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element_class_quadrangle_8.cc
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/**
* @file element_class_quadrangle_8.cc
* @author Nicolas Richart <nicolas.richart@epfl.ch>
* @date Tue May 17 18:35:04 2011
*
* @brief Specialization of the ElementClass for the _quadrangle_8
*
* @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/>.
*
* @verbatim
\eta
^
|
(-1,1) (0,1) (1,1)
x-------x-------x
| | |
| | |
| | |
(-1,0)| | |(1,0)
----x---------------X-----> \xi
| | |
| | |
| | |
| | |
x-------x-------x
(-1,-1) (0,-1) (1,-1)
|
@endverbatim
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{lll}
* N1 = (1 - \xi) (1 - \eta)(- 1 - \xi - \eta) / 4
* & \frac{\partial N1}{\partial \xi} = (1 - \eta)(2 \xi + \eta) / 4
* & \frac{\partial N1}{\partial \eta} = (1 - \xi)(\xi + 2 \eta) / 4 \\
* N2 = (1 + \xi) (1 - \eta)(- 1 + \xi - \eta) / 4 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta)(2 \xi - \eta) / 4
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi)(\xi - 2 \eta) / 4 \\
* N3 = (1 + \xi) (1 + \eta)(- 1 + \xi + \eta) / 4 \\
* & \frac{\partial N3}{\partial \xi} = (1 + \eta)(2 \xi + \eta) / 4
* & \frac{\partial N3}{\partial \eta} = (1 + \xi)(\xi + 2 \eta) / 4 \\
* N4 = (1 - \xi) (1 + \eta)(- 1 - \xi + \eta) / 4
* & \frac{\partial N4}{\partial \xi} = (1 + \eta)(2 \xi - \eta) / 4
* & \frac{\partial N4}{\partial \eta} = - (1 - \xi)(\xi - 2 \eta) / 4 \\
* N5 = (1 - \xi^2) (1 - \eta) / 2
* & \frac{\partial N1}{\partial \xi} = - \xi (1 - \eta)
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi^2) / 2 \\
* N6 = (1 + \xi) (1 - \eta^2) / 2 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta^2) / 2
* & \frac{\partial N2}{\partial \eta} = - \eta (1 + \xi) \\
* N7 = (1 - \xi^2) (1 + \eta) / 2 \\
* & \frac{\partial N3}{\partial \xi} = - \xi (1 + \eta)
* & \frac{\partial N3}{\partial \eta} = (1 - \xi^2) / 2 \\
* N8 = (1 - \xi) (1 - \eta^2) / 2
* & \frac{\partial N4}{\partial \xi} = - (1 - \eta^2) / 2
* & \frac{\partial N4}{\partial \eta} = - \eta (1 - \xi) \\
* \end{array}
* @f]
*
* @subsection quad_points Position of quadrature points
* @f{eqnarray*}{
* \xi_{q0} &=& 0 \qquad \eta_{q0} = 0
* @f}
*/
/* -------------------------------------------------------------------------- */
template<> UInt ElementClass<_quadrangle_8>::nb_nodes_per_element;
template<> UInt ElementClass<_quadrangle_8>::nb_quadrature_points;
template<> UInt ElementClass<_quadrangle_8>::spatial_dimension;
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_quadrangle_8>::computeShapes(const Real * natural_coords,
Real * shapes) {
/// Natural coordinates
const Real xi = natural_coords[0];
const Real eta = natural_coords[1];
shapes[0] = .25 * (1 - xi) * (1 - eta) * (- 1 - xi - eta);
shapes[1] = .25 * (1 + xi) * (1 - eta) * (- 1 + xi - eta);
shapes[2] = .25 * (1 + xi) * (1 + eta) * (- 1 + xi + eta);
shapes[3] = .25 * (1 - xi) * (1 + eta) * (- 1 - xi + eta);
shapes[4] = .5 * (1 - xi * xi) * (1 - eta );
shapes[5] = .5 * (1 + xi ) * (1 - eta * eta);
shapes[6] = .5 * (1 - xi * xi) * (1 + eta );
shapes[7] = .5 * (1 - xi ) * (1 - eta * eta);
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_quadrangle_8>::computeDNDS(const Real * natural_coords,
Real * dnds) {
const Real xi = natural_coords[0];
const Real eta = natural_coords[1];
/// dN/dxi
dnds[0] = .25 * (1 - eta) * (2 * xi + eta);
dnds[1] = .25 * (1 - eta) * (2 * xi - eta);
dnds[2] = .25 * (1 + eta) * (2 * xi + eta);
dnds[3] = .25 * (1 + eta) * (2 * xi - eta);
dnds[4] = - xi * (1 - eta);
dnds[5] = .5 * (1 - eta * eta);
dnds[6] = - xi * (1 + eta);
dnds[7] = - .5 * (1 - eta * eta);
/// dN/deta
dnds[8] = .25 * (1 - xi) * (2 * eta + xi);
dnds[9] = .25 * (1 + xi) * (2 * eta - xi);
dnds[10] = .25 * (1 + xi) * (2 * eta + xi);
dnds[11] = .25 * (1 - xi) * (2 * eta - xi);
dnds[12] = - .5 * (1 - xi * xi);
dnds[13] = - eta * (1 + xi);
dnds[14] = .5 * (1 - xi * xi);
dnds[15] = - eta * (1 - xi);
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_quadrangle_8>::computeJacobian(const Real * dxds,
const UInt dimension,
Real & jac){
if (dimension == spatial_dimension){
Real det_dxds = Math::det2(dxds);
jac = det_dxds;
} else {
AKANTU_DEBUG_TO_IMPLEMENT();
}
}
/* -------------------------------------------------------------------------- */
template<> inline Real ElementClass<_quadrangle_8>::getInradius(const Real * coord) {
Real a, b, h;
a = Math::distance_2d(coord + 0*2, coord + 4*2);
b = Math::distance_2d(coord + 4*2, coord + 1*2);
h = std::min(a, b);
a = Math::distance_2d(coord + 1*2, coord + 5*2);
b = Math::distance_2d(coord + 5*2, coord + 2*2);
h = std::min(h, std::min(a, b));
a = Math::distance_2d(coord + 2*2, coord + 6*2);
b = Math::distance_2d(coord + 6*2, coord + 3*2);
h = std::min(h, std::min(a, b));
a = Math::distance_2d(coord + 3*2, coord + 7*2);
b = Math::distance_2d(coord + 7*2, coord + 0*2);
h = std::min(h, std::min(a, b));
return h;
}
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