Programming Finite Element Computations - Programming the Finite Element Method - page 71

the calling programs by means of a simple USE statement at the beginning of the program
(see Chapter 1, Section 1.9.9).
A summary of these subroutines and functions is given in Appendices D, E, and F
where their actions and input/output parameters are described. Appendix D describes func-
tions and subroutines that appear in the main library, and describes “black box” routines
(concerned with some matrix operations), whose mode of action the reader need not nec-
essarily know in detail, and special purpose routines which are the basis of specific finite
element computations. Some of these routines should be thought of as an addition to
the intrinsic Fortran 95 library functions such as MATMUL or DOT_PRODUCT , and could
well be substituted with equivalents from a mathematical subroutine library, for example
Basic Linear Algebra Subroutine (BLAS), perhaps tuned to a specific machine. Appendix
E describes subroutines that appear in the geom library which holds customised routines,
usually for generating element nodal coordinates and numbering for some of the simple
geometries used with the specific examples described in this topic. Appendix F describes the
additional subroutines and functions needed by the Chapter 12 programs for their parallel
algorithms.

3.7.1 Black box routines

Readers are reminded of the much improved array handling facilities of Fortran 95, com-
pared with earlier FORTRANs. Chapter 1, Section 1.9 summarised such features as whole
array operations and intrinsic array procedures, which mean that most simple array manip-
ulations can be done using the power of the language itself and do not need to be
user-supplied.
In the programs which follow from Chapter 4 onwards, only three simple functions
or subroutines have been added to those provided as standard in the language. These are,
determinant , invert ,and cross_product
determinant returns the determinant of a 1

3 matrix (usually
the Jacobian matrix [ J ]), invert computes the inverse of a (small) square matrix, again
usually the Jacobian matrix (3.4) and cross_product computes the matrix result given
by the cross-product of two vectors.
A second batch of subroutines shown in Table 3.4 is concerned with the solution of lin-
ear algebraic equations. The subroutines have been split into factorisation and forward/back-
substitution phases.
Several subroutines are associated with eigenvalue and eigenvector determination; for
example for symmetric banded matrices, bandred tridiagonalises the matrix and bisect

× 1, 2 × 2, or 3 ×

Table 3.4 Subroutines for solution of linear algebraic equations

Method Gauss Cholesky Gauss Gauss

Storage Symmetric Symmetric Symmetric Unsymmetric

half-band skyline skyline full band

Factorisation banred

sparin sparin gauss gauss band

Substitution

bacsub spabac spabac gauss solve band