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rAKA/src/model/contact_mechanics/resolutions/resolution_penalty_quadratic.cc
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resolution_penalty_quadratic.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_quadratic.hh"
#include "element_class_helper.hh"
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
ResolutionPenaltyQuadratic::ResolutionPenaltyQuadratic(
ContactMechanicsModel & model, const ID & id)
: Parent(model, id) {
AKANTU_DEBUG_IN();
this->initialize();
AKANTU_DEBUG_OUT();
}
/* -------------------------------------------------------------------------- */
Real ResolutionPenaltyQuadratic::computeNormalTraction(const Real & gap) const {
return epsilon_n * (macaulay(gap) * macaulay(gap) + macaulay(gap));
}
/* -------------------------------------------------------------------------- */
void ResolutionPenaltyQuadratic::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) *
(2 * gap + 1) * nodal_area;
// construct the rotational part of the normal matrix
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() * n_outer_t * Aj) *
contravariant_metric_tensor(alpha, beta);
}
}
k_rot1 *= -epsilon_n * heaviside(gap) * (gap * gap + gap) * nodal_area;
k_rot2 *= -epsilon_n * heaviside(gap) * (gap * gap + gap) * nodal_area;
stiffness += k_main + k_rot1 + k_rot2;
}
INSTANTIATE_RESOLUTION(penalty_quadratic, ResolutionPenaltyQuadratic);
} // namespace akantu
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