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element_class_bernoulli_beam_inline_impl.cc
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element_class_bernoulli_beam_inline_impl.cc
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/**
* @file element_class_bernoulli_beam_inline_impl.cc
*
* @author Fabian Barras <fabian.barras@epfl.ch>
*
* @date creation: Fri Jul 15 2011
* @date last modification: Sun Oct 19 2014
*
* @brief Specialization of the element_class class for the type
_bernoulli_beam_2
*
* @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
--x-----q1----|----q2-----x---> x
-a 0 a
@endverbatim
*
* @subsection coords Nodes coordinates
*
* @f[
* \begin{array}{ll}
* x_{1} = -a & x_{2} = a
* \end{array}
* @f]
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{ll}
* N_1(x) &= \frac{1-x}{2a}\\
* N_2(x) &= \frac{1+x}{2a}
* \end{array}
*
* \begin{array}{ll}
* M_1(x) &= 1/4(x^{3}/a^{3}-3x/a+2)\\
* M_2(x) &= -1/4(x^{3}/a^{3}-3x/a-2)
* \end{array}
*
* \begin{array}{ll}
* L_1(x) &= a/4(x^{3}/a^{3}-x^{2}/a^{2}-x/a+1)\\
* L_2(x) &= a/4(x^{3}/a^{3}+x^{2}/a^{2}-x/a-1)
* \end{array}
*
* \begin{array}{ll}
* M'_1(x) &= 3/4a(x^{2}/a^{2}-1)\\
* M'_2(x) &= -3/4a(x^{2}/a^{2}-1)
* \end{array}
*
* \begin{array}{ll}
* L'_1(x) &= 1/4(3x^{2}/a^{2}-2x/a-1)\\
* L'_2(x) &= 1/4(3x^{2}/a^{2}+2x/a-1)
* \end{array}
*@f]
*
* @subsection dnds Shape derivatives
*
*@f[
* \begin{array}{ll}
* N'_1(x) &= -1/2a\\
* N'_2(x) &= 1/2a
* \end{array}]
*
* \begin{array}{ll}
* -M''_1(x) &= -3x/(2a^{3})\\
* -M''_2(x) &= 3x/(2a^{3})\\
* \end{array}
*
* \begin{array}{ll}
* -L''_1(x) &= -1/2a(3x/a-1)\\
* -L''_2(x) &= -1/2a(3x/a+1)
* \end{array}
*@f]
*
* @subsection quad_points Position of quadrature points
*
* @f[
* \begin{array}{ll}
* x_{q1} = -a/\sqrt{3} & x_{q2} = a/\sqrt{3}
* \end{array}
* @f]
*/
/* -------------------------------------------------------------------------- */
#include "aka_static_if.hh"
#include "element_class_structural.hh"
/* -------------------------------------------------------------------------- */
#ifndef __AKANTU_ELEMENT_CLASS_BERNOULLI_BEAM_INLINE_IMPL_CC__
#define __AKANTU_ELEMENT_CLASS_BERNOULLI_BEAM_INLINE_IMPL_CC__
namespace akantu {
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_STRUCTURAL_INTERPOLATION_TYPE_PROPERTY(_itp_bernoulli_beam_2,
_itp_lagrange_segment_2, 2,
3);
AKANTU_DEFINE_STRUCTURAL_INTERPOLATION_TYPE_PROPERTY(_itp_bernoulli_beam_3,
_itp_lagrange_segment_2, 3,
6);
AKANTU_DEFINE_STRUCTURAL_ELEMENT_CLASS_PROPERTY(_bernoulli_beam_2,
_gt_segment_2,
_itp_bernoulli_beam_2,
_segment_2, _ek_structural, 2,
_git_segment, 5);
AKANTU_DEFINE_STRUCTURAL_ELEMENT_CLASS_PROPERTY(_bernoulli_beam_3,
_gt_segment_2,
_itp_bernoulli_beam_3,
_segment_2, _ek_structural, 3,
_git_segment, 5);
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
namespace {
namespace details {
template <InterpolationType type>
void computeShapes(const Vector<Real> & natural_coords, Matrix<Real> & N,
const Matrix<Real> & real_coord) {
/// Compute the dimension of the beam
Vector<Real> x1 = real_coord(0);
Vector<Real> x2 = real_coord(1);
Real a = x1.distance(x2) / 2.;
/// natural coordinate
Real c = natural_coords(0);
auto N0 = (1 - c) / 2.;
auto N1 = (1 + c) / 2.;
auto M0 = (c * c * c - 3. * c + 2.) / 4.;
auto M1 = -(c * c * c - 3. * c - 2.) / 4.;
auto L0 = a * (c * c * c - c * c - c + 1.) / 4.;
auto L1 = a * (c * c * c + c * c - c - 1.) / 4.;
auto Mp0 = 3. / a * (c * c - 1.) / 4.;
auto Mp1 = -3. / a * (c * c - 1.) / 4.;
auto Lp0 = (3. * c * c - 2. * c - 1.) / 4.;
auto Lp1 = (3. * c * c + 2. * c - 1.) / 4.;
static_if(type == _itp_bernoulli_beam_2)
.then([&](auto && N) {
// clang-format off
// 0 1 2 3 4 5
N = {{N0, 0., 0., N1, 0., 0.},
{0., M0, L0, 0., M1, L1},
{0., Mp0, Lp0, 0., Mp1, Lp1}};
// clang-format on
})
.else_if(type == _itp_bernoulli_beam_3)
.then([&](auto && N) {
// clang-format off
// 0 1 2 3 4 5 6 7 8 9 10 11
N = {{N0, 0., 0., 0., 0., 0., N1, 0., 0., 0., 0., 0.},
{ 0., M0, 0., 0., 0., L0, 0., M1, 0., 0., 0., L1},
{ 0., 0., M0, 0., -L0, 0., 0., 0., M1, 0., -L1, 0.},
{ 0., 0., 0., N0, 0., 0., 0., 0., 0., N1, 0., 0.},
{ 0., 0., Mp0, 0., -Lp0, 0., 0., 0., Mp1, 0., -Lp1, 0.},
{ 0., Mp0, 0., 0., 0., Lp0, 0., Mp1, 0., 0., 0., Lp1}};
// clang-format on
})
.else_([](auto && /*unused*/) {
AKANTU_EXCEPTION("Should not be in this part of the code");
})(std::forward<decltype(N)>(N));
}
/* ---------------------------------------------------------------------- */
template <InterpolationType type>
void computeDNDS(const Vector<Real> & natural_coords, Matrix<Real> & B,
const Matrix<Real> & real_nodes_coord) {
/// Compute the dimension of the beam
Vector<Real> x1 = real_nodes_coord(0);
Vector<Real> x2 = real_nodes_coord(1);
Real a = .5 * x1.distance(x2);
/// natural coordinate
Real c = natural_coords(0) * a;
auto Np0 = -1. / (2. * a);
auto Np1 = 1. / (2. * a);
auto Mpp0 = -3. * c / (2. * pow(a, 3.));
auto Mpp1 = 3. * c / (2. * pow(a, 3.));
auto Lpp0 = -1. / (2. * a) * (3. * c / a - 1.);
auto Lpp1 = -1. / (2. * a) * (3. * c / a + 1.);
static_if(type == _itp_bernoulli_beam_2)
.then([&](auto && B) {
// clang-format off
// 0 1 2 3 4 5
B = {{Np0, 0., 0., Np1, 0., 0.},
{ 0., Mpp0, Lpp0, 0., Mpp1, Lpp1}};
// clang-format on
})
.else_if(type == _itp_bernoulli_beam_3)
.then([&](auto && B) {
// clang-format off
// 0 1 2 3 4 5 6 7 8 9 10 11
B = {{Np0, 0., 0., 0., 0., 0., Np1, 0., 0., 0., 0., 0.},
{ 0., Mpp0, 0., 0., 0., Lpp0, 0., Mpp1, 0., 0., 0., Lpp1},
{ 0., 0., Mpp0, 0., -Lpp0, 0., 0., 0., Mpp1, 0., -Lpp1, 0.},
{ 0., 0., 0., Np0, 0., 0., 0., 0., 0., Np1, 0., 0.}};
// clang-format on
})
.else_([](auto && /*unused*/) {
AKANTU_EXCEPTION("Should not be in this part of the code");
})(std::forward<decltype(B)>(B));
}
} // namespace details
} // namespace
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_bernoulli_beam_2, _itk_structural>::computeShapes(
const Vector<Real> & natural_coords, Matrix<Real> & N,
const Matrix<Real> & real_coord) {
details::computeShapes<_itp_bernoulli_beam_2>(natural_coords, N, real_coord);
}
template <>
inline void
InterpolationElement<_itp_bernoulli_beam_3, _itk_structural>::computeShapes(
const Vector<Real> & natural_coords, Matrix<Real> & N,
const Matrix<Real> & real_coord) {
details::computeShapes<_itp_bernoulli_beam_3>(natural_coords, N, real_coord);
}
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_bernoulli_beam_2, _itk_structural>::computeDNDS(
const Vector<Real> & natural_coords, Matrix<Real> & B,
const Matrix<Real> & real_nodes_coord) {
details::computeDNDS<_itp_bernoulli_beam_2>(natural_coords, B,
real_nodes_coord);
}
template <>
inline void
InterpolationElement<_itp_bernoulli_beam_3, _itk_structural>::computeDNDS(
const Vector<Real> & natural_coords, Matrix<Real> & B,
const Matrix<Real> & real_nodes_coord) {
details::computeDNDS<_itp_bernoulli_beam_3>(natural_coords, B,
real_nodes_coord);
}
} // namespace akantu
#endif /* __AKANTU_ELEMENT_CLASS_BERNOULLI_BEAM_INLINE_IMPL_CC__ */

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