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element_class_hermite_inline_impl.hh
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rAKA akantu
element_class_hermite_inline_impl.hh
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/**
* @file element_class_hermite_inline_impl.hh
*
* @author Fabian Barras <fabian.barras@epfl.ch>
* @author Lucas Frerot <lucas.frerot@epfl.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Fri Nov 10 2017
* @date last modification: Tue Feb 09 2021
*
* @brief Specialization of the element_class class for the type
* _hermite
*
*
* @section LICENSE
*
* Copyright (©) 2016-2021 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/>.
*
*/
/**
* @verbatim
--x-----q1----|----q2-----x---> x
-1 0 1
@endverbatim
*
* @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]
*
*
*@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_HH_
#define AKANTU_ELEMENT_CLASS_HERMITE_INLINE_IMPL_HH_
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
);
auto
xi2
=
xi
*
xi
;
auto
xi3
=
xi
*
xi
*
xi
;
// Cubic Hermite splines interpolating displacement
auto
M1
=
1.
/
4.
*
(
2.
-
3.
*
xi
+
xi3
);
auto
M2
=
1.
/
4.
*
(
2.
+
3.
*
xi
-
xi3
);
auto
L1
=
a
/
4.
*
(
1
-
xi
-
xi2
+
xi3
);
auto
L2
=
a
/
4.
*
(
-
1
-
xi
+
xi2
+
xi3
);
;
#if 1
// Version where we also interpolate the rotations
// Derivatives (with respect to x) of previous functions interpolating
// rotations
auto
M1_
=
3.
/
(
4.
*
a
)
*
(
xi2
-
1
);
auto
M2_
=
3.
/
(
4.
*
a
)
*
(
1
-
xi2
);
auto
L1_
=
1
/
4.
*
(
3
*
xi2
-
2
*
xi
-
1
);
auto
L2_
=
1
/
4.
*
(
3
*
xi2
+
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
=
1.
*
a
/
2.
*
(
3
*
xi
-
1
);
auto
L2
=
1.
*
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_HH_ */
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