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rAKA/fem/element_classes/element_class_quadrangle_4.cc
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element_class_quadrangle_4.cc
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/**
* @file element_class_quadrangle_4.cc
* @author Nicolas Richart <nicolas.richart@epfl.ch>
* @date Wed Oct 27 17:27:44 2010
*
* @brief Specialization of the element_class class for the type _quadrangle_4
*
* @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) | (1,1)
x---------x
| | |
| | |
--|---------|-----> \xi
| | |
| | |
x---------x
(-1,-1) | (1,-1)
@endverbatim
*
* @subsection shapes Shape functions
* \begin{array}{lll}
* N1 = (1 - \xi) (1 - \eta) / 4
* & \frac{\partial N1}{\partial \xi} = - (1 - \eta) / 4
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi) / 4 \\
* N2 = (1 + \xi) (1 - \eta) / 4 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta) / 4
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi) / 4 \\
* N3 = (1 + \xi) (1 + \eta) / 4 \\
* & \frac{\partial N3}{\partial \xi} = (1 + \eta) / 4
* & \frac{\partial N3}{\partial \eta} = (1 + \xi) / 4 \\
* N4 = (1 - \xi) (1 + \eta) / 4
* & \frac{\partial N4}{\partial \xi} = - (1 + \eta) / 4
* & \frac{\partial N4}{\partial \eta} = (1 - \xi) / 4 \\
* \end{array}
* @f}
*
* @subsection quad_points Position of quadrature points
* @f{eqnarray*}{
* \xi_{q0} &=& 0 \qquad \eta_{q0} = 0
* @f}
*/
/* -------------------------------------------------------------------------- */
template<> UInt ElementClass<_quadrangle_4>::nb_nodes_per_element;
template<> UInt ElementClass<_quadrangle_4>::nb_quadrature_points;
template<> UInt ElementClass<_quadrangle_4>::spatial_dimension;
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_quadrangle_4>::computeShapes(const Real * natural_coords,
Real * shapes) {
/// Natural coordinates
const Real * c = natural_coords;
shapes[0] = .25 * (1 - c[0]) * (1 - c[1]); /// N1(q_0)
shapes[1] = .25 * (1 + c[0]) * (1 - c[1]); /// N2(q_0)
shapes[2] = .25 * (1 + c[0]) * (1 + c[1]); /// N3(q_0)
shapes[3] = .25 * (1 - c[0]) * (1 + c[1]); /// N4(q_0)
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_quadrangle_4>::computeDNDS(const Real * natural_coords,
Real * dnds) {
/**
* @f[
* dnds = \left(
* \begin{array}{cccc}
* \frac{\partial N1}{\partial \xi} & \frac{\partial N2}{\partial \xi}
* & \frac{\partial N3}{\partial \xi} & \frac{\partial N4}{\partial \xi}\\
* \frac{\partial N1}{\partial \eta} & \frac{\partial N2}{\partial \eta}
* & \frac{\partial N3}{\partial \eta} & \frac{\partial N4}{\partial \eta}
* \end{array}
* \right)
* @f]
*/
const Real * c = natural_coords;
dnds[0] = - .25 * (1 - c[1]);
dnds[1] = .25 * (1 - c[1]);
dnds[2] = .25 * (1 + c[1]);
dnds[3] = - .25 * (1 + c[1]);
dnds[4] = - .25 * (1 - c[0]);
dnds[5] = - .25 * (1 + c[0]);
dnds[6] = .25 * (1 + c[0]);
dnds[7] = .25 * (1 - c[0]);
}
/* -------------------------------------------------------------------------- */
template <> inline void ElementClass<_quadrangle_4>::computeJacobian(const Real * dxds,
const UInt dimension,
Real & jac){
Real weight = 4.;
if (dimension == spatial_dimension){
Real det_dxds = Math::det2(dxds);
jac = det_dxds * weight;
} else {
AKANTU_DEBUG_TO_IMPLEMENT();
}
}
/* -------------------------------------------------------------------------- */
template<> inline Real ElementClass<_quadrangle_4>::getInradius(const Real * coord) {
Real a = Math::distance_2d(coord + 0, coord + 2);
Real b = Math::distance_2d(coord + 2, coord + 4);
Real c = Math::distance_2d(coord + 4, coord + 6);
Real d = Math::distance_2d(coord + 6, coord + 0);
// Real septimetre = (a + b + c + d) / 2.;
// Real p = Math::distance_2d(coord + 0, coord + 4);
// Real q = Math::distance_2d(coord + 2, coord + 6);
// Real area = sqrt(4*(p*p * q*q) - (a*a + b*b + c*c + d*d)*(a*a + c*c - b*b - d*d)) / 4.;
// Real h = sqrt(area); // to get a length
// Real h = area / septimetre; // formula of inradius for circumscritable quadrelateral
Real h = std::min(a, std::min(b, std::min(c, d)));
return h;
}
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