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element_class_triangle_6.cc
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/**
* @file element_class_triangle_6.cc
* @author Nicolas Richart <nicolas.richart@epfl.ch>
* @date Thu Jul 15 10:28:28 2010
*
* @brief Specialization of the element_class class for the type _triangle_6
*
* @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
\eta
^
|
x 2
| `
| `
| . `
| q2 `
5 x x 4
| `
| `
| .q0 q1. `
| `
x---------x---------x-----> \xi
0 3 1
@endverbatim
*
* @subsection coords Nodes coordinates
*
* @f[
* \begin{array}{ll}
* \xi_{0} = 0 & \eta_{0} = 0 \\
* \xi_{1} = 1 & \eta_{1} = 0 \\
* \xi_{2} = 0 & \eta_{2} = 1 \\
* \xi_{3} = 1/2 & \eta_{3} = 0 \\
* \xi_{4} = 1/2 & \eta_{4} = 1/2 \\
* \xi_{5} = 0 & \eta_{5} = 1/2
* \end{array}
* @f]
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{lll}
* N1 = -(1 - \xi - \eta) (1 - 2 (1 - \xi - \eta))
* & \frac{\partial N1}{\partial \xi} = 1 - 4(1 - \xi - \eta)
* & \frac{\partial N1}{\partial \eta} = 1 - 4(1 - \xi - \eta) \\
* N2 = - \xi (1 - 2 \xi)
* & \frac{\partial N1}{\partial \xi} = - 1 + 4 \xi
* & \frac{\partial N1}{\partial \eta} = 0 \\
* N3 = - \eta (1 - 2 \eta)
* & \frac{\partial N1}{\partial \xi} = 0
* & \frac{\partial N1}{\partial \eta} = - 1 + 4 \eta \\
* N4 = 4 \xi (1 - \xi - \eta)
* & \frac{\partial N1}{\partial \xi} = 4 (1 - 2 \xi - \eta)
* & \frac{\partial N1}{\partial \eta} = - 4 \eta \\
* N5 = 4 \xi \eta
* & \frac{\partial N1}{\partial \xi} = 4 \xi
* & \frac{\partial N1}{\partial \eta} = 4 \eta \\
* N6 = 4 \eta (1 - \xi - \eta)
* & \frac{\partial N1}{\partial \xi} = - 4 \xi
* & \frac{\partial N1}{\partial \eta} = 4 (1 - \xi - 2 \eta)
* \end{array}
* @f]
*
* @subsection quad_points Position of quadrature points
* @f{eqnarray*}{
* \xi_{q0} &=& 1/6 \qquad \eta_{q0} = 1/6 \\
* \xi_{q1} &=& 2/3 \qquad \eta_{q1} = 1/6 \\
* \xi_{q2} &=& 1/6 \qquad \eta_{q2} = 2/3
* @f}
*/
// /// quadrature point position
// quad[0] = 1./6.; /// q0_{\xi}
// quad[1] = 1./6.; /// q0_{\eta}
// quad[2] = 2./3.; /// q1_{\xi}
// quad[3] = 1./6.; /// q1_{\eta}
// quad[4] = 1./6.; /// q2_{\xi}
// quad[5] = 2./3.; /// q2_{\eta}
/* -------------------------------------------------------------------------- */
template<> UInt ElementClass<_triangle_6>::nb_nodes_per_element;
template<> UInt ElementClass<_triangle_6>::nb_quadrature_points;
template<> UInt ElementClass<_triangle_6>::spatial_dimension;
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_triangle_6>::computeShapes(const Real * natural_coords,
Real * shapes){
/// Natural coordinates
Real c0 = 1 - natural_coords[0] - natural_coords[1]; /// @f$ c0 = 1 - \xi - \eta @f$
Real c1 = natural_coords[0]; /// @f$ c1 = \xi @f$
Real c2 = natural_coords[1]; /// @f$ c2 = \eta @f$
shapes[0] = c0 * (2 * c0 - 1.);
shapes[1] = c1 * (2 * c1 - 1.);
shapes[2] = c2 * (2 * c2 - 1.);
shapes[3] = 4 * c0 * c1;
shapes[4] = 4 * c1 * c2;
shapes[5] = 4 * c2 * c0;
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_triangle_6>::computeDNDS(const Real * natural_coords,
Real * dnds){
/**
* @f[
* dnds = \left(
* \begin{array}{cccccc}
* \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 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}
* \end{array}
* \right) @f]
*/
/// Natural coordinates
Real c0 = 1 - natural_coords[0] - natural_coords[1]; /// @f$ c0 = 1 - \xi - \eta @f$
Real c1 = natural_coords[0]; /// @f$ c1 = \xi @f$
Real c2 = natural_coords[1]; /// @f$ c2 = \eta @f$
dnds[0] = 1 - 4 * c0;
dnds[1] = 4 * c1 - 1.;
dnds[2] = 0.;
dnds[3] = 4 * (c0 - c1);
dnds[4] = 4 * c2;
dnds[5] = - 4 * c2;
dnds[6] = 1 - 4 * c0;
dnds[7] = 0.;
dnds[8] = 4 * c2 - 1.;
dnds[9] = - 4 * c1;
dnds[10] = 4 * c1;
dnds[11] = 4 * (c0 - c2);
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_triangle_6>::computeJacobian(const Real * dxds,
const UInt dimension,
Real & jac){
// if element dimension is the same as the space dimension
// then jacobian factor is the determinent of dxds
if (dimension == spatial_dimension){
Real weight = 1./6.;
jac = Math::det2(dxds)*weight;
AKANTU_DEBUG_ASSERT(jac > 0,
"Negative jacobian computed, possible problem in the element node order.");
}
else {
AKANTU_DEBUG_ERROR("to implement");
}
}
/* -------------------------------------------------------------------------- */
template<> inline Real ElementClass<_triangle_6>::getInradius(const Real * coord) {
UInt triangles[4][3] = {
{0, 3, 5},
{3, 1, 4},
{3, 4, 5},
{5, 4, 2}
};
Real inradius = std::numeric_limits<Real>::max();
for (UInt t = 0; t < 4; t++) {
Real ir = Math::triangle_inradius(coord + triangles[t][0] * spatial_dimension,
coord + triangles[t][1] * spatial_dimension,
coord + triangles[t][2] * spatial_dimension);
inradius = ir < inradius ? ir : inradius;
}
return inradius;
}
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