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element_class_tetrahedron_10_inline_impl.cc
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/**
* @file element_class_tetrahedron_10_inline_impl.cc
*
* @author Peter Spijker <peter.spijker@epfl.ch>
*
* @date creation: Fri Jul 16 2010
* @date last modification: Sun Oct 19 2014
*
* @brief Specialization of the element_class class for the type _tetrahedron_10
*
* @section LICENSE
*
* Copyright (©) 2010-2012, 2014, 2015 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
^
|
(0,0,1)
x
|` .
| ` .
| ` .
| ` . (0,0.5,0.5)
| ` x.
| q4 o ` . \eta
| ` . -,
(0,0,0.5) x ` x (0.5,0,0.5) -
| ` x-(0,1,0)
| q3 o` - '
| (0,0.5,0) - ` '
| x- ` x (0.5,0.5,0)
| q1 o - o q2` '
| - ` '
| - ` '
x---------------x--------------` x-----> \xi
(0,0,0) (0.5,0,0) (1,0,0)
@endverbatim
*
* @subsection coords Nodes coordinates
*
* @f[
* \begin{array}{lll}
* \xi_{0} = 0 & \eta_{0} = 0 & \zeta_{0} = 0 \\
* \xi_{1} = 1 & \eta_{1} = 0 & \zeta_{1} = 0 \\
* \xi_{2} = 0 & \eta_{2} = 1 & \zeta_{2} = 0 \\
* \xi_{3} = 0 & \eta_{3} = 0 & \zeta_{3} = 1 \\
* \xi_{4} = 1/2 & \eta_{4} = 0 & \zeta_{4} = 0 \\
* \xi_{5} = 1/2 & \eta_{5} = 1/2 & \zeta_{5} = 0 \\
* \xi_{6} = 0 & \eta_{6} = 1/2 & \zeta_{6} = 0 \\
* \xi_{7} = 0 & \eta_{7} = 0 & \zeta_{7} = 1/2 \\
* \xi_{8} = 1/2 & \eta_{8} = 0 & \zeta_{8} = 1/2 \\
* \xi_{9} = 0 & \eta_{9} = 1/2 & \zeta_{9} = 1/2
* \end{array}
* @f]
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{llll}
* N1 = (1 - \xi - \eta - \zeta) (1 - 2 \xi - 2 \eta - 2 \zeta)
* & \frac{\partial N1}{\partial \xi} = 4 \xi + 4 \eta + 4 \zeta - 3
* & \frac{\partial N1}{\partial \eta} = 4 \xi + 4 \eta + 4 \zeta - 3
* & \frac{\partial N1}{\partial \zeta} = 4 \xi + 4 \eta + 4 \zeta - 3 \\
* N2 = \xi (2 \xi - 1)
* & \frac{\partial N2}{\partial \xi} = 4 \xi - 1
* & \frac{\partial N2}{\partial \eta} = 0
* & \frac{\partial N2}{\partial \zeta} = 0 \\
* N3 = \eta (2 \eta - 1)
* & \frac{\partial N3}{\partial \xi} = 0
* & \frac{\partial N3}{\partial \eta} = 4 \eta - 1
* & \frac{\partial N3}{\partial \zeta} = 0 \\
* N4 = \zeta (2 \zeta - 1)
* & \frac{\partial N4}{\partial \xi} = 0
* & \frac{\partial N4}{\partial \eta} = 0
* & \frac{\partial N4}{\partial \zeta} = 4 \zeta - 1 \\
* N5 = 4 \xi (1 - \xi - \eta - \zeta)
* & \frac{\partial N5}{\partial \xi} = 4 - 8 \xi - 4 \eta - 4 \zeta
* & \frac{\partial N5}{\partial \eta} = -4 \xi
* & \frac{\partial N5}{\partial \zeta} = -4 \xi \\
* N6 = 4 \xi \eta
* & \frac{\partial N6}{\partial \xi} = 4 \eta
* & \frac{\partial N6}{\partial \eta} = 4 \xi
* & \frac{\partial N6}{\partial \zeta} = 0 \\
* N7 = 4 \eta (1 - \xi - \eta - \zeta)
* & \frac{\partial N7}{\partial \xi} = -4 \eta
* & \frac{\partial N7}{\partial \eta} = 4 - 4 \xi - 8 \eta - 4 \zeta
* & \frac{\partial N7}{\partial \zeta} = -4 \eta \\
* N8 = 4 \zeta (1 - \xi - \eta - \zeta)
* & \frac{\partial N8}{\partial \xi} = -4 \zeta
* & \frac{\partial N8}{\partial \eta} = -4 \zeta
* & \frac{\partial N8}{\partial \zeta} = 4 - 4 \xi - 4 \eta - 8 \zeta \\
* N9 = 4 \zeta \xi
* & \frac{\partial N9}{\partial \xi} = 4 \zeta
* & \frac{\partial N9}{\partial \eta} = 0
* & \frac{\partial N9}{\partial \zeta} = 4 \xi \\
* N10 = 4 \eta \zeta
* & \frac{\partial N10}{\partial \xi} = 0
* & \frac{\partial N10}{\partial \eta} = 4 \zeta
* & \frac{\partial N10}{\partial \zeta} = 4 \eta \\
* \end{array}
* @f]
*
* @subsection quad_points Position of quadrature points
* @f[
* a = \frac{5 - \sqrt{5}}{20}\\
* b = \frac{5 + 3 \sqrt{5}}{20}
* \begin{array}{lll}
* \xi_{q_0} = a & \eta_{q_0} = a & \zeta_{q_0} = a \\
* \xi_{q_1} = b & \eta_{q_1} = a & \zeta_{q_1} = a \\
* \xi_{q_2} = a & \eta_{q_2} = b & \zeta_{q_2} = a \\
* \xi_{q_3} = a & \eta_{q_3} = a & \zeta_{q_3} = b
* \end{array}
* @f]
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_tetrahedron_10, _gt_tetrahedron_10, _itp_lagrange_tetrahedron_10, _ek_regular, 3,
_git_tetrahedron, 2);
AKANTU_DEFINE_SHAPE(_gt_tetrahedron_10, _gst_triangle);
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type>
inline void
InterpolationElement<_itp_lagrange_tetrahedron_10>::computeShapes(const vector_type & natural_coords,
vector_type & N) {
/// Natural coordinates
Real xi = natural_coords(0);
Real eta = natural_coords(1);
Real zeta = natural_coords(2);
Real sum = xi + eta + zeta;
Real c0 = 1 - sum;
Real c1 = 1 - 2*sum;
Real c2 = 2*xi - 1;
Real c3 = 2*eta - 1;
Real c4 = 2*zeta - 1;
/// Shape functions
N(0) = c0 * c1;
N(1) = xi * c2;
N(2) = eta * c3;
N(3) = zeta * c4;
N(4) = 4 * xi * c0;
N(5) = 4 * xi * eta;
N(6) = 4 * eta * c0;
N(7) = 4 * zeta * c0;
N(8) = 4 * xi * zeta;
N(9) = 4 * eta * zeta;
}
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type, class matrix_type>
inline void
InterpolationElement<_itp_lagrange_tetrahedron_10>::computeDNDS(const vector_type & natural_coords,
matrix_type & dnds) {
/**
* @f[
* dnds = \left(
* \begin{array}{cccccccccc}
* \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 N9}{\partial \xi} & \frac{\partial N10}{\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 N9}{\partial \eta} & \frac{\partial N10}{\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}
* & \frac{\partial N9}{\partial \zeta} & \frac{\partial N10}{\partial \zeta}
* \end{array}
* \right)
* @f]
*/
/// Natural coordinates
Real xi = natural_coords(0);
Real eta = natural_coords(1);
Real zeta = natural_coords(2);
Real sum = xi + eta + zeta;
/// \frac{\partial N_i}{\partial \xi}
dnds(0, 0) = 4 * sum - 3;
dnds(0, 1) = 4 * xi - 1;
dnds(0, 2) = 0;
dnds(0, 3) = 0;
dnds(0, 4) = 4 * (1 - sum - xi);
dnds(0, 5) = 4 * eta;
dnds(0, 6) = -4 * eta;
dnds(0, 7) = -4 * zeta;
dnds(0, 8) = 4 * zeta;
dnds(0, 9) = 0;
/// \frac{\partial N_i}{\partial \eta}
dnds(1, 0) = 4 * sum - 3;
dnds(1, 1) = 0;
dnds(1, 2) = 4 * eta - 1;
dnds(1, 3) = 0;
dnds(1, 4) = -4 * xi;
dnds(1, 5) = 4 * xi;
dnds(1, 6) = 4 * (1 - sum - eta);
dnds(1, 7) = -4 * zeta;
dnds(1, 8) = 0;
dnds(1, 9) = 4 * zeta;
/// \frac{\partial N_i}{\partial \zeta}
dnds(2, 0) = 4 * sum - 3;
dnds(2, 1) = 0;
dnds(2, 2) = 0;
dnds(2, 3) = 4 * zeta - 1;
dnds(2, 4) = -4 * xi;
dnds(2, 5) = 0;
dnds(2, 6) = -4 * eta;
dnds(2, 7) = 4 * (1 - sum - zeta);
dnds(2, 8) = 4 * xi;
dnds(2, 9) = 4 * eta;
}
/* -------------------------------------------------------------------------- */
template<>
inline Real
GeometricalElement<_gt_tetrahedron_10>::getInradius(const Matrix<Real> & coord) {
// Only take the four corner tetrahedra
UInt tetrahedra[4][4] = {
{0, 4, 6, 7},
{4, 1, 5, 8},
{6, 5, 2, 9},
{7, 8, 9, 3}
};
Real inradius = std::numeric_limits<Real>::max();
for (UInt t = 0; t < 4; t++) {
Real ir = Math::tetrahedron_inradius(coord(tetrahedra[t][0]).storage(),
coord(tetrahedra[t][1]).storage(),
coord(tetrahedra[t][2]).storage(),
coord(tetrahedra[t][3]).storage());
inradius = std::min(ir, inradius);
}
return inradius;
}
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