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element_class_quadrangle_8_inline_impl.hh
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element_class_quadrangle_8_inline_impl.hh
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/**
* Copyright (©) 2011-2023 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/>.
*/
/**
* @verbatim
\eta
^
|
(-1,1) (0,1) (1,1)
x-------x-------x
| | |
| | |
| | |
(-1,0)| | |(1,0)
----x---------------X-----> \xi
| | |
| | |
| | |
| | |
x-------x-------x
(-1,-1) (0,-1) (1,-1)
|
@endverbatim
*
* @f[
* \begin{array}{lll}
* N1 = (1 - \xi) (1 - \eta)(- 1 - \xi - \eta) / 4
* & \frac{\partial N1}{\partial \xi} = (1 - \eta)(2 \xi + \eta) / 4
* & \frac{\partial N1}{\partial \eta} = (1 - \xi)(\xi + 2 \eta) / 4 \\
* N2 = (1 + \xi) (1 - \eta)(- 1 + \xi - \eta) / 4 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta)(2 \xi - \eta) / 4
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi)(\xi - 2 \eta) / 4 \\
* N3 = (1 + \xi) (1 + \eta)(- 1 + \xi + \eta) / 4 \\
* & \frac{\partial N3}{\partial \xi} = (1 + \eta)(2 \xi + \eta) / 4
* & \frac{\partial N3}{\partial \eta} = (1 + \xi)(\xi + 2 \eta) / 4 \\
* N4 = (1 - \xi) (1 + \eta)(- 1 - \xi + \eta) / 4
* & \frac{\partial N4}{\partial \xi} = (1 + \eta)(2 \xi - \eta) / 4
* & \frac{\partial N4}{\partial \eta} = - (1 - \xi)(\xi - 2 \eta) / 4 \\
* N5 = (1 - \xi^2) (1 - \eta) / 2
* & \frac{\partial N1}{\partial \xi} = - \xi (1 - \eta)
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi^2) / 2 \\
* N6 = (1 + \xi) (1 - \eta^2) / 2 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta^2) / 2
* & \frac{\partial N2}{\partial \eta} = - \eta (1 + \xi) \\
* N7 = (1 - \xi^2) (1 + \eta) / 2 \\
* & \frac{\partial N3}{\partial \xi} = - \xi (1 + \eta)
* & \frac{\partial N3}{\partial \eta} = (1 - \xi^2) / 2 \\
* N8 = (1 - \xi) (1 - \eta^2) / 2
* & \frac{\partial N4}{\partial \xi} = - (1 - \eta^2) / 2
* & \frac{\partial N4}{\partial \eta} = - \eta (1 - \xi) \\
* \end{array}
* @f]
*
* @f{eqnarray*}{
* \xi_{q0} &=& 0 \qquad \eta_{q0} = 0
* @f}
*/
/* -------------------------------------------------------------------------- */
#include "element_class.hh"
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_quadrangle_8, _gt_quadrangle_8,
_itp_serendip_quadrangle_8, _ek_regular, 2,
_git_segment, 3);
/* -------------------------------------------------------------------------- */
template <>
template <class D1, class D2,
aka::enable_if_t<aka::are_vectors<D1, D2>::value> *>
inline void InterpolationElement<_itp_serendip_quadrangle_8>::computeShapes(
const Eigen::MatrixBase<D1> &c, Eigen::MatrixBase<D2> &N) {
/// Natural coordinates
const Real xi = c(0);
const Real eta = c(1);
N(0) = .25 * (1 - xi) * (1 - eta) * (-1 - xi - eta);
N(1) = .25 * (1 + xi) * (1 - eta) * (-1 + xi - eta);
N(2) = .25 * (1 + xi) * (1 + eta) * (-1 + xi + eta);
N(3) = .25 * (1 - xi) * (1 + eta) * (-1 - xi + eta);
N(4) = .5 * (1 - xi * xi) * (1 - eta);
N(5) = .5 * (1 + xi) * (1 - eta * eta);
N(6) = .5 * (1 - xi * xi) * (1 + eta);
N(7) = .5 * (1 - xi) * (1 - eta * eta);
}
/* -------------------------------------------------------------------------- */
template <>
template <class D1, class D2>
inline void InterpolationElement<_itp_serendip_quadrangle_8>::computeDNDS(
const Eigen::MatrixBase<D1> &c, Eigen::MatrixBase<D2> &dnds) {
const Real xi = c(0);
const Real eta = c(1);
/// dN/dxi
dnds(0, 0) = .25 * (1 - eta) * (2 * xi + eta);
dnds(0, 1) = .25 * (1 - eta) * (2 * xi - eta);
dnds(0, 2) = .25 * (1 + eta) * (2 * xi + eta);
dnds(0, 3) = .25 * (1 + eta) * (2 * xi - eta);
dnds(0, 4) = -xi * (1 - eta);
dnds(0, 5) = .5 * (1 - eta * eta);
dnds(0, 6) = -xi * (1 + eta);
dnds(0, 7) = -.5 * (1 - eta * eta);
/// dN/deta
dnds(1, 0) = .25 * (1 - xi) * (2 * eta + xi);
dnds(1, 1) = .25 * (1 + xi) * (2 * eta - xi);
dnds(1, 2) = .25 * (1 + xi) * (2 * eta + xi);
dnds(1, 3) = .25 * (1 - xi) * (2 * eta - xi);
dnds(1, 4) = -.5 * (1 - xi * xi);
dnds(1, 5) = -eta * (1 + xi);
dnds(1, 6) = .5 * (1 - xi * xi);
dnds(1, 7) = -eta * (1 - xi);
}
/* -------------------------------------------------------------------------- */
template <>
template <class D>
inline Real GeometricalElement<_gt_quadrangle_8>::getInradius(
const Eigen::MatrixBase<D> &coord) {
auto &&u0 = coord.col(0);
auto &&u1 = coord.col(1);
auto &&u2 = coord.col(2);
auto &&u3 = coord.col(3);
auto &&u4 = coord.col(4);
auto &&u5 = coord.col(5);
auto &&u6 = coord.col(6);
auto &&u7 = coord.col(7);
Real a = (u0 - u4).norm();
Real b = (u4 - u1).norm();
Real h = std::min(a, b);
a = (u1 - u5).norm();
b = (u5 - u2).norm();
h = std::min(h, std::min(a, b));
a = (u2 - u6).norm();
b = (u6 - u3).norm();
h = std::min(h, std::min(a, b));
a = (u3 - u7).norm();
b = (u7 - u0).norm();
h = std::min(h, std::min(a, b));
return h;
}
} // namespace akantu

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