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element_class_triangle_6_inline_impl.cc
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rAKA akantu
element_class_triangle_6_inline_impl.cc
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/**
* @file element_class_triangle_6_inline_impl.cc
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Fri Jul 16 2010
* @date last modification: Sun Oct 19 2014
*
* @brief Specialization of the element_class class for the type _triangle_6
*
* @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
\eta
^
|
x 2
| `
| `
| . `
| q2 `
5 x x 4
| `
| `
| .q0 q1. `
| `
x---------x---------x-----> \xi
0 3 1
@endverbatim
*
* @subsection coords Nodes coordinates
*
* @f[
* \begin{array}{ll}
* \xi_{0} = 0 & \eta_{0} = 0 \\
* \xi_{1} = 1 & \eta_{1} = 0 \\
* \xi_{2} = 0 & \eta_{2} = 1 \\
* \xi_{3} = 1/2 & \eta_{3} = 0 \\
* \xi_{4} = 1/2 & \eta_{4} = 1/2 \\
* \xi_{5} = 0 & \eta_{5} = 1/2
* \end{array}
* @f]
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{lll}
* N1 = -(1 - \xi - \eta) (1 - 2 (1 - \xi - \eta))
* & \frac{\partial N1}{\partial \xi} = 1 - 4(1 - \xi - \eta)
* & \frac{\partial N1}{\partial \eta} = 1 - 4(1 - \xi - \eta) \\
* N2 = - \xi (1 - 2 \xi)
* & \frac{\partial N2}{\partial \xi} = - 1 + 4 \xi
* & \frac{\partial N2}{\partial \eta} = 0 \\
* N3 = - \eta (1 - 2 \eta)
* & \frac{\partial N3}{\partial \xi} = 0
* & \frac{\partial N3}{\partial \eta} = - 1 + 4 \eta \\
* N4 = 4 \xi (1 - \xi - \eta)
* & \frac{\partial N4}{\partial \xi} = 4 (1 - 2 \xi - \eta)
* & \frac{\partial N4}{\partial \eta} = - 4 \xi \\
* N5 = 4 \xi \eta
* & \frac{\partial N5}{\partial \xi} = 4 \eta
* & \frac{\partial N5}{\partial \eta} = 4 \xi \\
* N6 = 4 \eta (1 - \xi - \eta)
* & \frac{\partial N6}{\partial \xi} = - 4 \eta
* & \frac{\partial N6}{\partial \eta} = 4 (1 - \xi - 2 \eta)
* \end{array}
* @f]
*
* @subsection quad_points Position of quadrature points
* @f{eqnarray*}{
* \xi_{q0} &=& 1/6 \qquad \eta_{q0} = 1/6 \\
* \xi_{q1} &=& 2/3 \qquad \eta_{q1} = 1/6 \\
* \xi_{q2} &=& 1/6 \qquad \eta_{q2} = 2/3
* @f}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY
(
_triangle_6
,
_gt_triangle_6
,
_itp_lagrange_triangle_6
,
_ek_regular
,
2
,
_git_triangle
,
2
);
AKANTU_DEFINE_SHAPE
(
_gt_triangle_6
,
_gst_triangle
);
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
>
inline
void
InterpolationElement
<
_itp_lagrange_triangle_6
>::
computeShapes
(
const
vector_type
&
natural_coords
,
vector_type
&
N
)
{
/// Natural coordinates
Real
c0
=
1
-
natural_coords
(
0
)
-
natural_coords
(
1
);
/// @f$ c0 = 1 - \xi - \eta @f$
Real
c1
=
natural_coords
(
0
);
/// @f$ c1 = \xi @f$
Real
c2
=
natural_coords
(
1
);
/// @f$ c2 = \eta @f$
N
(
0
)
=
c0
*
(
2
*
c0
-
1.
);
N
(
1
)
=
c1
*
(
2
*
c1
-
1.
);
N
(
2
)
=
c2
*
(
2
*
c2
-
1.
);
N
(
3
)
=
4
*
c0
*
c1
;
N
(
4
)
=
4
*
c1
*
c2
;
N
(
5
)
=
4
*
c2
*
c0
;
}
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
,
class
matrix_type
>
inline
void
InterpolationElement
<
_itp_lagrange_triangle_6
>::
computeDNDS
(
const
vector_type
&
natural_coords
,
matrix_type
&
dnds
)
{
/**
* @f[
* dnds = \left(
* \begin{array}{cccccc}
* \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 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}
* \end{array}
* \right)
* @f]
*/
/// Natural coordinates
Real
c0
=
1
-
natural_coords
(
0
)
-
natural_coords
(
1
);
/// @f$ c0 = 1 - \xi - \eta @f$
Real
c1
=
natural_coords
(
0
);
/// @f$ c1 = \xi @f$
Real
c2
=
natural_coords
(
1
);
/// @f$ c2 = \eta @f$
dnds
(
0
,
0
)
=
1
-
4
*
c0
;
dnds
(
0
,
1
)
=
4
*
c1
-
1.
;
dnds
(
0
,
2
)
=
0.
;
dnds
(
0
,
3
)
=
4
*
(
c0
-
c1
);
dnds
(
0
,
4
)
=
4
*
c2
;
dnds
(
0
,
5
)
=
-
4
*
c2
;
dnds
(
1
,
0
)
=
1
-
4
*
c0
;
dnds
(
1
,
1
)
=
0.
;
dnds
(
1
,
2
)
=
4
*
c2
-
1.
;
dnds
(
1
,
3
)
=
-
4
*
c1
;
dnds
(
1
,
4
)
=
4
*
c1
;
dnds
(
1
,
5
)
=
4
*
(
c0
-
c2
);
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_lagrange_triangle_6
>::
computeSpecialJacobian
(
const
Matrix
<
Real
>
&
J
,
Real
&
jac
){
Vector
<
Real
>
vprod
(
J
.
cols
());
Matrix
<
Real
>
Jt
(
J
.
transpose
(),
true
);
vprod
.
crossProduct
(
Jt
(
0
),
Jt
(
1
));
jac
=
vprod
.
norm
();
}
/* -------------------------------------------------------------------------- */
template
<>
inline
Real
GeometricalElement
<
_gt_triangle_6
>::
getInradius
(
const
Matrix
<
Real
>
&
coord
)
{
UInt
triangles
[
4
][
3
]
=
{
{
0
,
3
,
5
},
{
3
,
1
,
4
},
{
3
,
4
,
5
},
{
5
,
4
,
2
}
};
Real
inradius
=
std
::
numeric_limits
<
Real
>::
max
();
for
(
UInt
t
=
0
;
t
<
4
;
t
++
)
{
Real
ir
=
Math
::
triangle_inradius
(
coord
(
triangles
[
t
][
0
]).
storage
(),
coord
(
triangles
[
t
][
1
]).
storage
(),
coord
(
triangles
[
t
][
2
]).
storage
());
inradius
=
std
::
min
(
ir
,
inradius
);
}
return
inradius
;
}
/* -------------------------------------------------------------------------- */
// template<> inline bool ElementClass<_triangle_6>::contains(const Vector<Real> & natural_coords) {
// return ElementClass<_triangle_3>::contains(natural_coords);
// }
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