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element_class_hexahedron_8_inline_impl.hh
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/**
* @file element_class_hexahedron_8_inline_impl.hh
*
* @author Guillaume Anciaux <guillaume.anciaux@epfl.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
* @author Peter Spijker <peter.spijker@epfl.ch>
*
* @date creation: Mon Mar 14 2011
* @date last modification: Fri Feb 07 2020
*
* @brief Specialization of the element_class class for the type _hexahedron_8
*
*
* @section LICENSE
*
* Copyright (©) 2010-2021 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/>.
*
*/
/**
* @verbatim
\zeta
^
(-1,1,1) | (1,1,1)
7---|------6
/| | /|
/ | | / |
(-1,-1,1) 4----------5 | (1,-1,1)
| | | | |
| | | | |
| | +---|-------> \xi
| | / | |
(-1,1,-1) | 3-/-----|--2 (1,1,-1)
| / / | /
|/ / |/
0-/--------1
(-1,-1,-1) / (1,-1,-1)
/
\eta
@endverbatim
*
* \f[
* \begin{array}{llll}
* N1 = (1 - \xi) (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N1}{\partial \xi} = - (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi) (1 - \zeta) / 8
* & \frac{\partial N1}{\partial \zeta} = - (1 - \xi) (1 - \eta) / 8 \\
* N2 = (1 + \xi) (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N2}{\partial \xi} = (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi) (1 - \zeta) / 8
* & \frac{\partial N2}{\partial \zeta} = - (1 + \xi) (1 - \eta) / 8 \\
* N3 = (1 + \xi) (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N3}{\partial \xi} = (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N3}{\partial \eta} = (1 + \xi) (1 - \zeta) / 8
* & \frac{\partial N3}{\partial \zeta} = - (1 + \xi) (1 + \eta) / 8 \\
* N4 = (1 - \xi) (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N4}{\partial \xi} = - (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N4}{\partial \eta} = (1 - \xi) (1 - \zeta) / 8
* & \frac{\partial N4}{\partial \zeta} = - (1 - \xi) (1 + \eta) / 8 \\
* N5 = (1 - \xi) (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N5}{\partial \xi} = - (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N5}{\partial \eta} = - (1 - \xi) (1 + \zeta) / 8
* & \frac{\partial N5}{\partial \zeta} = (1 - \xi) (1 - \eta) / 8 \\
* N6 = (1 + \xi) (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N6}{\partial \xi} = (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N6}{\partial \eta} = - (1 + \xi) (1 + \zeta) / 8
* & \frac{\partial N6}{\partial \zeta} = (1 + \xi) (1 - \eta) / 8 \\
* N7 = (1 + \xi) (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N7}{\partial \xi} = (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N7}{\partial \eta} = (1 + \xi) (1 + \zeta) / 8
* & \frac{\partial N7}{\partial \zeta} = (1 + \xi) (1 + \eta) / 8 \\
* N8 = (1 - \xi) (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N8}{\partial \xi} = - (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N8}{\partial \eta} = (1 - \xi) (1 + \zeta) / 8
* & \frac{\partial N8}{\partial \zeta} = (1 - \xi) (1 + \eta) / 8 \\
* \end{array}
* \f]
*
* @f{eqnarray*}{
* \xi_{q0} &=& -1/\sqrt{3} \qquad \eta_{q0} = -1/\sqrt{3} \qquad \zeta_{q0} =
-1/\sqrt{3} \\
* \xi_{q1} &=& 1/\sqrt{3} \qquad \eta_{q1} = -1/\sqrt{3} \qquad \zeta_{q1} =
-1/\sqrt{3} \\
* \xi_{q2} &=& 1/\sqrt{3} \qquad \eta_{q2} = 1/\sqrt{3} \qquad \zeta_{q2} =
-1/\sqrt{3} \\
* \xi_{q3} &=& -1/\sqrt{3} \qquad \eta_{q3} = 1/\sqrt{3} \qquad \zeta_{q3} =
-1/\sqrt{3} \\
* \xi_{q4} &=& -1/\sqrt{3} \qquad \eta_{q4} = -1/\sqrt{3} \qquad \zeta_{q4} =
1/\sqrt{3} \\
* \xi_{q5} &=& 1/\sqrt{3} \qquad \eta_{q5} = -1/\sqrt{3} \qquad \zeta_{q5} =
1/\sqrt{3} \\
* \xi_{q6} &=& 1/\sqrt{3} \qquad \eta_{q6} = 1/\sqrt{3} \qquad \zeta_{q6} =
1/\sqrt{3} \\
* \xi_{q7} &=& -1/\sqrt{3} \qquad \eta_{q7} = 1/\sqrt{3} \qquad \zeta_{q7} =
1/\sqrt{3} \\
* @f}
*/
/* -------------------------------------------------------------------------- */
#include "element_class.hh"
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_hexahedron_8, _gt_hexahedron_8,
_itp_lagrange_hexahedron_8, _ek_regular, 3,
_git_segment, 2);
/* -------------------------------------------------------------------------- */
template <>
template <class D1, class D2,
aka::enable_if_t<aka::are_vectors<D1, D2>::value> *>
inline void InterpolationElement<_itp_lagrange_hexahedron_8>::computeShapes(
const Eigen::MatrixBase<D1> & c, Eigen::MatrixBase<D2> & N) {
/// Natural coordinates
N(0) = .125 * (1 - c(0)) * (1 - c(1)) * (1 - c(2)); /// N1(q_0)
N(1) = .125 * (1 + c(0)) * (1 - c(1)) * (1 - c(2)); /// N2(q_0)
N(2) = .125 * (1 + c(0)) * (1 + c(1)) * (1 - c(2)); /// N3(q_0)
N(3) = .125 * (1 - c(0)) * (1 + c(1)) * (1 - c(2)); /// N4(q_0)
N(4) = .125 * (1 - c(0)) * (1 - c(1)) * (1 + c(2)); /// N5(q_0)
N(5) = .125 * (1 + c(0)) * (1 - c(1)) * (1 + c(2)); /// N6(q_0)
N(6) = .125 * (1 + c(0)) * (1 + c(1)) * (1 + c(2)); /// N7(q_0)
N(7) = .125 * (1 - c(0)) * (1 + c(1)) * (1 + c(2)); /// N8(q_0)
}
/* -------------------------------------------------------------------------- */
template <>
template <class D1, class D2>
inline void InterpolationElement<_itp_lagrange_hexahedron_8>::computeDNDS(
const Eigen::MatrixBase<D1> & c, Eigen::MatrixBase<D2> & dnds) {
/**
* @f[
* dnds = \left(
* \begin{array}{cccccccc}
* \frac{\partial N1}{\partial \xi} & \frac{\partial N2}{\partial
* \xi}
* & \frac{\partial N3}{\partial \xi} & \frac{\partial
* N4}{\partial \xi}
* & \frac{\partial N5}{\partial \xi} & \frac{\partial
* N6}{\partial \xi}
* & \frac{\partial N7}{\partial \xi} & \frac{\partial
* N8}{\partial \xi}\\
* \frac{\partial N1}{\partial \eta} & \frac{\partial N2}{\partial
* \eta}
* & \frac{\partial N3}{\partial \eta} & \frac{\partial
* N4}{\partial \eta}
* & \frac{\partial N5}{\partial \eta} & \frac{\partial
* N6}{\partial \eta}
* & \frac{\partial N7}{\partial \eta} & \frac{\partial
* N8}{\partial \eta}\\
* \frac{\partial N1}{\partial \zeta} & \frac{\partial N2}{\partial
* \zeta}
* & \frac{\partial N3}{\partial \zeta} & \frac{\partial
* N4}{\partial \zeta}
* & \frac{\partial N5}{\partial \zeta} & \frac{\partial
* N6}{\partial \zeta}
* & \frac{\partial N7}{\partial \zeta} & \frac{\partial
* N8}{\partial \zeta}
* \end{array}
* \right)
* @f]
*/
dnds(0, 0) = -.125 * (1 - c(1)) * (1 - c(2));
dnds(0, 1) = .125 * (1 - c(1)) * (1 - c(2));
dnds(0, 2) = .125 * (1 + c(1)) * (1 - c(2));
dnds(0, 3) = -.125 * (1 + c(1)) * (1 - c(2));
dnds(0, 4) = -.125 * (1 - c(1)) * (1 + c(2));
dnds(0, 5) = .125 * (1 - c(1)) * (1 + c(2));
dnds(0, 6) = .125 * (1 + c(1)) * (1 + c(2));
dnds(0, 7) = -.125 * (1 + c(1)) * (1 + c(2));
dnds(1, 0) = -.125 * (1 - c(0)) * (1 - c(2));
dnds(1, 1) = -.125 * (1 + c(0)) * (1 - c(2));
dnds(1, 2) = .125 * (1 + c(0)) * (1 - c(2));
dnds(1, 3) = .125 * (1 - c(0)) * (1 - c(2));
dnds(1, 4) = -.125 * (1 - c(0)) * (1 + c(2));
dnds(1, 5) = -.125 * (1 + c(0)) * (1 + c(2));
dnds(1, 6) = .125 * (1 + c(0)) * (1 + c(2));
dnds(1, 7) = .125 * (1 - c(0)) * (1 + c(2));
dnds(2, 0) = -.125 * (1 - c(0)) * (1 - c(1));
dnds(2, 1) = -.125 * (1 + c(0)) * (1 - c(1));
dnds(2, 2) = -.125 * (1 + c(0)) * (1 + c(1));
dnds(2, 3) = -.125 * (1 - c(0)) * (1 + c(1));
dnds(2, 4) = .125 * (1 - c(0)) * (1 - c(1));
dnds(2, 5) = .125 * (1 + c(0)) * (1 - c(1));
dnds(2, 6) = .125 * (1 + c(0)) * (1 + c(1));
dnds(2, 7) = .125 * (1 - c(0)) * (1 + c(1));
}
/* -------------------------------------------------------------------------- */
template <>
template <class D>
constexpr inline Real GeometricalElement<_gt_hexahedron_8>::getInradius(
const Eigen::MatrixBase<D> & X) {
auto && a = (X(0) - X(1)).norm();
auto && b = (X(1) - X(2)).norm();
auto && c = (X(2) - X(3)).norm();
auto && d = (X(3) - X(0)).norm();
auto && e = (X(0) - X(4)).norm();
auto && f = (X(1) - X(5)).norm();
auto && g = (X(2) - X(6)).norm();
auto && h = (X(3) - X(7)).norm();
auto && i = (X(4) - X(5)).norm();
auto && j = (X(5) - X(6)).norm();
auto && k = (X(6) - X(7)).norm();
auto && l = (X(7) - X(4)).norm();
auto p = std::min({a, b, c, d, e, f, g, h, i, j, k, l});
return p;
}
} // namespace akantu
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