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rAKA/src/fe_engine/element_classes/element_class_triangle_6_inline_impl.cc
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element_class_triangle_6_inline_impl.cc
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/**
* @file element_class_triangle_6_inline_impl.cc
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Fri Jul 16 2010
* @date last modification: Wed Oct 11 2017
*
* @brief Specialization of the element_class class for the type _triangle_6
*
* @section LICENSE
*
* Copyright (©) 2010-2018 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 N2}{\partial \xi} = - 1 + 4 \xi
* & \frac{\partial N2}{\partial \eta} = 0 \\
* N3 = - \eta (1 - 2 \eta)
* & \frac{\partial N3}{\partial \xi} = 0
* & \frac{\partial N3}{\partial \eta} = - 1 + 4 \eta \\
* N4 = 4 \xi (1 - \xi - \eta)
* & \frac{\partial N4}{\partial \xi} = 4 (1 - 2 \xi - \eta)
* & \frac{\partial N4}{\partial \eta} = - 4 \xi \\
* N5 = 4 \xi \eta
* & \frac{\partial N5}{\partial \xi} = 4 \eta
* & \frac{\partial N5}{\partial \eta} = 4 \xi \\
* N6 = 4 \eta (1 - \xi - \eta)
* & \frac{\partial N6}{\partial \xi} = - 4 \eta
* & \frac{\partial N6}{\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}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_triangle_6, _gt_triangle_6,
_itp_lagrange_triangle_6, _ek_regular, 2,
_git_triangle, 2);
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type>
inline void InterpolationElement<_itp_lagrange_triangle_6>::computeShapes(
const vector_type & natural_coords, vector_type & N) {
/// 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$
N(0) = c0 * (2 * c0 - 1.);
N(1) = c1 * (2 * c1 - 1.);
N(2) = c2 * (2 * c2 - 1.);
N(3) = 4 * c0 * c1;
N(4) = 4 * c1 * c2;
N(5) = 4 * c2 * c0;
}
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type, class matrix_type>
inline void InterpolationElement<_itp_lagrange_triangle_6>::computeDNDS(
const vector_type & natural_coords, matrix_type & 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, 0) = 1 - 4 * c0;
dnds(0, 1) = 4 * c1 - 1.;
dnds(0, 2) = 0.;
dnds(0, 3) = 4 * (c0 - c1);
dnds(0, 4) = 4 * c2;
dnds(0, 5) = -4 * c2;
dnds(1, 0) = 1 - 4 * c0;
dnds(1, 1) = 0.;
dnds(1, 2) = 4 * c2 - 1.;
dnds(1, 3) = -4 * c1;
dnds(1, 4) = 4 * c1;
dnds(1, 5) = 4 * (c0 - c2);
}
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_lagrange_triangle_6>::computeSpecialJacobian(
const Matrix<Real> & J, Real & jac) {
Vector<Real> vprod(J.cols());
Matrix<Real> Jt(J.transpose(), true);
vprod.crossProduct(Jt(0), Jt(1));
jac = vprod.norm();
}
/* -------------------------------------------------------------------------- */
template <>
inline Real
GeometricalElement<_gt_triangle_6>::getInradius(const Matrix<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]).storage(),
coord(triangles[t][1]).storage(),
coord(triangles[t][2]).storage());
inradius = std::min(ir, inradius);
}
return 2. * inradius;
}
/* -------------------------------------------------------------------------- */
// template<> inline bool ElementClass<_triangle_6>::contains(const Vector<Real>
// & natural_coords) {
// return ElementClass<_triangle_3>::contains(natural_coords);
// }
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