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rAKA/src/fe_engine/element_classes/element_class_hermite_inline_impl.cc
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element_class_hermite_inline_impl.cc
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/**
* @file element_class_hermite_inline_impl.cc
*
* @author Fabian Barras <fabian.barras@epfl.ch>
* @author Lucas Frerot <lucas.frerot@epfl.ch>
*
* @date creation: Fri Nov 10 2017
* @date last modification: Mon Feb 19 2018
*
* @brief Specialization of the element_class class for the type
* _hermite
*
* @section LICENSE
*
* Copyright (©) 2016-2018 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
--x-----q1----|----q2-----x---> x
-1 0 1
@endverbatim
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{ll}
* M_1(\xi) &= 1/4(\xi^{3}/-3\xi+2)\\
* M_2(\xi) &= -1/4(\xi^{3}-3\xi-2)
* \end{array}
*
* \begin{array}{ll}
* L_1(\xi) &= 1/4(\xi^{3}-\xi^{2}-\xi+1)\\
* L_2(\xi) &= 1/4(\xi^{3}+\xi^{2}-\xi-1)
* \end{array}
*
* \begin{array}{ll}
* M'_1(\xi) &= 3/4(\xi^{2}-1)\\
* M'_2(\xi) &= -3/4(\xi^{2}-1)
* \end{array}
*
* \begin{array}{ll}
* L'_1(\xi) &= 1/4(3\xi^{2}-2\xi-1)\\
* L'_2(\xi) &= 1/4(3\xi^{2}+2\xi-1)
* \end{array}
*@f]
*
* @subsection dnds Shape derivatives
*
*@f[
* \begin{array}{ll}
* N'_1(\xi) &= -1/2\\
* N'_2(\xi) &= 1/2
* \end{array}]
*
* \begin{array}{ll}
* -M''_1(\xi) &= -3\xi/2\\
* -M''_2(\xi) &= 3\xi/2\\
* \end{array}
*
* \begin{array}{ll}
* -L''_1(\xi) &= -1/2a(3\xi/a-1)\\
* -L''_2(\xi) &= -1/2a(3\xi/a+1)
* \end{array}
*@f]
*
*/
/* -------------------------------------------------------------------------- */
#include "aka_static_if.hh"
#include "element_class_structural.hh"
/* -------------------------------------------------------------------------- */
#ifndef __AKANTU_ELEMENT_CLASS_HERMITE_INLINE_IMPL_CC__
#define __AKANTU_ELEMENT_CLASS_HERMITE_INLINE_IMPL_CC__
namespace akantu {
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_STRUCTURAL_INTERPOLATION_TYPE_PROPERTY(_itp_hermite_2,
_itp_lagrange_segment_2, 2,
1, 4);
/* -------------------------------------------------------------------------- */
namespace {
namespace details {
inline Real computeLength(const Matrix<Real> & real_coord) {
Vector<Real> x1 = real_coord(0);
Vector<Real> x2 = real_coord(1);
return x1.distance(x2);
}
inline void computeShapes(const Vector<Real> & natural_coords, Real a,
Matrix<Real> & N) {
/// natural coordinate
Real xi = natural_coords(0);
// Cubic Hermite splines interpolating displacement
auto M1 = 1. / 4. * Math::pow<2>(xi - 1) * (xi + 2);
auto M2 = 1. / 4. * Math::pow<2>(xi + 1) * (2 - xi);
auto L1 = a / 4. * Math::pow<2>(xi - 1) * (xi + 1);
auto L2 = a / 4. * Math::pow<2>(xi + 1) * (xi - 1);
#if 1 // Version where we also interpolate the rotations
// Derivatives (with respect to x) of previous functions interpolating
// rotations
auto M1_ = 3. / (4. * a) * (xi * xi - 1);
auto M2_ = 3. / (4. * a) * (1 - xi * xi);
auto L1_ = 1 / 4. * (3 * xi * xi - 2 * xi - 1);
auto L2_ = 1 / 4. * (3 * xi * xi + 2 * xi - 1);
// clang-format off
// v1 t1 v2 t2
N = {{M1 , L1 , M2 , L2}, // displacement interpolation
{M1_, L1_, M2_, L2_}}; // rotation interpolation
// clang-format on
#else // Version where we only interpolate displacements
// clang-format off
// v1 t1 v2 t2
N = {{M1, L1, M2, L2}};
// clang-format on
#endif
}
/* ---------------------------------------------------------------------- */
inline void computeDNDS(const Vector<Real> & natural_coords, Real a,
Matrix<Real> & B) {
// natural coordinate
Real xi = natural_coords(0);
// Derivatives with respect to xi for rotations
auto M1__ = 3. / 2. * xi;
auto M2__ = 3. / 2. * (-xi);
auto L1__ = a / 2. * (3 * xi - 1);
auto L2__ = a / 2. * (3 * xi + 1);
// v1 t1 v2 t2
B = {{M1__, L1__, M2__, L2__}}; // computing curvature : {chi} = [B]{d}
B /= a; // to account for first order deriv w/r to x
}
} // namespace details
} // namespace
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_hermite_2, _itk_structural>::computeShapes(
const Vector<Real> & natural_coords, const Matrix<Real> & real_coord,
Matrix<Real> & N) {
auto L = details::computeLength(real_coord);
details::computeShapes(natural_coords, L / 2, N);
}
/* -------------------------------------------------------------------------- */
template <>
inline void InterpolationElement<_itp_hermite_2, _itk_structural>::computeDNDS(
const Vector<Real> & natural_coords, const Matrix<Real> & real_coord,
Matrix<Real> & B) {
auto L = details::computeLength(real_coord);
details::computeDNDS(natural_coords, L / 2, B);
}
} // namespace akantu
#endif /* __AKANTU_ELEMENT_CLASS_HERMITE_INLINE_IMPL_CC__ */
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