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element_class_tetrahedron_10.cc
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element_class_tetrahedron_10.cc
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/**
* @file element_class_tetrahedron_10.cc
* @author Peter Spijker <peter.spijker@epfl.ch>
* @date Thu Dec 1 10:28:28 2010
*
* @brief Specialization of the element_class class for the type _tetrahedron_10
*
* @section LICENSE
*
* Copyright (©) 2010-2011 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]
*/
/* -------------------------------------------------------------------------- */
template<> UInt ElementClass<_tetrahedron_10>::nb_nodes_per_element;
template<> UInt ElementClass<_tetrahedron_10>::nb_quadrature_points;
template<> UInt ElementClass<_tetrahedron_10>::spatial_dimension;
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_tetrahedron_10>::computeShapes(const Real * natural_coords,
Real * shapes){
/// 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
shapes[0] = c0 * c1;
shapes[1] = xi * c2;
shapes[2] = eta * c3;
shapes[3] = zeta * c4;
shapes[4] = 4 * xi * c0;
shapes[5] = 4 * xi * eta;
shapes[6] = 4 * eta * c0;
shapes[7] = 4 * zeta * c0;
shapes[8] = 4 * xi * zeta;
shapes[9] = 4 * eta * zeta;
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_tetrahedron_10>::computeDNDS(__attribute__ ((unused)) const Real * natural_coords,
Real * 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;
/// dN/dxi
dnds[0] = 4 * sum - 3;
dnds[1] = 4 * xi - 1;
dnds[2] = 0;
dnds[3] = 0;
dnds[4] = 4 * (1 - sum - xi);
dnds[5] = 4 * eta;
dnds[6] = -4 * eta;
dnds[7] = -4 * zeta;
dnds[8] = 4 * zeta;
dnds[9] = 0;
/// dN/deta
dnds[10] = 4 * sum - 3;
dnds[11] = 0;
dnds[12] = 4 * eta - 1;
dnds[13] = 0;
dnds[14] = -4 * xi;
dnds[15] = 4 * xi;
dnds[16] = 4 * (1 - sum - eta);
dnds[17] = -4 * zeta;
dnds[18] = 0;
dnds[19] = 4 * zeta;
/// dN/dzeta
dnds[20] = 4 * sum - 3;
dnds[21] = 0;
dnds[22] = 0;
dnds[23] = 4 * zeta - 1;
dnds[24] = -4 * xi;
dnds[25] = 0;
dnds[26] = -4 * eta;
dnds[27] = 4 * (1 - sum - zeta);
dnds[28] = 4 * xi;
dnds[29] = 4 * eta;
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_tetrahedron_10>::computeJacobian(const Real * dxds,
const UInt dimension,
Real & jac) {
if (dimension == spatial_dimension){
Real weight = 1./24.;
Real det_dxds = Math::det3(dxds);
jac = det_dxds * weight;
}
else {
AKANTU_DEBUG_ERROR("to be implemented");
}
}
/* -------------------------------------------------------------------------- */
template<> inline Real ElementClass<_tetrahedron_10>::getInradius(const 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] * spatial_dimension,
coord + tetrahedra[t][1] * spatial_dimension,
coord + tetrahedra[t][2] * spatial_dimension,
coord + tetrahedra[t][3] * spatial_dimension);
inradius = ir < inradius ? ir : inradius;
}
return inradius;
}

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