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geometry_utils.cc
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/**
* @file geometry_utils.cc
*
* @author Mohit Pundir <mohit.pundir@epfl.ch>
*
* @date creation: Wed Oct 02 2019
* @date last modification: Thu Jun 24 2021
*
* @brief Implementation of various utilities needed for contact geometry
*
*
* @section LICENSE
*
* Copyright (©) 2018-2021 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 "geometry_utils.hh"
#include "element_class_helper.hh"
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
void GeometryUtils::normal(const Mesh & mesh, const Array<Real> & positions,
const Element & element, Vector<Real> & normal,
bool outward) {
Int spatial_dimension = mesh.getSpatialDimension();
UInt surface_dimension = spatial_dimension - 1;
UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(element.type);
Matrix<Real> coords(spatial_dimension, nb_nodes_per_element);
auto * elem_val = mesh.getConnectivity(element.type, _not_ghost).data();
mesh.extractNodalValuesFromElement(positions, coords.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
Matrix<Real> vectors(spatial_dimension, surface_dimension);
switch (spatial_dimension) {
case 1: {
normal[0] = 1;
break;
}
case 2: {
vectors(0) = Vector<Real>(coords(1)) - Vector<Real>(coords(0));
normal = Math::normal(vectors);
break;
}
case 3: {
vectors(0) = coords(1) - coords(0);
vectors(1) = coords(2) - coords(0);
normal = Math::normal(vectors(0), vectors(1));
break;
}
default: {
AKANTU_ERROR("Unknown dimension : " << spatial_dimension);
}
}
// to ensure that normal is always outwards from master surface
if (outward) {
const auto & element_to_subelement =
mesh.getElementToSubelement(element.type)(element.element);
Vector<Real> outside(spatial_dimension);
mesh.getBarycenter(element, outside);
// check if mesh facets exists for cohesive elements contact
Vector<Real> inside(spatial_dimension);
if (mesh.isMeshFacets()) {
mesh.getMeshParent().getBarycenter(element_to_subelement[0], inside);
} else {
mesh.getBarycenter(element_to_subelement[0], inside);
}
Vector<Real> inside_to_outside = outside - inside;
auto projection = inside_to_outside.dot(normal);
if (projection < 0) {
normal *= -1.0;
}
}
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::normal(const Mesh & mesh, const Element & element,
Matrix<Real> & tangents, Vector<Real> & normal,
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> inside_to_outside(spatial_dimension);
if (outward) {
const auto & element_to_subelement =
mesh.getElementToSubelement(element.type)(element.element);
Vector<Real> outside(spatial_dimension);
mesh.getBarycenter(element, outside);
// check if mesh facets exists for cohesive elements contact
Vector<Real> inside(spatial_dimension);
if (mesh.isMeshFacets()) {
mesh.getMeshParent().getBarycenter(element_to_subelement[0], inside);
} else {
mesh.getBarycenter(element_to_subelement[0], inside);
}
inside_to_outside = outside - inside;
// auto projection = inside_to_outside.dot(normal);
// if (projection < 0) {
// normal *= -1.0;
//}
}
// 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(0, 1);
normal[1] = tangents(0, 0);
auto ddot = inside_to_outside.dot(normal);
if (ddot < 0) {
tangents *= -1.0;
normal *= -1.0;
}
break;
}
case 3: {
auto tang_trans = tangents.transpose();
Vector<Real, 3> tang1 = tang_trans(0);
Vector<Real, 3> tang2 = tang_trans(1);
auto tang1_cross_tang2 = tang1.cross(tang2);
normal = tang1_cross_tang2 / tang1_cross_tang2.norm();
auto ddot = inside_to_outside.dot(normal);
if (ddot < 0) {
tang_trans(1) *= -1.0;
normal *= -1.0;
}
tangents = tang_trans.transpose();
break;
}
default:
break;
}
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::covariantBasis(const Mesh & mesh,
const Array<Real> & positions,
const Element & element,
const Vector<Real> & normal,
Vector<Real> & natural_coord,
Matrix<Real> & tangents) {
Int spatial_dimension = mesh.getSpatialDimension();
const auto type = element.type;
auto nb_nodes_per_element = mesh.getNbNodesPerElement(type);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
auto && dnds = ElementClassHelper<_ek_regular>::getDNDS(natural_coord, type);
tangents = dnds * nodes_coord.transpose();
auto temp_tangents = tangents.transpose();
for (UInt i = 0; i < spatial_dimension - 1; ++i) {
auto temp = Vector<Real>(temp_tangents(i));
temp_tangents(i) = temp.normalized();
}
tangents = temp_tangents.transpose();
// to ensure that direction of tangents are correct, cross product
// of tangents should give the normal vector computed earlier
switch (spatial_dimension) {
case 2: {
Vector<Real, 3> e_z;
e_z[0] = 0.;
e_z[1] = 0.;
e_z[2] = 1.;
Vector<Real, 3> tangent;
tangent[0] = tangents(0, 0);
tangent[1] = tangents(0, 1);
tangent[2] = 0.;
auto exp_normal = e_z.cross(tangent);
auto ddot = normal.dot(exp_normal);
if (ddot < 0) {
tangents *= -1.0;
}
break;
}
case 3: {
auto tang_trans = tangents.transpose();
Vector<Real, 3> tang1 = tang_trans(0);
Vector<Real, 3> tang2 = tang_trans(1);
auto tang1_cross_tang2 = tang1.cross(tang2);
auto exp_normal = tang1_cross_tang2 / tang1_cross_tang2.norm();
auto ddot = normal.dot(exp_normal);
if (ddot < 0) {
tang_trans(1) *= -1.0;
}
tangents = tang_trans.transpose();
break;
}
default:
break;
}
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::covariantBasis(const Mesh & mesh,
const Array<Real> & positions,
const Element & element,
Vector<Real> & natural_coord,
Matrix<Real> & tangents) {
auto spatial_dimension = mesh.getSpatialDimension();
auto type = element.type;
auto nb_nodes_per_element = mesh.getNbNodesPerElement(type);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
auto && dnds = ElementClassHelper<_ek_regular>::getDNDS(natural_coord, type);
tangents = dnds * nodes_coord.transpose();
auto temp_tangents = tangents.transpose();
for (Int i = 0; i < spatial_dimension - 1; ++i) {
auto temp = Vector<Real>(temp_tangents(i));
temp_tangents(i) = temp.normalized();
}
tangents = temp_tangents.transpose();
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::curvature(const Mesh & mesh, const Array<Real> & positions,
const Element & element,
const Vector<Real> & natural_coord,
Matrix<Real> & curvature) {
auto spatial_dimension = mesh.getSpatialDimension();
auto type = element.type;
auto nb_nodes_per_element = mesh.getNbNodesPerElement(type);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
auto && d2nds2 =
ElementClassHelper<_ek_regular>::getD2NDS2(natural_coord, type);
Matrix<Real> coords(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, coords.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
curvature = coords * d2nds2.transpose();
}
/* -------------------------------------------------------------------------- */
UInt GeometryUtils::orthogonalProjection(
const Mesh & mesh, const Array<Real> & positions,
const Vector<Real> & slave, const Array<Element> & elements, Real & gap,
Vector<Real> & natural_projection, Vector<Real> & normal, Real alpha,
UInt max_iterations, Real projection_tolerance, Real extension_tolerance) {
Int index = -1;
auto min_gap = std::numeric_limits<Real>::max();
auto spatial_dimension = mesh.getSpatialDimension();
auto surface_dimension = spatial_dimension - 1;
Int nb_same_sides{0};
Int nb_boundary_elements{0};
Int counter{0};
const auto & contact_group = mesh.getElementGroup("contact_surface");
for (const 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) == UInt(-1)) {
continue;
}
nb_boundary_elements++;
// find the natural coordinate corresponding to the minimum gap
// between slave node and master element
Vector<Real> master(spatial_dimension);
Vector<Real> xi(natural_projection.size());
GeometryUtils::naturalProjection(mesh, positions, element, slave, master,
xi, max_iterations, projection_tolerance);
Matrix<Real> tangent_ele(surface_dimension, spatial_dimension);
GeometryUtils::covariantBasis(mesh, positions, element, xi, tangent_ele);
Vector<Real> normal_ele(spatial_dimension);
GeometryUtils::normal(mesh, element, tangent_ele, normal_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 = master_to_slave / temp_gap;
}
// also the slave point should lie inside the master surface in
// case of explicit or outside in case of implicit, one way to
// check that is by checking the dot product of normal at each
// master element, if the values of all dot product is same then
// the slave point lies on the same side of each master element
// 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
Real direction_tolerance = 1e-8;
auto product = master_to_slave.dot(normal_ele);
auto variation = std::abs(product + alpha);
if (variation <= direction_tolerance and temp_gap <= min_gap and
GeometryUtils::isValidProjection(xi, extension_tolerance)) {
gap = -temp_gap;
min_gap = temp_gap;
index = counter;
natural_projection = xi;
normal = normal_ele;
}
if (temp_gap == 0 or variation <= direction_tolerance) {
nb_same_sides++;
}
counter++;
}
// if point is not on the same side of all the boundary elements
// than it is not consider even if the closet master element is
// found
if (nb_same_sides != nb_boundary_elements) {
index = UInt(-1);
}
return index;
}
/* -------------------------------------------------------------------------- */
UInt GeometryUtils::orthogonalProjection(
const Mesh & mesh, const Array<Real> & positions,
const Vector<Real> & slave, const Array<Element> & elements, Real & gap,
Vector<Real> & natural_projection, Vector<Real> & normal,
Matrix<Real> & tangent, Real /*alpha*/, UInt max_iterations,
Real projection_tolerance, Real extension_tolerance) {
UInt index = UInt(-1);
Real min_gap = std::numeric_limits<Real>::max();
Int spatial_dimension = mesh.getSpatialDimension();
UInt surface_dimension = spatial_dimension - 1;
const auto & contact_group = mesh.getElementGroup("contact_surface");
for (auto && tuple : enumerate(elements)) {
auto & counter = std::get<0>(tuple);
const auto & element = std::get<1>(tuple);
// 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) == UInt(-1)) {
continue;
}
Vector<Real> master(spatial_dimension);
Vector<Real> xi_ele(natural_projection.size());
GeometryUtils::naturalProjection(mesh, positions, element, slave, master,
xi_ele, max_iterations,
projection_tolerance);
Matrix<Real> tangent_ele(surface_dimension, spatial_dimension);
GeometryUtils::covariantBasis(mesh, positions, element, xi_ele,
tangent_ele);
Vector<Real> normal_ele(spatial_dimension);
GeometryUtils::normal(mesh, element, tangent_ele, normal_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;
index = counter;
natural_projection = xi_ele;
normal = normal_ele;
tangent = tangent_ele;
}
}
return index;
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::realProjection(const Mesh & mesh,
const Array<Real> & positions,
const Vector<Real> & slave,
const Element & element,
const Vector<Real> & normal,
Vector<Real> & projection) {
auto spatial_dimension = mesh.getSpatialDimension();
auto type = element.type;
auto nb_nodes_per_element = Mesh::getNbNodesPerElement(element.type);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
Vector<Real> point(nodes_coord(0));
auto alpha = (slave - point).dot(normal);
projection = slave - alpha * normal;
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::realProjection(const Mesh & mesh,
const Array<Real> & positions,
const Element & element,
const Vector<Real> & natural_coord,
Vector<Real> & projection) {
auto spatial_dimension = mesh.getSpatialDimension();
const auto & type = element.type;
auto nb_nodes_per_element = Mesh::getNbNodesPerElement(element.type);
auto shapes =
ElementClassHelper<_ek_regular>::getN(natural_coord, element.type);
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
mesh.extractNodalValuesFromElement(positions, nodes_coord.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
projection = nodes_coord * shapes;
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::naturalProjection(
const Mesh & mesh, const Array<Real> & positions, const Element & element,
const Vector<Real> & slave_coords, Vector<Real> & master_coords,
Vector<Real> & natural_projection, UInt max_iterations,
Real projection_tolerance) {
auto spatial_dimension = mesh.getSpatialDimension();
auto surface_dimension = spatial_dimension - 1;
auto type = element.type;
auto nb_nodes_per_element = mesh.getNbNodesPerElement(type);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
// initial guess
natural_projection.zero();
// obhjective function computed on the natural_guess
Matrix<Real> f(surface_dimension, 1);
// jacobian matrix computed on the natural_guess
Matrix<Real> J(surface_dimension, surface_dimension);
// dxi = \xi_{k+1} - \xi_{k} in the iterative process
Matrix<Real> dxi(surface_dimension, 1);
// 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,
nb_nodes_per_element);
auto compute_double_gradient = [&d2nds2, &nodes_coord, surface_dimension,
spatial_dimension](Int & alpha, Int & beta) {
auto index = alpha * surface_dimension + beta;
Vector<Real> d_alpha_beta(spatial_dimension);
auto d2nds2_transpose = d2nds2.transpose();
Vector<Real> d2nds2_alpha_beta(d2nds2_transpose(index));
d_alpha_beta = nodes_coord * d2nds2_alpha_beta;
return d_alpha_beta;
};
/* --------------------------- */
/* init before iteration loop */
/* --------------------------- */
// do interpolation
auto update_f = [&f, &master_coords, &natural_projection, &nodes_coord,
&slave_coords, &gradient, surface_dimension,
spatial_dimension, type]() {
// compute real coords on natural projection
auto && shapes =
ElementClassHelper<_ek_regular>::getN(natural_projection, type);
master_coords = nodes_coord * 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 * nodes_coord.transpose();
// gradient transpose at natural projection
Matrix<Real> gradient_transpose(surface_dimension, spatial_dimension);
gradient_transpose = gradient.transpose();
// loop over surface dimensions
for (auto alpha : arange(surface_dimension)) {
Vector<Real> gradient_alpha(gradient_transpose(alpha));
f(alpha, 0) = -2. * gradient_alpha.dot(distance);
}
// compute initial error
auto error = f.norm();
return error;
};
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(0);
projection_error = update_f();
iterations++;
}
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::contravariantBasis(const Matrix<Real> & covariant,
Matrix<Real> & contravariant) {
auto inv_A = GeometryUtils::contravariantMetricTensor(covariant);
contravariant = inv_A * covariant;
}
/* -------------------------------------------------------------------------- */
Matrix<Real>
GeometryUtils::covariantMetricTensor(const Matrix<Real> & covariant_bases) {
Matrix<Real> A(covariant_bases.rows(), covariant_bases.rows());
A = covariant_bases * covariant_bases.transpose();
return A;
}
/* -------------------------------------------------------------------------- */
Matrix<Real>
GeometryUtils::contravariantMetricTensor(const Matrix<Real> & covariant_bases) {
auto A = GeometryUtils::covariantMetricTensor(covariant_bases);
auto inv_A = A.inverse();
return inv_A;
}
/* -------------------------------------------------------------------------- */
Matrix<Real> GeometryUtils::covariantCurvatureTensor(
const Mesh & mesh, const Array<Real> & positions, const Element & element,
const Vector<Real> & natural_coord, const Vector<Real> & normal) {
auto spatial_dimension = mesh.getSpatialDimension();
auto surface_dimension = spatial_dimension - 1;
auto type = element.type;
auto nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
auto * elem_val = mesh.getConnectivity(type, _not_ghost).data();
auto && d2nds2 =
ElementClassHelper<_ek_regular>::getD2NDS2(natural_coord, type);
Matrix<Real> coords(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, coords.data(),
elem_val +
element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
Matrix<Real> curvature(spatial_dimension,
surface_dimension * surface_dimension);
curvature = coords * d2nds2.transpose();
Matrix<Real> H(surface_dimension, surface_dimension);
UInt i = 0;
for (auto alpha : arange(surface_dimension)) {
for (auto beta : arange(surface_dimension)) {
Vector<Real> temp(curvature(i));
H(alpha, beta) = temp.dot(normal);
i++;
}
}
return H;
}
} // namespace akantu

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