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element_class_tetrahedron_10_inline_impl.hh
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rAKA akantu
element_class_tetrahedron_10_inline_impl.hh
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/**
* @file element_class_tetrahedron_10_inline_impl.hh
*
* @author Guillaume Anciaux <guillaume.anciaux@epfl.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
* @author Peter Spijker <peter.spijker@epfl.ch>
*
* @date creation: Fri Jul 16 2010
* @date last modification: Fri Feb 07 2020
*
* @brief Specialization of the element_class class for the type
* _tetrahedron_10
*
*
* @section LICENSE
*
* Copyright (©) 2010-2021 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
\zeta
^
|
(0,0,1)
x
|` .
| ` .
| ` .
| ` . (0,0.5,0.5)
| ` x.
| q4 o ` . \eta
| ` . -,
(0,0,0.5) x ` x (0.5,0,0.5) -
| ` x-(0,1,0)
| q3 o` - '
| (0,0.5,0) - ` '
| x- ` x (0.5,0.5,0)
| q1 o - o q2` '
| - ` '
| - ` '
x---------------x--------------` x-----> \xi
(0,0,0) (0.5,0,0) (1,0,0)
@endverbatim
*
*
* @f[
* \begin{array}{lll}
* \xi_{0} = 0 & \eta_{0} = 0 & \zeta_{0} = 0 \\
* \xi_{1} = 1 & \eta_{1} = 0 & \zeta_{1} = 0 \\
* \xi_{2} = 0 & \eta_{2} = 1 & \zeta_{2} = 0 \\
* \xi_{3} = 0 & \eta_{3} = 0 & \zeta_{3} = 1 \\
* \xi_{4} = 1/2 & \eta_{4} = 0 & \zeta_{4} = 0 \\
* \xi_{5} = 1/2 & \eta_{5} = 1/2 & \zeta_{5} = 0 \\
* \xi_{6} = 0 & \eta_{6} = 1/2 & \zeta_{6} = 0 \\
* \xi_{7} = 0 & \eta_{7} = 0 & \zeta_{7} = 1/2 \\
* \xi_{8} = 1/2 & \eta_{8} = 0 & \zeta_{8} = 1/2 \\
* \xi_{9} = 0 & \eta_{9} = 1/2 & \zeta_{9} = 1/2
* \end{array}
* @f]
*
* @f[
* \begin{array}{llll}
* N1 = (1 - \xi - \eta - \zeta) (1 - 2 \xi - 2 \eta - 2 \zeta)
* & \frac{\partial N1}{\partial \xi} = 4 \xi + 4 \eta + 4 \zeta -
3
* & \frac{\partial N1}{\partial \eta} = 4 \xi + 4 \eta + 4 \zeta -
3
* & \frac{\partial N1}{\partial \zeta} = 4 \xi + 4 \eta + 4 \zeta -
3 \\
* N2 = \xi (2 \xi - 1)
* & \frac{\partial N2}{\partial \xi} = 4 \xi - 1
* & \frac{\partial N2}{\partial \eta} = 0
* & \frac{\partial N2}{\partial \zeta} = 0 \\
* N3 = \eta (2 \eta - 1)
* & \frac{\partial N3}{\partial \xi} = 0
* & \frac{\partial N3}{\partial \eta} = 4 \eta - 1
* & \frac{\partial N3}{\partial \zeta} = 0 \\
* N4 = \zeta (2 \zeta - 1)
* & \frac{\partial N4}{\partial \xi} = 0
* & \frac{\partial N4}{\partial \eta} = 0
* & \frac{\partial N4}{\partial \zeta} = 4 \zeta - 1 \\
* N5 = 4 \xi (1 - \xi - \eta - \zeta)
* & \frac{\partial N5}{\partial \xi} = 4 - 8 \xi - 4 \eta - 4
\zeta
* & \frac{\partial N5}{\partial \eta} = -4 \xi
* & \frac{\partial N5}{\partial \zeta} = -4 \xi \\
* N6 = 4 \xi \eta
* & \frac{\partial N6}{\partial \xi} = 4 \eta
* & \frac{\partial N6}{\partial \eta} = 4 \xi
* & \frac{\partial N6}{\partial \zeta} = 0 \\
* N7 = 4 \eta (1 - \xi - \eta - \zeta)
* & \frac{\partial N7}{\partial \xi} = -4 \eta
* & \frac{\partial N7}{\partial \eta} = 4 - 4 \xi - 8 \eta - 4
\zeta
* & \frac{\partial N7}{\partial \zeta} = -4 \eta \\
* N8 = 4 \zeta (1 - \xi - \eta - \zeta)
* & \frac{\partial N8}{\partial \xi} = -4 \zeta
* & \frac{\partial N8}{\partial \eta} = -4 \zeta
* & \frac{\partial N8}{\partial \zeta} = 4 - 4 \xi - 4 \eta - 8
\zeta \\
* N9 = 4 \zeta \xi
* & \frac{\partial N9}{\partial \xi} = 4 \zeta
* & \frac{\partial N9}{\partial \eta} = 0
* & \frac{\partial N9}{\partial \zeta} = 4 \xi \\
* N10 = 4 \eta \zeta
* & \frac{\partial N10}{\partial \xi} = 0
* & \frac{\partial N10}{\partial \eta} = 4 \zeta
* & \frac{\partial N10}{\partial \zeta} = 4 \eta \\
* \end{array}
* @f]
*
* @f[
* a = \frac{5 - \sqrt{5}}{20}\\
* b = \frac{5 + 3 \sqrt{5}}{20}
* \begin{array}{lll}
* \xi_{q_0} = a & \eta_{q_0} = a & \zeta_{q_0} = a \\
* \xi_{q_1} = b & \eta_{q_1} = a & \zeta_{q_1} = a \\
* \xi_{q_2} = a & \eta_{q_2} = b & \zeta_{q_2} = a \\
* \xi_{q_3} = a & \eta_{q_3} = a & \zeta_{q_3} = b
* \end{array}
* @f]
*/
/* -------------------------------------------------------------------------- */
#include "element_class.hh"
/* -------------------------------------------------------------------------- */
namespace
akantu
{
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY
(
_tetrahedron_10
,
_gt_tetrahedron_10
,
_itp_lagrange_tetrahedron_10
,
_ek_regular
,
3
,
_git_tetrahedron
,
2
);
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
>
inline
void
InterpolationElement
<
_itp_lagrange_tetrahedron_10
>::
computeShapes
(
const
vector_type
&
natural_coords
,
vector_type
&
N
)
{
/// Natural coordinates
Real
xi
=
natural_coords
(
0
);
Real
eta
=
natural_coords
(
1
);
Real
zeta
=
natural_coords
(
2
);
Real
sum
=
xi
+
eta
+
zeta
;
Real
c0
=
1
-
sum
;
Real
c1
=
1
-
2
*
sum
;
Real
c2
=
2
*
xi
-
1
;
Real
c3
=
2
*
eta
-
1
;
Real
c4
=
2
*
zeta
-
1
;
/// Shape functions
N
(
0
)
=
c0
*
c1
;
N
(
1
)
=
xi
*
c2
;
N
(
2
)
=
eta
*
c3
;
N
(
3
)
=
zeta
*
c4
;
N
(
4
)
=
4
*
xi
*
c0
;
N
(
5
)
=
4
*
xi
*
eta
;
N
(
6
)
=
4
*
eta
*
c0
;
N
(
7
)
=
4
*
zeta
*
c0
;
N
(
8
)
=
4
*
xi
*
zeta
;
N
(
9
)
=
4
*
eta
*
zeta
;
}
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
,
class
matrix_type
>
inline
void
InterpolationElement
<
_itp_lagrange_tetrahedron_10
>::
computeDNDS
(
const
vector_type
&
natural_coords
,
matrix_type
&
dnds
)
{
/**
* \f[
* dnds = \left(
* \begin{array}{cccccccccc}
* \frac{\partial N1}{\partial \xi} & \frac{\partial N2}{\partial
* \xi}
* & \frac{\partial N3}{\partial \xi} & \frac{\partial N4}{\partial
* \xi}
* & \frac{\partial N5}{\partial \xi} & \frac{\partial N6}{\partial
* \xi}
* & \frac{\partial N7}{\partial \xi} & \frac{\partial N8}{\partial
* \xi}
* & \frac{\partial N9}{\partial \xi} & \frac{\partial
* N10}{\partial \xi} \\
* \frac{\partial N1}{\partial \eta} & \frac{\partial N2}{\partial
* \eta}
* & \frac{\partial N3}{\partial \eta} & \frac{\partial N4}{\partial
* \eta}
* & \frac{\partial N5}{\partial \eta} & \frac{\partial N6}{\partial
* \eta}
* & \frac{\partial N7}{\partial \eta} & \frac{\partial N8}{\partial
* \eta}
* & \frac{\partial N9}{\partial \eta} & \frac{\partial
* N10}{\partial \eta} \\
* \frac{\partial N1}{\partial \zeta} & \frac{\partial N2}{\partial
* \zeta}
* & \frac{\partial N3}{\partial \zeta} & \frac{\partial N4}{\partial
* \zeta}
* & \frac{\partial N5}{\partial \zeta} & \frac{\partial N6}{\partial
* \zeta}
* & \frac{\partial N7}{\partial \zeta} & \frac{\partial N8}{\partial
* \zeta}
* & \frac{\partial N9}{\partial \zeta} & \frac{\partial
* N10}{\partial \zeta}
* \end{array}
* \right)
* \f]
*/
/// Natural coordinates
Real
xi
=
natural_coords
(
0
);
Real
eta
=
natural_coords
(
1
);
Real
zeta
=
natural_coords
(
2
);
Real
sum
=
xi
+
eta
+
zeta
;
/// \frac{\partial N_i}{\partial \xi}
dnds
(
0
,
0
)
=
4
*
sum
-
3
;
dnds
(
0
,
1
)
=
4
*
xi
-
1
;
dnds
(
0
,
2
)
=
0
;
dnds
(
0
,
3
)
=
0
;
dnds
(
0
,
4
)
=
4
*
(
1
-
sum
-
xi
);
dnds
(
0
,
5
)
=
4
*
eta
;
dnds
(
0
,
6
)
=
-
4
*
eta
;
dnds
(
0
,
7
)
=
-
4
*
zeta
;
dnds
(
0
,
8
)
=
4
*
zeta
;
dnds
(
0
,
9
)
=
0
;
/// \frac{\partial N_i}{\partial \eta}
dnds
(
1
,
0
)
=
4
*
sum
-
3
;
dnds
(
1
,
1
)
=
0
;
dnds
(
1
,
2
)
=
4
*
eta
-
1
;
dnds
(
1
,
3
)
=
0
;
dnds
(
1
,
4
)
=
-
4
*
xi
;
dnds
(
1
,
5
)
=
4
*
xi
;
dnds
(
1
,
6
)
=
4
*
(
1
-
sum
-
eta
);
dnds
(
1
,
7
)
=
-
4
*
zeta
;
dnds
(
1
,
8
)
=
0
;
dnds
(
1
,
9
)
=
4
*
zeta
;
/// \frac{\partial N_i}{\partial \zeta}
dnds
(
2
,
0
)
=
4
*
sum
-
3
;
dnds
(
2
,
1
)
=
0
;
dnds
(
2
,
2
)
=
0
;
dnds
(
2
,
3
)
=
4
*
zeta
-
1
;
dnds
(
2
,
4
)
=
-
4
*
xi
;
dnds
(
2
,
5
)
=
0
;
dnds
(
2
,
6
)
=
-
4
*
eta
;
dnds
(
2
,
7
)
=
4
*
(
1
-
sum
-
zeta
);
dnds
(
2
,
8
)
=
4
*
xi
;
dnds
(
2
,
9
)
=
4
*
eta
;
}
/* -------------------------------------------------------------------------- */
template
<>
inline
Real
GeometricalElement
<
_gt_tetrahedron_10
>::
getInradius
(
const
Matrix
<
Real
>
&
coord
)
{
// Only take the four corner tetrahedra
UInt
tetrahedra
[
4
][
4
]
=
{
{
0
,
4
,
6
,
7
},
{
4
,
1
,
5
,
8
},
{
6
,
5
,
2
,
9
},
{
7
,
8
,
9
,
3
}};
Real
inradius
=
std
::
numeric_limits
<
Real
>::
max
();
for
(
UInt
t
=
0
;
t
<
4
;
t
++
)
{
Real
ir
=
Math
::
tetrahedron_inradius
(
coord
(
tetrahedra
[
t
][
0
]).
storage
(),
coord
(
tetrahedra
[
t
][
1
]).
storage
(),
coord
(
tetrahedra
[
t
][
2
]).
storage
(),
coord
(
tetrahedra
[
t
][
3
]).
storage
());
inradius
=
std
::
min
(
ir
,
inradius
);
}
return
2.
*
inradius
;
}
}
// namespace akantu
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