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element_class_hermite_inline_impl.cc
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rAKA akantu
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 Frérot <lucas.frerot@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
_hermite
*
* @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
-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
,
1
,
4
,
4
);
/* -------------------------------------------------------------------------- */
namespace
{
namespace
details
{
template
<
InterpolationType
type
>
void
computeShapes
(
const
Vector
<
Real
>
&
natural_coords
,
Matrix
<
Real
>
&
N
)
{
/// natural coordinate
Real
xi
=
natural_coords
(
0
);
#if 0
// Version where we also interpolate the rotations
// 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 = 1. / 4. * Math::pow<2>(xi - 1) * (xi + 1);
auto L2 = 1. / 4. * Math::pow<2>(xi + 1) * (xi - 1);
// Derivatives of previous functions interpolating rotations
auto M1_ = 3. / 4. * (xi * xi - 1);
auto L1_ = 1. / 4. * (3 * xi * xi - 2 * xi - 1);
auto M2_ = 3. / 4. * (1 - xi * xi);
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
// 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 = 1. / 4. * Math::pow<2>(xi - 1) * (xi + 1);
auto L2 = 1. / 4. * Math::pow<2>(xi + 1) * (xi - 1);
// clang-format off
// v1 t1 v2 t2
N = {{M1, L1, M2, L2}};
// clang-format on
#endif
}
/* ---------------------------------------------------------------------- */
template
<
InterpolationType
type
>
void
computeDNDS
(
const
Vector
<
Real
>
&
natural_coords
,
Matrix
<
Real
>
&
B
)
{
// natural coordinate
Real
xi
=
natural_coords
(
0
);
// Derivatives for rotations
auto
M1__
=
3.
/
2.
*
xi
;
auto
L1__
=
1.
/
2.
*
(
3
*
xi
-
1
);
auto
M2__
=
3.
/
2.
*
(
-
xi
);
auto
L2__
=
1.
/
2.
*
(
3
*
xi
+
1
);
// v1 t1 v2 t2
B
=
{{
M1__
,
L1__
,
M2__
,
L2__
}};
// computing curvature : {chi} = [B]{d}
}
}
// namespace details
}
// namespace
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_hermite_2
,
_itk_structural
>::
computeShapes
(
const
Vector
<
Real
>
&
natural_coords
,
Matrix
<
Real
>
&
N
)
{
details
::
computeShapes
<
_itp_hermite_2
>
(
natural_coords
,
N
);
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_hermite_2
,
_itk_structural
>::
computeDNDS
(
const
Vector
<
Real
>
&
natural_coords
,
Matrix
<
Real
>
&
B
)
{
details
::
computeDNDS
<
_itp_hermite_2
>
(
natural_coords
,
B
);
}
}
// namespace akantu
#endif
/* __AKANTU_ELEMENT_CLASS_HERMITE_INLINE_IMPL_CC__ */
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