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resolution_penalty.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 "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(const Real & gap) const {
return epsilon_n * macaulay(gap);
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeNormalForce(const ContactElement & element,
Vector<Real> & force) {
const auto & gaps = model.getGaps();
const auto & projections = model.getProjections();
const auto & normals = model.getNormals();
const auto & nodal_area = model.getNodalArea();
auto surface_dimension = spatial_dimension - 1;
auto gap = gaps(element.slave);
auto && normal = normals.begin(spatial_dimension)[element.slave];
auto && projection = projections.begin(surface_dimension)[element.slave];
// compute normal traction
auto p_n = computeNormalTraction(gap) * nodal_area[element.slave];
auto shape_matrix =
ResolutionUtils::computeShapeFunctionMatrix(element, projection);
force = p_n * shape_matrix.transpose() * normal;
}
/* -------------------------------------------------------------------------- */
void ResolutionPenalty::computeTangentialForce(const ContactElement & element,
Vector<Real> & force) {
force.zero();
if (mu == 0) {
return;
}
auto surface_dimension = spatial_dimension - 1;
// compute covariant basis
auto && projection =
model.getProjections().begin(surface_dimension)[element.slave];
auto && covariant_basis = model.getTangents().begin(
spatial_dimension, surface_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();
auto && tangential_traction =
tangential_tractions.begin(surface_dimension)[element.slave];
this->computeTangentialTraction(element, covariant_basis,
tangential_traction);
auto shape_matrix =
ResolutionUtils::computeShapeFunctionMatrix(element, projection);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
const auto & nodal_area = model.getNodalArea();
for (auto && [alpha, tangent_alpha] : enumerate(covariant_basis)) {
for (auto && [beta, traction_beta] : enumerate(tangential_traction)) {
force += (traction_beta * shape_matrix.transpose() * tangent_alpha) *
contravariant_metric_tensor(alpha, beta) *
nodal_area[element.slave];
}
}
}
/* -------------------------------------------------------------------------- */
template <typename D>
void ResolutionPenalty::computeTangentialTraction(
const ContactElement & element, const Matrix<Real> & covariant_basis,
Eigen::MatrixBase<D> & traction_tangential) {
Int surface_dimension = spatial_dimension - 1;
const auto & gap = model.getGaps()(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 = std::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);
}
}
/* -------------------------------------------------------------------------- */
template <typename D>
void ResolutionPenalty::computeTrialTangentialTraction(
const ContactElement & element, const Matrix<Real> & current_tangent,
Eigen::MatrixBase<D> & 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(current_tangent);
auto & previous_tangential_tractions = model.getPreviousTangentialTractions();
auto && previous_traction(
previous_tangential_tractions.begin(surface_dimension)[element.slave]);
auto & previous_tangents = model.getPreviousTangents();
auto && previous_tangent = previous_tangents.begin(
spatial_dimension, surface_dimension)[element.slave];
auto previous_contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(previous_tangent);
for (auto alpha : arange(surface_dimension)) {
for (auto gamma : arange(surface_dimension)) {
for (auto beta : arange(surface_dimension)) {
auto t_alpha_t_beta =
previous_tangent(beta).dot(current_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];
auto coords = model.getMesh().extractNodalValuesFromElement(
model.getContactDetector().getPositions(), previous_element);
auto previous_real_projection = GeometryUtils::realProjection(
coords, previous_element, previous_projection);
auto current_real_projection =
GeometryUtils::realProjection(coords, element.master, current_projection);
auto increment_real = current_real_projection - previous_real_projection;
Vector<Real> increment_xi(surface_dimension);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(current_tangent);
// 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;
}
}
traction -= epsilon_t * covariant_metric_tensor * increment_xi;
}
/* -------------------------------------------------------------------------- */
template <typename D1, typename D2>
void ResolutionPenalty::computeStickTangentialTraction(
const ContactElement & /*element*/, Eigen::MatrixBase<D1> & traction_trial,
Eigen::MatrixBase<D2> & traction_tangential) {
traction_tangential = traction_trial;
}
/* -------------------------------------------------------------------------- */
template <typename D1, typename D2>
void ResolutionPenalty::computeSlipTangentialTraction(
const ContactElement & element, const Matrix<Real> & covariant_basis,
Eigen::MatrixBase<D1> & traction_trial,
Eigen::MatrixBase<D2> & traction_tangential) {
UInt surface_dimension = spatial_dimension - 1;
auto & gap = model.getGaps()(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);
Real p_n = computeNormalTraction(gap);
traction_tangential = traction_trial / traction_trial_norm * 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
auto A = ResolutionUtils::computeShapeFunctionMatrix(element, projection);
// construct the main part of normal matrix
Matrix<Real> k_main(A.cols(), A.cols());
Matrix<Real> n_outer_n(spatial_dimension, spatial_dimension);
n_outer_n = normal * normal.transpose();
k_main =
(A.transpose() * n_outer_n * A) * epsilon_n * heaviside(gap) * nodal_area;
// construct the rotational part of the normal matrix
const auto & tangents = model.getTangents();
auto && covariant_basis(
tangents.begin(spatial_dimension, surface_dimension)[element.slave]);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
// consists of 2 rotational parts
Matrix<Real> k_rot1(A.cols(), A.cols());
Matrix<Real> k_rot2(A.cols(), A.cols());
k_rot1.zero();
k_rot2.zero();
auto Ajs = ResolutionUtils::computeDerivativeShapeFunctionMatrix(element,
projection);
for (auto && [alpha, tangent] : enumerate(covariant_basis)) {
auto n_outer_t = normal * tangent.transpose();
auto t_outer_n = tangent * normal.transpose();
for (auto && [beta, Aj] : enumerate(Ajs)) {
// construct Aj from shape function wrt to jth natural
// coordinate
k_rot1 += (Aj.transpose() * n_outer_t * A) *
contravariant_metric_tensor(alpha, beta);
k_rot2 += (A.transpose() * t_outer_n * Aj) *
contravariant_metric_tensor(alpha, beta);
}
}
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
auto A = ResolutionUtils::computeShapeFunctionMatrix(element, projection);
// computing shape derivatives
auto Ajs = ResolutionUtils::computeDerivativeShapeFunctionMatrix(element,
projection);
// tangents should have been calculated in normal modulii
auto & tangents = model.getTangents();
auto && covariant_basis(
tangents.begin(spatial_dimension, surface_dimension)[element.slave]);
auto contravariant_metric_tensor =
GeometryUtils::contravariantMetricTensor(covariant_basis);
// construct 1st part of the stick modulii
Matrix<Real> k_main(A.cols(), A.cols());
k_main.zero();
for (auto && [alpha, tangent_alpha] : enumerate(covariant_basis)) {
for (auto && [beta, tangent_beta] : enumerate(covariant_basis)) {
auto t_outer_t = tangent_alpha * tangent_beta.transpose();
k_main += (A.transpose() * t_outer_t * A) *
contravariant_metric_tensor(alpha, beta);
}
}
k_main *= -epsilon_t;
// construct 2nd part of the stick modulii
auto & tangential_tractions = model.getTangentialTractions();
auto && tangential_traction =
tangential_tractions.begin(surface_dimension)[element.slave];
Matrix<Real> k_second(A.cols(), A.cols());
k_second.zero();
Matrix<Real> k_sum(A.cols(), A.cols());
for (auto alpha : arange(surface_dimension)) {
k_sum.zero();
for (auto && [beta, Aj] : enumerate(Ajs)) {
for (auto && [gamma, tangent_gamma] : enumerate(covariant_basis)) {
Matrix<Real> t_outer_t(spatial_dimension, spatial_dimension);
for (auto && values3 : enumerate(covariant_basis.transpose())) {
auto & theta = std::get<0>(values3);
auto & tangent_theta = std::get<1>(values3);
t_outer_t = tangent_gamma * tangent_theta.transpose();
k_sum += (A.transpose() * t_outer_t * Aj) *
contravariant_metric_tensor(alpha, theta) *
contravariant_metric_tensor(beta, gamma) +
(Aj.transpose() * t_outer_t * A) *
contravariant_metric_tensor(alpha, gamma) *
contravariant_metric_tensor(beta, theta);
}
}
}
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]);
// 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
auto A = ResolutionUtils::computeShapeFunctionMatrix(element, projection);
// computing shape derivatives
auto Ajs = ResolutionUtils::computeDerivativeShapeFunctionMatrix(element,
projection);
// tangents should have been calculated in normal modulii
auto && covariant_basis = model.getTangents().begin(
spatial_dimension, surface_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(A.cols(), A.cols());
Matrix<Real> k_second(A.cols(), A.cols());
Matrix<Real> k_third(A.cols(), A.cols());
Matrix<Real> k_fourth(A.cols(), A.cols());
k_first.zero();
k_second.zero();
k_first.zero();
k_fourth.zero();
Matrix<Real> t_outer_n(spatial_dimension, spatial_dimension);
Matrix<Real> t_outer_t(spatial_dimension, spatial_dimension);
for (auto && [alpha, tangent_alpha] : enumerate(covariant_basis)) {
for (auto && [beta, tangent_beta, Aj] :
zip(arange(surface_dimension), covariant_basis, Ajs)) {
// eq 107
t_outer_n = tangent_beta * normal.transpose();
k_first += (A.transpose() * t_outer_n * A) * epsilon_n * mu *
tangential_traction[alpha] *
contravariant_metric_tensor(alpha, beta) / traction_norm *
nodal_area;
// eq 107a
t_outer_t = tangent_alpha * tangent_beta.transpose();
k_second += (A.transpose() * t_outer_t * A) * epsilon_t * mu * p_n *
contravariant_metric_tensor(alpha, beta) / traction_norm *
nodal_area;
for (auto && values3 : enumerate(covariant_basis)) {
auto & gamma = std::get<0>(values3);
auto & tangent_gamma = std::get<1>(values3);
for (auto && values4 : enumerate(covariant_basis)) {
auto & theta = std::get<0>(values4);
auto & tangent_theta = std::get<1>(values4);
t_outer_t = tangent_gamma * tangent_theta.transpose();
// eq 107b
k_third += (A.transpose() * t_outer_t * A) * epsilon_t * mu * p_n *
tangential_traction[alpha] * tangential_traction[beta] *
contravariant_metric_tensor(alpha, gamma) *
contravariant_metric_tensor(beta, theta) /
pow(traction_norm, 3) * nodal_area;
// eq 107c
k_fourth += ((A.transpose() * t_outer_t * Aj) *
contravariant_metric_tensor(alpha, theta) *
contravariant_metric_tensor(beta, gamma) +
(Aj.transpose() * t_outer_t * A) *
contravariant_metric_tensor(alpha, gamma) *
contravariant_metric_tensor(beta, theta)) *
nodal_area * mu * p_n * tangential_traction[alpha] /
traction_norm;
}
}
}
}
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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