType & type,
			GhostType ghost_type) {
			auto & stress = this->stress(type, ghost_type);
			auto & sigma_p = this->pre_stress(type, ghost_type);

			for (auto && tuple : zip(stress, sigma_p)) {
			std::get<0>(tuple) += std::get<1>(tuple);
			}
			}

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::assembleInternalForces(
			const ElementType & type, GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			Mesh & mesh = emodel.getMesh();

			for (auto && mesh_type : mesh.elementTypes(dim, ghost_type)) {
			assembleInternalForcesInterface(type, mesh_type, ghost_type);
			}

			AKANTU_DEBUG_OUT();
			}

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

			/**
			* Computes and assemble the residual. Residual in reinforcement is computed as:
			*
			* \f[
			* \vec{r} = A_s \int_S{\mathbf{B}^T\mathbf{C}^T \vec{\sigma_s}\,\mathrm{d}s}
			* \f]
			*/
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::assembleInternalForcesInterface(
			const ElementType & interface_type, const ElementType & background_type,
			GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			UInt voigt_size = VoigtHelper<dim>::size;

			FEEngine & interface_engine = emodel.getFEEngine("EmbeddedInterfaceFEEngine");

			Array<UInt> & elem_filter = this->element_filter(interface_type, ghost_type);

			UInt nodes_per_background_e = Mesh::getNbNodesPerElement(background_type);
			UInt nb_quadrature_points =
			interface_engine.getNbIntegrationPoints(interface_type, ghost_type);
			UInt nb_element = elem_filter.size();

			UInt back_dof = dim * nodes_per_background_e;

			Array<Real> & shapesd = (*shape_derivatives(interface_type, ghost_type))(
			background_type, ghost_type);

			Array<Real> integrant(nb_quadrature_points * nb_element, back_dof,
			"integrant");

			auto integrant_it = integrant.begin(back_dof);
			auto integrant_end = integrant.end(back_dof);

			Array<Real>::matrix_iterator B_it =
			shapesd.begin(dim, nodes_per_background_e);
			auto C_it = directing_cosines(interface_type, ghost_type).begin(voigt_size);

			auto sigma_it = this->stress(interface_type, ghost_type).begin();

			Matrix<Real> Bvoigt(voigt_size, back_dof);

			for (; integrant_it != integrant_end;
			++integrant_it, ++B_it, ++C_it, ++sigma_it) {
			VoigtHelper<dim>::transferBMatrixToSymVoigtBMatrix(*B_it, Bvoigt,
			nodes_per_background_e);
			Vector<Real> & C = *C_it;
			Vector<Real> & BtCt_sigma = *integrant_it;

			BtCt_sigma.mul<true>(Bvoigt, C);
			BtCt_sigma = sigma_it * area;
			}

			Array<Real> residual_interface(nb_element, back_dof, "residual_interface");
			interface_engine.integrate(integrant, residual_interface, back_dof,
			interface_type, ghost_type, elem_filter);
			integrant.resize(0);

			Array<UInt> background_filter(nb_element, 1, "background_filter");

			auto & filter =
			getBackgroundFilter(interface_type, background_type, ghost_type);

			emodel.getDOFManager().assembleElementalArrayLocalArray(
			residual_interface, emodel.getInternalForce(), background_type,
			ghost_type, -1., filter);

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::computeDirectingCosines(
			const ElementType & type, GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			Mesh & interface_mesh = emodel.getInterfaceMesh();

			const UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
			const UInt steel_dof = dim * nb_nodes_per_element;
			const UInt voigt_size = VoigtHelper<dim>::size;
			const UInt nb_quad_points = emodel.getFEEngine("EmbeddedInterfaceFEEngine")
			.getNbIntegrationPoints(type, ghost_type);

			Array<Real> node_coordinates(this->element_filter(type, ghost_type).size(),
			steel_dof);

			this->emodel.getFEEngine().template extractNodalToElementField<Real>(
			interface_mesh, interface_mesh.getNodes(), node_coordinates, type,
			ghost_type, this->element_filter(type, ghost_type));

			Array<Real>::matrix_iterator directing_cosines_it =
			directing_cosines(type, ghost_type).begin(1, voigt_size);

			Array<Real>::matrix_iterator node_coordinates_it =
			node_coordinates.begin(dim, nb_nodes_per_element);
			Array<Real>::matrix_iterator node_coordinates_end =
			node_coordinates.end(dim, nb_nodes_per_element);

			for (; node_coordinates_it != node_coordinates_end; ++node_coordinates_it) {
			for (UInt i = 0; i < nb_quad_points; i++, ++directing_cosines_it) {
			Matrix<Real> & nodes = *node_coordinates_it;
			Matrix<Real> & cosines = *directing_cosines_it;

			computeDirectingCosinesOnQuad(nodes, cosines);
			}
			}

			// Mauro: the directing_cosines internal is defined on the quadrature points
			// of each element

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::assembleStiffnessMatrix(
			const ElementType & type, GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			Mesh & mesh = emodel.getMesh();

			for (auto && mesh_type : mesh.elementTypes(dim, ghost_type)) {
			assembleStiffnessMatrixInterface(type, mesh_type, ghost_type);
			}

			AKANTU_DEBUG_OUT();
			}

			/* -------------------------------------------------------------------------- */
			/**
			* Computes the reinforcement stiffness matrix (Gomes & Awruch, 2001)
			* \f[
			* \mathbf{K}_e = \sum_{i=1}^R{A_i\int_{S_i}{\mathbf{B}^T
			* \mathbf{C}_i^T \mathbf{D}_{s, i} \mathbf{C}_i \mathbf{B}\,\mathrm{d}s}}
			* \f]
			*/
			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::assembleStiffnessMatrixInterface(
			const ElementType & interface_type, const ElementType & background_type,
			GhostType ghost_type) {
			AKANTU_DEBUG_IN();

			UInt voigt_size = VoigtHelper<dim>::size;

			FEEngine & interface_engine = emodel.getFEEngine("EmbeddedInterfaceFEEngine");

			Array<UInt> & elem_filter = this->element_filter(interface_type, ghost_type);
			Array<Real> & grad_u = gradu_embedded(interface_type, ghost_type);

			UInt nb_element = elem_filter.size();
			UInt nodes_per_background_e = Mesh::getNbNodesPerElement(background_type);
			UInt nb_quadrature_points =
			interface_engine.getNbIntegrationPoints(interface_type, ghost_type);

			UInt back_dof = dim * nodes_per_background_e;

			UInt integrant_size = back_dof;

			grad_u.resize(nb_quadrature_points * nb_element);

			Array<Real> tangent_moduli(nb_element * nb_quadrature_points, 1,
			"interface_tangent_moduli");
			this->computeTangentModuli(interface_type, tangent_moduli, ghost_type);

			Array<Real> & shapesd = (*shape_derivatives(interface_type, ghost_type))(
			background_type, ghost_type);

			Array<Real> integrant(nb_element * nb_quadrature_points,
			integrant_size * integrant_size, "B^tC^tDCB");

			/// Temporary matrices for integrant product
			Matrix<Real> Bvoigt(voigt_size, back_dof);
			Matrix<Real> DCB(1, back_dof);
			Matrix<Real> CtDCB(voigt_size, back_dof);

			Array<Real>::scalar_iterator D_it = tangent_moduli.begin();
			Array<Real>::scalar_iterator D_end = tangent_moduli.end();

			Array<Real>::matrix_iterator C_it =
			directing_cosines(interface_type, ghost_type).begin(1, voigt_size);
			Array<Real>::matrix_iterator B_it =
			shapesd.begin(dim, nodes_per_background_e);
			Array<Real>::matrix_iterator integrant_it =
			integrant.begin(integrant_size, integrant_size);

			for (; D_it != D_end; ++D_it, ++C_it, ++B_it, ++integrant_it) {
			Real & D = *D_it;
			Matrix<Real> & C = *C_it;
			Matrix<Real> & B = *B_it;
			Matrix<Real> & BtCtDCB = *integrant_it;

			VoigtHelper<dim>::transferBMatrixToSymVoigtBMatrix(B, Bvoigt,
			nodes_per_background_e);

			DCB.mul<false, false>(C, Bvoigt);
			DCB = D area;
			CtDCB.mul<true, false>(C, DCB);
			BtCtDCB.mul<true, false>(Bvoigt, CtDCB);
			}

			tangent_moduli.resize(0);

			Array<Real> K_interface(nb_element, integrant_size * integrant_size,
			"K_interface");
			interface_engine.integrate(integrant, K_interface,
			integrant_size * integrant_size, interface_type,
			ghost_type, elem_filter);

			integrant.resize(0);

			// Mauro: Here K_interface contains the local stiffness matrices,
			// directing_cosines contains the information about the orientation
			// of the reinforcements, any rotation of the local stiffness matrix
			// can be done here

			auto & filter =
			getBackgroundFilter(interface_type, background_type, ghost_type);

			emodel.getDOFManager().assembleElementalMatricesToMatrix(
			"K", "displacement", K_interface, background_type, ghost_type, _symmetric,
			filter);

			AKANTU_DEBUG_OUT();
			}

			/* -------------------------------------------------------------------------- */
			template <class Mat, UInt dim>
			Real MaterialReinforcement<Mat, dim>::getEnergy(const std::string & id) {
			AKANTU_DEBUG_IN();
			if (id == "potential") {
			Real epot = 0.;

			this->computePotentialEnergyByElements();

			for (auto && type :
			this->element_filter.elementTypes(this->spatial_dimension)) {
			FEEngine & interface_engine =
			emodel.getFEEngine("EmbeddedInterfaceFEEngine");
			epot += interface_engine.integrate(
			this->potential_energy(type, _not_ghost), type, _not_ghost,
			this->element_filter(type, _not_ghost));
			epot *= area;
			}

			return epot;
			}

			AKANTU_DEBUG_OUT();
			return 0;
			}

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

			template <class Mat, UInt dim>
			void MaterialReinforcement<Mat, dim>::computeGradU(
			const ElementType & interface_type, GhostType ghost_type) {
			// Looping over background types
			for (auto && bg_type :
			background_filter(interface_type, ghost_type)->elementTypes(dim)) {

			const UInt nodes_per_background_e = Mesh::getNbNodesPerElement(bg_type);
			const UInt voigt_size = VoigtHelper<dim>::size;

			auto & bg_shapesd =
			(*shape_derivatives(interface_type, ghost_type))(bg_type, ghost_type);

			auto & filter = getBackgroundFilter(interface_type, bg_type, ghost_type);

			Array<Real> disp_per_element(0, dim * nodes_per_background_e, "disp_elem");

			FEEngine::extractNodalToElementField(
			emodel.getMesh(), emodel.getDisplacement(), disp_per_element, bg_type,
			ghost_type, filter);

			Matrix<Real> concrete_du(dim, dim);
			Matrix<Real> epsilon(dim, dim);
			Vector<Real> evoigt(voigt_size);

			for (auto && tuple :
			zip(make_view(disp_per_element, dim, nodes_per_background_e),
			make_view(bg_shapesd, dim, nodes_per_background_e),
			this->gradu(interface_type, ghost_type),
			make_view(this->directing_cosines(interface_type, ghost_type),
			voigt_size))) {
			auto & u = std::get<0>(tuple);
			auto & B = std::get<1>(tuple);
			auto & du = std::get<2>(tuple);
			auto & C = std::get<3>(tuple);

			concrete_du.mul<false, true>(u, B);
			auto epsilon = 0.5 * (concrete_du + concrete_du.transpose());
			strainTensorToVoigtVector(epsilon, evoigt);
			du = C.dot(evoigt);
			}
			}
			}

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

			/**
			* The structure of the directing cosines matrix is :
			* \f{eqnarray*}{
			* C_{1,\cdot} & = & (l^2, m^2, n^2, mn, ln, lm) \\
			* C_{i,j} & = & 0
			* \f}
			*
			* with :
			* \f[
			* (l, m, n) = \frac{1}{\\|\frac{\mathrm{d}\vec{r}(s)}{\mathrm{d}s}\\|} \cdot
			* \frac{\mathrm{d}\vec{r}(s)}{\mathrm{d}s}
			* \f]
			*/
			template <class Mat, UInt dim>
			inline void MaterialReinforcement<Mat, dim>::computeDirectingCosinesOnQuad(
			const Matrix<Real> & nodes, Matrix<Real> & cosines) {
			AKANTU_DEBUG_IN();

			AKANTU_DEBUG_ASSERT(nodes.cols() == 2,
			"Higher order reinforcement elements not implemented");

			const Vector<Real> a = nodes(0), b = nodes(1);
			Vector<Real> delta = b - a;

			Real sq_length = 0.;
			for (UInt i = 0; i < dim; i++) {
			sq_length += delta(i) * delta(i);
			}

			if (dim == 2) {
			cosines(0, 0) = delta(0) * delta(0); // l^2
			cosines(0, 1) = delta(1) * delta(1); // m^2
			cosines(0, 2) = delta(0) * delta(1); // lm
			} else if (dim == 3) {
			cosines(0, 0) = delta(0) * delta(0); // l^2
			cosines(0, 1) = delta(1) * delta(1); // m^2
			cosines(0, 2) = delta(2) * delta(2); // n^2

			cosines(0, 3) = delta(1) * delta(2); // mn
			cosines(0, 4) = delta(0) * delta(2); // ln
			cosines(0, 5) = delta(0) * delta(1); // lm
			}

			cosines /= sq_length;

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			inline void MaterialReinforcement<Mat, dim>::stressTensorToVoigtVector(
			const Matrix<Real> & tensor, Vector<Real> & vector) {
			AKANTU_DEBUG_IN();

			for (UInt i = 0; i < dim; i++) {
			vector(i) = tensor(i, i);
			}

			if (dim == 2) {
			vector(2) = tensor(0, 1);
			} else if (dim == 3) {
			vector(3) = tensor(1, 2);
			vector(4) = tensor(0, 2);
			vector(5) = tensor(0, 1);
			}

			AKANTU_DEBUG_OUT();
			}

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

			template <class Mat, UInt dim>
			inline void MaterialReinforcement<Mat, dim>::strainTensorToVoigtVector(
			const Matrix<Real> & tensor, Vector<Real> & vector) {
			AKANTU_DEBUG_IN();

			for (UInt i = 0; i < dim; i++) {
			vector(i) = tensor(i, i);
			}

			if (dim == 2) {
			vector(2) = 2 * tensor(0, 1);
			} else if (dim == 3) {
			vector(3) = 2 * tensor(1, 2);
			vector(4) = 2 * tensor(0, 2);
			vector(5) = 2 * tensor(0, 1);
			}

			AKANTU_DEBUG_OUT();
			}

			} // namespace akantu

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rAKA/src/model/solid_mechanics/materials/material_embedded/material_reinforcement_tmpl.hh
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