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element_class_bernoulli_beam_2_inline_impl.cc
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rAKA akantu
element_class_bernoulli_beam_2_inline_impl.cc
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/**
* @file element_class_bernoulli_beam_2.cc
* @author Fabian Barras <fabian.barras@epfl.ch>
* @date Thu Mar 31 14:02:22 2011
*
* @brief Specialization of the element_class class for the type _bernoulli_beam_2
*
* @section LICENSE
*
* Copyright (©) 2010-2011 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
--x-----q1----|----q2-----x---> x
-a 0 a
@endverbatim
*
* @subsection coords Nodes coordinates
*
* @f[
* \begin{array}{ll}
* x_{1} = -a & x_{2} = a
* \end{array}
* @f]
*
* @subsection shapes Shape functions
* @f[
* \begin{array}{ll}
* N_1(x) &= \frac{1-x}{2a}\\
* N_2(x) &= \frac{1+x}{2a}
* \end{array}
*
* \begin{array}{ll}
* M_1(x) &= 1/4(x^{3}/a^{3}-3x/a+2)\\
* M_2(x) &= -1/4(x^{3}/a^{3}-3x/a-2)
* \end{array}
*
* \begin{array}{ll}
* L_1(x) &= a/4(x^{3}/a^{3}-x^{2}/a^{2}-x/a+1)\\
* L_2(x) &= a/4(x^{3}/a^{3}+x^{2}/a^{2}-x/a-1)
* \end{array}
*
* \begin{array}{ll}
* M'_1(x) &= 3/4a(x^{2}/a^{2}-1)\\
* M'_2(x) &= -3/4a(x^{2}/a^{2}-1)
* \end{array}
*
* \begin{array}{ll}
* L'_1(x) &= 1/4(3x^{2}/a^{2}-2x/a-1)\\
* L'_2(x) &= 1/4(3x^{2}/a^{2}+2x/a-1)
* \end{array}
*@f]
*
* @subsection dnds Shape derivatives
*
*@f[
* \begin{array}{ll}
* N'_1(x) &= -1/2a\\
* N'_2(x) &= 1/2a
* \end{array}]
*
* \begin{array}{ll}
* -M''_1(x) &= -3x/(2a^{3})\\
* -M''_2(x) &= 3x/(2a^{3})\\
* \end{array}
*
* \begin{array}{ll}
* -L''_1(x) &= -1/2a(3x/a-1)\\
* -L''_2(x) &= -1/2a(3x/a+1)
* \end{array}
*@f]
*
* @subsection quad_points Position of quadrature points
*
* @f[
* \begin{array}{ll}
* x_{q1} = -a/\sqrt{3} & x_{q2} = a/\sqrt{3}
* \end{array}
* @f]
*/
/* -------------------------------------------------------------------------- */
template
<>
UInt
ElementClass
<
_bernoulli_beam_2
>::
nb_nodes_per_element
;
template
<>
UInt
ElementClass
<
_bernoulli_beam_2
>::
nb_quadrature_points
;
template
<>
UInt
ElementClass
<
_bernoulli_beam_2
>::
spatial_dimension
;
/* -------------------------------------------------------------------------- */
template
<>
inline
void
ElementClass
<
_bernoulli_beam_2
>::
computeShapes
(
const
Real
*
natural_coords
,
Real
*
shapes
,
const
Real
*
local_coord
,
UInt
id
)
{
/// Compute the dimension of the beam
Real
a
=
.5
*
Math
::
distance_2d
(
local_coord
,
local_coord
+
2
);
/// natural coordinate
Real
c
=
(
*
natural_coords
)
*
a
;
switch
(
id
)
{
case
0
:
shapes
[
0
]
=
0.5
*
(
1
-
c
/
a
);
shapes
[
1
]
=
0.5
*
(
1
+
c
/
a
);
break
;
case
1
:
shapes
[
0
]
=
0.25
*
(
pow
(
c
,
3
)
/
pow
(
a
,
3
)
-
3
*
c
/
a
+
2
);
shapes
[
1
]
=-
0.25
*
(
pow
(
c
,
3
)
/
pow
(
a
,
3
)
-
3
*
c
/
a
-
2
);
break
;
case
2
:
shapes
[
0
]
=
0.25
*
a
*
(
pow
(
c
,
3
)
/
pow
(
a
,
3
)
-
c
*
c
/
(
a
*
a
)
-
c
/
a
+
1
);
shapes
[
1
]
=
0.25
*
a
*
(
pow
(
c
,
3
)
/
pow
(
a
,
3
)
+
c
*
c
/
(
a
*
a
)
-
c
/
a
-
1
);
break
;
case
3
:
shapes
[
0
]
=
0.75
/
a
*
(
c
*
c
/
(
a
*
a
)
-
1
);
shapes
[
1
]
=-
0.75
/
a
*
(
c
*
c
/
(
a
*
a
)
-
1
);
break
;
case
4
:
shapes
[
0
]
=
0.25
*
(
3
*
c
*
c
/
(
a
*
a
)
-
2
*
c
/
a
-
1
);
shapes
[
1
]
=
0.25
*
(
3
*
c
*
c
/
(
a
*
a
)
+
2
*
c
/
a
-
1
);
break
;
}
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
ElementClass
<
_bernoulli_beam_2
>::
computeShapeDerivatives
(
const
Real
*
natural_coords
,
Real
*
shape_deriv
,
const
Real
*
local_coord
,
UInt
id
)
{
/// Compute the dimension of the beam
Real
a
=
0.5
*
Math
::
distance_2d
(
local_coord
,
local_coord
+
2
);
Real
x1
=*
local_coord
;
Real
y1
=*
(
local_coord
+
1
);
Real
x2
=*
(
local_coord
+
2
);
Real
y2
=*
(
local_coord
+
3
);
Real
tetha
=
std
::
atan
((
y2
-
y1
)
/
(
x2
-
x1
));
Real
pi
=
std
::
atan
(
1.0
)
*
4
;
if
((
x2
-
x1
)
<
0
)
{
tetha
+=
pi
;
}
/// natural coordinate
Real
c
=
(
*
natural_coords
)
*
a
;
/// Definition of the rotation matrix
Real
T
[
4
];
T
[
0
]
=
cos
(
tetha
);
T
[
1
]
=
sin
(
tetha
);
T
[
2
]
=-
T
[
1
];
T
[
3
]
=
T
[
0
];
// B archetype
Real
shape_deriv_arch
[
4
];
switch
(
id
)
{
case
0
:
shape_deriv_arch
[
0
]
=-
0.5
/
a
;
shape_deriv_arch
[
1
]
=
0
;
shape_deriv_arch
[
2
]
=
0.5
/
a
;
shape_deriv_arch
[
3
]
=
0
;
Math
::
matrix_matrix
(
2
,
2
,
2
,
shape_deriv_arch
,
T
,
shape_deriv
);
break
;
case
1
:
shape_deriv_arch
[
0
]
=
0.
;
shape_deriv_arch
[
1
]
=-
3.
*
c
/
(
2.
*
pow
(
a
,
3
));
shape_deriv_arch
[
2
]
=
0.
;
shape_deriv_arch
[
3
]
=
3.
*
c
/
(
2.
*
pow
(
a
,
3
));
Math
::
matrix_matrix
(
2
,
2
,
2
,
shape_deriv_arch
,
T
,
shape_deriv
);
break
;
case
2
:
shape_deriv
[
0
]
=-
0.5
/
a
*
(
3
*
c
/
a
-
1
);
shape_deriv
[
1
]
=-
0.5
/
a
*
(
3
*
c
/
a
+
1
);
shape_deriv
[
2
]
=
0
;
shape_deriv
[
3
]
=
0
;
break
;
}
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
ElementClass
<
_bernoulli_beam_2
>::
computeJacobian
(
const
Real
*
coord
,
const
UInt
nb_points
,
__attribute__
((
unused
))
const
UInt
dimension
,
Real
*
jac
){
Real
a
=
0.5
*
Math
::
distance_2d
(
coord
,
coord
+
2
);
for
(
UInt
p
=
0
;
p
<
nb_points
;
++
p
)
{
jac
[
p
]
=
a
;
}
}
/* -------------------------------------------------------------------------- */
template
<>
inline
Real
ElementClass
<
_bernoulli_beam_2
>::
getInradius
(
const
Real
*
coord
)
{
return
Math
::
distance_2d
(
coord
,
coord
+
2
);
}
/* -------------------------------------------------------------------------- */
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