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element_class_bernoulli_beam_inline_impl.cc
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rAKA akantu
element_class_bernoulli_beam_inline_impl.cc
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/**
* @file element_class_bernoulli_beam_inline_impl.cc
*
* @author Fabian Barras <fabian.barras@epfl.ch>
*
* @date creation: Fri Jul 15 2011
* @date last modification: Sun Oct 19 2014
*
* @brief Specialization of the element_class class for the type _bernoulli_beam_2
*
* @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
--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]
*/
/* -------------------------------------------------------------------------- */
AKANTU_DEFINE_STRUCTURAL_ELEMENT_CLASS_PROPERTY
(
_bernoulli_beam_2
,
_gt_segment_2
,
_itp_bernoulli_beam
,
_segment_2
,
_ek_structural
,
2
,
_git_segment
,
4
);
AKANTU_DEFINE_STRUCTURAL_ELEMENT_CLASS_PROPERTY
(
_bernoulli_beam_3
,
_gt_segment_2
,
_itp_bernoulli_beam
,
_segment_2
,
_ek_structural
,
3
,
_git_segment
,
4
);
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_bernoulli_beam
>::
computeShapes
(
const
Vector
<
Real
>
&
natural_coords
,
Vector
<
Real
>
&
N
,
const
Matrix
<
Real
>
&
real_coord
,
UInt
id
)
{
/// Compute the dimension of the beam
Vector
<
Real
>
x1
=
real_coord
(
0
);
Vector
<
Real
>
x2
=
real_coord
(
1
);
Real
a
=
.5
*
x1
.
distance
(
x2
);
/// natural coordinate
Real
c
=
natural_coords
(
0
);
switch
(
id
)
{
case
0
:
{
// N
N
(
0
)
=
0.5
*
(
1
-
c
);
N
(
1
)
=
0.5
*
(
1
+
c
);
break
;
}
case
1
:
{
// M
N
(
0
)
=
0.25
*
(
c
*
c
*
c
-
3
*
c
+
2
);
N
(
1
)
=
-
0.25
*
(
c
*
c
*
c
-
3
*
c
-
2
);
break
;
}
case
2
:
{
// L
N
(
0
)
=
0.25
*
a
*
(
c
*
c
*
c
-
c
*
c
-
c
+
1
);
N
(
1
)
=
0.25
*
a
*
(
c
*
c
*
c
+
c
*
c
-
c
-
1
);
break
;
}
case
3
:
{
// M'
N
(
0
)
=
0.75
/
a
*
(
c
*
c
-
1
);
N
(
1
)
=
-
0.75
/
a
*
(
c
*
c
-
1
);
break
;
}
case
4
:
{
// L'
N
(
0
)
=
0.25
*
(
3
*
c
*
c
-
2
*
c
-
1
);
N
(
1
)
=
0.25
*
(
3
*
c
*
c
+
2
*
c
-
1
);
break
;
}
}
}
/* -------------------------------------------------------------------------- */
template
<>
inline
void
InterpolationElement
<
_itp_bernoulli_beam
>::
computeDNDS
(
const
Vector
<
Real
>
&
natural_coords
,
Matrix
<
Real
>
&
dnds
,
const
Matrix
<
Real
>
&
real_nodes_coord
,
UInt
id
)
{
/// Compute the dimension of the beam
Vector
<
Real
>
x1
=
real_nodes_coord
(
0
);
Vector
<
Real
>
x2
=
real_nodes_coord
(
1
);
Real
a
=
.5
*
x1
.
distance
(
x2
);
/// natural coordinate
Real
c
=
natural_coords
(
0
)
*
a
;
switch
(
id
)
{
case
0
:
{
// N'
dnds
(
0
,
0
)
=
-
0.5
/
a
;
dnds
(
0
,
1
)
=
0.5
/
a
;
break
;
}
case
1
:
{
// M''
dnds
(
0
,
0
)
=
-
3.
*
c
/
(
2.
*
pow
(
a
,
3
));
dnds
(
0
,
1
)
=
3.
*
c
/
(
2.
*
pow
(
a
,
3
));
break
;
}
case
2
:
{
// L''
dnds
(
0
,
0
)
=
-
0.5
/
a
*
(
3
*
c
/
a
-
1
);
dnds
(
0
,
1
)
=-
0.5
/
a
*
(
3
*
c
/
a
+
1
);
break
;
}
}
}
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