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element_class_quadrangle_8_inline_impl.cc
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rAKA akantu
element_class_quadrangle_8_inline_impl.cc
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/**
* @file element_class_quadrangle_8_inline_impl.cc
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Wed May 18 2011
* @date last modification: Tue Apr 07 2015
*
* @brief Specialization of the ElementClass for the _quadrangle_8
*
* @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
^
|
(-1,1) (0,1) (1,1)
x-------x-------x
| | |
| | |
| | |
(-1,0)| | |(1,0)
----x---------------X-----> \xi
| | |
| | |
| | |
| | |
x-------x-------x
(-1,-1) (0,-1) (1,-1)
|
@endverbatim
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{lll}
* N1 = (1 - \xi) (1 - \eta)(- 1 - \xi - \eta) / 4
* & \frac{\partial N1}{\partial \xi} = (1 - \eta)(2 \xi + \eta) / 4
* & \frac{\partial N1}{\partial \eta} = (1 - \xi)(\xi + 2 \eta) / 4 \\
* N2 = (1 + \xi) (1 - \eta)(- 1 + \xi - \eta) / 4 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta)(2 \xi - \eta) / 4
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi)(\xi - 2 \eta) / 4 \\
* N3 = (1 + \xi) (1 + \eta)(- 1 + \xi + \eta) / 4 \\
* & \frac{\partial N3}{\partial \xi} = (1 + \eta)(2 \xi + \eta) / 4
* & \frac{\partial N3}{\partial \eta} = (1 + \xi)(\xi + 2 \eta) / 4 \\
* N4 = (1 - \xi) (1 + \eta)(- 1 - \xi + \eta) / 4
* & \frac{\partial N4}{\partial \xi} = (1 + \eta)(2 \xi - \eta) / 4
* & \frac{\partial N4}{\partial \eta} = - (1 - \xi)(\xi - 2 \eta) / 4 \\
* N5 = (1 - \xi^2) (1 - \eta) / 2
* & \frac{\partial N1}{\partial \xi} = - \xi (1 - \eta)
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi^2) / 2 \\
* N6 = (1 + \xi) (1 - \eta^2) / 2 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta^2) / 2
* & \frac{\partial N2}{\partial \eta} = - \eta (1 + \xi) \\
* N7 = (1 - \xi^2) (1 + \eta) / 2 \\
* & \frac{\partial N3}{\partial \xi} = - \xi (1 + \eta)
* & \frac{\partial N3}{\partial \eta} = (1 - \xi^2) / 2 \\
* N8 = (1 - \xi) (1 - \eta^2) / 2
* & \frac{\partial N4}{\partial \xi} = - (1 - \eta^2) / 2
* & \frac{\partial N4}{\partial \eta} = - \eta (1 - \xi) \\
* \end{array}
* @f]
*
* @subsection quad_points Position of quadrature points
* @f{eqnarray*}{
* \xi_{q0} &=& 0 \qquad \eta_{q0} = 0
* @f}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY
(
_quadrangle_8
,
_gt_quadrangle_8
,
_itp_serendip_quadrangle_8
,
_ek_regular
,
2
,
_git_segment
,
3
);
AKANTU_DEFINE_SHAPE
(
_gt_quadrangle_8
,
_gst_square
);
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
>
inline
void
InterpolationElement
<
_itp_serendip_quadrangle_8
>::
computeShapes
(
const
vector_type
&
c
,
vector_type
&
N
)
{
/// Natural coordinates
const
Real
xi
=
c
(
0
);
const
Real
eta
=
c
(
1
);
N
(
0
)
=
.25
*
(
1
-
xi
)
*
(
1
-
eta
)
*
(
-
1
-
xi
-
eta
);
N
(
1
)
=
.25
*
(
1
+
xi
)
*
(
1
-
eta
)
*
(
-
1
+
xi
-
eta
);
N
(
2
)
=
.25
*
(
1
+
xi
)
*
(
1
+
eta
)
*
(
-
1
+
xi
+
eta
);
N
(
3
)
=
.25
*
(
1
-
xi
)
*
(
1
+
eta
)
*
(
-
1
-
xi
+
eta
);
N
(
4
)
=
.5
*
(
1
-
xi
*
xi
)
*
(
1
-
eta
);
N
(
5
)
=
.5
*
(
1
+
xi
)
*
(
1
-
eta
*
eta
);
N
(
6
)
=
.5
*
(
1
-
xi
*
xi
)
*
(
1
+
eta
);
N
(
7
)
=
.5
*
(
1
-
xi
)
*
(
1
-
eta
*
eta
);
}
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
,
class
matrix_type
>
inline
void
InterpolationElement
<
_itp_serendip_quadrangle_8
>::
computeDNDS
(
const
vector_type
&
c
,
matrix_type
&
dnds
)
{
const
Real
xi
=
c
(
0
);
const
Real
eta
=
c
(
1
);
/// dN/dxi
dnds
(
0
,
0
)
=
.25
*
(
1
-
eta
)
*
(
2
*
xi
+
eta
);
dnds
(
0
,
1
)
=
.25
*
(
1
-
eta
)
*
(
2
*
xi
-
eta
);
dnds
(
0
,
2
)
=
.25
*
(
1
+
eta
)
*
(
2
*
xi
+
eta
);
dnds
(
0
,
3
)
=
.25
*
(
1
+
eta
)
*
(
2
*
xi
-
eta
);
dnds
(
0
,
4
)
=
-
xi
*
(
1
-
eta
);
dnds
(
0
,
5
)
=
.5
*
(
1
-
eta
*
eta
);
dnds
(
0
,
6
)
=
-
xi
*
(
1
+
eta
);
dnds
(
0
,
7
)
=
-
.5
*
(
1
-
eta
*
eta
);
/// dN/deta
dnds
(
1
,
0
)
=
.25
*
(
1
-
xi
)
*
(
2
*
eta
+
xi
);
dnds
(
1
,
1
)
=
.25
*
(
1
+
xi
)
*
(
2
*
eta
-
xi
);
dnds
(
1
,
2
)
=
.25
*
(
1
+
xi
)
*
(
2
*
eta
+
xi
);
dnds
(
1
,
3
)
=
.25
*
(
1
-
xi
)
*
(
2
*
eta
-
xi
);
dnds
(
1
,
4
)
=
-
.5
*
(
1
-
xi
*
xi
);
dnds
(
1
,
5
)
=
-
eta
*
(
1
+
xi
);
dnds
(
1
,
6
)
=
.5
*
(
1
-
xi
*
xi
);
dnds
(
1
,
7
)
=
-
eta
*
(
1
-
xi
);
}
/* -------------------------------------------------------------------------- */
template
<>
inline
Real
GeometricalElement
<
_gt_quadrangle_8
>::
getInradius
(
const
Matrix
<
Real
>
&
coord
)
{
Real
a
,
b
,
h
;
Vector
<
Real
>
u0
=
coord
(
0
);
Vector
<
Real
>
u1
=
coord
(
1
);
Vector
<
Real
>
u2
=
coord
(
2
);
Vector
<
Real
>
u3
=
coord
(
3
);
Vector
<
Real
>
u4
=
coord
(
4
);
Vector
<
Real
>
u5
=
coord
(
5
);
Vector
<
Real
>
u6
=
coord
(
6
);
Vector
<
Real
>
u7
=
coord
(
7
);
a
=
u0
.
distance
(
u4
);
b
=
u4
.
distance
(
u1
);
h
=
std
::
min
(
a
,
b
);
a
=
u1
.
distance
(
u5
);
b
=
u5
.
distance
(
u2
);
h
=
std
::
min
(
h
,
std
::
min
(
a
,
b
));
a
=
u2
.
distance
(
u6
);
b
=
u6
.
distance
(
u3
);
h
=
std
::
min
(
h
,
std
::
min
(
a
,
b
));
a
=
u3
.
distance
(
u7
);
b
=
u7
.
distance
(
u0
);
h
=
std
::
min
(
h
,
std
::
min
(
a
,
b
));
return
h
;
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_serendip_quadrangle_8
>::
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
();
}
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