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rAKA/src/fe_engine/element_classes/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]
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_STRUCTURAL_ELEMENT_CLASS_PROPERTY(_bernoulli_beam_2,
_gt_segment_2,
_itp_bernoulli_beam,
_segment_2,
_ek_structural,
2,
_git_segment, 4);
AKANTU_DEFINE_STRUCTURAL_ELEMENT_CLASS_PROPERTY(_bernoulli_beam_3,
_gt_segment_2,
_itp_bernoulli_beam,
_segment_2,
_ek_structural,
3,
_git_segment, 4);
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_bernoulli_beam>::computeShapes(const Vector<Real> & natural_coords,
Vector<Real> & N,
const Matrix<Real> & real_coord,
UInt id) {
/// Compute the dimension of the beam
Vector<Real> x1 = real_coord(0);
Vector<Real> x2 = real_coord(1);
Real a = .5 * x1.distance(x2);
/// natural coordinate
Real c = natural_coords(0);
switch (id) {
case 0: { // N
N(0) = 0.5*(1 - c);
N(1) = 0.5*(1 + c);
break;
}
case 1: { // M
N(0) = 0.25 * (c*c*c - 3*c + 2);
N(1) = -0.25 * (c*c*c - 3*c - 2);
break;
}
case 2: { // L
N(0) = 0.25*a * (c*c*c - c*c - c + 1);
N(1) = 0.25*a * (c*c*c + c*c - c - 1);
break;
}
case 3: { // M'
N(0) = 0.75/a * (c*c - 1);
N(1) = -0.75/a * (c*c - 1);
break;
}
case 4: { // L'
N(0) = 0.25 * (3*c*c - 2*c - 1);
N(1) = 0.25 * (3*c*c + 2*c - 1);
break;
}
}
}
/* -------------------------------------------------------------------------- */
template <>
inline void
InterpolationElement<_itp_bernoulli_beam>::computeDNDS(const Vector<Real> & natural_coords,
Matrix<Real> & dnds,
const Matrix<Real> & real_nodes_coord,
UInt id) {
/// 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;
switch (id) {
case 0: { // N'
dnds(0, 0) = -0.5/a;
dnds(0, 1) = 0.5/a;
break;
}
case 1: { // M''
dnds(0, 0) = -3.*c/(2.*pow(a,3));
dnds(0, 1) = 3.*c/(2.*pow(a,3));
break;
}
case 2: { // L''
dnds(0, 0) = -0.5/a * (3*c/a - 1);
dnds(0, 1) =- 0.5/a * (3*c/a + 1);
break;
}
}
}
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