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rAKA/src/fe_engine/element_classes/element_class_quadrangle_8_inline_impl.cc
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element_class_quadrangle_8_inline_impl.cc
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/**
* @file element_class_quadrangle_8_inline_impl.cc
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Wed May 18 2011
* @date last modification: Tue Apr 07 2015
*
* @brief Specialization of the ElementClass for the _quadrangle_8
*
* @section LICENSE
*
* Copyright (©) 2010-2012, 2014, 2015 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
^
|
(-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}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_quadrangle_8, _gt_quadrangle_8, _itp_serendip_quadrangle_8, _ek_regular, 2,
_git_segment, 3);
AKANTU_DEFINE_SHAPE(_gt_quadrangle_8, _gst_square);
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type>
inline void
InterpolationElement<_itp_serendip_quadrangle_8>::computeShapes(const vector_type & c,
vector_type & N) {
/// Natural coordinates
const Real xi = c(0);
const Real eta = c(1);
N(0) = .25 * (1 - xi) * (1 - eta) * (- 1 - xi - eta);
N(1) = .25 * (1 + xi) * (1 - eta) * (- 1 + xi - eta);
N(2) = .25 * (1 + xi) * (1 + eta) * (- 1 + xi + eta);
N(3) = .25 * (1 - xi) * (1 + eta) * (- 1 - xi + eta);
N(4) = .5 * (1 - xi * xi) * (1 - eta );
N(5) = .5 * (1 + xi ) * (1 - eta * eta);
N(6) = .5 * (1 - xi * xi) * (1 + eta );
N(7) = .5 * (1 - xi ) * (1 - eta * eta);
}
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type, class matrix_type>
inline void
InterpolationElement<_itp_serendip_quadrangle_8>::computeDNDS(const vector_type & c,
matrix_type & dnds) {
const Real xi = c(0);
const Real eta = c(1);
/// dN/dxi
dnds(0, 0) = .25 * (1 - eta) * (2 * xi + eta);
dnds(0, 1) = .25 * (1 - eta) * (2 * xi - eta);
dnds(0, 2) = .25 * (1 + eta) * (2 * xi + eta);
dnds(0, 3) = .25 * (1 + eta) * (2 * xi - eta);
dnds(0, 4) = - xi * (1 - eta);
dnds(0, 5) = .5 * (1 - eta * eta);
dnds(0, 6) = - xi * (1 + eta);
dnds(0, 7) = - .5 * (1 - eta * eta);
/// dN/deta
dnds(1, 0) = .25 * (1 - xi) * (2 * eta + xi);
dnds(1, 1) = .25 * (1 + xi) * (2 * eta - xi);
dnds(1, 2) = .25 * (1 + xi) * (2 * eta + xi);
dnds(1, 3) = .25 * (1 - xi) * (2 * eta - xi);
dnds(1, 4) = - .5 * (1 - xi * xi);
dnds(1, 5) = - eta * (1 + xi);
dnds(1, 6) = .5 * (1 - xi * xi);
dnds(1, 7) = - eta * (1 - xi);
}
/* -------------------------------------------------------------------------- */
template<>
inline Real
GeometricalElement<_gt_quadrangle_8>::getInradius(const Matrix<Real> & coord) {
Real a, b, h;
Vector<Real> u0 = coord(0);
Vector<Real> u1 = coord(1);
Vector<Real> u2 = coord(2);
Vector<Real> u3 = coord(3);
Vector<Real> u4 = coord(4);
Vector<Real> u5 = coord(5);
Vector<Real> u6 = coord(6);
Vector<Real> u7 = coord(7);
a = u0.distance(u4);
b = u4.distance(u1);
h = std::min(a, b);
a = u1.distance(u5);
b = u5.distance(u2);
h = std::min(h, std::min(a, b));
a = u2.distance(u6);
b = u6.distance(u3);
h = std::min(h, std::min(a, b));
a = u3.distance(u7);
b = u7.distance(u0);
h = std::min(h, std::min(a, b));
return h;
}
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_serendip_quadrangle_8>::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();
}
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