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rAKA/src/model/contact_mechanics/resolutions/resolution_penalty.cc
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/**
* @file resolution_penalty.cc
*
* @author Mohit Pundir <mohit.pundir@epfl.ch>
*
* @date creation: Thu Jan 17 2019
* @date last modification: Wed Jun 09 2021
*
* @brief Specialization of the resolution class for the penalty method
*
*
* @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 "resolution_penalty.hh"
#include "element_class_helper.hh"
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
ResolutionPenalty::ResolutionPenalty(ContactMechanicsModel & model,
const ID & id)
: Resolution(model, id) {
AKANTU_DEBUG_IN();
this->initialize();
AKANTU_DEBUG_OUT();
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::initialize() {
this->registerParam("epsilon_n", epsilon_n, Real(0.),
_pat_parsable | _pat_modifiable,
"Normal penalty parameter");
this->registerParam("epsilon_t", epsilon_t, Real(0.),
_pat_parsable | _pat_modifiable,
"Tangential penalty parameter");
}
/* -------------------------------------------------------------------------- */
Real ResolutionPenalty::computeNormalTraction(Real & gap) const {
return epsilon_n * macaulay(gap);
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeNormalForce(const ContactElement & element,
Vector<Real> & force) {
force.zero();
auto & gaps = model.getGaps();
auto & projections = model.getProjections();
auto & normals = model.getNormals();
auto surface_dimension = spatial_dimension - 1;
Real gap(gaps.begin()[element.slave]);
Vector<Real> normal(normals.begin(spatial_dimension)[element.slave]);
Vector<Real> projection(projections.begin(surface_dimension)[element.slave]);
auto & nodal_area = const_cast<Array<Real> &>(model.getNodalArea());
// compute normal traction
Real p_n = computeNormalTraction(gap);
p_n *= nodal_area[element.slave];
UInt nb_nodes_per_contact = element.getNbNodes();
Matrix<Real> shape_matric(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
ResolutionUtils::computeShapeFunctionMatric(element, projection,
shape_matric);
force.mul<true>(shape_matric, normal, p_n);
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeTangentialForce(const ContactElement & element,
Vector<Real> & force) {
if (mu == 0) {
return;
}
force.zero();
UInt surface_dimension = spatial_dimension - 1;
// compute covariant basis
auto & projections = model.getProjections();
Vector<Real> projection(projections.begin(surface_dimension)[element.slave]);
auto & normals = model.getNormals();
Vector<Real> normal(normals.begin(spatial_dimension)[element.slave]);
auto & tangents = model.getTangents();
Matrix<Real> covariant_basis(
tangents.begin(surface_dimension, spatial_dimension)[element.slave]);
// check for no-contact to contact condition
// need a better way to check if new node added is not presnt in the
// previous master elemets
auto & previous_master_elements = model.getPreviousMasterElements();
if (element.slave >= previous_master_elements.size()) {
return;
}
auto & previous_element = previous_master_elements[element.slave];
if (previous_element.type == _not_defined) {
return;
}
// compute tangential traction using return map algorithm
auto & tangential_tractions = model.getTangentialTractions();
Vector<Real> tangential_traction(
tangential_tractions.begin(surface_dimension)[element.slave]);
this->computeTangentialTraction(element, covariant_basis,
tangential_traction);
UInt nb_nodes_per_contact = element.getNbNodes();
Matrix<Real> shape_matric(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
ResolutionUtils::computeShapeFunctionMatric(element, projection,
shape_matric);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
auto & nodal_area = const_cast<Array<Real> &>(model.getNodalArea());
for (auto && values1 : enumerate(covariant_basis.transpose())) {
auto & alpha = std::get<0>(values1);
auto & tangent_alpha = std::get<1>(values1);
for (auto && values2 : enumerate(tangential_traction)) {
auto & beta = std::get<0>(values2);
auto & traction_beta = std::get<1>(values2);
Vector<Real> tmp(force.size());
tmp.mul<true>(shape_matric, tangent_alpha, traction_beta);
tmp *=
contravariant_metric_tensor(alpha, beta) * nodal_area[element.slave];
force += tmp;
}
}
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeTangentialTraction(
const ContactElement & element, const Matrix<Real> & covariant_basis,
Vector<Real> & traction_tangential) {
UInt surface_dimension = spatial_dimension - 1;
auto & gaps = model.getGaps();
auto & gap = gaps.begin()[element.slave];
// Return map algorithm is employed
// compute trial traction
Vector<Real> traction_trial(surface_dimension);
this->computeTrialTangentialTraction(element, covariant_basis,
traction_trial);
// compute norm of trial traction
Real traction_trial_norm = 0;
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
for (auto i : arange(surface_dimension)) {
for (auto j : arange(surface_dimension)) {
traction_trial_norm += traction_trial[i] * traction_trial[j] *
contravariant_metric_tensor(i, j);
}
}
traction_trial_norm = sqrt(traction_trial_norm);
// check stick or slip condition
auto & contact_state = model.getContactState();
auto & state = contact_state.begin()[element.slave];
Real p_n = computeNormalTraction(gap);
bool stick = (traction_trial_norm <= mu * p_n);
if (stick) {
state = ContactState::_stick;
computeStickTangentialTraction(element, traction_trial,
traction_tangential);
} else {
state = ContactState::_slip;
computeSlipTangentialTraction(element, covariant_basis, traction_trial,
traction_tangential);
}
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeTrialTangentialTraction(
const ContactElement & element, const Matrix<Real> & covariant_basis,
Vector<Real> & traction) {
UInt surface_dimension = spatial_dimension - 1;
auto & projections = model.getProjections();
Vector<Real> current_projection(
projections.begin(surface_dimension)[element.slave]);
auto & previous_projections = model.getPreviousProjections();
Vector<Real> previous_projection(
previous_projections.begin(surface_dimension)[element.slave]);
// method from Laursen et. al.
/*auto covariant_metric_tensor =
GeometryUtils::covariantMetricTensor(covariant_basis); auto
increment_projection = current_projection - previous_projection;
traction.mul<false>(covariant_metric_tensor, increment_projection, epsilon_t);
auto & previous_tangential_tractions = model.getPreviousTangentialTractions();
Vector<Real>
previous_traction(previous_tangential_tractions.begin(surface_dimension)[element.slave]);
traction = previous_traction + traction;*/
// method from Schweizerhof
auto covariant_metric_tensor =
GeometryUtils::covariantMetricTensor(covariant_basis);
auto & previous_tangential_tractions = model.getPreviousTangentialTractions();
Vector<Real> previous_traction(
previous_tangential_tractions.begin(surface_dimension)[element.slave]);
auto & previous_tangents = model.getPreviousTangents();
Matrix<Real> previous_covariant_basis(previous_tangents.begin(
surface_dimension, spatial_dimension)[element.slave]);
auto previous_contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(previous_covariant_basis);
auto current_tangent = covariant_basis.transpose();
auto previous_tangent = previous_covariant_basis.transpose();
for (auto alpha : arange(surface_dimension)) {
Vector<Real> tangent_alpha(current_tangent(alpha));
for (auto gamma : arange(surface_dimension)) {
for (auto beta : arange(surface_dimension)) {
Vector<Real> tangent_beta(previous_tangent(beta));
auto t_alpha_t_beta = tangent_beta.dot(tangent_alpha);
traction[alpha] += previous_traction[gamma] *
previous_contravariant_metric_tensor(gamma, beta) *
t_alpha_t_beta;
}
}
}
auto & previous_master_elements = model.getPreviousMasterElements();
auto & previous_element = previous_master_elements[element.slave];
Vector<Real> previous_real_projection(spatial_dimension);
GeometryUtils::realProjection(
model.getMesh(), model.getContactDetector().getPositions(),
previous_element, previous_projection, previous_real_projection);
Vector<Real> current_real_projection(spatial_dimension);
GeometryUtils::realProjection(
model.getMesh(), model.getContactDetector().getPositions(),
element.master, current_projection, current_real_projection);
auto increment_real = current_real_projection - previous_real_projection;
Vector<Real> increment_xi(surface_dimension);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
// increment in natural coordinate
for (auto beta : arange(surface_dimension)) {
for (auto gamma : arange(surface_dimension)) {
auto temp = increment_real.dot(current_tangent(gamma));
temp *= contravariant_metric_tensor(beta, gamma);
increment_xi[beta] += temp;
}
}
Vector<Real> temp(surface_dimension);
temp.mul<false>(covariant_metric_tensor, increment_xi, epsilon_t);
traction -= temp;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeStickTangentialTraction(
const ContactElement & /*element*/, Vector<Real> & traction_trial,
Vector<Real> & traction_tangential) {
traction_tangential = traction_trial;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeSlipTangentialTraction(
const ContactElement & element, const Matrix<Real> & covariant_basis,
Vector<Real> & traction_trial, Vector<Real> & traction_tangential) {
UInt surface_dimension = spatial_dimension - 1;
auto & gaps = model.getGaps();
auto & gap = gaps.begin()[element.slave];
// compute norm of trial traction
Real traction_trial_norm = 0;
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
for (auto alpha : arange(surface_dimension)) {
for (auto beta : arange(surface_dimension)) {
traction_trial_norm += traction_trial[alpha] * traction_trial[beta] *
contravariant_metric_tensor(alpha, beta);
}
}
traction_trial_norm = sqrt(traction_trial_norm);
auto slip_direction = traction_trial;
slip_direction /= traction_trial_norm;
Real p_n = computeNormalTraction(gap);
traction_tangential = slip_direction;
traction_tangential *= mu * p_n;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeNormalModuli(const ContactElement & element,
Matrix<Real> & stiffness) {
auto surface_dimension = spatial_dimension - 1;
auto & gaps = model.getGaps();
Real gap(gaps.begin()[element.slave]);
auto & projections = model.getProjections();
Vector<Real> projection(projections.begin(surface_dimension)[element.slave]);
auto & nodal_areas = model.getNodalArea();
auto & nodal_area = nodal_areas.begin()[element.slave];
auto & normals = model.getNormals();
Vector<Real> normal(normals.begin(spatial_dimension)[element.slave]);
// method from Schweizerhof and A. Konyukhov, K. Schweizerhof
// DOI 10.1007/s00466-004-0616-7 and DOI 10.1007/s00466-003-0515-3
// construct A matrix
const ElementType & type = element.master.type;
auto && shapes = ElementClassHelper<_ek_regular>::getN(projection, type);
UInt nb_nodes_per_contact = element.getNbNodes();
Matrix<Real> A(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
for (auto i : arange(nb_nodes_per_contact)) {
for (auto j : arange(spatial_dimension)) {
if (i == 0) {
A(j, i * spatial_dimension + j) = 1;
continue;
}
A(j, i * spatial_dimension + j) = -shapes[i - 1];
}
}
// construct the main part of normal matrix
Matrix<Real> k_main(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
Matrix<Real> n_outer_n(spatial_dimension, spatial_dimension);
Matrix<Real> mat_n(normal.storage(), normal.size(), 1.);
n_outer_n.mul<false, true>(mat_n, mat_n);
Matrix<Real> tmp(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
tmp.mul<false, false>(n_outer_n, A);
k_main.mul<true, false>(A, tmp);
k_main *= epsilon_n * heaviside(gap) * nodal_area;
// construct the rotational part of the normal matrix
auto & tangents = model.getTangents();
Matrix<Real> covariant_basis(
tangents.begin(surface_dimension, spatial_dimension)[element.slave]);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
// computing shape derivatives
auto && shape_derivatives =
ElementClassHelper<_ek_regular>::getDNDS(projection, type);
// consists of 2 rotational parts
Matrix<Real> k_rot1(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
Matrix<Real> k_rot2(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
Matrix<Real> Aj(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
auto construct_Aj = [&](auto && dnds) {
for (auto i : arange(nb_nodes_per_contact)) {
for (auto j : arange(spatial_dimension)) {
if (i == 0) {
Aj(j, i * spatial_dimension + j) = 0;
continue;
}
Aj(j, i * spatial_dimension + j) = dnds(i - 1);
}
}
};
for (auto && values1 : enumerate(covariant_basis.transpose())) {
auto & alpha = std::get<0>(values1);
auto & tangent = std::get<1>(values1);
Matrix<Real> n_outer_t(spatial_dimension, spatial_dimension);
Matrix<Real> mat_t(tangent.storage(), tangent.size(), 1.);
n_outer_t.mul<false, true>(mat_n, mat_t);
Matrix<Real> t_outer_n(spatial_dimension, spatial_dimension);
t_outer_n.mul<false, true>(mat_t, mat_n);
for (auto && values2 : enumerate(shape_derivatives.transpose())) {
auto & beta = std::get<0>(values2);
auto & dnds = std::get<1>(values2);
// construct Aj from shape function wrt to jth natural
// coordinate
construct_Aj(dnds);
Matrix<Real> tmp(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
Matrix<Real> tmp1(nb_nodes_per_contact * spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
tmp.mul<false, false>(n_outer_t, A);
tmp1.mul<true, false>(Aj, tmp);
tmp1 *= contravariant_metric_tensor(alpha, beta);
k_rot1 += tmp1;
tmp.mul<false, false>(t_outer_n, Aj);
tmp1.mul<true, false>(A, tmp);
tmp1 *= contravariant_metric_tensor(alpha, beta);
k_rot2 += tmp1;
}
}
k_rot1 *= -epsilon_n * heaviside(gap) * gap * nodal_area;
k_rot2 *= -epsilon_n * heaviside(gap) * gap * nodal_area;
stiffness += k_main + k_rot1 + k_rot2;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeTangentialModuli(const ContactElement & element,
Matrix<Real> & stiffness) {
if (mu == 0) {
return;
}
stiffness.zero();
auto & contact_state = model.getContactState();
auto state = contact_state.begin()[element.slave];
switch (state) {
case ContactState::_stick: {
computeStickModuli(element, stiffness);
break;
}
case ContactState::_slip: {
computeSlipModuli(element, stiffness);
break;
}
default:
break;
}
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeStickModuli(const ContactElement & element,
Matrix<Real> & stiffness) {
auto surface_dimension = spatial_dimension - 1;
auto & projections = model.getProjections();
Vector<Real> projection(projections.begin(surface_dimension)[element.slave]);
auto & nodal_areas = model.getNodalArea();
auto & nodal_area = nodal_areas.begin()[element.slave];
// method from Schweizerhof and A. Konyukhov, K. Schweizerhof
// DOI 10.1007/s00466-004-0616-7 and DOI 10.1007/s00466-003-0515-3
// construct A matrix
const ElementType & type = element.master.type;
auto && shapes = ElementClassHelper<_ek_regular>::getN(projection, type);
UInt nb_nodes_per_contact = element.getNbNodes();
Matrix<Real> A(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
for (auto i : arange(nb_nodes_per_contact)) {
for (auto j : arange(spatial_dimension)) {
if (i == 0) {
A(j, i * spatial_dimension + j) = 1;
continue;
}
A(j, i * spatial_dimension + j) = -shapes[i - 1];
}
}
// computing shape derivatives
auto && shape_derivatives =
ElementClassHelper<_ek_regular>::getDNDS(projection, type);
Matrix<Real> Aj(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
auto construct_Aj = [&](auto && dnds) {
for (auto i : arange(nb_nodes_per_contact)) {
for (auto j : arange(spatial_dimension)) {
if (i == 0) {
Aj(j, i * spatial_dimension + j) = 0;
continue;
}
Aj(j, i * spatial_dimension + j) = dnds(i - 1);
}
}
};
// tangents should have been calculated in normal modulii
auto & tangents = model.getTangents();
Matrix<Real> covariant_basis(
tangents.begin(surface_dimension, spatial_dimension)[element.slave]);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
// construct 1st part of the stick modulii
Matrix<Real> k_main(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
for (auto && values1 : enumerate(covariant_basis.transpose())) {
auto & alpha = std::get<0>(values1);
auto & tangent_alpha = std::get<1>(values1);
Matrix<Real> t_outer_t(spatial_dimension, spatial_dimension);
Matrix<Real> mat_t_alpha(tangent_alpha.storage(), tangent_alpha.size(), 1.);
for (auto && values2 : enumerate(covariant_basis.transpose())) {
auto & beta = std::get<0>(values2);
auto & tangent_beta = std::get<1>(values2);
Matrix<Real> mat_t_beta(tangent_beta.storage(), tangent_beta.size(), 1.);
t_outer_t.mul<false, true>(mat_t_alpha, mat_t_beta);
Matrix<Real> tmp(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
Matrix<Real> tmp1(nb_nodes_per_contact * spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
tmp.mul<false, false>(t_outer_t, A);
tmp1.mul<true, false>(A, tmp);
tmp1 *= contravariant_metric_tensor(alpha, beta);
k_main += tmp1;
}
}
k_main *= -epsilon_t;
// construct 2nd part of the stick modulii
auto & tangential_tractions = model.getTangentialTractions();
Vector<Real> tangential_traction(
tangential_tractions.begin(surface_dimension)[element.slave]);
Matrix<Real> k_second(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
for (auto alpha : arange(surface_dimension)) {
Matrix<Real> k_sum(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
for (auto && values1 : enumerate(shape_derivatives.transpose())) {
auto & beta = std::get<0>(values1);
auto & dnds = std::get<1>(values1);
// construct Aj from shape function wrt to jth natural
// coordinate
construct_Aj(dnds);
for (auto && values2 : enumerate(covariant_basis.transpose())) {
auto & gamma = std::get<0>(values2);
auto & tangent_gamma = std::get<1>(values2);
Matrix<Real> t_outer_t(spatial_dimension, spatial_dimension);
Matrix<Real> mat_t_gamma(tangent_gamma.storage(), tangent_gamma.size(),
1.);
for (auto && values3 : enumerate(covariant_basis.transpose())) {
auto & theta = std::get<0>(values3);
auto & tangent_theta = std::get<1>(values3);
Matrix<Real> mat_t_theta(tangent_theta.storage(),
tangent_theta.size(), 1.);
t_outer_t.mul<false, true>(mat_t_gamma, mat_t_theta);
Matrix<Real> tmp(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
Matrix<Real> tmp1(nb_nodes_per_contact * spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
tmp.mul<false, false>(t_outer_t, Aj);
tmp1.mul<true, false>(A, tmp);
tmp1 *= contravariant_metric_tensor(alpha, theta) *
contravariant_metric_tensor(beta, gamma);
Matrix<Real> tmp2(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
Matrix<Real> tmp3(nb_nodes_per_contact * spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
tmp2.mul<false, false>(t_outer_t, A);
tmp3.mul<true, false>(Aj, tmp2);
tmp3 *= contravariant_metric_tensor(alpha, gamma) *
contravariant_metric_tensor(beta, theta);
k_sum += tmp1 + tmp3;
}
}
}
k_second += tangential_traction[alpha] * k_sum;
}
stiffness += k_main * nodal_area - k_second * nodal_area;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeSlipModuli(const ContactElement & element,
Matrix<Real> & stiffness) {
auto surface_dimension = spatial_dimension - 1;
auto & gaps = model.getGaps();
Real gap(gaps.begin()[element.slave]);
auto & nodal_areas = model.getNodalArea();
auto & nodal_area = nodal_areas.begin()[element.slave];
// compute normal traction
Real p_n = computeNormalTraction(gap);
auto & projections = model.getProjections();
Vector<Real> projection(projections.begin(surface_dimension)[element.slave]);
auto & normals = model.getNormals();
Vector<Real> normal(normals.begin(spatial_dimension)[element.slave]);
// restructure normal as a matrix for an outer product
Matrix<Real> mat_n(normal.storage(), normal.size(), 1.);
// method from Schweizerhof and A. Konyukhov, K. Schweizerhof
// DOI 10.1007/s00466-004-0616-7 and DOI 10.1007/s00466-003-0515-3
// construct A matrix
const ElementType & type = element.master.type;
auto && shapes = ElementClassHelper<_ek_regular>::getN(projection, type);
UInt nb_nodes_per_contact = element.getNbNodes();
Matrix<Real> A(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
for (auto i : arange(nb_nodes_per_contact)) {
for (auto j : arange(spatial_dimension)) {
if (i == 0) {
A(j, i * spatial_dimension + j) = 1;
continue;
}
A(j, i * spatial_dimension + j) = -shapes[i - 1];
}
}
// computing shape derivatives
auto && shape_derivatives =
ElementClassHelper<_ek_regular>::getDNDS(projection, type);
Matrix<Real> Aj(spatial_dimension, spatial_dimension * nb_nodes_per_contact);
auto construct_Aj = [&](auto && dnds) {
for (auto i : arange(nb_nodes_per_contact)) {
for (auto j : arange(spatial_dimension)) {
if (i == 0) {
Aj(j, i * spatial_dimension + j) = 0;
continue;
}
Aj(j, i * spatial_dimension + j) = dnds(i - 1);
}
}
};
// tangents should have been calculated in normal modulii
auto & tangents = model.getTangents();
Matrix<Real> covariant_basis(
tangents.begin(surface_dimension, spatial_dimension)[element.slave]);
auto & tangential_tractions = model.getTangentialTractions();
Vector<Real> tangential_traction(
tangential_tractions.begin(surface_dimension)[element.slave]);
// compute norm of trial traction
Real traction_norm = 0;
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
for (auto i : arange(surface_dimension)) {
for (auto j : arange(surface_dimension)) {
traction_norm += tangential_traction[i] * tangential_traction[j] *
contravariant_metric_tensor(i, j);
}
}
traction_norm = sqrt(traction_norm);
// construct four parts of stick modulii (eq 107,107a-c)
Matrix<Real> k_first(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
Matrix<Real> k_second(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
Matrix<Real> k_third(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
Matrix<Real> k_fourth(nb_nodes_per_contact * spatial_dimension,
nb_nodes_per_contact * spatial_dimension);
for (auto && values1 : enumerate(covariant_basis.transpose())) {
auto & alpha = std::get<0>(values1);
auto & tangent_alpha = std::get<1>(values1);
Matrix<Real> mat_t_alpha(tangent_alpha.storage(), tangent_alpha.size(), 1.);
Matrix<Real> t_outer_n(spatial_dimension, spatial_dimension);
Matrix<Real> t_outer_t(spatial_dimension, spatial_dimension);
for (auto && values2 :
zip(arange(surface_dimension), covariant_basis.transpose(),
shape_derivatives.transpose())) {
auto & beta = std::get<0>(values2);
auto & tangent_beta = std::get<1>(values2);
auto & dnds = std::get<2>(values2);
// construct Aj from shape function wrt to jth natural
// coordinate
construct_Aj(dnds);
// eq 107
Matrix<Real> mat_t_beta(tangent_beta.storage(), tangent_beta.size(), 1.);
t_outer_n.mul<false, true>(mat_t_beta, mat_n);
Matrix<Real> tmp(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
Matrix<Real> tmp1(nb_nodes_per_contact * spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
tmp.mul<false, false>(t_outer_n, A);
tmp1.mul<true, false>(A, tmp);
tmp1 *= epsilon_n * mu * tangential_traction[alpha] *
contravariant_metric_tensor(alpha, beta);
tmp1 /= traction_norm;
k_first += tmp1 * nodal_area;
// eq 107a
t_outer_t.mul<false, true>(mat_t_alpha, mat_t_beta);
tmp.mul<false, false>(t_outer_t, A);
tmp1.mul<true, false>(A, tmp);
tmp1 *= epsilon_t * mu * p_n * contravariant_metric_tensor(alpha, beta);
tmp1 /= traction_norm;
k_second += tmp1 * nodal_area;
for (auto && values3 : enumerate(covariant_basis.transpose())) {
auto & gamma = std::get<0>(values3);
auto & tangent_gamma = std::get<1>(values3);
Matrix<Real> mat_t_gamma(tangent_gamma.storage(), tangent_gamma.size(),
1.);
for (auto && values4 : enumerate(covariant_basis.transpose())) {
auto & theta = std::get<0>(values4);
auto & tangent_theta = std::get<1>(values4);
Matrix<Real> mat_t_theta(tangent_theta.storage(),
tangent_theta.size(), 1.);
t_outer_t.mul<false, true>(mat_t_gamma, mat_t_theta);
// eq 107b
tmp.mul<false, false>(t_outer_t, A);
tmp1.mul<true, false>(A, tmp);
tmp1 *= epsilon_t * mu * p_n * tangential_traction[alpha] *
tangential_traction[beta];
tmp1 *= contravariant_metric_tensor(alpha, gamma) *
contravariant_metric_tensor(beta, theta);
tmp1 /= pow(traction_norm, 3);
k_third += tmp1 * nodal_area;
// eq 107c
tmp.mul<false, false>(t_outer_t, Aj);
tmp1.mul<true, false>(A, tmp);
tmp1 *= contravariant_metric_tensor(alpha, theta) *
contravariant_metric_tensor(beta, gamma);
tmp1 *= mu * p_n * tangential_traction[alpha];
tmp1 /= traction_norm;
Matrix<Real> tmp2(spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
Matrix<Real> tmp3(nb_nodes_per_contact * spatial_dimension,
spatial_dimension * nb_nodes_per_contact);
tmp2.mul<false, false>(t_outer_t, A);
tmp3.mul<true, false>(Aj, tmp2);
tmp3 *= contravariant_metric_tensor(alpha, gamma) *
contravariant_metric_tensor(beta, theta);
tmp3 *= mu * p_n * tangential_traction[alpha];
tmp3 /= traction_norm;
k_fourth += (tmp1 + tmp3) * nodal_area;
}
}
}
}
stiffness += k_third + k_fourth - k_first - k_second;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::beforeSolveStep() {}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::afterSolveStep(__attribute__((unused)) bool converged) {
}
INSTANTIATE_RESOLUTION(penalty_linear, ResolutionPenalty);
} // namespace akantu
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