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TET10 - Finite Element Library Shape Function

$\textstyle \parbox{125mm}{
NOTE: before using this routine, please read the
app...
...ath{\ bold \ italicised}$\ terms and other implementation-
dependent details.
}$

1. Purpose

TET10 returns the values of shape functions and their derivatives at a specified point for a ten-noded tetrahedral element; the approximated function will be continuous across element boundaries.

2. Specification

 SUBROUTINE TET10(FUN,IFUN,DER,IDER,JDER,XI,ETA,ZETA,ITEST)
 INTEGER IFUN,IDER,JDER,ITEST
  $\boldmath {real}$ XI,ETA,ZETA
  $\boldmath {real}$ FUN(IFUN),DER(IDER,JDER)

3. Description

This routine returns the values of the ten shape functions and their derivatives associated with a ten-noded tetrahedral element. The shape functions are specified in terms of local coordinates, the origin being at the centroid of the element. The element, node numbering and local coordinates are shown in Figure 1.

Figure 1: TET10 element description
\begin{figure}
\vspace*{70mm}
\special{psfile=tet10.ps vscale=60 hscale=60 hoffset=-20 voffset=260 angle=-90}
\end{figure}

The shape functions in terms of the local coordinates are:

Typical corner node:

\begin{eqnarray*}
N_3(\xi,\eta,\zeta)&=&(2L_2-1)L_2
\end{eqnarray*}

Typical midside node:

\begin{eqnarray*}
N_4(\xi,\eta,\zeta)&=&4L_2L_3
\end{eqnarray*}

where $L_1, L_2, L_3$ and $L_4$ are the volume coordinates of the standard tetrahedral. These in terms of the local cartesian coordinates are:

\begin{eqnarray*}
L_1(\xi,\eta,\zeta)& = &\frac{1}{12}(3+8\xi-2\sqrt2\zeta)\\
L...
...rt2\zeta)\\
L_4(\xi,\eta,\zeta)& = &\frac{1}{4}(1+2\sqrt2\zeta)
\end{eqnarray*}

When these shape functions are use to approximate a three-dimensional problem, the resulting elements will give continuity in the approximated function across element boundaries.

Any variable$\phi$ defined on the element may be approximated by

\begin{displaymath}
\phi^\dagger = \sum_{i=1}^{10} N_i(\xi,\eta,\zeta) \phi_i
\end{displaymath}

where $\phi_i$ are the nodal values of the variable $\phi$.

4. References

[1]
ZIENKIEWICZ, O.C.
The Finite Element Method, pp 172-174.
McGraw-Hill, London, 1977.
5. Parameters
FUN -   $\boldmath {real}$ array of DIMENSION (IFUN) where IFUN $\leq$ 10.
On successful exit, FUN($i$) contains the value the shape function $N_i(\xi,\eta,\zeta)$ at the the specified point (XI,ETA,ZETA), for $i$=1,2,...,10.

IFUN -   INTEGER.
On entry, IFUN specifies the dimension of array FUN as declared in the calling (sub)program.
Unchanged on exit.

DER -   $\boldmath {real}$ array of DIMENSION (IDER,JDER) where IDER $\leq$ 3 and JDER $\leq$ 10.
On successful exit, DER($i,j$) contains the value of the derivative of the shape function $N_i(\xi,\eta,\zeta)$ with respect to the $i$the coordinate at the specified point (XI,ETA,ZETA), for $i$=1,2,3 and $j$=1,2,...,10.

IDER -   INTEGER.
On entry, IDER specifies the first dimension of array DER as declared in the calling (sub)program.
Unchanged on exit.

JDER -   INTEGER.
On entry, JDER specifies the second dimension of array DER as declared in the calling (sub)program.
Unchanged on exit.

XI -   $\boldmath {real}$.
On entry, XI specifies the value of the local coordinate $\xi$ at which the function and derivative values are required.
Unchanged on exit.

ETA -   $\boldmath {real}$.
On entry, ETA specifies the value of the local coordinate $\eta$ at which the function and derivative values are required.
Unchanged on exit.

ZETA -   $\boldmath {real}$.
On entry, ZETA spcifies the value of the local coordinate $\zeta$ at which the function and derivative values are required.
Unchanged on exit.

ITEST -   INTEGER.
Before entry, ITEST must be set to 0, 1 or -1.
Users not familiar with this parameter (described in routine ERRMES) are advised to assign ITEST to 0.
Unless the routine detects an error (see next section), ITEST is set to 0 on exit.

6. Error Indicators and Warnings

ITEST=1   On entry IFUN $<$ 10.
ITEST=2   On entry IDER $<$ 3 or JDER $<$ 10.
ITEST=3   On entry (XI,ETA,ZETA) outside range of definition.

7. Auxiliary Rouines

This routine uses the Level 0 Library routine ERRMES.

8. Timing

Not available.

9. Storage

There are no internally declared arrays.

10. Accuracy

$\boldmath {Basic~precision}$ arithmetic is used.

11. Further Comments

None.


next up previous
Next: TET20 Up: Contents Previous: SURBRK
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Chris Greenough (c.greenough@rl.ac.uk): October 2000