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geometry_utils_inline_impl.hh
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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

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