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rAKA/src/fe_engine/element_classes/element_class_quadrangle_4_inline_impl.cc
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element_class_quadrangle_4_inline_impl.cc
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/**
* @file element_class_quadrangle_4_inline_impl.cc
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Mon Dec 13 2010
* @date last modification: Wed Oct 11 2017
*
* @brief Specialization of the element_class class for the type _quadrangle_4
*
* @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
^
(-1,1) | (1,1)
x---------x
| | |
| | |
--|---------|-----> \xi
| | |
| | |
x---------x
(-1,-1) | (1,-1)
@endverbatim
*
* @subsection shapes Shape functions
* @f[
* \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}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY(_quadrangle_4, _gt_quadrangle_4,
_itp_lagrange_quadrangle_4, _ek_regular, 2,
_git_segment, 2);
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type>
inline void InterpolationElement<_itp_lagrange_quadrangle_4>::computeShapes(
const vector_type & c, vector_type & N) {
N(0) = 1. / 4. * (1. - c(0)) * (1. - c(1)); /// N1(q_0)
N(1) = 1. / 4. * (1. + c(0)) * (1. - c(1)); /// N2(q_0)
N(2) = 1. / 4. * (1. + c(0)) * (1. + c(1)); /// N3(q_0)
N(3) = 1. / 4. * (1. - c(0)) * (1. + c(1)); /// N4(q_0)
}
/* -------------------------------------------------------------------------- */
template <>
template <class vector_type, class matrix_type>
inline void InterpolationElement<_itp_lagrange_quadrangle_4>::computeDNDS(
const vector_type & c, matrix_type & 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]
*/
dnds(0, 0) = -1. / 4. * (1. - c(1));
dnds(0, 1) = 1. / 4. * (1. - c(1));
dnds(0, 2) = 1. / 4. * (1. + c(1));
dnds(0, 3) = -1. / 4. * (1. + c(1));
dnds(1, 0) = -1. / 4. * (1. - c(0));
dnds(1, 1) = -1. / 4. * (1. + c(0));
dnds(1, 2) = 1. / 4. * (1. + c(0));
dnds(1, 3) = 1. / 4. * (1. - c(0));
}
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_lagrange_quadrangle_4>::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_quadrangle_4>::getInradius(const Matrix<Real> & coord) {
Vector<Real> u0 = coord(0);
Vector<Real> u1 = coord(1);
Vector<Real> u2 = coord(2);
Vector<Real> u3 = coord(3);
Real a = u0.distance(u1);
Real b = u1.distance(u2);
Real c = u2.distance(u3);
Real d = u3.distance(u0);
// 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, b, c, d});
return h;
}
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