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element_class_quadrangle_4_inline_impl.cc
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rAKA akantu
element_class_quadrangle_4_inline_impl.cc
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/**
* @file element_class_quadrangle_4_inline_impl.cc
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date creation: Mon Dec 13 2010
* @date last modification: Mon Dec 08 2014
*
* @brief Specialization of the element_class class for the type _quadrangle_4
*
* @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) | (1,1)
x---------x
| | |
| | |
--|---------|-----> \xi
| | |
| | |
x---------x
(-1,-1) | (1,-1)
@endverbatim
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{lll}
* N1 = (1 - \xi) (1 - \eta) / 4
* & \frac{\partial N1}{\partial \xi} = - (1 - \eta) / 4
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi) / 4 \\
* N2 = (1 + \xi) (1 - \eta) / 4 \\
* & \frac{\partial N2}{\partial \xi} = (1 - \eta) / 4
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi) / 4 \\
* N3 = (1 + \xi) (1 + \eta) / 4 \\
* & \frac{\partial N3}{\partial \xi} = (1 + \eta) / 4
* & \frac{\partial N3}{\partial \eta} = (1 + \xi) / 4 \\
* N4 = (1 - \xi) (1 + \eta) / 4
* & \frac{\partial N4}{\partial \xi} = - (1 + \eta) / 4
* & \frac{\partial N4}{\partial \eta} = (1 - \xi) / 4 \\
* \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_4
,
_gt_quadrangle_4
,
_itp_lagrange_quadrangle_4
,
_ek_regular
,
2
,
_git_segment
,
2
);
AKANTU_DEFINE_SHAPE
(
_gt_quadrangle_4
,
_gst_square
);
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
>
inline
void
InterpolationElement
<
_itp_lagrange_quadrangle_4
>::
computeShapes
(
const
vector_type
&
c
,
vector_type
&
N
)
{
N
(
0
)
=
1.
/
4.
*
(
1.
-
c
(
0
))
*
(
1.
-
c
(
1
));
/// N1(q_0)
N
(
1
)
=
1.
/
4.
*
(
1.
+
c
(
0
))
*
(
1.
-
c
(
1
));
/// N2(q_0)
N
(
2
)
=
1.
/
4.
*
(
1.
+
c
(
0
))
*
(
1.
+
c
(
1
));
/// N3(q_0)
N
(
3
)
=
1.
/
4.
*
(
1.
-
c
(
0
))
*
(
1.
+
c
(
1
));
/// N4(q_0)
}
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
,
class
matrix_type
>
inline
void
InterpolationElement
<
_itp_lagrange_quadrangle_4
>::
computeDNDS
(
const
vector_type
&
c
,
matrix_type
&
dnds
)
{
/**
* @f[
* dnds = \left(
* \begin{array}{cccc}
* \frac{\partial N1}{\partial \xi} & \frac{\partial N2}{\partial \xi}
* & \frac{\partial N3}{\partial \xi} & \frac{\partial N4}{\partial \xi}\\
* \frac{\partial N1}{\partial \eta} & \frac{\partial N2}{\partial \eta}
* & \frac{\partial N3}{\partial \eta} & \frac{\partial N4}{\partial \eta}
* \end{array}
* \right)
* @f]
*/
dnds
(
0
,
0
)
=
-
1.
/
4.
*
(
1.
-
c
(
1
));
dnds
(
0
,
1
)
=
1.
/
4.
*
(
1.
-
c
(
1
));
dnds
(
0
,
2
)
=
1.
/
4.
*
(
1.
+
c
(
1
));
dnds
(
0
,
3
)
=
-
1.
/
4.
*
(
1.
+
c
(
1
));
dnds
(
1
,
0
)
=
-
1.
/
4.
*
(
1.
-
c
(
0
));
dnds
(
1
,
1
)
=
-
1.
/
4.
*
(
1.
+
c
(
0
));
dnds
(
1
,
2
)
=
1.
/
4.
*
(
1.
+
c
(
0
));
dnds
(
1
,
3
)
=
1.
/
4.
*
(
1.
-
c
(
0
));
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_lagrange_quadrangle_4
>::
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_quadrangle_4
>::
getInradius
(
const
Matrix
<
Real
>
&
coord
)
{
Vector
<
Real
>
u0
=
coord
(
0
);
Vector
<
Real
>
u1
=
coord
(
1
);
Vector
<
Real
>
u2
=
coord
(
2
);
Vector
<
Real
>
u3
=
coord
(
3
);
Real
a
=
u0
.
distance
(
u1
);
Real
b
=
u1
.
distance
(
u2
);
Real
c
=
u2
.
distance
(
u3
);
Real
d
=
u3
.
distance
(
u0
);
// Real septimetre = (a + b + c + d) / 2.;
// Real p = Math::distance_2d(coord + 0, coord + 4);
// Real q = Math::distance_2d(coord + 2, coord + 6);
// Real area = sqrt(4*(p*p * q*q) - (a*a + b*b + c*c + d*d)*(a*a + c*c - b*b - d*d)) / 4.;
// Real h = sqrt(area); // to get a length
// Real h = area / septimetre; // formula of inradius for circumscritable quadrelateral
Real
h
=
std
::
min
(
a
,
std
::
min
(
b
,
std
::
min
(
c
,
d
)));
return
h
;
}
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