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rAKA/src/model/contact_mechanics/geometry_utils.cc
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geometry_utils.cc
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/**
* @file geometry_utils.cc
*
* @author Mohit Pundir <mohit.pundir@epfl.ch>
*
* @date creation: Mon Mmay 20 2019
* @date last modification: Mon May 20 2019
*
* @brief Implementation of various utilities needed for contact geometry
*
* @section LICENSE
*
* Copyright (©) 2010-2018 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 "geometry_utils.hh"
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
void GeometryUtils::normal(const Mesh & mesh, const Array<Real> & positions,
const Element & element, Vector<Real> & normal, bool outward) {
UInt 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);
UInt * elem_val = mesh.getConnectivity(element.type, _not_ghost).storage();
mesh.extractNodalValuesFromElement(positions, coords.storage(),
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));
Math::normal2(vectors.storage(), normal.storage());
break;
}
case 3: {
vectors(0) = Vector<Real>(coords(1)) - Vector<Real>(coords(0));
vectors(1) = Vector<Real>(coords(2)) - Vector<Real>(coords(0));
Math::normal3(vectors(0).storage(), vectors(1).storage(), normal.storage());
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::covariantBasis(const Mesh & mesh, const Array<Real> & positions,
const Element & element, const Vector<Real> & normal,
Vector<Real> & natural_coord,
Matrix<Real> & tangents) {
UInt spatial_dimension = mesh.getSpatialDimension();
const ElementType & type = element.type;
UInt nb_nodes_per_element = mesh.getNbNodesPerElement(type);
UInt * elem_val = mesh.getConnectivity(type, _not_ghost).storage();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.storage(),
elem_val + element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
UInt surface_dimension = spatial_dimension - 1;
Matrix<Real> dnds(surface_dimension, nb_nodes_per_element);
#define GET_SHAPE_DERIVATIVES_NATURAL(type) \
ElementClass<type>::computeDNDS(natural_coord, dnds)
AKANTU_BOOST_ALL_ELEMENT_SWITCH(GET_SHAPE_DERIVATIVES_NATURAL);
#undef GET_SHAPE_DERIVATIVES_NATURAL
tangents.mul<false, true>(dnds, nodes_coord);
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.normalize();
}
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> e_z(3);
e_z[0] = 0.;
e_z[1] = 0.;
e_z[2] = 1.;
Vector<Real> tangent(3);
tangent[0] = tangents(0, 0);
tangent[1] = tangents(0, 1);
tangent[2] = 0.;
auto exp_normal = e_z.crossProduct(tangent);
auto ddot = normal.dot(exp_normal);
if (ddot < 0) {
tangents *= -1.0;
}
break;
}
case 3: {
auto tang_trans = tangents.transpose();
auto tang1 = Vector<Real>(tang_trans(0));
auto tang2 = Vector<Real>(tang_trans(1));
auto tang1_cross_tang2 = tang1.crossProduct(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::curvature(const Mesh & mesh, const Array<Real> & positions,
const Element & element, const Vector<Real> & natural_coord,
Matrix<Real> & curvature) {
UInt spatial_dimension = mesh.getSpatialDimension();
auto surface_dimension = spatial_dimension - 1;
const ElementType & type = element.type;
UInt nb_nodes_per_element = mesh.getNbNodesPerElement(type);
UInt * elem_val = mesh.getConnectivity(type, _not_ghost).storage();
Matrix<Real> dn2ds2(surface_dimension * surface_dimension,
Mesh::getNbNodesPerElement(type));
#define GET_SHAPE_SECOND_DERIVATIVES_NATURAL(type) \
ElementClass<type>::computeDN2DS2(natural_coord, dn2ds2)
AKANTU_BOOST_ALL_ELEMENT_SWITCH(GET_SHAPE_SECOND_DERIVATIVES_NATURAL);
#undef GET_SHAPE_SECOND_DERIVATIVES_NATURAL
Matrix<Real> coords(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, coords.storage(),
elem_val + element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
curvature.mul<false, true>(coords, dn2ds2);
}
/* -------------------------------------------------------------------------- */
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,
Real projection_tolerance,
Real extension_tolerance) {
UInt index = UInt(-1);
Real min_gap = std::numeric_limits<Real>::max();
UInt spatial_dimension = mesh.getSpatialDimension();
UInt nb_same_sides{0};
UInt nb_boundary_elements{0};
UInt counter{0};
for (auto & element : elements) {
if (!GeometryUtils::isBoundaryElement(mesh, element))
continue;
nb_boundary_elements++;
Vector<Real> normal_ele(spatial_dimension);
GeometryUtils::normal(mesh, positions, element, normal_ele);
Vector<Real> master(spatial_dimension);
GeometryUtils::realProjection(mesh, positions, slave, element, normal_ele, master);
Vector<Real> xi(natural_projection.size());
GeometryUtils::naturalProjection(mesh, positions, element, master, xi,
projection_tolerance);
// 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;
Real temp_gap = master_to_slave.norm();
if (temp_gap != 0)
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;
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::realProjection(const Mesh & mesh, const Array<Real> & positions,
const Vector<Real> & slave, const Element & element,
const Vector<Real> & normal,
Vector<Real> & projection) {
UInt spatial_dimension = mesh.getSpatialDimension();
const ElementType & type = element.type;
UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(element.type);
UInt * elem_val = mesh.getConnectivity(type, _not_ghost).storage();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.storage(),
elem_val + element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
Vector<Real> point(nodes_coord(0));
Real 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) {
UInt spatial_dimension = mesh.getSpatialDimension();
const ElementType & type = element.type;
UInt nb_nodes_per_element = mesh.getNbNodesPerElement(element.type);
Vector<Real> shapes(nb_nodes_per_element);
#define GET_SHAPE_NATURAL(type) \
ElementClass<type>::computeShapes(natural_coord, shapes)
AKANTU_BOOST_ALL_ELEMENT_SWITCH(GET_SHAPE_NATURAL);
#undef GET_SHAPE_NATURAL
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
UInt * elem_val = mesh.getConnectivity(type, _not_ghost).storage();
mesh.extractNodalValuesFromElement(positions, nodes_coord.storage(),
elem_val + element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
projection.mul<false>(nodes_coord, shapes);
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::naturalProjection(const Mesh & mesh, const Array<Real> & positions,
const Element & element,
Vector<Real> & real_projection,
Vector<Real> & natural_projection,
Real projection_tolerance) {
UInt spatial_dimension = mesh.getSpatialDimension();
const ElementType & type = element.type;
UInt nb_nodes_per_element = mesh.getNbNodesPerElement(type);
UInt * elem_val = mesh.getConnectivity(type, _not_ghost).storage();
Matrix<Real> nodes_coord(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, nodes_coord.storage(),
elem_val + element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
#define GET_NATURAL_COORDINATE(type) \
ElementClass<type>::inverseMap(real_projection, nodes_coord, \
natural_projection, projection_tolerance)
AKANTU_BOOST_ALL_ELEMENT_SWITCH(GET_NATURAL_COORDINATE);
#undef GET_NATURAL_COORDINATE
}
/* -------------------------------------------------------------------------- */
void GeometryUtils::contravariantBasis(const Matrix<Real> & covariant,
Matrix<Real> & contravariant) {
auto inv_A = GeometryUtils::contravariantMetricTensor(covariant);
contravariant.mul<false, false>(inv_A, covariant);
}
/* -------------------------------------------------------------------------- */
Matrix<Real> GeometryUtils::covariantMetricTensor(const Matrix<Real> & covariant_bases) {
Matrix<Real> A(covariant_bases.rows(), covariant_bases.rows());
A.mul<false, true>(covariant_bases, covariant_bases);
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) {
UInt spatial_dimension = mesh.getSpatialDimension();
auto surface_dimension = spatial_dimension - 1;
const ElementType & type = element.type;
UInt nb_nodes_per_element = Mesh::getNbNodesPerElement(type);
UInt * elem_val = mesh.getConnectivity(type, _not_ghost).storage();
Matrix<Real> dn2ds2(surface_dimension * surface_dimension,
nb_nodes_per_element);
#define GET_SHAPE_SECOND_DERIVATIVES_NATURAL(type) \
ElementClass<type>::computeDN2DS2(natural_coord, dn2ds2)
AKANTU_BOOST_ALL_ELEMENT_SWITCH(GET_SHAPE_SECOND_DERIVATIVES_NATURAL);
#undef GET_SHAPE_SECOND_DERIVATIVES_NATURAL
Matrix<Real> coords(spatial_dimension, nb_nodes_per_element);
mesh.extractNodalValuesFromElement(positions, coords.storage(),
elem_val + element.element * nb_nodes_per_element,
nb_nodes_per_element, spatial_dimension);
Matrix<Real> curvature(spatial_dimension, surface_dimension*surface_dimension);
curvature.mul<false, true>(coords, dn2ds2);
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;
}
}
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