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rAKA/src/fe_engine/element_classes/element_class_hexahedron_8_inline_impl.cc
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element_class_hexahedron_8_inline_impl.cc
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/**
* @file element_class_hexahedron_8_inline_impl.cc
*
* @author Peter Spijker <peter.spijker@epfl.ch>
*
* @date creation: Mon Mar 14 2011
* @date last modification: Fri Jun 13 2014
*
* @brief Specialization of the element_class class for the type _hexahedron_8
*
* @section LICENSE
*
* Copyright (©) 2014 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/>.
*
* @section DESCRIPTION
*
* @verbatim
\zeta
^
(-1,1,1) | (1,1,1)
8---|------7
/| | /|
/ | | / |
(-1,-1,1) 5----------6 | (1,-1,1)
| | | | |
| | | | |
| | +---|-------> \xi
| | / | |
(-1,1,-1) | 4-/-----|--3 (1,1,-1)
| / / | /
|/ / |/
1-/--------2
(-1,-1,-1) / (1,-1,-1)
/
\eta
@endverbatim
*
* @subsection shapes Shape functions
* @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]
*
* @subsection quad_points Position of quadrature points
* @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}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_hexahedron_8,
_gt_hexahedron_8,
_itp_lagrange_hexahedron_8,
_ek_regular,
3,
_git_segment, 2);
AKANTU_DEFINE_SHAPE(_gt_hexahedron_8, _gst_square);
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type>
inline void
InterpolationElement<_itp_lagrange_hexahedron_8>::computeShapes(const vector_type & c,
vector_type & 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 vector_type, class matrix_type>
inline void
InterpolationElement<_itp_lagrange_hexahedron_8>::computeDNDS(const vector_type & c,
matrix_type & 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<>
inline Real
GeometricalElement<_gt_hexahedron_8>::getInradius(const Matrix<Real> & coord) {
Vector<Real> u0 = coord(0);
Vector<Real> u1 = coord(1);
Vector<Real> u2 = coord(2);
Vector<Real> u3 = coord(3);
Vector<Real> u4 = coord(4);
Vector<Real> u5 = coord(5);
Vector<Real> u6 = coord(6);
Vector<Real> u7 = coord(7);
Real a = u0.distance(u1);
Real b = u1.distance(u2);
Real c = u2.distance(u3);
Real d = u3.distance(u0);
Real e = u0.distance(u4);
Real f = u1.distance(u5);
Real g = u2.distance(u6);
Real h = u3.distance(u7);
Real i = u4.distance(u5);
Real j = u5.distance(u6);
Real k = u6.distance(u7);
Real l = u7.distance(u4);
Real x = std::min(a, std::min(b, std::min(c, d)));
Real y = std::min(e, std::min(f, std::min(g, h)));
Real z = std::min(i, std::min(j, std::min(k, l)));
Real p = std::min(x, std::min(y, z));
return p;
}
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