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ElementQuad4.hpp
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rGOOSEFEM GooseFEM
ElementQuad4.hpp
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/**
Implementation of ElementQuad4.h
\file ElementQuad4.hpp
\copyright Copyright 2017. Tom de Geus. All rights reserved.
\license This project is released under the GNU Public License (GPLv3).
*/
#ifndef GOOSEFEM_ELEMENTQUAD4_HPP
#define GOOSEFEM_ELEMENTQUAD4_HPP
#include "ElementQuad4.h"
#include "detail.hpp"
namespace
GooseFEM
{
namespace
Element
{
namespace
Quad4
{
namespace
Gauss
{
inline
size_t
nip
()
{
return
4
;
}
inline
xt
::
xtensor
<
double
,
2
>
xi
()
{
size_t
nip
=
4
;
size_t
ndim
=
2
;
xt
::
xtensor
<
double
,
2
>
xi
=
xt
::
empty
<
double
>
({
nip
,
ndim
});
xi
(
0
,
0
)
=
-
1.0
/
std
::
sqrt
(
3.0
);
xi
(
0
,
1
)
=
-
1.0
/
std
::
sqrt
(
3.0
);
xi
(
1
,
0
)
=
+
1.0
/
std
::
sqrt
(
3.0
);
xi
(
1
,
1
)
=
-
1.0
/
std
::
sqrt
(
3.0
);
xi
(
2
,
0
)
=
+
1.0
/
std
::
sqrt
(
3.0
);
xi
(
2
,
1
)
=
+
1.0
/
std
::
sqrt
(
3.0
);
xi
(
3
,
0
)
=
-
1.0
/
std
::
sqrt
(
3.0
);
xi
(
3
,
1
)
=
+
1.0
/
std
::
sqrt
(
3.0
);
return
xi
;
}
inline
xt
::
xtensor
<
double
,
1
>
w
()
{
size_t
nip
=
4
;
xt
::
xtensor
<
double
,
1
>
w
=
xt
::
empty
<
double
>
({
nip
});
w
(
0
)
=
1.0
;
w
(
1
)
=
1.0
;
w
(
2
)
=
1.0
;
w
(
3
)
=
1.0
;
return
w
;
}
}
// namespace Gauss
namespace
Nodal
{
inline
size_t
nip
()
{
return
4
;
}
inline
xt
::
xtensor
<
double
,
2
>
xi
()
{
size_t
nip
=
4
;
size_t
ndim
=
2
;
xt
::
xtensor
<
double
,
2
>
xi
=
xt
::
empty
<
double
>
({
nip
,
ndim
});
xi
(
0
,
0
)
=
-
1.0
;
xi
(
0
,
1
)
=
-
1.0
;
xi
(
1
,
0
)
=
+
1.0
;
xi
(
1
,
1
)
=
-
1.0
;
xi
(
2
,
0
)
=
+
1.0
;
xi
(
2
,
1
)
=
+
1.0
;
xi
(
3
,
0
)
=
-
1.0
;
xi
(
3
,
1
)
=
+
1.0
;
return
xi
;
}
inline
xt
::
xtensor
<
double
,
1
>
w
()
{
size_t
nip
=
4
;
xt
::
xtensor
<
double
,
1
>
w
=
xt
::
empty
<
double
>
({
nip
});
w
(
0
)
=
1.0
;
w
(
1
)
=
1.0
;
w
(
2
)
=
1.0
;
w
(
3
)
=
1.0
;
return
w
;
}
}
// namespace Nodal
namespace
MidPoint
{
inline
size_t
nip
()
{
return
1
;
}
inline
xt
::
xtensor
<
double
,
2
>
xi
()
{
size_t
nip
=
1
;
size_t
ndim
=
2
;
xt
::
xtensor
<
double
,
2
>
xi
=
xt
::
empty
<
double
>
({
nip
,
ndim
});
xi
(
0
,
0
)
=
0.0
;
xi
(
0
,
1
)
=
0.0
;
return
xi
;
}
inline
xt
::
xtensor
<
double
,
1
>
w
()
{
size_t
nip
=
1
;
xt
::
xtensor
<
double
,
1
>
w
=
xt
::
empty
<
double
>
({
nip
});
w
(
0
)
=
1.0
;
return
w
;
}
}
// namespace MidPoint
inline
Quadrature
::
Quadrature
(
const
xt
::
xtensor
<
double
,
3
>&
x
)
:
Quadrature
(
x
,
Gauss
::
xi
(),
Gauss
::
w
())
{
}
inline
Quadrature
::
Quadrature
(
const
xt
::
xtensor
<
double
,
3
>&
x
,
const
xt
::
xtensor
<
double
,
2
>&
xi
,
const
xt
::
xtensor
<
double
,
1
>&
w
)
{
size_t
nip
=
w
.
size
();
xt
::
xtensor
<
double
,
2
>
N
=
xt
::
empty
<
double
>
({
nip
,
m_nne
});
xt
::
xtensor
<
double
,
3
>
dNxi
=
xt
::
empty
<
double
>
({
nip
,
m_nne
,
m_ndim
});
for
(
size_t
q
=
0
;
q
<
nip
;
++
q
)
{
N
(
q
,
0
)
=
0.25
*
(
1.0
-
xi
(
q
,
0
))
*
(
1.0
-
xi
(
q
,
1
));
N
(
q
,
1
)
=
0.25
*
(
1.0
+
xi
(
q
,
0
))
*
(
1.0
-
xi
(
q
,
1
));
N
(
q
,
2
)
=
0.25
*
(
1.0
+
xi
(
q
,
0
))
*
(
1.0
+
xi
(
q
,
1
));
N
(
q
,
3
)
=
0.25
*
(
1.0
-
xi
(
q
,
0
))
*
(
1.0
+
xi
(
q
,
1
));
}
for
(
size_t
q
=
0
;
q
<
nip
;
++
q
)
{
// - dN / dxi_0
dNxi
(
q
,
0
,
0
)
=
-
0.25
*
(
1.0
-
xi
(
q
,
1
));
dNxi
(
q
,
1
,
0
)
=
+
0.25
*
(
1.0
-
xi
(
q
,
1
));
dNxi
(
q
,
2
,
0
)
=
+
0.25
*
(
1.0
+
xi
(
q
,
1
));
dNxi
(
q
,
3
,
0
)
=
-
0.25
*
(
1.0
+
xi
(
q
,
1
));
// - dN / dxi_1
dNxi
(
q
,
0
,
1
)
=
-
0.25
*
(
1.0
-
xi
(
q
,
0
));
dNxi
(
q
,
1
,
1
)
=
-
0.25
*
(
1.0
+
xi
(
q
,
0
));
dNxi
(
q
,
2
,
1
)
=
+
0.25
*
(
1.0
+
xi
(
q
,
0
));
dNxi
(
q
,
3
,
1
)
=
+
0.25
*
(
1.0
-
xi
(
q
,
0
));
}
this
->
initQuadratureBaseCartesian
(
x
,
xi
,
w
,
N
,
dNxi
);
}
inline
void
Quadrature
::
compute_dN
()
{
#pragma omp parallel
{
xt
::
xtensor
<
double
,
2
>
J
=
xt
::
empty
<
double
>
({
2
,
2
});
xt
::
xtensor
<
double
,
2
>
Jinv
=
xt
::
empty
<
double
>
({
2
,
2
});
#pragma omp for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
x
=
xt
::
adapt
(
&
m_x
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
dNxi
=
xt
::
adapt
(
&
m_dNxi
(
q
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
auto
dNx
=
xt
::
adapt
(
&
m_dNx
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
// J(i,j) += dNxi(m,i) * x(m,j);
J
(
0
,
0
)
=
dNxi
(
0
,
0
)
*
x
(
0
,
0
)
+
dNxi
(
1
,
0
)
*
x
(
1
,
0
)
+
dNxi
(
2
,
0
)
*
x
(
2
,
0
)
+
dNxi
(
3
,
0
)
*
x
(
3
,
0
);
J
(
0
,
1
)
=
dNxi
(
0
,
0
)
*
x
(
0
,
1
)
+
dNxi
(
1
,
0
)
*
x
(
1
,
1
)
+
dNxi
(
2
,
0
)
*
x
(
2
,
1
)
+
dNxi
(
3
,
0
)
*
x
(
3
,
1
);
J
(
1
,
0
)
=
dNxi
(
0
,
1
)
*
x
(
0
,
0
)
+
dNxi
(
1
,
1
)
*
x
(
1
,
0
)
+
dNxi
(
2
,
1
)
*
x
(
2
,
0
)
+
dNxi
(
3
,
1
)
*
x
(
3
,
0
);
J
(
1
,
1
)
=
dNxi
(
0
,
1
)
*
x
(
0
,
1
)
+
dNxi
(
1
,
1
)
*
x
(
1
,
1
)
+
dNxi
(
2
,
1
)
*
x
(
2
,
1
)
+
dNxi
(
3
,
1
)
*
x
(
3
,
1
);
double
Jdet
=
detail
::
tensor
<
2
>::
inv
(
J
,
Jinv
);
// dNx(m,i) += Jinv(i,j) * dNxi(m,j);
for
(
size_t
m
=
0
;
m
<
m_nne
;
++
m
)
{
dNx
(
m
,
0
)
=
Jinv
(
0
,
0
)
*
dNxi
(
m
,
0
)
+
Jinv
(
0
,
1
)
*
dNxi
(
m
,
1
);
dNx
(
m
,
1
)
=
Jinv
(
1
,
0
)
*
dNxi
(
m
,
0
)
+
Jinv
(
1
,
1
)
*
dNxi
(
m
,
1
);
}
m_vol
(
e
,
q
)
=
m_w
(
q
)
*
Jdet
;
}
}
}
}
inline
void
Quadrature
::
interp_N_vector
(
const
xt
::
xtensor
<
double
,
3
>&
elemvec
,
xt
::
xtensor
<
double
,
3
>&
qvector
)
const
{
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
elemvec
,
{
m_nelem
,
m_nne
,
m_ndim
}));
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
qvector
,
{
m_nelem
,
m_nip
,
m_ndim
}));
qvector
.
fill
(
0.0
);
#pragma omp parallel for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
u
=
xt
::
adapt
(
&
elemvec
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
N
=
xt
::
adapt
(
&
m_N
(
q
,
0
),
xt
::
xshape
<
m_nne
>
());
auto
ui
=
xt
::
adapt
(
&
qvector
(
e
,
q
,
0
),
xt
::
xshape
<
m_ndim
>
());
ui
(
0
)
=
N
(
0
)
*
u
(
0
,
0
)
+
N
(
1
)
*
u
(
1
,
0
)
+
N
(
2
)
*
u
(
2
,
0
)
+
N
(
3
)
*
u
(
3
,
0
);
ui
(
1
)
=
N
(
0
)
*
u
(
0
,
1
)
+
N
(
1
)
*
u
(
1
,
1
)
+
N
(
2
)
*
u
(
2
,
1
)
+
N
(
3
)
*
u
(
3
,
1
);
}
}
}
inline
void
Quadrature
::
gradN_vector
(
const
xt
::
xtensor
<
double
,
3
>&
elemvec
,
xt
::
xtensor
<
double
,
4
>&
qtensor
)
const
{
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
elemvec
,
{
m_nelem
,
m_nne
,
m_ndim
}));
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
qtensor
,
{
m_nelem
,
m_nip
,
m_ndim
,
m_ndim
}));
#pragma omp parallel for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
u
=
xt
::
adapt
(
&
elemvec
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
dNx
=
xt
::
adapt
(
&
m_dNx
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
auto
gradu
=
xt
::
adapt
(
&
qtensor
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_ndim
,
m_ndim
>
());
// gradu(i,j) += dNx(m,i) * u(m,j)
gradu
(
0
,
0
)
=
dNx
(
0
,
0
)
*
u
(
0
,
0
)
+
dNx
(
1
,
0
)
*
u
(
1
,
0
)
+
dNx
(
2
,
0
)
*
u
(
2
,
0
)
+
dNx
(
3
,
0
)
*
u
(
3
,
0
);
gradu
(
0
,
1
)
=
dNx
(
0
,
0
)
*
u
(
0
,
1
)
+
dNx
(
1
,
0
)
*
u
(
1
,
1
)
+
dNx
(
2
,
0
)
*
u
(
2
,
1
)
+
dNx
(
3
,
0
)
*
u
(
3
,
1
);
gradu
(
1
,
0
)
=
dNx
(
0
,
1
)
*
u
(
0
,
0
)
+
dNx
(
1
,
1
)
*
u
(
1
,
0
)
+
dNx
(
2
,
1
)
*
u
(
2
,
0
)
+
dNx
(
3
,
1
)
*
u
(
3
,
0
);
gradu
(
1
,
1
)
=
dNx
(
0
,
1
)
*
u
(
0
,
1
)
+
dNx
(
1
,
1
)
*
u
(
1
,
1
)
+
dNx
(
2
,
1
)
*
u
(
2
,
1
)
+
dNx
(
3
,
1
)
*
u
(
3
,
1
);
}
}
}
inline
void
Quadrature
::
gradN_vector_T
(
const
xt
::
xtensor
<
double
,
3
>&
elemvec
,
xt
::
xtensor
<
double
,
4
>&
qtensor
)
const
{
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
elemvec
,
{
m_nelem
,
m_nne
,
m_ndim
}));
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
qtensor
,
{
m_nelem
,
m_nip
,
m_ndim
,
m_ndim
}));
#pragma omp parallel for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
u
=
xt
::
adapt
(
&
elemvec
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
dNx
=
xt
::
adapt
(
&
m_dNx
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
auto
gradu
=
xt
::
adapt
(
&
qtensor
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_ndim
,
m_ndim
>
());
// gradu(j,i) += dNx(m,i) * u(m,j)
gradu
(
0
,
0
)
=
dNx
(
0
,
0
)
*
u
(
0
,
0
)
+
dNx
(
1
,
0
)
*
u
(
1
,
0
)
+
dNx
(
2
,
0
)
*
u
(
2
,
0
)
+
dNx
(
3
,
0
)
*
u
(
3
,
0
);
gradu
(
1
,
0
)
=
dNx
(
0
,
0
)
*
u
(
0
,
1
)
+
dNx
(
1
,
0
)
*
u
(
1
,
1
)
+
dNx
(
2
,
0
)
*
u
(
2
,
1
)
+
dNx
(
3
,
0
)
*
u
(
3
,
1
);
gradu
(
0
,
1
)
=
dNx
(
0
,
1
)
*
u
(
0
,
0
)
+
dNx
(
1
,
1
)
*
u
(
1
,
0
)
+
dNx
(
2
,
1
)
*
u
(
2
,
0
)
+
dNx
(
3
,
1
)
*
u
(
3
,
0
);
gradu
(
1
,
1
)
=
dNx
(
0
,
1
)
*
u
(
0
,
1
)
+
dNx
(
1
,
1
)
*
u
(
1
,
1
)
+
dNx
(
2
,
1
)
*
u
(
2
,
1
)
+
dNx
(
3
,
1
)
*
u
(
3
,
1
);
}
}
}
inline
void
Quadrature
::
symGradN_vector
(
const
xt
::
xtensor
<
double
,
3
>&
elemvec
,
xt
::
xtensor
<
double
,
4
>&
qtensor
)
const
{
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
elemvec
,
{
m_nelem
,
m_nne
,
m_ndim
}));
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
qtensor
,
{
m_nelem
,
m_nip
,
m_ndim
,
m_ndim
}));
#pragma omp parallel for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
u
=
xt
::
adapt
(
&
elemvec
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
dNx
=
xt
::
adapt
(
&
m_dNx
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
auto
eps
=
xt
::
adapt
(
&
qtensor
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_ndim
,
m_ndim
>
());
// gradu(i,j) += dNx(m,i) * u(m,j)
// eps(j,i) = 0.5 * (gradu(i,j) + gradu(j,i))
eps
(
0
,
0
)
=
dNx
(
0
,
0
)
*
u
(
0
,
0
)
+
dNx
(
1
,
0
)
*
u
(
1
,
0
)
+
dNx
(
2
,
0
)
*
u
(
2
,
0
)
+
dNx
(
3
,
0
)
*
u
(
3
,
0
);
eps
(
1
,
1
)
=
dNx
(
0
,
1
)
*
u
(
0
,
1
)
+
dNx
(
1
,
1
)
*
u
(
1
,
1
)
+
dNx
(
2
,
1
)
*
u
(
2
,
1
)
+
dNx
(
3
,
1
)
*
u
(
3
,
1
);
eps
(
0
,
1
)
=
0.5
*
(
dNx
(
0
,
0
)
*
u
(
0
,
1
)
+
dNx
(
1
,
0
)
*
u
(
1
,
1
)
+
dNx
(
2
,
0
)
*
u
(
2
,
1
)
+
dNx
(
3
,
0
)
*
u
(
3
,
1
)
+
dNx
(
0
,
1
)
*
u
(
0
,
0
)
+
dNx
(
1
,
1
)
*
u
(
1
,
0
)
+
dNx
(
2
,
1
)
*
u
(
2
,
0
)
+
dNx
(
3
,
1
)
*
u
(
3
,
0
));
eps
(
1
,
0
)
=
eps
(
0
,
1
);
}
}
}
inline
void
Quadrature
::
int_N_scalar_NT_dV
(
const
xt
::
xtensor
<
double
,
2
>&
qscalar
,
xt
::
xtensor
<
double
,
3
>&
elemmat
)
const
{
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
qscalar
,
{
m_nelem
,
m_nip
}));
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
elemmat
,
{
m_nelem
,
m_nne
*
m_ndim
,
m_nne
*
m_ndim
}));
elemmat
.
fill
(
0.0
);
#pragma omp parallel for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
M
=
xt
::
adapt
(
&
elemmat
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
*
m_ndim
,
m_nne
*
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
N
=
xt
::
adapt
(
&
m_N
(
q
,
0
),
xt
::
xshape
<
m_nne
>
());
auto
&
vol
=
m_vol
(
e
,
q
);
auto
&
rho
=
qscalar
(
e
,
q
);
// M(m*ndim+i,n*ndim+i) += N(m) * scalar * N(n) * dV
for
(
size_t
m
=
0
;
m
<
m_nne
;
++
m
)
{
for
(
size_t
n
=
0
;
n
<
m_nne
;
++
n
)
{
M
(
m
*
m_ndim
+
0
,
n
*
m_ndim
+
0
)
+=
N
(
m
)
*
rho
*
N
(
n
)
*
vol
;
M
(
m
*
m_ndim
+
1
,
n
*
m_ndim
+
1
)
+=
N
(
m
)
*
rho
*
N
(
n
)
*
vol
;
}
}
}
}
}
inline
void
Quadrature
::
int_gradN_dot_tensor2_dV
(
const
xt
::
xtensor
<
double
,
4
>&
qtensor
,
xt
::
xtensor
<
double
,
3
>&
elemvec
)
const
{
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
qtensor
,
{
m_nelem
,
m_nip
,
m_ndim
,
m_ndim
}));
GOOSEFEM_ASSERT
(
xt
::
has_shape
(
elemvec
,
{
m_nelem
,
m_nne
,
m_ndim
}));
elemvec
.
fill
(
0.0
);
#pragma omp parallel for
for
(
size_t
e
=
0
;
e
<
m_nelem
;
++
e
)
{
auto
f
=
xt
::
adapt
(
&
elemvec
(
e
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
for
(
size_t
q
=
0
;
q
<
m_nip
;
++
q
)
{
auto
dNx
=
xt
::
adapt
(
&
m_dNx
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_nne
,
m_ndim
>
());
auto
sig
=
xt
::
adapt
(
&
qtensor
(
e
,
q
,
0
,
0
),
xt
::
xshape
<
m_ndim
,
m_ndim
>
());
auto
&
vol
=
m_vol
(
e
,
q
);
for
(
size_t
m
=
0
;
m
<
m_nne
;
++
m
)
{
f
(
m
,
0
)
+=
(
dNx
(
m
,
0
)
*
sig
(
0
,
0
)
+
dNx
(
m
,
1
)
*
sig
(
1
,
0
))
*
vol
;
f
(
m
,
1
)
+=
(
dNx
(
m
,
0
)
*
sig
(
0
,
1
)
+
dNx
(
m
,
1
)
*
sig
(
1
,
1
))
*
vol
;
}
}
}
}
}
// namespace Quad4
}
// namespace Element
}
// namespace GooseFEM
#endif
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