An interactive graphics software package which allows users to display the non-zero structure of large sparse symmetric materials was described and methods used to implement it as a portable FORTRAN callable subroutine were summarized. In particular, the system permits the display of the resulting matrix after reordering the rows and columns, with the reordering scheme either defined by the user or automatically generated by the program with the aim of reducing matrix bandwidth and profile. Although the primary application of the package has been to the finite element analysis of structures, it is equally well suited to the many other areas of engineering and science which use sparse matrices

Boisvert, R. F.

Poole, W. G., Jr.

English

NASA Technical Reports Server

AN INTERACTIVE GRAPHICS PACKAGE FOR THE
AUTOMATIC NODE RENUMBERING OF FINITE ELEMENT MATRICES*
Ronald F. Boisvert and William G. Poole, Jr.
College of William and Mary
and
Institute for Computer Applications in Science and Engineering (ICASE)
SUMMARY
An interactive graphics software package vhich allows users to display
the non-zero structure of large sparse symmetric matrices is described and
methods used to implement it as a portable FORTRAN callable subroutine are
summarized. In particular, the system permits the display of the resulting
matrix after reordering the rows and columns, with the reordering scheme
either defined by the user or automatically generated by the program with the
aim of reducing matrix bandwidth and profile. Although the primary applica-
tion of the package has been to the finite element analysis of structures, it
is equally well suited to the many other areas of engineering and science
which use sparse matrices.
INTRODUCTION
Frequently, in many areas of application, we must solve the linear alge-
braic system of equations represented by
Ax = b (1)
where A is a non-singular n x n symmetric matrix and x and b are n-
vectors. Here we assume that n is moderately large (from about one hundred
to several thousand) and that the matrix A is sparse; that is, the number of
non-zero elements in the matrix is small compared to n^.
In order to describe the non-zero structure of sparse matrices the con-
cepts of bandwidth and profile are helpful. The bandwidth of a matrix A is
defined as b = max, |i-j|, which is simply the radius of the smallest band
aiy
about the diagonal which includes all non-zero components of the matrix. The
This paper is a result of work performed, in part, under NASA Grant NCR
U7-102-001 while in residence at ICASE, NASA Langley Research Center. The
work of the second author also was partially supported by the Office of
Naval Research Contract N0001^ -73-A-037U-0001, NR OUU-1*59. The first
author is currently at the Computer Science Department, Purdue University.
605
https://ntrs.nasa.gov/search.jsp?R=19760009755 2020-03-22T16:31:45+00:00Z
n
profile is defined as p = Z d., where d = i - min {j : a. 1 0}, that is,
i=l 3 J ij
the sum of the distances from the main diagonal of the leftmost non-zero com-
ponent in each row. The profile is exactly the number of non-zeroes in the
lower triangular factor of the decomposition of A whose calculation is typi-
cally the first step in solving (l) by Gaussian elimination.
The non-zero structure of large sparse matrices is often used to reduce
the otherwise restrictive storage and computational requirements for solving
the linear system. Storage schemes for sparse matrices abound (references 1
and 2) with the band scheme being among the simplest to use. In this case,
when the bandwidth of A is small, we can eliminate most zero components by
simply storing only the diagonal bands of A. Other techniques incur even
larger savings by taking advantage of the profile of the matrix. The non-zero
structure of the matrix also greatly influences the computation time required
to solve (l). The time required to perform Gaussian elimination on a full
matrix is proportional to n3 while methods for band matrices typically re-
quire computation times proportional to nb . Once again, this can be reduced
further by exploiting profile.
It is clear that for matrices with small bandwidth and profile consider-
able savings can be realized and, in fact, many codes are currently available
for band or profile decomposition. Thus, the engineer wishing to solve (l)
would normally pay particular attention to the non-zero structure of the
matrix and, whenever possible, would seek that representation of the matrix
which minimizes bandwidth and profile. Although band and profile schemes are
not always the best way to treat sparse matrices, many of the matrices encoun-
tered in real applications, in particular, finite element approximations in
structural engineering, are quite appropriate for the use of band or profile
schemes.
The sparse matrix graphics package, by allowing the display of a repre-
sentation of the non-zero structure of the matrix, gives the user a unique
visual aid in the evaluation of various methods of storage and solution of the
particular problem in question. In addition, by providing a matrix bandwidth
and profile reducer as part of the package, the user may visually evaluate the
worth of a renumbering of the rows and columns of the matrix which reduces
bandwidth and profile to nearly minimal levels.
An example of the use of the .graphics program is given in the following
section with a description of the full capabilities and implementation of the
graphics system following that. In the last two sections the algorithm used
for matrix bandwidth and profile reduction is outlined and the results of the
author's experience with the program are discussed.
EXAMPLE OF USE
The following scenario describes a typical use of the system. Suppose
a design engineer, seated at a graphics terminal, is using the finite element
606
method for analysing some structure. After generating the elements, the desig-
ner numbers the nodes in some order, often an order which is easy to describe,
and then generates a table defining the location of the non-zero components in
the associated sparse matrix which represents properties of the structure being
analysed. Because of interest in the effect of the nodal numbering on the non-
zero structure of the matrix, the designer now invokes the sparse matrix gra-
phics program via a subroutine call in the control program, causing the infor-
mation at the top of Figure 1 to be displayed at the terminal. The design
engineer then continues with the interaction shown in Figure 1, informing the
program that the first display should be that of the input matrix shown in
Figure 2.
Somewhat dissatisfied with his/her own nodal numbering, the designer
might then request the program to generate a new numbering which reduces band-
width and profile, thus producing the display of Figure 3. As a final compari-
son of the two numberings, the engineer may finally request the lower triangu-
lar halves of both matrices to be displayed at the same time, generating the
display of Figure k.
Not only does the design engineer get a nodal numbering which produces a
small bandwidth and profile, but he also now has a visual conception of how
effective the original numbering was.
IMPLEMENTATION OF THE DISPLAY SYSTEM
The sparse matrix graphics program is a highly portable FORTRAN program,
requiring minimal local computing facilities. As demonstrated in the pre-
vious section, the graphics program has the following capabilities:
* Display the original matrix.
* Generate a reordering of the rows and columns which reduces bandwidth
and profile, displaying the resultant matrix.
* Display the matrix after reordering the rows and columns according to
a scheme input by the user.
* Compare any two of the above, displaying them at the same time on the
screen.
* Matrices of any order may be displayed regardless of screen size.
* The size of the matrix display may be varied by the user.
* Either the full matrix or the lower triangular half may be displayed.
The package is invoked by a call to the subroutine SPARSE, with the gene-
rated renumbering returned as a parameter if calculated during the interactive
session. In this way the package is easily interfaced with existing FORTRAN
607
coded interactive design systems.
The data structure chosen to represent the sparse matrix is the connec-
tion table, one frequently used in bandwidth and profile schemes (reference 3).
The table has n rows and m columns, where n is the number of rows in the
actual sparse matrix and m is the maximum number of off-diagonal non-zero
components to be found in any row. The ith row of the table contains the
column indices of all off-diagonal non-zero components in the ith row of the
actual matrix. These column indices may be in any order. The values of the
non-zero components themselves are never needed, only their indices. The re-
numbering schemes are represented as permutation vectors, whose entries show
the order of the old rows and columns in the new matrix. For example, if the
reordering vector is p, then the ith row in the original matrix becomes the
p(i)-th row in the reordered matrix.
Due to the simplicity of the displays generated, rather minimal local
graphics software capabilities are required to support the system. Indeed,
any computation facility supporting interactive graphics terminals which pro-
vides FORTRAN callable routines to display a symbol and move the beam invis-
ibly can easily implement the package on their system. These capabilities are
interfaced with the sparse matrix graphics program via two user supplied rou-
tines, one which plots a row of user chosen symbols representing non-zero
components at coordinates which are given, and one which simply moves the beam
invisibly to a given location.
Displaying large matrices on most screens is a problem; for instance, on
a terminal with a resolution of 102^  points by 780 points, a 1000 x 1000 matrix
could not be displayed, even if each element were represented by a single
point. To overcome this, the graphics program instead displays a related par-
titioned matrix; specifically, each symbol displayed represents an r x r block
of elements of the original matrix in which a non-zero occurs. The blocking
factor r is chosen by the program and depends on the order of the original
matrix and the display size requested by the user. The resulting display thus
yields a visual description of the placement of non-zeroes, even when the
actual matrix could not be displayed.
In order for the program to generate the coordinates of the symbols to be
displayed, the calling program provides information concerning the dimensions
of the screen, the size of the square symbol which the user has chosen to repre-
sent non-zeroes, and the size of the displayed characters resulting from ordi-
nary FORTRAN write statements. The program then supplies display coordinates
to the user-supplied graphics routines in the same units as the user implicitly
defines the screen dimension information.
To run the program, some 7n+2m core storage locations are required, in
addition to the storage for the connection table (nm locations) and for the
optional user supplied permutation (n locations). Thus the solution of large
problems may be restricted by the core memory size. The largest part of these
storage requirements is needed to implement the bandwidth and profile reduc-
tion subroutine, REDUCE.
608
REDUCING BANDWIDTH AND PROFILE
The algorithm for bandwidth and profile reduction made available by the
sparse matrix graphics program is one recently proposed by Gibbs, Poole, and
Stockmeyer (reference U). The actual FORTRAN implementation used, REDUCE, is
detailed in reference 5« REDUCE has been found to be considerably faster than
all other reduction codes in widespread use, generally superior for bandwidth
reduction, and generally as successful as any other for profile reduction.
The algorithm can best be described in terms of the related adjacency
graph, G, which has the property that there is an edge in G between vertices
v. and v if and only if a.. / 0 and i ^  j.
*J J- J
Step 1: Find the endpoints of a pseudo-diameter of the graph, that is,
a pair of vertices that are at nearly maximal distance apart. This is done by
a finite, iterative process of determining a vertex that is a maximum distance
away from a given vertex.
Step 2: Given the pseudo-diameter endpoints u and v of distance k
apart, partition the set of vertices into levels L, ,Lp,...,L. such that
adjacent vertices in G are in the same or adjacent levels and such that
maxJL I is nearly minimized,
i i
Step 3: Number the vertices of G, level by level, beginning at an
endpoint of the pseudo-diameter.
A detailed description of the algorithm may be found in reference U.
EXPERIENCE
The sparse matrix graphics program was developed on a PRIME 300, a multi-
programmed minicomputer with 6UK of core storage at ICASE, NASA Langley
Research Center (LRC). The program is interfaced with the University of
Michigan Graph-Pack Library and has been run using primarily a Tektronix
UOlU-1 graphics terminal. The program is also running on the CDC 6600 at LRC
under the KRONOS time-sharing operating system, again utilizing the Michigan
graphics library, using a Tektronix model ^ 015-1 display terminal. Earlier
batch versions of the program were run on an IBM 360/50 computer with a
Calcomp model 765 plotter as well as on a CDC 6hOO system with both Calcomp
and Varian type plotters.
The package has been tested using most of the finite element matrices
used to test subroutine REDUCE in reference k, as well as on matrices generat-
ed in finite element work at LRC. In general, response times for the genera-
tion of displays (especially on the minicomputer system) were more than ade-
quate for an interactive design environment. The block matrix scheme for dis-
playing otherwise unrepresentable matrices was found to be most acceptable.
609
Although very large sparse matrices do not in general appear as sparse tinder
this scheme, a good representation of the distribution of non-zero elements
in the matrices is nevertheless provided.
REFERENCES
1. Rose, Donald J. and Willoughby, Ralph A. (editors): Sparse Matrices and
Their Applications. Plenum Press, New York, 1972.
2. Tewarson, Reginald P.: Sparse Matrices. Academic Press, New York, 1973.
3. Everstine, G. C.: The BANDIT Computer Program for the Reduction of Matrix
Bandwidth for NASTRAN. NSRDC Report 3827, 1972.
k. Gibbs, N. E., Poole, W. G., Jr. and Stockmeyer, P. K.: An Algorithm for
Reducing the Bandwidth and Profile of a Sparse Matrix. ICASE Report,
July 22, 197^ - To appear in SIAM Journal on Numerical Analysis,
volume 13, 1976.
5. Crane, H. L., Gibbs, N. E., Poole, W. G., Jr. and Stockmeyer, P. K.:
Matrix Bandwidth and Profile Reduction. ICASE Report 75-9, April ll*,
1975. To appear in ACM Transactions on Mathematical Software, volume
2, 1976.
610
SPARSE MATRIX GRAPHICS PROGRAM t UERSION IB * 18/75
CMMRACTERISTICS OF INPUT HATRIX
ORDER • 319
MAX NUMBER OFF-DIAGONAL ELEMENTS IN ANV «ou • le
BANDUIDTH • 3«2 PROFILE • 23357
(FOR ALL VES-NO QUESTIONS REPLY 1 FOR VES. 0 FOR'NO)
UOULD VOU LIKE THE ORIGINAL MATRIX DISPLAYED 7
/ UOULD VOU LIKE A PERMUTED MATRIX DISPLAYED ON THE SAflC SCREEN '
•
/ UHAT SIZE BOX UOULD VOU LIKE THE PICTURE FIT INTO '/ nAxinun SIZE is 0 «64«e «3
.6C4E3
' ACTUAL DISPLAY MATPIX SIZE IS • .5167E 93
/ DISPLAY BftTPlX PARTITIONED INTO »LOC*S Of SIZE 3
' IF YOU UOULD LIKE ONLY THE LOUEP TRIANGULAR HALF DISPLAYED ENTER 9
/ OTHERUISE ENTER 1
1
/ TO BEGIN DISPLAY. HIT ANV KEY. CLEAR SCREEN. AND RETURN
Figure 1.- Typical interaction with sparse matrix graphics program.
611
s t. i a t L~
N*Tf>|X
MNWIOTH* MI
Figure 2.- Display of input matrix.
612
14
Figure 3.- Display of matrix after renumbering.
613
-3
e.3
 O
w
III
&s§0>X•H1MI
It
^^«I
*
J
-
(U
614


An interactive graphics package for the automatic node renumbering of finite element matrices

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server