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# TET10 - Finite Element Library Shape Function

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
XI,ETA,ZETA
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.

The shape functions in terms of the local coordinates are:

Typical corner node:

Typical midside node:

where and are the volume coordinates of the standard tetrahedral. These in terms of the local cartesian coordinates are:

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 defined on the element may be approximated by

where are the nodal values of the variable .

4. References

[1]
ZIENKIEWICZ, O.C.
The Finite Element Method, pp 172-174.
McGraw-Hill, London, 1977.
5. Parameters
FUN -   array of DIMENSION (IFUN) where IFUN 10.
On successful exit, FUN() contains the value the shape function at the the specified point (XI,ETA,ZETA), for =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 -   array of DIMENSION (IDER,JDER) where IDER 3 and JDER 10.
On successful exit, DER() contains the value of the derivative of the shape function with respect to the the coordinate at the specified point (XI,ETA,ZETA), for =1,2,3 and =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 -   .
On entry, XI specifies the value of the local coordinate at which the function and derivative values are required.
Unchanged on exit.

ETA -   .
On entry, ETA specifies the value of the local coordinate at which the function and derivative values are required.
Unchanged on exit.

ZETA -   .
On entry, ZETA spcifies the value of the local coordinate 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

arithmetic is used.