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geometry_utils_inline_impl.cc
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			/**
			* Copyright (©) 2019-2023 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 "element_class_helper.hh"
			#include "geometry_utils.hh"
			/* -------------------------------------------------------------------------- */

			#ifndef __AKANTU_GEOMETRY_UTILS_INLINE_IMPL_CC__
			#define __AKANTU_GEOMETRY_UTILS_INLINE_IMPL_CC__

			namespace akantu {

			/* -------------------------------------------------------------------------- */
			inline bool GeometryUtils::isBoundaryElement(const Mesh & mesh,
			const Element & subelement) {
			const auto & element_to_subelement =
			mesh.getElementToSubelement(subelement.type)(subelement.element);

			// for regular boundary elements when akantu::SurfaceSelector is set to
			// physical surfaces, the mesh contains only 1 element attached to a
			// boundary sub-element
			if (element_to_subelement.size() == 1 and
			element_to_subelement[0].kind() == _ek_regular) {
			return true;
			}

			// for cohesive interface elements when akantu::SurfaceSelector is set
			// either cohesive surface selector or all surface selector, in this
			// case mesh passed is actually mesh_facet and for boundary or
			// cohesive interface 2 elements are associated to a sub-element
			// we want only one regular element attached to the sub-element

			Int nb_elements_regular{0};
			// Int nb_elements_cohesive{0};

			for (auto elem : element_to_subelement) {
			if (elem == ElementNull) {
			continue;
			}

			if (elem.kind() == _ek_regular) {
			++nb_elements_regular;
			}

			// if (elem.kind() == _ek_cohesive) {
			// ++nb_elements_cohesive;
			// }
			}

			Int nb_elements = element_to_subelement.size();
			return nb_elements_regular < nb_elements;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived>
			inline bool
			GeometryUtils::isValidProjection(const Eigen::MatrixBase<Derived> & projection,
			Real extension_tolerance) {
			Int nb_xi_inside = 0;
			for (auto xi : projection) {
			if (xi >= -1.0 - extension_tolerance and xi <= 1.0 + extension_tolerance) {
			nb_xi_inside++;
			}
			}

			return nb_xi_inside == projection.size();
			}

			/* -------------------------------------------------------------------------- */
			inline Vector<Real> GeometryUtils::outsideDirection(const Mesh & mesh,
			const Element & element) {
			const auto & element_to_subelement = mesh.getElementToSubelement()(element);

			Vector<Real> outside = mesh.getBarycenter(element);

			// check if mesh facets exists for cohesive elements contact
			Vector<Real> inside;
			if (mesh.isMeshFacets()) {
			inside = mesh.getMeshParent().getBarycenter(element_to_subelement[0]);
			} else {
			inside = mesh.getBarycenter(element_to_subelement[0]);
			}

			return (outside - inside);
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived>
			Vector<Real> GeometryUtils::normal(const Mesh & mesh,
			const Eigen::MatrixBase<Derived> & coords,
			const Element & element, bool outward) {
			Int spatial_dimension = coords.rows();
			Vector<Real> normal(spatial_dimension);

			switch (spatial_dimension) {
			case 1: {
			normal[0] = 1;
			break;
			}
			case 2: {
			normal = Math::normal(coords(1) - coords(0));
			break;
			}
			case 3: {
			normal = Math::normal(coords(1) - coords(0), coords(2) - coords(0));
			break;
			}
			default: {
			AKANTU_ERROR("Unknown dimension : " << spatial_dimension);
			}
			}

			// to ensure that normal is always outwards from master surface
			if (outward) {
			auto projection = outsideDirection(mesh, element).dot(normal);
			if (projection < 0) {
			normal *= -1.0;
			}
			}
			return normal;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived>
			Vector<Real> GeometryUtils::normal(const Mesh & mesh, const Element & element,
			Eigen::MatrixBase<Derived> & tangents,
			bool outward) {
			auto spatial_dimension = mesh.getSpatialDimension();
			// to ensure that normal is always outwards from master surface we
			// compute a direction vector form inside of element attached to the
			// suurface element

			Vector<Real> normal(spatial_dimension);
			// to ensure that direction of tangents are correct, cross product
			// of tangents should give be in the same direction as of inside to outside
			switch (spatial_dimension) {
			case 2: {
			normal(0) = -tangents(1, 0);
			normal(1) = tangents(0, 0);
			break;
			}
			case 3: {
			VectorProxy<Real, 3> tangent1(tangents(0).data());
			VectorProxy<Real, 3> tangent2(tangents(1).data());
			normal = (tangent1.cross(tangent2)).normalized();
			break;
			}
			default:
			break;
			}

			if (outward) {
			auto ddot = outsideDirection(mesh, element).dot(normal);
			if (ddot < 0) {
			tangents *= -1.0;
			normal *= -1.0;
			}
			}
			return normal;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2>
			inline Matrix<Real>
			GeometryUtils::covariantBasis(const Eigen::MatrixBase<Derived1> & coords,
			const Element & element,
			Eigen::MatrixBase<Derived2> & natural_coord) {
			auto && dnds =
			ElementClassHelper<_ek_regular>::getDNDS(natural_coord, element.type);

			Matrix<Real> tangents_transpose = coords * dnds.transpose();
			for (auto && vect : tangents_transpose) {
			vect = vect.normalized();
			}

			return tangents_transpose;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2, class Derived3>
			inline Matrix<Real>
			GeometryUtils::covariantBasis(const Eigen::MatrixBase<Derived1> & coords,
			const Element & element,
			const Eigen::MatrixBase<Derived2> & normal,
			Eigen::MatrixBase<Derived3> & natural_coord) {
			auto tangents = covariantBasis(coords, element, natural_coord);

			// to ensure that direction of tangents are correct, cross product
			// of tangents should give the normal vector computed earlier
			Int spatial_dimension = coords.rows();
			Vector<Real, 3> exp_normal;

			switch (spatial_dimension) {
			case 2: {
			Vector<Real, 3> e_z{0., 0., 1.};

			Vector<Real, 3> tangent;
			tangent[0] = tangents(0, 0);
			tangent[1] = tangents(1, 0);
			tangent[2] = 0.;

			exp_normal = e_z.cross(tangent);
			break;
			}
			case 3: {
			VectorProxy<Real, 3> tangent1(tangents(0).data());
			VectorProxy<Real, 3> tangent2(tangents(1).data());
			exp_normal = (tangent1.cross(tangent2)).normalized();
			break;
			}
			default:
			AKANTU_TO_IMPLEMENT();
			}

			auto ddot = normal.dot(exp_normal);
			if (ddot < 0) {
			tangents(1) *= -1.0;
			}

			return tangents;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2>
			inline Matrix<Real>
			GeometryUtils::curvature(const Eigen::MatrixBase<Derived1> & coords,
			const Element & element,
			const Eigen::MatrixBase<Derived2> & natural_coord) {
			auto && d2nds2 =
			ElementClassHelper<_ek_regular>::getD2NDS2(natural_coord, element.type);
			return coords * d2nds2.transpose();
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2, class Derived3, class Derived4,
			class ElementList>
			Element GeometryUtils::orthogonalProjection(
			const Mesh & mesh, const Array<Real> & positions,
			const Eigen::MatrixBase<Derived1> & slave, const ElementList & elements,
			Real & gap, Eigen::MatrixBase<Derived2> & natural_projection,
			Eigen::MatrixBase<Derived3> & normal, Eigen::MatrixBase<Derived4> & tangent,
			Real /alpha/, Int max_iterations, Real projection_tolerance,
			Real extension_tolerance) {

			auto found_element = ElementNull;
			auto min_gap = std::numeric_limits<Real>::max();

			const auto & contact_group = mesh.getElementGroup("contact_surface");

			for (auto && element : elements) {
			// filter out elements which are not there in the element group
			// contact surface created by the surface selector and is stored
			// in the mesh or mesh_facet, if a element is not there it
			// returnas UInt(-1)

			const auto & elements_of_type = contact_group.getElements(element.type);
			if (elements_of_type.find(element.element) == -1) {
			continue;
			}

			auto coords = mesh.extractNodalValuesFromElement(positions, element);

			auto && [xi_ele, master] = GeometryUtils::naturalProjection(
			coords, element, slave, max_iterations, projection_tolerance);

			auto && tangent_ele =
			GeometryUtils::covariantBasis(coords, element, xi_ele);

			auto && normal_ele = GeometryUtils::normal(mesh, element, tangent_ele);

			// if gap between master projection and slave point is zero, then
			// it means that slave point lies on the master element, hence the
			// normal from master to slave cannot be computed in that case
			auto master_to_slave = (slave - master).eval();
			auto temp_gap = master_to_slave.norm();

			if (temp_gap != 0) {
			master_to_slave /= temp_gap;
			}

			// A alpha parameter is introduced which is 1 in case of explicit
			// and -1 in case of implicit, therefor the variation (dot product
			// + alpha) should be close to zero (within tolerance) for both
			// cases
			auto product = master_to_slave.dot(normal_ele);

			if (product < 0 and temp_gap <= min_gap and
			GeometryUtils::isValidProjection(xi_ele, extension_tolerance)) {
			gap = -temp_gap;
			min_gap = temp_gap;
			found_element = element;
			natural_projection = xi_ele;
			normal = normal_ele;
			tangent = tangent_ele;
			}
			}

			return found_element;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2, class Derived3>
			Vector<Real>
			GeometryUtils::realProjection(const Eigen::MatrixBase<Derived1> & coords,
			const Eigen::MatrixBase<Derived2> & slave,
			const Eigen::MatrixBase<Derived3> & normal) {
			auto alpha = (slave - coords(0)).dot(normal);
			return slave - alpha * normal;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2>
			Vector<Real> GeometryUtils::realProjection(
			const Eigen::MatrixBase<Derived1> & coords, const Element & element,
			const Eigen::MatrixBase<Derived2> & natural_coord) {
			auto shapes =
			ElementClassHelper<_ek_regular>::getN(natural_coord, element.type);
			return coords * shapes;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2>
			std::pair<Vector<Real>, Vector<Real>> GeometryUtils::naturalProjection(
			const Eigen::MatrixBase<Derived1> & coords, const Element & element,
			const Eigen::MatrixBase<Derived2> & slave_coords, Int max_iterations,
			Real projection_tolerance) {

			auto spatial_dimension = coords.rows();
			auto surface_dimension = spatial_dimension - 1;

			auto type = element.type;

			Vector<Real> master_coords(spatial_dimension);
			Vector<Real> natural_projection(surface_dimension);

			// initial guess
			natural_projection.zero();

			// obhjective function computed on the natural_guess
			Vector<Real> f(surface_dimension);

			// jacobian matrix computed on the natural_guess
			Matrix<Real> J(surface_dimension, surface_dimension);

			// dxi = \xi_{k+1} - \xi_{k} in the iterative process
			Vector<Real> dxi(surface_dimension);

			// gradient at natural projection
			Matrix<Real> gradient(surface_dimension, spatial_dimension);

			// second derivative at natural peojection
			Matrix<Real> double_gradient(surface_dimension, surface_dimension);

			// second derivative of shape function at natural projection
			Matrix<Real> d2nds2(surface_dimension * surface_dimension, coords.cols());

			auto compute_double_gradient = [&d2nds2, &coords, surface_dimension,
			spatial_dimension](Int & alpha, Int & beta) {
			auto index = alpha * surface_dimension + beta;
			Vector<Real> d_alpha_beta(spatial_dimension);

			d_alpha_beta = coords * d2nds2.transpose()(index);

			return d_alpha_beta;
			};

			/* --------------------------- */
			/* init before iteration loop */
			/* --------------------------- */
			// do interpolation
			auto update_f = [&f, &master_coords, &natural_projection, &coords,
			&slave_coords, &gradient, surface_dimension, type]() {
			// compute real coords on natural projection
			auto && shapes =
			ElementClassHelper<_ek_regular>::getN(natural_projection, type);

			master_coords = coords * shapes;
			auto distance = slave_coords - master_coords;

			// first derivative of shape function at natural projection
			auto && dnds =
			ElementClassHelper<_ek_regular>::getDNDS(natural_projection, type);
			gradient = dnds * coords.transpose();

			// loop over surface dimensions
			for (auto alpha : arange(surface_dimension)) {
			f(alpha) = -2. * gradient.transpose()(alpha).dot(distance);
			}

			// compute initial error
			return f.norm();
			};

			auto projection_error = update_f();

			/* --------------------------- */
			/* iteration loop */
			/* --------------------------- */
			Int iterations{0};
			while (projection_tolerance < projection_error and
			iterations < max_iterations) {

			// compute covariant components of metric tensor
			auto a = GeometryUtils::covariantMetricTensor(gradient);

			// computing second derivative at natural projection
			d2nds2 =
			ElementClassHelper<_ek_regular>::getD2NDS2(natural_projection, type);

			// real coord - physical guess
			auto distance = slave_coords - master_coords;

			// computing Jacobian J
			for (auto alpha : arange(surface_dimension)) {
			for (auto beta : arange(surface_dimension)) {
			auto dgrad_alpha_beta = compute_double_gradient(alpha, beta);
			J(alpha, beta) = 2. * (a(alpha, beta) - dgrad_alpha_beta.dot(distance));
			}
			}

			// compute increment
			dxi = -1 * J.inverse() * f;

			// update our guess
			natural_projection += dxi;

			projection_error = update_f();
			iterations++;
			}

			return std::make_pair(natural_projection, master_coords);
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived>
			Matrix<Real> GeometryUtils::contravariantBasis(
			const Eigen::MatrixBase<Derived> & covariant) {
			auto && inv_A = GeometryUtils::contravariantMetricTensor(covariant);
			return inv_A * covariant;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived>
			Matrix<Real> GeometryUtils::covariantMetricTensor(
			const Eigen::MatrixBase<Derived> & covariant_bases) {
			auto A = covariant_bases.transpose() * covariant_bases;
			return A;
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived>
			Matrix<Real> GeometryUtils::contravariantMetricTensor(
			const Eigen::MatrixBase<Derived> & covariant_bases) {
			Matrix<Real> A_inv = GeometryUtils::covariantMetricTensor(covariant_bases);
			return A_inv.inverse();
			}

			/* -------------------------------------------------------------------------- */
			template <class Derived1, class Derived2, class Derived3>
			Matrix<Real> GeometryUtils::covariantCurvatureTensor(
			const Eigen::MatrixBase<Derived1> & coords, const Element & element,
			const Eigen::MatrixBase<Derived2> & natural_coord,
			const Eigen::MatrixBase<Derived3> & normal) {

			auto spatial_dimension = coords.rows();
			auto surface_dimension = spatial_dimension - 1;

			auto type = element.type;

			auto && d2nds2 =
			ElementClassHelper<_ek_regular>::getD2NDS2(natural_coord, type);

			Matrix<Real> curvature = coords * d2nds2.transpose();
			Matrix<Real> H(surface_dimension, surface_dimension);

			Int i = 0;
			for (auto alpha : arange(surface_dimension)) {
			for (auto beta : arange(surface_dimension)) {
			H(alpha, beta) = curvature(i).dot(normal);
			i++;
			}
			}

			return H;
			}

			} // namespace akantu

			#endif

rAKA/src/model/contact_mechanics/geometry_utils_inline_impl.ccb35bf118a6e3releases/10.1007/s40571…

geometry_utils_inline_impl.ccView Options

rAKA/src/model/contact_mechanics/geometry_utils_inline_impl.cc
b35bf118a6e3releases/10.1007/s40571…

geometry_utils_inline_impl.cc
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