Page Menu
Home
c4science
Search
Configure Global Search
Log In
Files
F80754837
manual-appendix-elements.tex
No One
Temporary
Actions
Download File
Edit File
Delete File
View Transforms
Subscribe
Mute Notifications
Award Token
Subscribers
None
File Metadata
Details
File Info
Storage
Attached
Created
Mon, Sep 2, 10:40
Size
30 KB
Mime Type
text/x-tex
Expires
Wed, Sep 4, 10:40 (2 d)
Engine
blob
Format
Raw Data
Handle
18285691
Attached To
rAKA akantu
manual-appendix-elements.tex
View Options
\chapter
{
Shape Functions
\index
{
Elements
}}
\label
{
app:elements
}
Schmatic overview of all the element types defined in
\akantu
is described in
Section~
\ref
{
sec:elements
}
. In this appendix, more detailed information (shape
function, location of Gaussian quadrature points, and so on) of each of these
types is listed. For each element type, the coordinates of the nodes are given
in the isoparametric frame of reference, together with the shape functions (and
their derivatives) on these respective nodes. Also all the Gaussian quadrature
points within each element are assigned (together with the weight that is
applied on these points). The graphical representations of all the element types
can be found in Section~
\ref
{
sec:elements
}
.
%%%%%%%%%% 1D %%%%%%%%%
%\section{Isoparametric Elements in 1D\index{Elements!1D}}
\section
{
1D-Shape Functions
\index
{
Elements!1D
}}
\subsection
{
Segment 2
\index
{
Elements!1D!Segment 2
}}
\begin
{
Element
}{
1D
}
1
&
\inelemone
{
-1
}
&
$
\half\left
(
1
-
\xi\right
)
$
&
\inelemone
{
-
\half
}
\\
\elemline
2
&
\inelemone
{
1
}
&
$
\half\left
(
1
+
\xi\right
)
$
&
\inelemone
{
\half
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lc
}
Coord.
\elemcooroned
&
0
\\
\elemline
Weight
&
2
\\
\end
{
QuadPoints
}
\subsection
{
Segment 3
\index
{
Elements!1D!Segment 3
}}
\begin
{
Element
}{
1D
}
1
&
\inelemone
{
-1
}
&
$
\half\xi\left
(
\xi
-
1
\right
)
$
&
\inelemone
{
\xi
-
\half
}
\\
\elemline
2
&
\inelemone
{
1
}
&
$
\half\xi\left
(
\xi
+
1
\right
)
$
&
\inelemone
{
\xi
+
\half
}
\\
\elemline
3
&
\inelemone
{
0
}
&
$
1
-
\xi
^{
2
}
$
&
\inelemone
{
-2
\xi
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lcc
}
Coord.
\elemcooroned
&
$
-
1
/
\sqrt
{
3
}
$
&
$
1
/
\sqrt
{
3
}
$
\\
\elemline
Weight
&
1
&
1
\\
\end
{
QuadPoints
}
%%%%%%%%%% 2D %%%%%%%%%
%\section{Isoparametric Elements in 2D\index{Elements!2D}}
\section
{
2D-Shape Functions
\index
{
Elements!2D
}}
\subsection
{
Triangle 3
\index
{
Elements!2D!Triangle 3
}}
\begin
{
Element
}{
2D
}
1
&
\inelemtwo
{
0
}{
0
}
&
$
1
-
\xi
-
\eta
$
&
\inelemtwo
{
-1
}{
-1
}
\\
\elemline
2
&
\inelemtwo
{
1
}{
0
}
&
$
\xi
$
&
\inelemtwo
{
1
}{
0
}
\\
\elemline
3
&
\inelemtwo
{
0
}{
1
}
&
$
\eta
$
&
\inelemtwo
{
0
}{
1
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lc
}
Coord.
\elemcoortwod
&
\inquadtwo
{
\third
}{
\third
}
\\
\elemline
Weight
&
\half
\\
\end
{
QuadPoints
}
\subsection
{
Triangle 6
\index
{
Elements!2D!Triangle 6
}}
\begin
{
Element
}{
2D
}
1
&
\inelemtwo
{
0
}{
0
}
&
$
-
\left
(
1
-
\xi
-
\eta\right
)
\left
(
1
-
2
\left
(
1
-
\xi
-
\eta\right
)
\right
)
$
&
\inelemtwo
{
1-4
\left
(1-
\xi
-
\eta\right
)
}{
1-4
\left
(1-
\xi
-
\eta\right
)
}
\\
\elemline
2
&
\inelemtwo
{
1
}{
0
}
&
$
-
\xi\left
(
1
-
2
\xi\right
)
$
&
\inelemtwo
{
4
\xi
-1
}{
0
}
\\
\elemline
3
&
\inelemtwo
{
0
}{
1
}
&
$
-
\eta\left
(
1
-
2
\eta\right
)
$
&
\inelemtwo
{
0
}{
4
\eta
-1
}
\\
\elemline
4
&
\inelemtwo
{
\half
}{
0
}
&
$
4
\xi\left
(
1
-
\xi
-
\eta\right
)
$
&
\inelemtwo
{
4
\left
(1-2
\xi
-
\eta\right
)
}{
-4
\xi
}
\\
\elemline
5
&
\inelemtwo
{
\half
}{
\half
}
&
$
4
\xi\eta
$
&
\inelemtwo
{
4
\eta
}{
4
\xi
}
\\
\elemline
6
&
\inelemtwo
{
0
}{
\half
}
&
$
4
\eta\left
(
1
-
\xi
-
\eta\right
)
$
&
\inelemtwo
{
-4
\eta
}{
4
\left
(1-
\xi
-2
\eta\right
)
}
\\
\elemline
\end
{
Element
}
\begin
{
QuadPoints
}{
lccc
}
Coord.
\elemcoortwod
&
\inquadtwo
{
\sixth
}{
\sixth
}
&
\inquadtwo
{
\twothird
}{
\sixth
}
&
\inquadtwo
{
\sixth
}{
\twothird
}
\\
\elemline
Weight
&
\sixth
&
\sixth
&
\sixth
\\
\end
{
QuadPoints
}
\clearpage
\subsection
{
Quadrangle 4
\index
{
Elements!2D!Quadrangle 4
}}
\begin
{
Element
}{
2D
}
1
&
\inelemtwo
{
-1
}{
-1
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
$
&
\inelemtwo
{
-
\quart\left
(1-
\eta\right
)
}{
-
\quart\left
(1-
\xi\right
)
}
\\
\elemline
2
&
\inelemtwo
{
1
}{
-1
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
$
&
\inelemtwo
{
\quart\left
(1-
\eta\right
)
}{
-
\quart\left
(1+
\xi\right
)
}
\\
\elemline
3
&
\inelemtwo
{
1
}{
1
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
$
&
\inelemtwo
{
\quart\left
(1+
\eta\right
)
}{
\quart\left
(1+
\xi\right
)
}
\\
\elemline
4
&
\inelemtwo
{
-1
}{
1
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
$
&
\inelemtwo
{
-
\quart\left
(1+
\eta\right
)
}{
\quart\left
(1-
\xi\right
)
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lcccc
}
\elemcoortwod
&
\inquadtwo
{
-
\invsqrtthree
}{
-
\invsqrtthree
}
&
\inquadtwo
{
\invsqrtthree
}{
-
\invsqrtthree
}
&
\inquadtwo
{
\invsqrtthree
}{
\invsqrtthree
}
&
\inquadtwo
{
-
\invsqrtthree
}{
\invsqrtthree
}
\\
\elemline
Weight
&
1
&
1
&
1
&
1
\\
\end
{
QuadPoints
}
\subsection
{
Quadrangle 8
\index
{
Elements!2D!Quadrangle 8
}}
\begin
{
Element
}{
2D
}
1
&
\inelemtwo
{
-1
}{
-1
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(-
1
-
\xi
-
\eta\right
)
$
&
\inelemtwo
{
\quart\left
(1-
\eta\right
)
\left
(2
\xi
+
\eta\right
)
}
{
\quart\left
(1-
\xi\right
)
\left
(
\xi
+2
\eta\right
)
}
\\
\elemline
2
&
\inelemtwo
{
1
}{
-1
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(-
1
+
\xi
-
\eta\right
)
$
&
\inelemtwo
{
\quart\left
(1-
\eta\right
)
\left
(2
\xi
-
\eta\right
)
}
{
-
\quart\left
(1+
\xi\right
)
\left
(
\xi
-2
\eta\right
)
}
\\
\elemline
3
&
\inelemtwo
{
1
}{
1
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(-
1
+
\xi
+
\eta\right
)
$
&
\inelemtwo
{
\quart\left
(1+
\eta\right
)
\left
(2
\xi
+
\eta\right
)
}
{
\quart\left
(1+
\xi\right
)
\left
(
\xi
+2
\eta\right
)
}
\\
\elemline
4
&
\inelemtwo
{
-1
}{
1
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(-
1
-
\xi
+
\eta\right
)
$
&
\inelemtwo
{
\quart\left
(1+
\eta\right
)
\left
(2
\xi
-
\eta\right
)
}
{
-
\quart\left
(1-
\xi\right
)
\left
(
\xi
-2
\eta\right
)
}
\\
\elemline
5
&
\inelemtwo
{
0
}{
-1
}
&
$
\half\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
-
\eta\right
)
$
&
\inelemtwo
{
-
\xi\left
(1-
\eta\right
)
}
{
-
\half\left
(1-
\xi
^{
2
}
\right
)
}
\\
\elemline
6
&
\inelemtwo
{
1
}{
0
}
&
$
\half\left
(
1
+
\xi\right
)
\left
(
1
-
\eta
^{
2
}
\right
)
$
&
\inelemtwo
{
\half\left
(1-
\eta
^{
2
}
\right
)
}
{
-
\eta\left
(1+
\xi\right
)
}
\\
\elemline
7
&
\inelemtwo
{
0
}{
1
}
&
$
\half\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
+
\eta\right
)
$
&
\inelemtwo
{
-
\xi\left
(1+
\eta\right
)
}
{
\half\left
(1-
\xi
^{
2
}
\right
)
}
\\
\elemline
8
&
\inelemtwo
{
-1
}{
0
}
&
$
\half\left
(
1
-
\xi\right
)
\left
(
1
-
\eta
^{
2
}
\right
)
$
&
\inelemtwo
{
-
\half\left
(1-
\eta
^{
2
}
\right
)
}
{
-
\eta\left
(1-
\xi\right
)
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lccccc
}
Coord.
\elemcoortwod
&
\inquadtwo
{
0
}{
0
}
&
\inquadtwo
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadtwo
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadtwo
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadtwo
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
\\
\elemline
Weight
&
64/81
&
25/81
&
25/81
&
25/81
&
25/81
\\
\elemline
Coord.
\elemcoortwod
&
\inquadtwo
{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadtwo
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
&
\inquadtwo
{
0
}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadtwo
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
&
\\
\elemline
Weight
&
40/81
&
40/81
&
40/81
&
40/81
&
\\
\end
{
QuadPoints
}
\clearpage
%%%%%%%%%% 3D %%%%%%%%%
\section
{
3D-Shape Functions
\index
{
Elements!3D
}}
%\section{Isoparametric Elements in 3D\index{Elements!3D}}
\subsection
{
Tetrahedron 4
\index
{
Elements!3D!Tetrahedron 4
}}
\begin
{
Element
}{
3D
}
1
&
\inelemthree
{
0
}{
0
}{
0
}
&
$
1
-
\xi
-
\eta
-
\zeta
$
&
\inelemthree
{
-1
}{
-1
}{
-1
}
\\
\elemline
2
&
\inelemthree
{
1
}{
0
}{
0
}
&
$
\xi
$
&
\inelemthree
{
1
}{
0
}{
0
}
\\
\elemline
3
&
\inelemthree
{
0
}{
1
}{
0
}
&
$
\eta
$
&
\inelemthree
{
0
}{
1
}{
0
}
\\
\elemline
4
&
\inelemthree
{
0
}{
0
}{
1
}
&
$
\zeta
$
&
\inelemthree
{
0
}{
0
}{
1
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lc
}
Coord.
\elemcoorthreed
&
\inquadthree
{
\quart
}{
\quart
}{
\quart
}
\\
\elemline
Weight
&
\sixth
\\
\end
{
QuadPoints
}
\subsection
{
Tetrahedron 10
\index
{
Elements!3D!Tetrahedron 10
}}
\begin
{
Element
}{
3D
}
1
&
\inelemthree
{
0
}{
0
}{
0
}
&
$
\left
(
1
-
\xi
-
\eta
-
\zeta\right
)
\left
(
1
-
2
\xi
-
2
\eta
-
2
\zeta\right
)
$
&
\inelemthreecolumn
{
4
\xi
+4
\eta
+4
\zeta
-3
}{
4
\xi
+4
\eta
+4
\zeta
-3
}{
4
\xi
+4
\eta
+4
\zeta
-3
}
\\
\elemline
2
&
\inelemthree
{
1
}{
0
}{
0
}
&
$
\xi\left
(
2
\xi
-
1
\right
)
$
&
\inelemthree
{
4
\xi
-1
}{
0
}{
0
}
\\
\elemline
3
&
\inelemthree
{
0
}{
1
}{
0
}
&
$
\eta\left
(
2
\eta
-
1
\right
)
$
&
\inelemthree
{
0
}{
4
\eta
-1
}{
0
}
\\
\elemline
4
&
\inelemthree
{
0
}{
0
}{
1
}
&
$
\zeta\left
(
2
\zeta
-
1
\right
)
$
&
\inelemthree
{
0
}{
0
}{
4
\zeta
-1
}
\\
\elemline
5
&
\inelemthree
{
\half
}{
0
}{
0
}
&
$
4
\xi\left
(
1
-
\xi
-
\eta
-
\zeta\right
)
$
&
\inelemthree
{
4-8
\xi
-4
\eta
-4
\zeta
}{
-4
\xi
}{
-4
\xi
}
\\
\elemline
6
&
\inelemthree
{
\half
}{
\half
}{
0
}
&
$
4
\xi\eta
$
&
\inelemthree
{
4
\eta
}{
4
\xi
}{
0
}
\\
\elemline
7
&
\inelemthree
{
0
}{
\half
}{
0
}
&
$
4
\eta\left
(
1
-
\xi
-
\eta
-
\zeta\right
)
$
&
\inelemthree
{
-4
\eta
}{
4-4
\xi
-8
\eta
-4
\zeta
}{
-4
\eta
}
\\
\elemline
8
&
\inelemthree
{
0
}{
0
}{
\half
}
&
$
4
\zeta\left
(
1
-
\xi
-
\eta
-
\zeta\right
)
$
&
\inelemthree
{
-4
\zeta
}{
-4
\zeta
}{
4-4
\xi
-4
\eta
-8
\zeta
}
\\
\elemline
9
&
\inelemthree
{
\half
}{
0
}{
\half
}
&
$
4
\xi\zeta
$
&
\inelemthree
{
4
\zeta
}{
0
}{
4
\xi
}
\\
\elemline
10
&
\inelemthree
{
0
}{
\half
}{
\half
}
&
$
4
\eta\zeta
$
&
\inelemthree
{
0
}{
4
\zeta
}{
4
\eta
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lcc
}
Coord.
\elemcoorthreed
&
\inquadthree
{
\quada
}{
\quada
}{
\quada
}
&
\inquadthree
{
\quadb
}{
\quada
}{
\quada
}
\\
\elemline
Weight
&
1/24
&
1/24
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
\quada
}{
\quadb
}{
\quada
}
&
\inquadthree
{
\quada
}{
\quada
}{
\quadb
}
\\
\elemline
Weight
&
1/24
&
1/24
\\
\end
{
QuadPoints
}
\clearpage
\subsection
{
Hexahedron 8
\index
{
Elements!3D!Hexahedron 8
}}
\begin
{
Element
}{
3D
}
1
&
\inelemthree
{
-1
}{
-1
}{
-1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta\right
)
$
&
\inelemthree
{
-
\eighth\left
(1-
\eta\right
)
\left
(1-
\zeta\right
)
}
{
-
\eighth\left
(1-
\xi\right
)
\left
(1-
\zeta\right
)
}
{
-
\eighth\left
(1-
\xi\right
)
\left
(1-
\eta\right
)
}
\\
\elemline
2
&
\inelemthree
{
1
}{
-1
}{
-1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta\right
)
$
&
\inelemthree
{
\eighth\left
(1-
\eta\right
)
\left
(1-
\zeta\right
)
}
{
-
\eighth\left
(1+
\xi\right
)
\left
(1-
\zeta\right
)
}
{
-
\eighth\left
(1+
\xi\right
)
\left
(1-
\eta\right
)
}
\\
\elemline
3
&
\inelemthree
{
1
}{
1
}{
-1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta\right
)
$
&
\inelemthree
{
\eighth\left
(1+
\eta\right
)
\left
(1-
\zeta\right
)
}
{
\eighth\left
(1+
\xi\right
)
\left
(1-
\zeta\right
)
}
{
-
\eighth\left
(1+
\xi\right
)
\left
(1+
\eta\right
)
}
\\
\elemline
4
&
\inelemthree
{
-1
}{
1
}{
-1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta\right
)
$
&
\inelemthree
{
-
\eighth\left
(1+
\eta\right
)
\left
(1-
\zeta\right
)
}
{
\eighth\left
(1-
\xi\right
)
\left
(1-
\zeta\right
)
}
{
-
\eighth\left
(1-
\xi\right
)
\left
(1+
\eta\right
)
}
\\
\elemline
5
&
\inelemthree
{
-1
}{
-1
}{
1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
+
\zeta\right
)
$
&
\inelemthree
{
-
\eighth\left
(1-
\eta\right
)
\left
(1+
\zeta\right
)
}
{
-
\eighth\left
(1-
\xi\right
)
\left
(1+
\zeta\right
)
}
{
\eighth\left
(1-
\xi\right
)
\left
(1-
\eta\right
)
}
\\
\elemline
6
&
\inelemthree
{
1
}{
-1
}{
1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
+
\zeta\right
)
$
&
\inelemthree
{
\eighth\left
(1-
\eta\right
)
\left
(1+
\zeta\right
)
}
{
-
\eighth\left
(1+
\xi\right
)
\left
(1+
\zeta\right
)
}
{
\eighth\left
(1+
\xi\right
)
\left
(1-
\eta\right
)
}
\\
\elemline
7
&
\inelemthree
{
1
}{
1
}{
1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
+
\zeta\right
)
$
&
\inelemthree
{
\eighth\left
(1+
\eta\right
)
\left
(1+
\zeta\right
)
}
{
\eighth\left
(1+
\xi\right
)
\left
(1+
\zeta\right
)
}
{
\eighth\left
(1+
\xi\right
)
\left
(1+
\eta\right
)
}
\\
\elemline
8
&
\inelemthree
{
-1
}{
1
}{
1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
+
\zeta\right
)
$
&
\inelemthree
{
-
\eighth\left
(1+
\eta\right
)
\left
(1+
\zeta\right
)
}
{
\eighth\left
(1-
\xi\right
)
\left
(1+
\zeta\right
)
}
{
\eighth\left
(1-
\xi\right
)
\left
(1+
\eta\right
)
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lcccc
}
\elemcoorthreed
&
\inquadthree
{
-
\invsqrtthree
}{
-
\invsqrtthree
}{
-
\invsqrtthree
}
&
\inquadthree
{
\invsqrtthree
}{
-
\invsqrtthree
}{
-
\invsqrtthree
}
&
\inquadthree
{
\invsqrtthree
}{
\invsqrtthree
}{
-
\invsqrtthree
}
&
\inquadthree
{
-
\invsqrtthree
}{
\invsqrtthree
}{
-
\invsqrtthree
}
\\
\elemline
Weight
&
1
&
1
&
1
&
1
\\
\elemline
\elemcoorthreed
&
\inquadthree
{
-
\invsqrtthree
}{
-
\invsqrtthree
}{
\invsqrtthree
}
&
\inquadthree
{
\invsqrtthree
}{
-
\invsqrtthree
}{
\invsqrtthree
}
&
\inquadthree
{
\invsqrtthree
}{
\invsqrtthree
}{
\invsqrtthree
}
&
\inquadthree
{
-
\invsqrtthree
}{
\invsqrtthree
}{
\invsqrtthree
}
\\
\elemline
Weight
&
1
&
1
&
1
&
1
\\
\end
{
QuadPoints
}
\subsection
{
Pentahedron 6
\index
{
Elements!3D!Pentahedron 6
}}
\begin
{
Element
}{
3D
}
1
&
\inelemthree
{
-1
}{
1
}{
0
}
&
$
\half\left
(
1
-
\xi\right
)
\eta
$
&
\inelemthree
{
-
\half\eta
}
{
\half\left
(1-
\xi\right
)
}
{
0.0
}
\\
\elemline
2
&
\inelemthree
{
-1
}{
0
}{
1
}
&
$
\half\left
(
1
-
\xi\right
)
\zeta
$
&
\inelemthree
{
-
\half\zeta
}
{
0.0
}
{
\half\left
(1-
\xi\right
)
}
\\
\elemline
3
&
\inelemthree
{
-1
}{
0
}{
0
}
&
$
\half\left
(
1
-
\xi\right
)
\left
(
1
-
\eta
-
\zeta\right
)
$
&
\inelemthree
{
-
\half\left
(1-
\eta
-
\zeta\right
)
}
{
-
\half\left
(1-
\xi\right
)
}
{
-
\half\left
(1-
\xi\right
)
}
\\
\elemline
4
&
\inelemthree
{
1
}{
1
}{
0
}
&
$
\half\left
(
1
+
\xi\right
)
\eta
$
&
\inelemthree
{
\half\eta
}
{
\half\left
(1+
\xi\right
)
}
{
0.0
}
\\
\elemline
5
&
\inelemthree
{
1
}{
0
}{
1
}
&
$
\half\left
(
1
+
\xi\right
)
\zeta
$
&
\inelemthree
{
\half\zeta
}
{
0.0
}
{
\half\left
(1+
\xi\right
)
}
\\
\elemline
6
&
\inelemthree
{
1
}{
0
}{
0
}
&
$
\half\left
(
1
+
\xi\right
)
\left
(
1
-
\eta
-
\zeta\right
)
$
&
\inelemthree
{
\half\left
(1-
\eta
-
\zeta\right
)
}
{
-
\half\left
(1+
\xi\right
)
}
{
-
\half\left
(1+
\xi\right
)
}
\\
\end
{
Element
}
\begin
{
QuadPoints
}{
lcccccc
}
\elemcoorthreed
&
\inquadthree
{
-
\invsqrtthree
}{
0.5
}{
0.5
}
&
\inquadthree
{
-
\invsqrtthree
}{
0.0
}{
0.5
}
&
\inquadthree
{
-
\invsqrtthree
}{
0.5
}{
0.0
}
&
\inquadthree
{
\invsqrtthree
}{
0.5
}{
0.5
}
&
\inquadthree
{
\invsqrtthree
}{
0.0
}{
0.5
}
&
\inquadthree
{
\invsqrtthree
}{
0.5
}{
0.0
}
\\
\elemline
Weight
&
1/6
&
1/6
&
1/6
&
1/6
&
1/6
&
1/6
\\
\end
{
QuadPoints
}
\clearpage
\subsection
{
Hexahedron 20
\index
{
Elements!3D!Hexahedron 20
}}
\begin
{
Element
_
part1
}{
3D
}
1
&
\inelemthree
{
-1
}{
-1
}{
-1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta\right
)
\left
(-
2
-
\xi
-
\eta
-
\zeta\right
)
$
\\
\elemline
2
&
\inelemthree
{
1
}{
-1
}{
-1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta\right
)
\left
(-
2
+
\xi
-
\eta
-
\zeta\right
)
$
\\
\elemline
3
&
\inelemthree
{
1
}{
1
}{
-1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta\right
)
\left
(-
2
+
\xi
+
\eta
-
\zeta\right
)
$
\\
\elemline
4
&
\inelemthree
{
-1
}{
1
}{
-1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta\right
)
\left
(-
2
-
\xi
+
\eta
-
\zeta\right
)
$
\\
\elemline
5
&
\inelemthree
{
-1
}{
-1
}{
1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
+
\zeta\right
)
\left
(-
2
-
\xi
-
\eta
+
\zeta\right
)
$
\\
\elemline
6
&
\inelemthree
{
1
}{
-1
}{
1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
+
\zeta\right
)
\left
(-
2
+
\xi
-
\eta
+
\zeta\right
)
$
\\
\elemline
7
&
\inelemthree
{
1
}{
1
}{
1
}
&
$
\eighth\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
+
\zeta\right
)
\left
(-
2
+
\xi
+
\eta
+
\zeta\right
)
$
\\
\elemline
8
&
\inelemthree
{
-1
}{
1
}{
1
}
&
$
\eighth\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
+
\zeta\right
)
\left
(-
2
-
\xi
+
\eta
+
\zeta\right
)
$
\\
\elemline
9
&
\inelemthree
{
0
}{
-1
}{
-1
}
&
$
\quart\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta\right
)
$
\\
\elemline
10
&
\inelemthree
{
1
}{
0
}{
-1
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
-
\eta
^{
2
}
\right
)
\left
(
1
-
\zeta\right
)
$
\\
\elemline
11
&
\inelemthree
{
0
}{
1
}{
-1
}
&
$
\quart\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta\right
)
$
\\
\elemline
12
&
\inelemthree
{
-1
}{
0
}{
-1
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
-
\eta
^{
2
}
\right
)
\left
(
1
-
\zeta\right
)
$
\\
\elemline
13
&
\inelemthree
{
-1
}{
-1
}{
0
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta
^{
2
}
\right
)
$
\\
\elemline
14
&
\inelemthree
{
1
}{
-1
}{
0
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
-
\eta\right
)
\left
(
1
-
\zeta
^{
2
}
\right
)
$
\\
\elemline
15
&
\inelemthree
{
1
}{
1
}{
0
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta
^{
2
}
\right
)
$
\\
\elemline
16
&
\inelemthree
{
-1
}{
1
}{
0
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
+
\eta\right
)
\left
(
1
-
\zeta
^{
2
}
\right
)
$
\\
\elemline
17
&
\inelemthree
{
0
}{
-1
}{
1
}
&
$
\quart\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
-
\eta\right
)
\left
(
1
+
\zeta\right
)
$
\\
\elemline
18
&
\inelemthree
{
1
}{
0
}{
1
}
&
$
\quart\left
(
1
+
\xi\right
)
\left
(
1
-
\eta
^{
2
}
\right
)
\left
(
1
+
\zeta\right
)
$
\\
\elemline
19
&
\inelemthree
{
0
}{
1
}{
1
}
&
$
\quart\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
+
\eta\right
)
\left
(
1
+
\zeta\right
)
$
\\
\elemline
20
&
\inelemthree
{
-1
}{
0
}{
1
}
&
$
\quart\left
(
1
-
\xi\right
)
\left
(
1
-
\eta
^{
2
}
\right
)
\left
(
1
+
\zeta\right
)
$
\\
\end
{
Element
_
part1
}
\clearpage
\begin
{
Element
_
part2
}{
3D
}
1
&
\inelemthree
{
\quart\left
(
\xi
+
\half\left
(
\eta
+
\zeta
+1
\right
)
\right
)
\left
(
\eta
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\eta
+
\half\left
(
\xi
+
\zeta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\zeta
+
\half\left
(
\xi
+
\eta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
2
&
\inelemthree
{
\quart\left
(
\xi
-
\half\left
(
\eta
+
\zeta
+1
\right
)
\right
)
\left
(
\eta
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\eta
-
\half\left
(
\xi
-
\zeta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\zeta
-
\half\left
(
\xi
-
\eta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
3
&
\inelemthree
{
-
\quart\left
(
\xi
+
\half\left
(
\eta
-
\zeta
-1
\right
)
\right
)
\left
(
\eta
+1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\eta
+
\half\left
(
\xi
-
\zeta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\zeta
-
\half\left
(
\xi
+
\eta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
4
&
\inelemthree
{
-
\quart\left
(
\xi
-
\half\left
(
\eta
-
\zeta
-1
\right
)
\right
)
\left
(
\eta
+1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\eta
-
\half\left
(
\xi
+
\zeta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\zeta
+
\half\left
(
\xi
-
\eta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
5
&
\inelemthree
{
-
\quart\left
(
\xi
+
\half\left
(
\eta
-
\zeta
+1
\right
)
\right
)
\left
(
\eta
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\eta
+
\half\left
(
\xi
-
\zeta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\zeta
-
\half\left
(
\xi
+
\eta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
6
&
\inelemthree
{
-
\quart\left
(
\xi
-
\half\left
(
\eta
-
\zeta
+1
\right
)
\right
)
\left
(
\eta
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\eta
-
\half\left
(
\xi
+
\zeta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\zeta
+
\half\left
(
\xi
-
\eta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
7
&
\inelemthree
{
\quart\left
(
\xi
+
\half\left
(
\eta
+
\zeta
-1
\right
)
\right
)
\left
(
\eta
+1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\eta
+
\half\left
(
\xi
+
\zeta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\zeta
+
\half\left
(
\xi
+
\eta
-1
\right
)
\right
)
\left
(
\xi
+1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
8
&
\inelemthree
{
\quart\left
(
\xi
-
\half\left
(
\eta
+
\zeta
-1
\right
)
\right
)
\left
(
\eta
+1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\eta
-
\half\left
(
\xi
-
\zeta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\zeta
-
\half\left
(
\xi
-
\eta
+1
\right
)
\right
)
\left
(
\xi
-1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
9
&
\inelemthree
{
-
\half\xi\left
(
\eta
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
10
&
\inelemthree
{
\quart\left
(
\eta
^{
2
}
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\half\eta\left
(
\xi
+1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\xi
+1
\right
)
\left
(
\eta
^{
2
}
-1
\right
)
}
\\
\elemline
11
&
\inelemthree
{
\half\xi\left
(
\eta
+1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
12
&
\inelemthree
{
-
\quart\left
(
\eta
^{
2
}
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\half\eta\left
(
\xi
-1
\right
)
\left
(
\zeta
-1
\right
)
}
{
-
\quart\left
(
\xi
-1
\right
)
\left
(
\eta
^{
2
}
-1
\right
)
}
\\
\elemline
13
&
\inelemthree
{
-
\quart\left
(
\eta
-1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
-
\quart\left
(
\xi
-1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
-
\half\zeta\left
(
\xi
-1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
14
&
\inelemthree
{
\quart\left
(
\eta
-1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
\quart\left
(
\xi
+1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
\half\zeta\left
(
\xi
+1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
15
&
\inelemthree
{
-
\quart\left
(
\eta
+1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
-
\quart\left
(
\xi
+1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
-
\half\zeta\left
(
\xi
+1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
16
&
\inelemthree
{
\quart\left
(
\eta
+1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
\quart\left
(
\xi
-1
\right
)
\left
(
\zeta
^{
2
}
-1
\right
)
}
{
\half\zeta\left
(
\xi
-1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
17
&
\inelemthree
{
\half\xi\left
(
\eta
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\eta
-1
\right
)
}
\\
\elemline
18
&
\inelemthree
{
-
\quart\left
(
\eta
^{
2
}
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\half\eta\left
(
\xi
+1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\xi
+1
\right
)
\left
(
\eta
^{
2
}
-1
\right
)
}
\\
\elemline
19
&
\inelemthree
{
-
\half\xi\left
(
\eta
+1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
-
\quart\left
(
\xi
^{
2
}
-1
\right
)
\left
(
\eta
+1
\right
)
}
\\
\elemline
20
&
\inelemthree
{
\quart\left
(
\eta
^{
2
}
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\half\eta\left
(
\xi
-1
\right
)
\left
(
\zeta
+1
\right
)
}
{
\quart\left
(
\xi
-1
\right
)
\left
(
\eta
^{
2
}
-1
\right
)
}
\\
\end
{
Element
_
part2
}
\begin
{
QuadPoints
}{
lcccc
}
Coord.
\elemcoorthreed
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
{
0
}
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
\\
\elemline
Weight
&
125/729
&
200/729
&
125/729
&
200/729
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}{
0
}
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
\\
\elemline
Weight
&
320/729
&
200/729
&
125/729
&
200/729
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
0
}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
0
}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
&
\inquadthree
{
0
}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
\\
\elemline
Weight
&
125/729
&
200/729
&
320/729
&
200/729
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
0
}{
0
}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
0
}{
0
}{
0
}
&
\inquadthree
{
0
}{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
\\
\elemline
Weight
&
320/729
&
512/729
&
320/729
&
200/729
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
&
\inquadthree
{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
\\
\elemline
Weight
&
320/729
&
200/729
&
125/729
&
200/729
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}{
0
}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
\\
\elemline
Weight
&
125/729
&
200/729
&
320/729
&
200/729
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
-
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
0
}
&
\inquadthree
{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}{
\sqrt
{
\tfrac
{
3
}{
5
}}}
&
\\
\elemline
Weight
&
125/729
&
200/729
&
125/729
&
\\
\end
{
QuadPoints
}
\clearpage
\subsection
{
Pentahedron 15
\index
{
Elements!3D!Pentahedron 15
}}
\begin
{
Element
_
part1
}{
3D
}
1
&
\inelemthree
{
-1
}{
1
}{
0
}
&
$
\half\eta\left
(
1
-
\xi\right
)
\left
(
2
\eta
-
2
-
\xi\right
)
$
\\
\elemline
2
&
\inelemthree
{
-1
}{
0
}{
1
}
&
$
\half\zeta\left
(
1
-
\xi\right
)
\left
(
2
\zeta
-
2
-
\xi\right
)
$
\\
\elemline
3
&
\inelemthree
{
-1
}{
0
}{
0
}
&
$
\half\left
(
\xi
-
1
\right
)
\left
(
1
-
\eta
-
\zeta\right
)
\left
(
\xi
+
2
\eta
+
2
\zeta\right
)
$
\\
\elemline
4
&
\inelemthree
{
1
}{
1
}{
0
}
&
$
\half\eta\left
(
1
+
\xi\right
)
\left
(
2
\eta
-
2
+
\xi\right
)
$
\\
\elemline
5
&
\inelemthree
{
1
}{
0
}{
1
}
&
$
\half\zeta\left
(
1
+
\xi\right
)
\left
(
2
\zeta
-
2
+
\xi\right
)
$
\\
\elemline
6
&
\inelemthree
{
1
}{
0
}{
0
}
&
$
\half\left
(-
\xi
-
1
\right
)
\left
(
1
-
\eta
-
\zeta\right
)
\left
(-
\xi
+
2
\eta
+
2
\zeta\right
)
$
\\
\elemline
7
&
\inelemthree
{
-1
}{
0.5
}{
0.5
}
&
$
2
\eta\zeta\left
(
1
-
\xi\right
)
$
\\
\elemline
8
&
\inelemthree
{
-1
}{
0
}{
0.5
}
&
$
2
\zeta\left
(
1
-
\eta
-
\zeta\right
)
\left
(
1
-
\xi\right
)
$
\\
\elemline
9
&
\inelemthree
{
-1
}{
0.5
}{
0
}
&
$
2
\eta\left
(
1
-
\xi\right
)
\left
(
1
-
\eta
-
\zeta\right
)
$
\\
\elemline
10
&
\inelemthree
{
0
}{
1
}{
0
}
&
$
\eta\left
(
1
-
\xi
^{
2
}
\right
)
$
\\
\elemline
11
&
\inelemthree
{
0
}{
0
}{
1
}
&
$
\zeta\left
(
1
-
\xi
^{
2
}
\right
)
$
\\
\elemline
12
&
\inelemthree
{
0
}{
0
}{
0
}
&
$
\left
(
1
-
\xi
^{
2
}
\right
)
\left
(
1
-
\eta
-
\zeta\right
)
$
\\
\elemline
13
&
\inelemthree
{
1
}{
0.5
}{
0.5
}
&
$
2
\eta\zeta\left
(
1
+
\xi\right
)
$
\\
\elemline
14
&
\inelemthree
{
1
}{
0
}{
0.5
}
&
$
2
\zeta\left
(
1
+
\xi\right
)
\left
(
1
-
\eta
-
\zeta\right
)
$
\\
\elemline
15
&
\inelemthree
{
1
}{
0.5
}{
0
}
&
$
2
\eta\left
(
1
+
\xi\right
)
\left
(
1
-
\eta
-
\zeta\right
)
$
\\
\end
{
Element
_
part1
}
\clearpage
\begin
{
Element
_
part2
}{
3D
}
1
&
\inelemthree
{
\half\eta\left
(2
\xi
-2
\eta
+1
\right
)
}
{
-
\half\left
(
\xi
-1
\right
)
\left
(4
\eta
-
\xi
-2
\right
)
}
{
0.0
}
\\
\elemline
2
&
\inelemthree
{
\half\zeta\left
(2
\xi
-2
\zeta
+1
\right
)
}
{
0.0
}
{
-
\half\left
(
\xi
-1
\right
)
\left
(4
\zeta
-
\xi
-2
\right
)
}
\\
\elemline
3
&
\inelemthree
{
-
\half\left
(2
\xi
+2
\eta
+2
\zeta
-1
\right
)
\left
(
\eta
+
\zeta
-1
\right
)
}
{
-
\half\left
(
\xi
-1
\right
)
\left
(4
\eta
+
\xi
+2
\left
(2
\zeta
-1
\right
)
\right
)
}
{
-
\half\left
(
\xi
-1
\right
)
\left
(4
\zeta
+
\xi
+2
\left
(2
\eta
-1
\right
)
\right
)
}
\\
\elemline
4
&
\inelemthree
{
\half\eta\left
(2
\xi
+2
\eta
-1
\right
)
}
{
\half\left
(
\xi
+1
\right
)
\left
(4
\eta
+
\xi
-2
\right
)
}
{
0.0
}
\\
\elemline
5
&
\inelemthree
{
\half\zeta\left
(2
\xi
+2
\zeta
-1
\right
)
}
{
0.0
}
{
\half\left
(
\xi
+1
\right
)
\left
(4
\zeta
+
\xi
-2
\right
)
}
\\
\elemline
6
&
\inelemthree
{
-
\half\left
(
\eta
+
\zeta
-1
\right
)
\left
(2
\xi
-2
\eta
-2
\zeta
+1
\right
)
}
{
\half\left
(
\xi
+1
\right
)
\left
(4
\eta
-
\xi
+2
\left
(2
\zeta
-1
\right
)
\right
)
}
{
\half\left
(
\xi
+1
\right
)
\left
(4
\zeta
-
\xi
+2
\left
(2
\eta
-1
\right
)
\right
)
}
\\
\elemline
7
&
\inelemthree
{
-2
\eta\zeta
}
{
-2
\left
(
\xi
-1
\right
)
\zeta
}
{
-2
\left
(
\xi
-1
\right
)
\eta
}
\\
\elemline
8
&
\inelemthree
{
2
\zeta\left
(
\eta
+
\zeta
-1
\right
)
}
{
2
\zeta
-
\left
(
\xi
-1
\right
)
}
{
2
\left
(
\xi
-1
\right
)
\left
(2
\zeta
+
\eta
-1
\right
)
}
\\
\elemline
9
&
\inelemthree
{
2
\eta\left
(
\eta
+
\zeta
-1
\right
)
}
{
2
\left
(2
\eta
+
\zeta
-1
\right
)
\left
(
\xi
-1
\right
)
}
{
2
\eta\left
(
\xi
-1
\right
)
}
\\
\elemline
10
&
\inelemthree
{
-2
\xi\eta
}
{
-
\left
(
\xi
^{
2
}
-1
\right
)
}
{
0.0
}
\\
\elemline
11
&
\inelemthree
{
-2
\xi\zeta
}
{
0.0
}
{
-
\left
(
\xi
^{
2
}
-1
\right
)
}
\\
\elemline
12
&
\inelemthree
{
2
\xi\left
(
\eta
+
\zeta
-1
\right
)
}
{
\left
(
\xi
^{
2
}
-1
\right
)
}
{
\left
(
\xi
^{
2
}
-1
\right
)
}
\\
\elemline
13
&
\inelemthree
{
2
\eta\zeta
}
{
2
\zeta\left
(
\xi
+1
\right
)
}
{
2
\eta\left
(
\xi
+1
\right
)
}
\\
\elemline
14
&
\inelemthree
{
-2
\zeta\left
(
\eta
+
\zeta
-1
\right
)
}
{
-2
\zeta\left
(
\xi
+1
\right
)
}
{
-2
\left
(
\xi
+1
\right
)
\left
(2
\zeta
+
\eta
-1
\right
)
}
\\
\elemline
15
&
\inelemthree
{
-2
\eta\left
(
\eta
+
\zeta
-1
\right
)
}
{
-2
\left
(2
\eta
+
\zeta
-1
\right
)
\left
(
\xi
+1
\right
)
}
{
-2
\eta\left
(
\xi
+1
\right
)
}
\\
\end
{
Element
_
part2
}
\begin
{
QuadPoints
}{
lcccc
}
Coord.
\elemcoorthreed
&
\inquadthree
{
-
{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
\tfrac
{
1
}{
3
}}{
\tfrac
{
1
}{
3
}}
&
\inquadthree
{
-
{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
0.6
}{
0.2
}
&
\inquadthree
{
-
{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
0.2
}{
0.6
}
&
\inquadthree
{
-
{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
0.2
}{
0.2
}
\\
\elemline
Weight
&
-27/96
&
25/96
&
25/96
&
25/96
\\
\elemline
Coord.
\elemcoorthreed
&
\inquadthree
{{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
\tfrac
{
1
}{
3
}}{
\tfrac
{
1
}{
3
}}
&
\inquadthree
{{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
0.6
}{
0.2
}
&
\inquadthree
{{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
0.2
}{
0.6
}
&
\inquadthree
{{
\tfrac
{
1
}{
\sqrt
{
3
}}}}{
0.2
}{
0.2
}
\\
\elemline
Weight
&
-27/96
&
25/96
&
25/96
&
25/96
\\
\end
{
QuadPoints
}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "manual"
%%% End:
Event Timeline
Log In to Comment