phase_field_model.cc
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Created: Wed, Jul 10, 04:21

phase_field_model.cc
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	/**
	* Copyright (©) 2018-2023 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 "phase_field_model.hh"
	#include "aka_common.hh"
	#include "dumpable_inline_impl.hh"
	#include "element_synchronizer.hh"
	#include "fe_engine_template.hh"
	#include "generalized_trapezoidal.hh"
	#include "group_manager_inline_impl.hh"
	#include "integrator_gauss.hh"
	#include "mesh.hh"
	#include "parser.hh"
	#include "shape_lagrange.hh"
	/* -------------------------------------------------------------------------- */
	#include "dumper_element_partition.hh"
	#include "dumper_elemental_field.hh"
	#include "dumper_internal_material_field.hh"
	#include "dumper_iohelper_paraview.hh"
	#include <utility>
	/* -------------------------------------------------------------------------- */
	namespace akantu {

	/* -------------------------------------------------------------------------- */
	PhaseFieldModel::PhaseFieldModel(Mesh & mesh, Int dim, const ID & id,
	std::shared_ptr<DOFManager> dof_manager,
	ModelType model_type)
	: Model(mesh, model_type, dim, id),
	phasefield_index("phasefield index", id),
	phasefield_local_numbering("phasefield local numbering", id) {

	AKANTU_DEBUG_IN();

	this->initDOFManager(std::move(dof_manager));

	this->registerFEEngineObject<FEEngineType>("PhaseFieldFEEngine", mesh,
	Model::spatial_dimension);

	this->mesh.registerDumper<DumperParaview>("phase_field", id, true);
	this->mesh.addDumpMesh(mesh, Model::spatial_dimension, _not_ghost,
	_ek_regular);

	phasefield_selector =
	std::make_shared<DefaultPhaseFieldSelector>(phasefield_index);

	this->registerDataAccessor(*this);

	if (this->mesh.isDistributed()) {
	auto & synchronizer = this->mesh.getElementSynchronizer();
	this->registerSynchronizer(synchronizer,
	SynchronizationTag::_phasefield_id);
	this->registerSynchronizer(synchronizer, SynchronizationTag::_pfm_damage);
	this->registerSynchronizer(synchronizer, SynchronizationTag::_for_dump);
	}

	AKANTU_DEBUG_OUT();
	}

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

	/* -------------------------------------------------------------------------- */
	MatrixType PhaseFieldModel::getMatrixType(const ID & matrix_id) const {
	if (matrix_id == "K") {
	return _symmetric;
	}
	return _mt_not_defined;
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::initModel() {
	auto & fem = this->getFEEngine();
	fem.initShapeFunctions(_not_ghost);
	fem.initShapeFunctions(_ghost);
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::initFullImpl(const ModelOptions & options) {
	phasefield_index.initialize(mesh, _element_kind = _ek_not_defined,
	_default_value = Idx(-1),
	_with_nb_element = true);
	phasefield_local_numbering.initialize(mesh, _element_kind = _ek_not_defined,
	_with_nb_element = true);

	Model::initFullImpl(options);

	// initialize the phasefields
	if (!this->parser.getLastParsedFile().empty()) {
	this->instantiatePhaseFields();
	this->initPhaseFields();
	}

	this->initBC(this, damage, *external_force);
	}

	/* -------------------------------------------------------------------------- */
	PhaseField &
	PhaseFieldModel::registerNewPhaseField(const ParserSection & section) {
	std::string phase_name;
	std::string phase_type = section.getName();
	std::string opt_param = section.getOption();

	try {
	std::string tmp = section.getParameter("name");
	phase_name = tmp; /** this can seam weird, but there is an ambiguous
	* operator overload that i couldn't solve. @todo remove
	* the weirdness of this code
	*/
	} catch (debug::Exception &) {
	AKANTU_ERROR("A phasefield of type \'"
	<< phase_type
	<< "\' in the input file has been defined without a name!");
	}
	PhaseField & phase =
	this->registerNewPhaseField(phase_name, phase_type, opt_param);

	phase.parseSection(section);

	return phase;
	}

	/* -------------------------------------------------------------------------- */
	PhaseField & PhaseFieldModel::registerNewPhaseField(const ID & phase_name,
	const ID & phase_type,
	const ID & opt_param) {
	AKANTU_DEBUG_ASSERT(phasefields_names_to_id.find(phase_name) ==
	phasefields_names_to_id.end(),
	"A phasefield with this name '"
	<< phase_name << "' has already been registered. "
	<< "Please use unique names for phasefields");

	Int phase_count = phasefields.size();
	phasefields_names_to_id[phase_name] = phase_count;

	std::stringstream sstr_phase;
	sstr_phase << this->id << ":" << phase_count << ":" << phase_type;
	ID mat_id = sstr_phase.str();

	std::unique_ptr<PhaseField> phase = PhaseFieldFactory::getInstance().allocate(
	phase_type, opt_param, *this, mat_id);

	phasefields.push_back(std::move(phase));

	return *(phasefields.back());
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::instantiatePhaseFields() {

	ParserSection model_section;
	bool is_empty;
	std::tie(model_section, is_empty) = this->getParserSection();

	if (not is_empty) {
	auto model_phasefields =
	model_section.getSubSections(ParserType::_phasefield);
	for (const auto & section : model_phasefields) {
	this->registerNewPhaseField(section);
	}
	}

	auto sub_sections = this->parser.getSubSections(ParserType::_phasefield);
	for (const auto & section : sub_sections) {
	this->registerNewPhaseField(section);
	}

	if (phasefields.empty()) {
	AKANTU_EXCEPTION("No phasefields where instantiated for the model"
	<< getID());
	}
	are_phasefields_instantiated = true;
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::initPhaseFields() {
	AKANTU_DEBUG_ASSERT(phasefields.size() != 0, "No phasefield to initialize !");

	if (!are_phasefields_instantiated) {
	instantiatePhaseFields();
	}

	this->assignPhaseFieldToElements();

	for (auto & phasefield : phasefields) {
	/// init internals properties
	phasefield->initPhaseField();
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::assignPhaseFieldToElements(
	const ElementTypeMapArray<Idx> * filter) {

	for_each_element(
	mesh,
	[&](auto && element) {
	Int phase_index = (*phasefield_selector)(element);
	AKANTU_DEBUG_ASSERT(
	phase_index < Int(phasefields.size()),
	"The phasefield selector returned an index that does not exists");
	phasefield_index(element) = phase_index;
	},
	_element_filter = filter, _ghost_type = _not_ghost);

	for_each_element(
	mesh,
	[&](auto && element) {
	auto phase_index = phasefield_index(element);
	auto index = phasefields[phase_index]->addElement(element);
	phasefield_local_numbering(element) = index;
	},
	_element_filter = filter, _ghost_type = _not_ghost);

	// synchronize the element phasefield arrays
	this->synchronize(SynchronizationTag::_phasefield_id);
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::assembleMatrix(const ID & matrix_id) {

	if (matrix_id == "K") {
	this->assembleStiffnessMatrix();
	} else {
	AKANTU_ERROR("Unknown Matrix ID for PhaseFieldModel : " << matrix_id);
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::predictor() {
	// AKANTU_TO_IMPLEMENT();
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::corrector() {
	// AKANTU_TO_IMPLEMENT();
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::initSolver(TimeStepSolverType time_step_solver_type,
	NonLinearSolverType /unused/) {
	DOFManager & dof_manager = this->getDOFManager();

	this->allocNodalField(this->damage, 1, "damage");
	this->allocNodalField(this->external_force, 1, "external_force");
	this->allocNodalField(this->internal_force, 1, "internal_force");
	this->allocNodalField(this->blocked_dofs, 1, "blocked_dofs");
	this->allocNodalField(this->previous_damage, 1, "previous_damage");

	if (!dof_manager.hasDOFs("damage")) {
	dof_manager.registerDOFs("damage", *this->damage, _dst_nodal);
	dof_manager.registerBlockedDOFs("damage", *this->blocked_dofs);
	dof_manager.registerDOFsPrevious("damage", *this->previous_damage);
	}

	if (time_step_solver_type == TimeStepSolverType::_dynamic) {
	AKANTU_TO_IMPLEMENT();
	}
	}

	/* -------------------------------------------------------------------------- */
	FEEngine & PhaseFieldModel::getFEEngineBoundary(const ID & name) {
	return dynamic_cast<FEEngine &>(getFEEngineClassBoundary<FEEngineType>(name));
	}

	/* -------------------------------------------------------------------------- */
	TimeStepSolverType PhaseFieldModel::getDefaultSolverType() const {
	return TimeStepSolverType::_static;
	}

	/* -------------------------------------------------------------------------- */
	std::tuple<ID, TimeStepSolverType>
	PhaseFieldModel::getDefaultSolverID(const AnalysisMethod & method) {
	switch (method) {
	case _explicit_lumped_mass: {
	return std::make_tuple("explicit_lumped",
	TimeStepSolverType::_dynamic_lumped);
	}
	case _explicit_consistent_mass: {
	return std::make_tuple("explicit", TimeStepSolverType::_dynamic);
	}
	case _static: {
	return std::make_tuple("static", TimeStepSolverType::_static);
	}
	case _implicit_dynamic: {
	return std::make_tuple("implicit", TimeStepSolverType::_dynamic);
	}
	default:
	return std::make_tuple("unknown", TimeStepSolverType::_not_defined);
	}
	}

	/* -------------------------------------------------------------------------- */
	ModelSolverOptions PhaseFieldModel::getDefaultSolverOptions(
	const TimeStepSolverType & type) const {
	ModelSolverOptions options;

	switch (type) {
	case TimeStepSolverType::_dynamic_lumped: {
	options.non_linear_solver_type = NonLinearSolverType::_lumped;
	options.integration_scheme_type["damage"] =
	IntegrationSchemeType::_central_difference;
	options.solution_type["damage"] = IntegrationScheme::_acceleration;
	break;
	}
	case TimeStepSolverType::_static: {
	options.non_linear_solver_type = NonLinearSolverType::_newton_raphson;
	options.integration_scheme_type["damage"] =
	IntegrationSchemeType::_pseudo_time;
	options.solution_type["damage"] = IntegrationScheme::_not_defined;
	break;
	}
	case TimeStepSolverType::_dynamic: {
	options.non_linear_solver_type = NonLinearSolverType::_newton_raphson;
	options.integration_scheme_type["damage"] =
	IntegrationSchemeType::_backward_euler;
	options.solution_type["damage"] = IntegrationScheme::_damage;
	break;
	}
	default:
	AKANTU_EXCEPTION(type << " is not a valid time step solver type");
	}

	return options;
	}

	/* -------------------------------------------------------------------------- */
	Real PhaseFieldModel::getEnergy() {
	AKANTU_DEBUG_IN();

	Real energy = 0.;
	for (auto & phasefield : phasefields) {
	energy += phasefield->getEnergy();
	}

	/// reduction sum over all processors
	mesh.getCommunicator().allReduce(energy, SynchronizerOperation::_sum);

	AKANTU_DEBUG_OUT();
	return energy;
	}

	/* -------------------------------------------------------------------------- */
	Real PhaseFieldModel::getEnergy(ElementType type, Idx index) {
	AKANTU_DEBUG_IN();

	Idx phase_index = this->phasefield_index(type, _not_ghost)(index);
	Idx phase_loc_num = this->phasefield_local_numbering(type, _not_ghost)(index);
	Real energy = this->phasefields[phase_index]->getEnergy(
	Element{type, phase_loc_num, _not_ghost});

	AKANTU_DEBUG_OUT();
	return energy;
	}

	/* -------------------------------------------------------------------------- */
	Real PhaseFieldModel::getEnergy(const ID & group_id) {
	auto && group = mesh.getElementGroup(group_id);
	auto energy = 0.;
	for (auto && type : group.elementTypes()) {
	for (auto el : group.getElementsIterable(type)) {
	energy += getEnergy(el);
	}
	}

	/// reduction sum over all processors
	mesh.getCommunicator().allReduce(energy, SynchronizerOperation::_sum);

	return energy;
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::beforeSolveStep() {
	for (auto & phasefield : phasefields) {
	phasefield->beforeSolveStep();
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::afterSolveStep(bool converged) {
	if (not converged) {
	return;
	}

	for (auto && values : zip(damage, previous_damage)) {
	auto & dam = std::get<0>(values);
	auto & prev_dam = std::get<1>(values);

	prev_dam = dam;
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::assembleStiffnessMatrix() {
	AKANTU_DEBUG_INFO("Assemble the new stiffness matrix");

	if (!this->getDOFManager().hasMatrix("K")) {
	this->getDOFManager().getNewMatrix("K", getMatrixType("K"));
	}

	this->getDOFManager().zeroMatrix("K");

	for (auto & phasefield : phasefields) {
	phasefield->assembleStiffnessMatrix(_not_ghost);
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::assembleResidual() {
	this->assembleInternalForces();

	this->getDOFManager().assembleToResidual("damage", *this->external_force, 1);
	this->getDOFManager().assembleToResidual("damage", *this->internal_force, 1);
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::assembleInternalForces() {
	AKANTU_DEBUG_INFO("Assemble the internal forces");

	this->internal_force->zero();

	this->synchronize(SynchronizationTag::_pfm_damage);

	for (auto & phasefield : phasefields) {
	phasefield->computeAllDrivingForces(_not_ghost);
	}

	// assemble the forces due to local driving forces
	AKANTU_DEBUG_INFO("Assemble residual for local elements");
	for (auto & phasefield : phasefields) {
	phasefield->assembleInternalForces(_not_ghost);
	}

	// assemble the forces due to local driving forces
	AKANTU_DEBUG_INFO("Assemble residual for ghost elements");
	for (auto & phasefield : phasefields) {
	phasefield->assembleInternalForces(_ghost);
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::assembleLumpedMatrix(const ID & /matrix_id/) {}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::setTimeStep(Real time_step, const ID & solver_id) {
	Model::setTimeStep(time_step, solver_id);

	this->mesh.getDumper("phase_field").setTimeStep(time_step);
	}

	/* -------------------------------------------------------------------------- */
	Int PhaseFieldModel::getNbData(const Array<Element> & elements,
	const SynchronizationTag & tag) const {
	Int size = 0;
	Int nb_nodes_per_element = 0;

	for (const Element & el : elements) {
	nb_nodes_per_element += Mesh::getNbNodesPerElement(el.type);
	}

	switch (tag) {
	case SynchronizationTag::_phasefield_id: {
	size += elements.size() * sizeof(Int);
	break;
	}
	case SynchronizationTag::_for_dump: {
	// damage
	size += nb_nodes_per_element * sizeof(Real);
	break;
	}
	case SynchronizationTag::_pfm_damage: {
	size += nb_nodes_per_element * sizeof(Real);
	break;
	}
	default: {
	AKANTU_ERROR("Unknown ghost synchronization tag : " << tag);
	}
	}

	return size;
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::packData(CommunicationBuffer & buffer,
	const Array<Element> & elements,
	const SynchronizationTag & tag) const {
	switch (tag) {
	case SynchronizationTag::_phasefield_id: {
	packElementalDataHelper(phasefield_index, buffer, elements, false,
	getFEEngine());
	break;
	}
	case SynchronizationTag::_for_dump: {
	packNodalDataHelper(*damage, buffer, elements, mesh);
	break;
	}
	case SynchronizationTag::_pfm_damage: {
	packNodalDataHelper(*damage, buffer, elements, mesh);
	break;
	}
	default: {
	AKANTU_ERROR("Unknown ghost synchronization tag : " << tag);
	}
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::unpackData(CommunicationBuffer & buffer,
	const Array<Element> & elements,
	const SynchronizationTag & tag) {
	AKANTU_DEBUG_IN();

	switch (tag) {
	case SynchronizationTag::_phasefield_id: {
	for (auto && element : elements) {
	Idx recv_phase_index;
	buffer >> recv_phase_index;
	Idx & phase_index = phasefield_index(element);
	if (phase_index != Idx(-1)) {
	continue;
	}

	// add ghosts element to the correct phasefield
	phase_index = recv_phase_index;
	Idx index = phasefields[phase_index]->addElement(element);
	phasefield_local_numbering(element) = index;
	}
	break;
	}
	case SynchronizationTag::_for_dump: {
	unpackNodalDataHelper(*damage, buffer, elements, mesh);
	break;
	}
	case SynchronizationTag::_pfm_damage: {
	unpackNodalDataHelper(*damage, buffer, elements, mesh);
	break;
	}
	default: {
	AKANTU_ERROR("Unknown ghost synchronization tag : " << tag);
	}
	}

	AKANTU_DEBUG_OUT();
	}

	/* -------------------------------------------------------------------------- */
	Int PhaseFieldModel::getNbData(const Array<Idx> & indexes,
	const SynchronizationTag & tag) const {
	Int size = 0;
	Int nb_nodes = indexes.size();

	switch (tag) {
	case SynchronizationTag::_for_dump: {
	size += nb_nodes * sizeof(Real);
	break;
	}
	case SynchronizationTag::_pfm_damage: {
	size += nb_nodes * sizeof(Real);
	break;
	}
	default: {
	AKANTU_ERROR("Unknown ghost synchronization tag : " << tag);
	}
	}
	return size;
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::packData(CommunicationBuffer & buffer,
	const Array<Idx> & indexes,
	const SynchronizationTag & tag) const {
	switch (tag) {
	case SynchronizationTag::_for_dump: {
	packDOFDataHelper(*damage, buffer, indexes);
	break;
	}
	case SynchronizationTag::_pfm_damage: {
	packDOFDataHelper(*damage, buffer, indexes);
	break;
	}
	default: {
	AKANTU_ERROR("Unknown ghost synchronization tag : " << tag);
	}
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::unpackData(CommunicationBuffer & buffer,
	const Array<Idx> & indexes,
	const SynchronizationTag & tag) {
	switch (tag) {
	case SynchronizationTag::_for_dump: {
	unpackDOFDataHelper(*damage, buffer, indexes);
	break;
	}
	case SynchronizationTag::_pfm_damage: {
	unpackDOFDataHelper(*damage, buffer, indexes);
	break;
	}
	default: {
	AKANTU_ERROR("Unknown ghost synchronization tag : " << tag);
	}
	}
	}

	/* -------------------------------------------------------------------------- */
	std::shared_ptr<dumpers::Field>
	PhaseFieldModel::createNodalFieldBool(const std::string & field_name,
	const std::string & group_name,
	bool /unused/) {
	std::map<std::string, Array<bool> *> uint_nodal_fields;
	uint_nodal_fields["blocked_dofs"] = blocked_dofs.get();

	return mesh.createNodalField(uint_nodal_fields[field_name], group_name);

	std::shared_ptr<dumpers::Field> field;
	return field;
	}

	/* -------------------------------------------------------------------------- */
	std::shared_ptr<dumpers::Field>
	PhaseFieldModel::createNodalFieldReal(const std::string & field_name,
	const std::string & group_name,
	bool /unused/) {
	std::map<std::string, Array<Real> *> real_nodal_fields;
	real_nodal_fields["damage"] = damage.get();
	real_nodal_fields["external_force"] = external_force.get();
	real_nodal_fields["internal_force"] = internal_force.get();

	return mesh.createNodalField(real_nodal_fields[field_name], group_name);

	std::shared_ptr<dumpers::Field> field;
	return field;
	}

	/* -------------------------------------------------------------------------- */
	std::shared_ptr<dumpers::Field> PhaseFieldModel::createElementalField(
	const std::string & field_name, const std::string & group_name,
	bool /unused/, Int /unused/, ElementKind element_kind) {

	if (field_name == "partitions") {
	return mesh.createElementalField<Int, dumpers::ElementPartitionField>(
	mesh.getConnectivities(), group_name, this->spatial_dimension,
	element_kind);
	}

	std::shared_ptr<dumpers::Field> field;
	return field;
	}

	/* -------------------------------------------------------------------------- */
	ElementTypeMapArray<Real> &
	PhaseFieldModel::flattenInternal(const std::string & field_name,
	ElementKind kind, const GhostType ghost_type) {
	auto key = std::make_pair(field_name, kind);

	ElementTypeMapArray<Real> * internal_flat;

	auto it = this->registered_internals.find(key);
	if (it == this->registered_internals.end()) {
	auto internal =
	std::make_unique<ElementTypeMapArray<Real>>(field_name, this->id);

	internal_flat = internal.get();
	this->registered_internals[key] = std::move(internal);
	} else {
	internal_flat = it->second.get();
	}

	for (auto type :
	mesh.elementTypes(Model::spatial_dimension, ghost_type, kind)) {
	if (internal_flat->exists(type, ghost_type)) {
	auto & internal = (*internal_flat)(type, ghost_type);
	internal.resize(0);
	}
	}

	for (auto & phasefield : phasefields) {
	if (phasefield->isInternal<Real>(field_name, kind)) {
	phasefield->flattenInternal(field_name, *internal_flat, ghost_type, kind);
	}
	}

	return *internal_flat;
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::inflateInternal(const std::string & field_name,
	const ElementTypeMapArray<Real> & field,
	ElementKind kind, GhostType ghost_type) {

	for (auto & phasefield : phasefields) {
	if (phasefield->isInternal<Real>(field_name, kind)) {
	phasefield->inflateInternal(field_name, field, ghost_type, kind);
	} else {
	AKANTU_ERROR("A internal of name \'"
	<< field_name
	<< "\' has not been defined in the phasefield");
	}
	}
	}

	/* -------------------------------------------------------------------------- */
	void PhaseFieldModel::printself(std::ostream & stream, int indent) const {
	std::string space(indent, AKANTU_INDENT);

	stream << space << "Phase Field Model [" << std::endl;
	stream << space << " + id : " << id << std::endl;
	stream << space << " + spatial dimension : " << Model::spatial_dimension
	<< std::endl;
	stream << space << " + fem [" << std::endl;
	getFEEngine().printself(stream, indent + 2);
	stream << space << AKANTU_INDENT << "]" << std::endl;
	stream << space << " + nodals information [" << std::endl;
	damage->printself(stream, indent + 2);
	external_force->printself(stream, indent + 2);
	internal_force->printself(stream, indent + 2);
	blocked_dofs->printself(stream, indent + 2);
	stream << space << AKANTU_INDENT << "]" << std::endl;

	stream << space << " + phasefield information [" << std::endl;
	stream << space << AKANTU_INDENT << "]" << std::endl;

	stream << space << "]" << std::endl;
	}

	} // namespace akantu

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