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element_class_hexahedron_8_inline_impl.cc
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rAKA akantu
element_class_hexahedron_8_inline_impl.cc
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/**
* @file element_class_hexahedron_8_inline_impl.cc
*
* @author Peter Spijker <peter.spijker@epfl.ch>
*
* @date creation: Mon Mar 14 2011
* @date last modification: Fri Jun 13 2014
*
* @brief Specialization of the element_class class for the type _hexahedron_8
*
* @section LICENSE
*
* Copyright (©) 2014 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
\zeta
^
(-1,1,1) | (1,1,1)
8---|------7
/| | /|
/ | | / |
(-1,-1,1) 5----------6 | (1,-1,1)
| | | | |
| | | | |
| | +---|-------> \xi
| | / | |
(-1,1,-1) | 4-/-----|--3 (1,1,-1)
| / / | /
|/ / |/
1-/--------2
(-1,-1,-1) / (1,-1,-1)
/
\eta
@endverbatim
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{llll}
* N1 = (1 - \xi) (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N1}{\partial \xi} = - (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N1}{\partial \eta} = - (1 - \xi) (1 - \zeta) / 8
* & \frac{\partial N1}{\partial \zeta} = - (1 - \xi) (1 - \eta) / 8 \\
* N2 = (1 + \xi) (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N2}{\partial \xi} = (1 - \eta) (1 - \zeta) / 8
* & \frac{\partial N2}{\partial \eta} = - (1 + \xi) (1 - \zeta) / 8
* & \frac{\partial N2}{\partial \zeta} = - (1 + \xi) (1 - \eta) / 8 \\
* N3 = (1 + \xi) (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N3}{\partial \xi} = (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N3}{\partial \eta} = (1 + \xi) (1 - \zeta) / 8
* & \frac{\partial N3}{\partial \zeta} = - (1 + \xi) (1 + \eta) / 8 \\
* N4 = (1 - \xi) (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N4}{\partial \xi} = - (1 + \eta) (1 - \zeta) / 8
* & \frac{\partial N4}{\partial \eta} = (1 - \xi) (1 - \zeta) / 8
* & \frac{\partial N4}{\partial \zeta} = - (1 - \xi) (1 + \eta) / 8 \\
* N5 = (1 - \xi) (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N5}{\partial \xi} = - (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N5}{\partial \eta} = - (1 - \xi) (1 + \zeta) / 8
* & \frac{\partial N5}{\partial \zeta} = (1 - \xi) (1 - \eta) / 8 \\
* N6 = (1 + \xi) (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N6}{\partial \xi} = (1 - \eta) (1 + \zeta) / 8
* & \frac{\partial N6}{\partial \eta} = - (1 + \xi) (1 + \zeta) / 8
* & \frac{\partial N6}{\partial \zeta} = (1 + \xi) (1 - \eta) / 8 \\
* N7 = (1 + \xi) (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N7}{\partial \xi} = (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N7}{\partial \eta} = (1 + \xi) (1 + \zeta) / 8
* & \frac{\partial N7}{\partial \zeta} = (1 + \xi) (1 + \eta) / 8 \\
* N8 = (1 - \xi) (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N8}{\partial \xi} = - (1 + \eta) (1 + \zeta) / 8
* & \frac{\partial N8}{\partial \eta} = (1 - \xi) (1 + \zeta) / 8
* & \frac{\partial N8}{\partial \zeta} = (1 - \xi) (1 + \eta) / 8 \\
* \end{array}
* @f]
*
* @subsection quad_points Position of quadrature points
* @f{eqnarray*}{
* \xi_{q0} &=& -1/\sqrt{3} \qquad \eta_{q0} = -1/\sqrt{3} \qquad \zeta_{q0} = -1/\sqrt{3} \\
* \xi_{q1} &=& 1/\sqrt{3} \qquad \eta_{q1} = -1/\sqrt{3} \qquad \zeta_{q1} = -1/\sqrt{3} \\
* \xi_{q2} &=& 1/\sqrt{3} \qquad \eta_{q2} = 1/\sqrt{3} \qquad \zeta_{q2} = -1/\sqrt{3} \\
* \xi_{q3} &=& -1/\sqrt{3} \qquad \eta_{q3} = 1/\sqrt{3} \qquad \zeta_{q3} = -1/\sqrt{3} \\
* \xi_{q4} &=& -1/\sqrt{3} \qquad \eta_{q4} = -1/\sqrt{3} \qquad \zeta_{q4} = 1/\sqrt{3} \\
* \xi_{q5} &=& 1/\sqrt{3} \qquad \eta_{q5} = -1/\sqrt{3} \qquad \zeta_{q5} = 1/\sqrt{3} \\
* \xi_{q6} &=& 1/\sqrt{3} \qquad \eta_{q6} = 1/\sqrt{3} \qquad \zeta_{q6} = 1/\sqrt{3} \\
* \xi_{q7} &=& -1/\sqrt{3} \qquad \eta_{q7} = 1/\sqrt{3} \qquad \zeta_{q7} = 1/\sqrt{3} \\
* @f}
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_ELEMENT_CLASS_PROPERTY
(
_hexahedron_8
,
_gt_hexahedron_8
,
_itp_lagrange_hexahedron_8
,
_ek_regular
,
3
,
_git_segment
,
2
);
AKANTU_DEFINE_SHAPE
(
_gt_hexahedron_8
,
_gst_square
);
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
>
inline
void
InterpolationElement
<
_itp_lagrange_hexahedron_8
>::
computeShapes
(
const
vector_type
&
c
,
vector_type
&
N
)
{
/// Natural coordinates
N
(
0
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
-
c
(
1
))
*
(
1
-
c
(
2
));
/// N1(q_0)
N
(
1
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
-
c
(
1
))
*
(
1
-
c
(
2
));
/// N2(q_0)
N
(
2
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
+
c
(
1
))
*
(
1
-
c
(
2
));
/// N3(q_0)
N
(
3
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
+
c
(
1
))
*
(
1
-
c
(
2
));
/// N4(q_0)
N
(
4
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
-
c
(
1
))
*
(
1
+
c
(
2
));
/// N5(q_0)
N
(
5
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
-
c
(
1
))
*
(
1
+
c
(
2
));
/// N6(q_0)
N
(
6
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
+
c
(
1
))
*
(
1
+
c
(
2
));
/// N7(q_0)
N
(
7
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
+
c
(
1
))
*
(
1
+
c
(
2
));
/// N8(q_0)
}
/* -------------------------------------------------------------------------- */
template
<>
template
<
class
vector_type
,
class
matrix_type
>
inline
void
InterpolationElement
<
_itp_lagrange_hexahedron_8
>::
computeDNDS
(
const
vector_type
&
c
,
matrix_type
&
dnds
)
{
/**
* @f[
* dnds = \left(
* \begin{array}{cccccccc}
* \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 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 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}
* \end{array}
* \right)
* @f]
*/
dnds
(
0
,
0
)
=
-
.125
*
(
1
-
c
(
1
))
*
(
1
-
c
(
2
));;
dnds
(
0
,
1
)
=
.125
*
(
1
-
c
(
1
))
*
(
1
-
c
(
2
));;
dnds
(
0
,
2
)
=
.125
*
(
1
+
c
(
1
))
*
(
1
-
c
(
2
));;
dnds
(
0
,
3
)
=
-
.125
*
(
1
+
c
(
1
))
*
(
1
-
c
(
2
));;
dnds
(
0
,
4
)
=
-
.125
*
(
1
-
c
(
1
))
*
(
1
+
c
(
2
));;
dnds
(
0
,
5
)
=
.125
*
(
1
-
c
(
1
))
*
(
1
+
c
(
2
));;
dnds
(
0
,
6
)
=
.125
*
(
1
+
c
(
1
))
*
(
1
+
c
(
2
));;
dnds
(
0
,
7
)
=
-
.125
*
(
1
+
c
(
1
))
*
(
1
+
c
(
2
));;
dnds
(
1
,
0
)
=
-
.125
*
(
1
-
c
(
0
))
*
(
1
-
c
(
2
));;
dnds
(
1
,
1
)
=
-
.125
*
(
1
+
c
(
0
))
*
(
1
-
c
(
2
));;
dnds
(
1
,
2
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
-
c
(
2
));;
dnds
(
1
,
3
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
-
c
(
2
));;
dnds
(
1
,
4
)
=
-
.125
*
(
1
-
c
(
0
))
*
(
1
+
c
(
2
));;
dnds
(
1
,
5
)
=
-
.125
*
(
1
+
c
(
0
))
*
(
1
+
c
(
2
));;
dnds
(
1
,
6
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
+
c
(
2
));;
dnds
(
1
,
7
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
+
c
(
2
));;
dnds
(
2
,
0
)
=
-
.125
*
(
1
-
c
(
0
))
*
(
1
-
c
(
1
));;
dnds
(
2
,
1
)
=
-
.125
*
(
1
+
c
(
0
))
*
(
1
-
c
(
1
));;
dnds
(
2
,
2
)
=
-
.125
*
(
1
+
c
(
0
))
*
(
1
+
c
(
1
));;
dnds
(
2
,
3
)
=
-
.125
*
(
1
-
c
(
0
))
*
(
1
+
c
(
1
));;
dnds
(
2
,
4
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
-
c
(
1
));;
dnds
(
2
,
5
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
-
c
(
1
));;
dnds
(
2
,
6
)
=
.125
*
(
1
+
c
(
0
))
*
(
1
+
c
(
1
));;
dnds
(
2
,
7
)
=
.125
*
(
1
-
c
(
0
))
*
(
1
+
c
(
1
));;
}
/* -------------------------------------------------------------------------- */
template
<>
inline
Real
GeometricalElement
<
_gt_hexahedron_8
>::
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
);
Vector
<
Real
>
u4
=
coord
(
4
);
Vector
<
Real
>
u5
=
coord
(
5
);
Vector
<
Real
>
u6
=
coord
(
6
);
Vector
<
Real
>
u7
=
coord
(
7
);
Real
a
=
u0
.
distance
(
u1
);
Real
b
=
u1
.
distance
(
u2
);
Real
c
=
u2
.
distance
(
u3
);
Real
d
=
u3
.
distance
(
u0
);
Real
e
=
u0
.
distance
(
u4
);
Real
f
=
u1
.
distance
(
u5
);
Real
g
=
u2
.
distance
(
u6
);
Real
h
=
u3
.
distance
(
u7
);
Real
i
=
u4
.
distance
(
u5
);
Real
j
=
u5
.
distance
(
u6
);
Real
k
=
u6
.
distance
(
u7
);
Real
l
=
u7
.
distance
(
u4
);
Real
x
=
std
::
min
(
a
,
std
::
min
(
b
,
std
::
min
(
c
,
d
)));
Real
y
=
std
::
min
(
e
,
std
::
min
(
f
,
std
::
min
(
g
,
h
)));
Real
z
=
std
::
min
(
i
,
std
::
min
(
j
,
std
::
min
(
k
,
l
)));
Real
p
=
std
::
min
(
x
,
std
::
min
(
y
,
z
));
return
p
;
}
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