The dual reciprocity boundary element method for the eigenvalue analysis of Helmhoitz problems

D. P. N. K O N T O N I

Department of Civil Engineering, University of Patras, GR-26110 Patras, Greece.

P. W. P A R T R I D G E AND C. A. BREBBIA

Computational Mechanics Institute, Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton S04 2AA, England

The dual reciprocity method (DRM) is a general technique for taking domain integrals to the bound- ary in BEM analysis. In this paper it is applied to the eigenvalue analysis of Helmholtz problems. A solu- tion procedure is presented which avoids the complex eigenvalues usually associated with the non-sym- metric BEM matrices and which is at the same time easy to implement. Characteristic numerical exam- pies are used to illustrate the proposed method.

Key Words: Boundary element method (BEM), dual reciprocity method (DRM), Helmholtz equation, eigen- value analysis.

INTRODUCTION

In early BEM analysis there existed only two alternative solution procedures: either internal cells had to be de- fined, or a fundamental solution had to be found which took into account all the terms in the governing equation of the problem under consideration. With the recent developments of the method, fundamental solutions to many different equations have been found 1'2, however this does not ensure that one is available for any given case. The use of cells to evaluate a domain integral implies an interior discretization which considerably increases the amount of data needed to run the program and hence the method loses some of its attraction in relation to finite elements or other techniques.

In order to avoid the problems mentioned above it is desirable to have available a boundary element technique that

a) Enables a "boundary only" solution to be ob- tained, i.e. without discretizing the domain into cells;

b) That does not depend on obtaining a new funda- mental solution for each case;

Paper accepted September 1990. Discussion closes June 1991.

c) That can be applied using a similar approach in each case.

Although several techniques have been proposed which aim to satisfy the above conditions the dual reciprocity method is the only alternative which can be used in the general case. The method was proposed by Nardini and Brebbia 5 in 1982 and can be used with any type of fundamental solution without the need to define internal cells. The method has now been extended to a wide range of engineering applications 3''~.

In this paper the dual reciprocity method is applied to the solution of the Helmholtz equation. The determi- nation of the eigenvalues and eigenmodes of the Helm- holtz equation is a subject of interest. These have importance in a number of physical problems, e.g., for mechanical systems undergoing periodic forced motion, eigenvalues will correspond to natural frequencies or resonances which are, in general, to be avoided, for nonperiodic motion, solutions for these systems may be developed as expansions in series of these eigenmodes. Typical mechanical systems in these cases include vibrat- ing membranes, acoustic waves in two- and three-dimen- sional containers, surface gravity water waves in closed basins of constant depth, etc.

MATHEMATICAL DEVELOPMENT OF THE DRM

The BEMoDRM method will be explained with reference to a basic, Poisson-type equation of the form

V2u = b (1)

b will be considered to be a function of the potential and position, i.e. b = b(x, y, u).

The solution to equation (I) can be expressed as the sum of a solution of a homogeneous, Laplace's equation and a particular solution a, such as

V2t~ = b (2)

2 Adv. Eng. St?/?ware, 1991, VoL 13, No. 1 ~', 1991 Computational Mechanics Publications

It is generally difficult to find a solution fi that satisfies the above, particularly in the case of non-linear or time- dependent problems. The dual reciprocity method pro- poses the use of a series of particular solutions fit instead of a single function a. The number of ~t used is equal to the total number of nodes in the problem. If there are N boundary nodes and L internal nodes, there will be N + L values of ~j, see Fig. 1.

The following approximation for b is then proposed:

N + L

b ~ Z ~tjfj (3) j = l

where the atj are a set of initially unknown coefficients and the f~ are approximating functions. The particular solutions, ~j, and the approximating functions, fj are such that:

V2fit = fj (4)

The functions fj in (3) can be compared to the usual interpolation functions ~0~ in expansions such as,

u = Y' ~0,u, (5)

which is used on the boundary elements themselves and in Finite Element analysis. Equation (3) is exact at the node points, as is equation (5).

The expansion (3) may then be considered as applying over the whole problem domain as in the case of one large superelement. The functions fj are geometry-depen- dent functions, as are the ~0~ in equation (5). At present, no restriction will be placed on these functions, and in fact many different types may be used, each of which results in a different function a t as determined from (4). The question of which type of function fi to use will be considered in Section 3.

Substituting equation (4) in (3) one obtains

N + L

b = ~ ~9(V2t~j) (6) j = l

Equation (6) can be substituted into the original equa- tion (1) to give the following expression:

N + L

VZu = ~ ~j(VZt~t) (7) j f l

In equation (7) the source term b has been replaced by a summation of products of the Laplacian of the original

Fig. 1.

D o

Boundary and internal nodes

equation, operating on the particular solutions tit and the coefficients ~9-

The usual procedure for developing the Boundary Element Method will now be applied. Equation (7) can be multiplied by the fundamental solution and integrated over the domain, producing

f (V2u)u,d~ yL f (v%)u*an m j = 1 ~J (8)

Integrating by parts produces the following integral equation for each source node i,

c,u, + f q*udr- f u*qdr

-- ~=i ~j@a. + f q*ajdr- f ur4jdI') (9)

The term qj in equation (9) is analogous to q = Ou/On, where n is the outward normal i.e.

0 = no ~-- (10)

which can be expanded to

0fi 0x 0fi 0y - Ox Dn + Oy On (11)

Note that equation (9) involves no domain integrals. The source term b in (1) has been substituted by equiva- lent boundary integrals. This was done by first trans- forming the source term which appeared in the original equation, and then expressing both right and left hand sides of the resulting expression as boundary integrals using a weighted residual technique. The same result may be achieved using Green's second identity of the reci- procity principle. It is this operation which gives the name to the method: reciprocity has been applied on both sides of (8) to take all terms to the boundary, hence dual reciprocity method.

The next step is to express equation (9) in a discretized form, with summations over the boundary elements replacing the integrals. One then obtains for a source node i the expression

N N

ciui + ~. H,kuJ,- ~. Gikqk k = l k = l

= E ~ j i C i U i j "~- Hik~kj- Gikt~kt (12) t = l k \ ' k=1 k=1

The index k is used for the boundary nodes which are the field points. After application to all boundary nodes using a collocation technique, equation (12) can be expressed in matrix form as

N + L

Hu - Gq = ~ ~j(H~j -- G:]j) (13) j=l

In equation (13) the terms ci have been incorporated onto the principal diagonal of H.

If each of the vectors t~ t and 0j is considered to be one column of the matrices U and Q respectively, then

Adv. Eng. Software, 1991, Vol. 13, No. 1 3

equation (13) may be written without the summation to produce

n u - Gq = ( H U - GO_)~ (14)

Equation (14) is the basis for the application of the boundary element dual reciprocity method and involves discretization of the boundary only.

Internal nodes may be defined in the number and at the locations desired by the user, this is generally done at points where it is desirable to know the interior solution. If the exact locations of the internal nodes are not important another possibility is for the computer code to choose them according to some rules presented at the end of this paper.

The definition of internal nodes is not normally a necessary condition to obtain a boundary solution, how- ever, the solution may not be sufficiently accurate if a number of such nodes is not used.

When internal nodes are defined, each one is indepen- dently placed, and they do not form part of any element or cell, thus only the coordinates are needed as input data. Hence these nodes may be defined in any order.

The e vector in equation (14) will now be considered. It was seen in equation (3) that

N + L

b = Z fJ~J (15) j=1

By taking the value of b at (N + L) different points then (15) may be expressed in matrix form as

b = Fe (16)

where each column of F consists of a vector f j which contains the values of the function fj at the (N + L) DRM collocation points. Equation (16) may be inverted to produce

= F - t b (17)

and equation (14) becomes

n u - Gq = ( n ( J - G Q . ) F - t b (18)

Matrix F depends only on geometric data and has no relation to either governing equations or boundary con- ditions. It may be calculated and inverted once and stored in a data file for use with all subsequent computer runs involving the same discretization. Defining

S = ( H U - GO.)F -1 (19)

then equation (18) becomes

H u - Gq = Sb (20)

The calculation of S is done by multiplying known matrices.

The foregoing is the basis of the application of the dual reciprocity method to solve equations of type (1). The only difference in each new case will be a new vector b. In a next section, equation (20) will be specialized for the case of the Helmholtz equation.

DIFFERENT f EXPANSIONS The particular solution, t~, its normal derivative 0 and the corresponding approximating functions f used in DRM

analysis are not limited by the formulation except that the resulting F matrix, equation (17), should be non- singular.

In order to define these functions it is customary to propose an expansion for f and then compute fi and 4 using equations (4) and (I0).

Nardini and Brebbia, the originators of the method. have proposed the following types of function for f5.6

(i) Elements of Pascal's triangle. (ii) Trigonometric series, and

(iii) The distance function r used in the definition of the fundamental solution.

Many other types of functions may be proposed. The r function was adopted first by Nardini and Brebbia and then by most researchers as the simplest and most accurate alternative.

If f = r, then it can easily be shown that the corre- sponding fi function for two-dimensional problems is r ' /9 , remembering the definition of r

r2 2 2 = r x + r r (21)

where r x and rr are the components of r in the direction of the x and y axes.

The corresponding function 0 will be given by

r {rxcos(n, x) + rycos(n, y)} (22) 0=5 In the above, the direction cosines refer to the outward normal at the boundary with respect to the x and y axes. Formula (22) may be easily obtained using (11) and remembering that 8r/Ox = rx/r and t?r/c?y = rv/r.

Recent work 7's, suggests that f = r is in fact one component of the series

f = 1 + r + r 2 + . . . r " (23)

The fi and 0 functions for two-dimensional cases corresponding to (23) are:

r 2 r 3 rra + 2

t~=-~+~-+. . . +(m+2) ~ (24)

( c~x Oy'](1 r r" ) 0 = rx + r, + 5 + ' + + 2----) (25)

In principle, any combination of terms may be selected from (23). In what follows, f = 1 + r was adopted which has been shown 7.s, to be the simplest and most accurate alternative.

Notice that in some cases when f is defined as a distance function it may be better to scale r by a distance to avoid numerical overflow.

TH E H E L M H O L T Z EQUATION

The solution to the Helmholtz equation provides the natural or fundamental frequencies and vibration modes for a system. The equation in its usual form is given by

V2H -{- ,[/2U = 0 (26)

where the coefficient #2 is related to the natural frequenc3, and u represents displacements.

~.2.

O F C

F 1

B

Vibratin9 fixed-free rod

Consider the vibrating rod shown in Fig. 2. In this case .2 _ 2 _2 2 ( / 9 ) where 0 and E are material /d ~ tU " t o ~ O) " properties and w are the natural frequencies 9'~ o.

In the case of harbour resonance problems the equa- tion becomes ~

(./)2 V2H + ~-g H = 0 (27)

where h is the mean water depth from a reference datum, o~ is the natural frequency as before and H = (h + t/), i.e. mean water depth plus free surface elevation, t/.

The Helmholtz equation also has important applica- tions in vibrating membranes ~°, acoustics, and other areas.

BEM-DRM FORMULATIONS FOR THE H E L M H O L T Z EQUATION

The Helmholtz equation can be modelled with DRM noticing that the last term in (1) is now

b = -/~2u (28)

such that (17) becomes

= _ _ / t 2 F - I u (29)

This gives the matrix equation

Hu - Gq = - / a2 (H0 - GQ)F- lu (30)

or using the S matrix (equation (19))

Hu - Gq = - # 2 S u (3t)

This equation has been studied by Loeffier 1:, who partitioned the matrices as shown below to eliminate the terms in q which are unknown on part of the boundary.

H2, HnJ u= LG=, onJ q2

2 FS11 S121 [Ul ] = Ls , (32)

Considering the problem of the vibrating rod shown in Fig. 2, on the part A-D of the boundary or F~, the rod is clamped so u = 0. On F 2 which is the free boundary

ABCD in Fig. 2, q = 0, such that equation (32) may be rewritten

[ H~I H~21 [G11 Gx21 LG21 G2zJ

o r

H12 u - Gllq = -~2Sa2u (34)

H 2 2 u - G21 q = -/u2S221J

Eliminating q between equations (33) gives

(H22 - G21G~11HI2)u = -/a2($22 - GzIG~,XS12)u (35)

o r

Ku = 1~2Mu (36)

where K and M signify stiffness and mass matrices and are defined from equation (35). The above procedure is the same as that used in condensation in finite element analysis. Equation (36) is now in the usual form for eigenvalue/eigenvector analysis. The matrix M is in- verted to procedure Au =/~2u where A = M-~K. The eigenvalues and eigenvectors are then extracted by analy- sis of the matrix A as was proposed by Brebbia and Nardini for elasticity problems s'6. This solution proce- dure may produce difficulties however as the matrix A is non-symmetric, and as a consequence, some of the eigen- values will be complex. Those authors found that the complex eigenvalues are the higher ones, which are normally of less or no interest. In any case, problems with higher order vibration modes are inevitable when repre- senting a continuum problem by a finite number of degrees of freedom.

A simple solution procedure for Helmholtz problems using the boundary element method avoiding complex eigenvalues will now be given.

Instead of obtaining the first n eigenvalues and asso- ciated eigenvectors all at once, as is usually done when solving the standard eigenvalue equation (36), each is obtained separately by incrementing /~ in small steps from 0. At each step, the right hand side of equation (31) is multiplied by a different value of/z 2 and the boundary conditions are applied to obtain the usual equation

Ax = y (37)

which is then solved. If/~ is not near a natural frequency, the displacements obtained at the nodes are small. As/~ is increased the maximum values of displacements occur as the natural frequencies are approached. Matrices H, G and S are all unaltered by the increments of/~ and may be calculated and stored once. If desired, a variable step in/a may be used.

If the usual homogeneous boundary conditions u = 0 on F 1, q = 0 on F 2 are imposed, only the trivial solution u = q = 0 everywhere will be obtained. However, this difficulty may be overcome by assigning a small rigid body displacement, uo, along the F: boundary (or at least one node of the F~ boundary). The exact value of Uo is of no importance, and will not affect the results. This is because the magnitudes of the displacements in any

Adv. Eng. Software. 1991, Vol. 13, No. 1 5

resonance problem are always multiplied by an arbitrary constant. It is the deformed shape which is important, the exact magnitudes will only ix given by a full dynamic analysis, and will depend on the type and size of excita- tion.

The discontinuity in q at A and D may ix taken into account by imposing the boundary conditions q before the node at A and q after the node at D as described in Ref. 2.

Equation (31) is solved starting from # = 0 using a variable step on #. The program starts with A# = 0.1; if, for two successive iterations, the mean increase in u at all nodes was more than 40% then A/~ was decreased to 0.01. When mean values of u started to drop, A# was increased to 0.1, etc.

NUMERICAL EXAMPLES

This section describes the detailed numerical solution of representative numerical examples in order to illustrate the use of the proposed method and demonstrate its advantages and disadvantages.

E x a m p l e 1 - - D R M resu l t s f o r v ibra t ino rods

Consider the free vibration of the rods shown in Fig. 3. The exact solutions for the natural frequencies oJ. and the normal modes U . ( x ) are given by 9'1°

q=O

Fixed-Free Rod

q=O

o c;v

q=O

Fixed-Fixed Rod

0 I$

~ . 3 .

q--O

Free-Free Rod 0

II o '

q--O

Vibra t ino rods wi th d i f f e ren t b o u n d a r y cond i t ions

{( {( fixed-free rod co. = # . . c o = n - c o U . ( x ) = C . sin n - for n = 1, 2 . . . . (38a)

fixed-fixed rod in. =/z..Co = T .co U . ( x ) = C . sin --7- for n = 1, 2 . . . . (38b)

(7) C x) free-free rod ~o. = # . . Co = • co U . ( x ) = C . cos T for n = 1, 2 . . . . (38c)

where co = x / ~ and n is an integer which takes values up to the order desired. In (38) I is the length of the rod and C an arbitrary constant. In what follows l = 0.9 m and C = 1.

The boundary conditions are shown in Fig. 3. More- over, a small arbitrary displacement Uo, (e.g. uo = 0.1), was imposed at the boundary node in the middle of the left short side for the fixed-free and fixed-fixed rod case, while a small arbitrary value qo (e.g. qo = 0.1) was imposed at the node for the free-free rod case.

To test sensitivity to the number of boundary ele- ments/side and also to the number of internal nodes, different discretizations were tested, some of which are shown in Fig. 4, for the fixed-free rod case.

BEM-DRM results for #i are shown in Tables 1.1, 1.2 and 1.3 for the fixed-free, fixed-fixed and free-free rod case, respectively, and are compared with the exact solutions calculated from equations (38). Moreover re- sults are represented for two values of the width: b = 0.2 m and b = 0.4 m.

Analysing these tables, it may be seen that results corresponding to discretizations with I0 nodes/long side are less accurate than those with 19 nodes/long side.

Moreover, especially for higher vibration modes, a fair density of internal nodes is advisable. The results show that it is important to include approximately L = N/2 internal nodes.

In Figs 5, 6 and 7 the first four normal modes of the rods of Fig. 3 with b = 0.2 m have been plotted. In each plot, four sets of data are presented: the exact results obtained from equations (38) with C = 1 and the BEM- DRM results obtained with (N = 40, L = 1), (N = 40, L = 8) and (N = 40, L = 24). (All BEM-DRM results are normalised, i.e., have been divided by the maximum value obtained. Results have been plotted for boundary nodes 1-19, those at remaining nodes can be obtained by symmetry). Note that the correspondence between the DRM results and the exact solution is very good. Other natural frequencies may be obtained, but this requires the use of a finer discretization. It may be noted in Fig. 6(d) that the fourth vibration mode contains 8 quarter wave- lengths. Employing 19 nodes per side, there are thus 19/8 nodes per quarter wavelength for this mode and the discretization has thus reached its limits. If/z 5 is to be obtained, a finer discretization must be used. For a good representation of the deformed shape it is evident that

) I 10

II b=0.2

2_ Zl IZ

I=o.9

I 19

. . . . . . . . . . . . . . . . . t .T 20 b-0.2

39 21

! I :o.9

1 . . . . . . . . . 19

. . . . . . t20 l

pX ) 1 19

20 b-O.2

39 Z1

p,X

) 1 19

2.(> b-0.2

. . . . . L 39 21

[pX

iiii ii t: b'4 F' ... . . . . . . . . . . . . o l

Fi9. 4. Different B E M - D R M discretizations o f afixed-free rod

h X

~ X

~sr

Table 1.1

FIXED-FREE ROD

b = 0.2 m b = 0.4 m #i

N = 22 N = 40 N = 26 N = 44 (10 nodes/long side) (19 nodes/long side) (10 nodes/long side) (19 nodes/long side)

EXACT L = 1 L = 8 L = 2 4 L = I L = 8 L = 2 4 1.745 1.75 1.75 1.75 1.74 1.74 1.74 5.24 5.26 5.26 5.26 5.22 5.23 5.24 8.73 9.03 8.88 8.86 8.84 8.76 8.75

12.22 13.30 12.76 12.61 12.77 12.44 12.36

L = I L = 8 L = 2 4 L = I L = 8 L = 2 4 1.75 1.75 1.75 1.75 1.75 1.74 5.38 5.28 5.26 5.35 5.25 5.24 9.71 9.07 8.87 9.52 8.96 8.80

14.70 13.40 12.73 14.13 13.10 12.56

Table 1.2

FIXED-FIXED ROD

b = 0 . 2 m b = 0 . 4 m

N = 2 2 N = 4 0 N = 2 6 N = 4 4 (10 nodes/long side) (19 nodes/long side) (10 nodes/long side) (19 nodes/long side)

EXACT L = 1 L = 8 L = 2 4 L = I L = 8 L = 2 4 L = 1 L = 8 L = 2 4 L = I L = 8 L = 2 4 3.49 3.49 3.50 3.50 3.48 3.49 3.49 3.52 3.50 3.50 3.51 3.48 3.49 6.98 7.12 7.05 7.05 6.98 6.99 6.99 7.68 7.14 7.04 7.56 7.08 7.01

10.47 11.13 10.79 10.70 10.80 10.58 10.54 12.38 11.23 10.76 12.00 11.04 10.65 13.96 16.09 14.86 14.54 t5.14 14.37 14.20 18.17 16.03 14.78 17.12 15.52 14.53

Adv. Eng. Software, 1991, Vol. 13, No. I 7

Table 1.3

FREE-FREE ROD

b = 0 . 2 m b=0 .4m

N = 22 N = 40 N = 26 N = 44 ( 10 nodes/long side) ( 19 nodes/long side) ( 10 nodes, long side) ( 19 nodes 1o n g side)

EXACT L= 1 L = 8 L = 2 4 L= 1 L = 8 L = 2 4 3.49 3.49 3.50 3.50 3.48 3.48 3.49 6.98 7.08 7.05 7.05 6.99 6.98 6.98

10.47 11.03 10.77 10.7 l 10.71 10.56 10.54 13.96 15.40 14.84 14.61 14.67 14.35 14.23

L = I L = 8 L = 2 4 L = I L = 8 L = 2 4 3.52 3.50 3.50 3.51 3.49 3.49 7.22 7.11 7.05 7.15 7.05 7.01

1 t,91 I 1,06 10.75 11.59 10,89 10.64 16.67 15.64 14.80 15.95 15.21 14.54

there should be at least 3 nodes per quarter wavelength. It must be noted that using this formulation the

problem of working with complex eigenvalues is avoided. Any vibration mode may be obtained if an adequate discretization is used. Extremely fine discretizations must be employed if results for high order vibration modes are desired, following the rule of approximately three nodes per side for each quarter wavelength for the highest mode to be obtained. This is also true for Finite Element analysis or any other discrete numerical method. In many cases in engineering practice, high order vibration modes are of a lesser importance.

Example 2 - - D R M results f o r rectangular ribrating membranes

Consider the free vibration of a membrane of length a, width b and fixed along all edges.

For a given value of n, m, the vibrational frequency ~ , , , and the normal modes IV... are given by z°

{C; ~ ~ ~ • + "co (39a) OJnm = ~nr,, " Co = ~ 2 2

THE F I ~ S T NOflMAL MODE OF A F I X E O - F ~ E E

o g a

0,?o

O~H3

o ~

040

O ~

0; :0,

OllO ,

0 .00

AO0

( exact ) x / l

THE S E C 0 N 0 NOQMAL MODE OF A ~IXED-~QEE Q00

t.2O

1.00,

O,~O,

0.~0,

0 . , ~ .

0.;'0 ,

-'Q.40

-O.M

-OAIO

-t.O0

-! .20

@ o

. , , , t , , l , , , !

+

( - - - - exact ) x/1

THE T H I R O NORMAL MODE OF A FIXEO-FAEE R0D THE ~ 0 U R T H NORMAL MODE OF ~ F I X E D - F ~ E E ~00

/ I O N

-o. ,~

t.O0

0.80

O ~

O.,tO

0.00

-0.20

-0.40

-0 .~

-O.m

-1 .~'0

+

I t

( e x i t . t ) x / ] ( . . . . e x a c t l x / ]

Fig. 5. The f i rs t f o u r normal modes ~?/" a)qxed- free rod ( N = 40, • • • L = I, o o o L = 8, x x x L = 24)

8 Adv. En.q. So[?ware, 1991, Vol. 13, No. 1

I I0

1.00

090

070

0~0

0.50

0.40

030

O ~

0 10

O.O~

THE ~ I R S T NORMAL NODE OF A FIXED-FIXED ROD

( , . ' xac t ) x / }

THE S E C O N O NORMAL NOOE OF A FIXED-FIXED ROD

0 . ~

0~0

0.40

0 . ~

0.00

-O.L~

-O.SO

- 0 . ~

'*' I~ llI II IR m I~ 8 8 s~

( ~ exact ) x/]

THE T H I R D NORMAL MODE OF A FIXED-FIXED ROD THE F O U R T H NORNAL NODE OF A F I X E D - F I X E D ROD

t . ~

0 . N

0.M

0 ~ 0

0.00 . . .

-0.~

-0.~

-0.e0

- t .M

~- ~ IR Q ~ S ~ 8 8 8 2

?; Z 0 . ~

-0,10

-O,M

Z " Ig II~ II ~ 8 £ 8 8 • ~.

{ e~act ) s/l ( - - exact ) x/}

Fig. 6. The first four normal modes o f a f ixed-f ixed rod ( N = 40, • • • L = 1, o o o L = 8, x x x L = 24)

Numerical results will be given for two cases:

i) square membrane with a = 1.0 m ii) rectanguhtr membrane with a = 2.5 m, b = 1.1 m

The boundary conditions are u = 0 along all edges. Moreover, a small arbitrary displacement uo, was im- posed at one boundary node.

In order to test sensitivity to the number of boundary elements and also to the number of internal nodes, different discrel:izations were tested.

BEM-DRM results for/a,", corresponding to the first four vibration modes, are given for the unit square membrane in Table 2.1, while results for the rectangular membrane wit]a a = 2.5 m and b = 1.1 m are given in Table 2.2.

In Figs 8 and 9, mode shapes of the square and the rectangular membrane, respectively, are plotted. Note that the correspondence between the DRM results and the exact solution is very good.

Analysing Tables 2.1 and 2.2, it may be seen that, due to the boundary conditions (u = 0) of the membranes, an increase in the number of boundary nodes N does not improve the accuracy of the results. As it is obvious from equation (31), for problems with fixed boundaries, a number of internal nodes is necessary to be employed in

order to obtain an accurate solution. For the internal nodes the next rule must be followed in both the x- and the y-direction: approximately three nodes for each half wavelength for the highest mode to be obtained in this direction. For example, for the square membrane with n = m = 3 : L > 7 × 7 = 4 9 or better L = 9 × 9 = 8 1 ; for the rectangular membrane with n = 3, m = 1: L > 7 x 3 = 2 1 , w i t h n = 3 , m = 2 : L > 7 × " 5 = 3 5 , etc.

Example 3 - - D R M results f o r rectangular cavities Consider the wave propagation in a rectangular cavity

of dimensions a in the x-direction and b in the y-direction under Newmann boundary conditions.

The exact eigenvalues co,,.z and the corresponding eigenmodes W,m are given by

( .19n" . = / . l n " . = 1"~. 2 . - ~ . C 2 '

m, n = O, 1, 2 . . . . (40a)

(7} (7} I4I.,, = cos x • cos y , m, n = O, 1, 2 . . . . (40b)

with c denoting the sound velocity.

Adt'. Eng. Software, 1991, Vol. 13, No. 1 9

THE F I ~ $ Y NORMAL MODE OF A F~EE-WREE ~00 T~E S E C 0 N 0 NORMAL WOOE 0F { ¢(~EE-CF~EE ~O0

O N

0 I t 0

0 ~ 0

0 ~

0 ,00

- 0 M

"1 0(I

- I .L~

7

o a n, ~ o

( exact 1 x / l

O . N

°! \ / (L IO

0 .L~

000 . ,

.... \_ /

( _ _ exact ) ' ( / 1

THE T H r R O NORMAL MODE OF A FREE-FREE ~I00 THE F 0 U FI I" H NORMAL MOOE OF A ~REE-FHEE A00

tO

t 0 0 ~

O W

O M

O ~

O ~

O ~

-o ,o , :r -0 M .

- 0 1 m .

O . m !

/ \ / 0 . 4 0 "

0120 - - . - . : : . . . . . . ~ _

( ...... exact ) xll ( ~, e x a c t ) x/l

Fio. 7. The f i rs t f o u r normal modes o f a free-free rod ( N = 40, • • • L ~- 1, o o o L = 8, x x x L = 24)

Table 2.1

SQUARE MEMBRANE (a = 1.0 m)

~ n m

(n, m) EXACT

N f 2 0 (6 nodes~ide)

L = 9 L = 25 L = 49 L = 81 ( = 3 x 3) ( = 5 x 5 ) ( = T x 7) ( = 9 x 9 )

( I, 1 ) 4.44 4.56 4.49 4.44 4.43 (2, 1) 7.02 7.75 7.38 7.08 7.03 (2, 2) 8.89 10.53 9.70 9.02 8.90 (3, 1 ) 9.93 12.27 11.26 10.23 10.06

N = 40 (11 nodes/side)

L - - 9 L = 2 5 L = 4 9 L = 8 1 ( = 3 x 3) ( = 5 x 5) ( = 7 × 7) ( = 9 x 9 )

4.58 4.51 4.46 4.45 7.82 7.42 7. l 2 7.07

10.68 9.85 9,13 9,0 12.36 11.36 10.31 10.12

Table 2.2

RECTANGULAR MEMBRANE (a = 2.5 m, b = 1.I m)

N = 20(84 n°des/l°ng side a ~ nodes/short side b ]

EXACT (n, m) L - 15 L = 27 L = 36 L -- 60 L ~- 65

( = 5 x 3 ) ( = 9 × 3 ) ( = 9 x 4 ) ( = 1 2 x 5 ) ( = 1 3 x 5 )

(1, I) 3.12 3.24 3.15 3.14 3,13 3,13 (2, 1 ) 3.80 4.02 3.85 3.82 3.80 3,80 (3, I ) 4.73 5.23 4.85 4.80 4.74 4,73 ( I, 2) 5.85 6.69 6.08 5.98 5.84 5.83

{ 15 nodes/long side a'~ N = 40 k, 7 nodes/short side b /

3.25 3.16 3.15 3.14 3.13 4.07 3.89 3.88 3.83 3.83 5.32 4.91 4.86 4.79 4.78 6.83 6.17 6.07 5.92 5.90

L = I 5 L = 2 7 L - 3 6 L = 6 0 L = 6 5 ( = 5 x 3) ( = 9 × 3) ( = 9 x 4) ( = 12 × 5) ( = 13 × 5)

THE MODE SHAPES OF A $OuARE MEMBRANE FO~ N-M- ! THE MODE SHAPES OF A SQUARE MEMBRANE FOB H=2 . M=I

0 ~

OgO

z O. 70

i 0.50

0 . 5 0 m

~ 0 4 0 ,

o . ~ .

0 2 0 .

O. lO

OPO

100

[ - - exact ) y - 0 . 5 x . . . . •

- t . 00

o

I 0 0

0 6 0

0 . ~

0 ~

0.20 uJ

-~.&O

THE MODE SHARES OF 4 RECTANGULAR MENBRANI~ FOR N - ! , H - !

( exact ) y - 0 5 x .... >

Fi9. 8. Mode shapes o f a square membrane ( N = 20, o o o L = 9, x x x L = 49)

1.10 . .

090

O N

0 7O

POO

020

OOO

( - - e x a c t ) y - 0 5 5 x - - - >

- t~ Pt af IR at ~ aB IR S m I~

THE MODE SHAPES OF A RECTANGULAR MEMBRAN~ FOR N-% . M - |

| . 10 . . . . . . i . , . . . , J , , , ,

I .W .

0.I0

0,70.

~ 0 . ~

~ O.4O

0.30

0 . ~

0,10

( ~ e x a c t ) x - 1 . 2 5 y - - - >

THE MOOE SHAPES OF A RECTANGULA~ MEMBQANE FOR N ' 2 . N ' i THE NODE SHAPES OF A RECTANGULAR MEMBRANE FOR N - 3 , M - !

t20

100

o00

o ~

0 in z

~ O O O

~ 0 OO _ _ J

c ~ -O 20

- 0 OO

-t oo

-t 2 0 . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :. . . . . . . . 4

_ _ e x a c t ) v - 0 5~ x - - - >

t.eo i i

O ~

0 . I0

0 . ~ , ,

8 1~ I1' 3 IB S ~ ~ 3 | S ~ g 8

( e x a c t 1 y - 0 S~ x - - - >

Fig. 9. Mode shapes o f a rectanoular membrane ( N = 20, o o o L = 27, x x x L = 65)

Adv. Eng. Software, 1991, Vol. 13, No. 1 11

Numer ica l results will be given for two cases:

i) square cavi ty with a = 1.0 m ii) r ec tangu la r cavi ty with a = 2.5 m, b = 1.1 m.

The b o u n d a r y cond i t ions are q = 0 a long all. edges. Moreover , a small a rb i t r a ry value qo, was imposed at one b o u n d a r y node.

In o rde r to test sensi t ivi ty to the number of b o u n d a r y e lements and also to the n u m b e r of in ternal nodes, different d iscre t iza t ions were tested.

B E M - D R M results for # , , , co r r e spond ing to the first four v ib ra t ion modes , a re given for the unit square cavi ty in Table 3.1, while results for the r ec t angu la r cavi ty with a = 2.5 m and b = 1.1 m are given in Tab le 3.2.

In Figs 10 and 11, m o d e shapes of the square and the r ec tangu la r cavity, respectively, a re plot ted. No te tha t the co r respondence between the D R M results and the exact so lu t ion is very good.

As it is obv ious from equa t ion (31), due to the b o u n d - ary cond i t ions (q = 0) of these p rob lems , the b o u n d a r y nodes signif icantly affect the results. As a rule app rox i - mate ly three b o u n d a r y nodes for each qua r t e r wave- length for the highest m o d e to be ob t a ined in this d i rec t ion mus t be used.

In o rde r to increase the accuracy of the results, in ternal nodes mus t be used fol lowing the rule o f a p p r o x i m a t e l y three nodes for each half wavelength for the highest m o d e to be ob t a ined in this direct ion.

Example 4- - -DRM results for circular vibratin# membranes

Cons ide r the free v ib ra t ion of a c i rcular m e m b r a n e of rad ius R = 1.0 m and fixed a long its per imeter .

Z

<

e C3

100

O H

O.H

O,H

-O.H

-O.H

-O.W

-t.2O

THE MODE SHAPES OF A SOUA~E CAVITY ~OR N-1 ~ - 0

i

( - - exact ) y-0.5 x - - - >

@ o

THE MODE SHAPES OF A SQUARE CAVITY FO@ N = I . M=l

7

c l

I . H

0 .W

0 . 6 0 .

0 ~ O ,

0 . H

"0.~0

- 0 , , 1 0

" 0 6 0 .

~ N

- ! ,0a

- i . z ¢

/

[ e~act ] a = l S - + - >

Fig. 10. Mode shapes o f a square cavity ( N = 40, o o o L = 9, x x x L = 49)

Table 3.1

SQUARE CAVITY (a = 1.0 m)

#,m N = 20 (6 nodes/side) N = 40 (11 nodes/side) N = 80 (21 nodes/side)

EXACT (n, m) L = 9 L = 25 L = 49 L = 81 L -- 9 L -- 25 L = 49 L = 81 L = 49 L = 81

(--3 x3) ( = 5 x 5 ) ( = 7 x 7 ) ( = 9 x 9 ) ( = 3 x 3 ) ( = 5 x 5 ) ( = 7 x 7) ( = 9 x 9 ) ( = 7 x 7) ( = 9 × 9 )

(1, O) 3.14 3.18 3.16 3.15 3.15 3.18 3.16 3.15 3.14 3.15 3.14 ( 1, I ) 4.44 4.52 4.47 4.46 4.45 4.49 4.47 4.45 4.45 4.45 4.44 (2, 0) 6.28 6.70 6.45 6.37 6.33 6.63 6.44 6.36 6.33 6.35 6.32 (2, 1) 7.02 7.53 7.23 7.17 7.12 7.34 7.18 7.10 7.07 7.08 7.06

Table 3.2

RECTANGULAR CAVITY (a = 2.5 m, b = 1.1 m)

/in., /'8 nodes/long side a \

N = 20 L4 nodes/short side b) /15 nodes/long side a \

N = 40 L7 nodes/short side b)

EXACT (n,m) L = 15 L=27 L=36 L = 6 0 L=65 L--15 L=27 L = 3 6 L = 6 0 L=65

( = 5 x 3) ( = 9 x 3) ( = 9 x 4) (=12 x 5) (=13 x 5) ( = 5 x 3) (=9 x 3) (=9 x 4) (=12 x 5) (=13 x 5)

( 1, O) 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 (2, o) 2.51 2.55 2.54 2.54 2.53 2.53 2.54 2.53 2.53 2.52 2.52 (0, I ) 2.86 2.92 2.89 2.88 2.87 2.86 2.92 2.89 2.88 2.87 2.86 (1, I) 3.12 3.19 3.16 3.15 3.13 3.13 3.17 3.15 3.14 3.13 3.13

12 Adv. En,q. Software, 1991, Vol. 13, No. 1

THE MODE SHARES OF A RECTANGULAR CAVITY FOP N ' I , M'O THE MODE SHAPES OF A RECTANGULAR CAVITY FOR N*2 . M-O

+

0 . 1 1 0

~ 0 . ~

" l L ~

[ exact ) a -2+S b=%.1 S - - - >

t,20

t,O0

°.aDo

0.50

0.40

~ . 6 0

( e x a c t ) a - 2 . 5 b - t . t S - - - >

THE MODE SHAPES OF A RECTANGULAR CAVITY FOR N-O , M ' l THE MODE SHAPES OF A RECTANGULAR CAVITY ~OR N - ] , M - |

• 0 . N

<

" 0 + ~ .

" 0 . 4 0 .

' - 0 . I 0

' -O . i lO

-~ .oo I I I • t~ 3 S ~ S • 8

o . ~

o . m

o . . - . . . . . . : . . . . : . . . . . . . . . : . . . . . . . : . . . . . . . . ; . . . . . . . . . : . . . . : . . . . . . . . . . . . . . : . . . . . . . .

I I • i i I ~ I t I I I I • R • •

( e x a c t ) a - 2 + 5 b-~+l S - - - > ( _ _ _ _ exact ) 0 " ~ 5 b ' t . l S - - - >

Fig. l l ( a ) . Mode shapes of a rectangular cavity (N = 40, o o o L = 27, x x x L = 65) ( N = 2 0 , o o o L = 2 7 , * * * L = 6 5 )

THE MODE SHAPES OF A RECTANGULAR CAVITY FOR N - I . M-0 THE M00E SHAPES OF A RECTANGULAR CAVITY FOR N-2 . M-0

e l K + .

0 . ~ .

.-.0 I 0

- 0 ~

- i . 0 0 .

( e x a c t } y - O + 5 5 X - - - >

0 . ~ .

o . I o ,

0 . ~ .

0 . 0 0 . , +, , J , , ,

+++1 - . . - . - - - - . . : . . . . . . . ; . . . . . . . : + : . . . . . :+ ++:+ . : . . . . . +++ : + . . . . . . . . . . . . . . . . : . , .

( e ~ a ~ t ) y ' O 55 X - - - >

Fig. l l ( b ). Mode shapes of a rectangular cavity ( N = 40, ooo L = 27, x x x L = 65)

Adv. Eng. Software, 1991, Vol. 13, No. 1 13

The natural frequencies o~.,. and the normal modes IV.= of a circular membrane are given by1°:

oJ,,, = i~.,., co (n = 0, 1 . . . . . m = 1, 2 . . . . ) (41a)

W,m = J,(l~,, d • r) . cos{n0} (41b)

where the #,,, are assigned as follows: For a given value of n, m = 1 corresponds to the first root of J,(/~,= • R) = 0, m = 2 is the second root, and so forth.

The boundary condit ions are u = 0 a long the peri- meter. Moreover , a small arbi t rary displacement Uo, was imposed at one boundary node.

To test sensitivity to the number of Boundary Ele- ments and also to the number of interior nodes, different discretizations were tested.

D E M - D R M results for /am, corresponding to the first four vibrat ion modes are given in Table 4, while in Fig. 12 mode shapes of the diameter are plotted.

It is seen again that, due to the boundary condit ions ( u - - 0 ) of the circular membrane, an increase in the number of boundary nodes N does not improve the accuracy of the results. As it is again obvious from equat ion (31) , a number of internal nodes is necessary to be employed in order to obtain an accurate solution. For the internal nodes the rule of "approximate ly three nodes for each half wavelength for the highest mode to be obtained in this direct ion" can be followed. While for rectangular geometries there are only two (the x- and y-) directions, for circular geometries each diameter is a direction and approximate ly 8 directions should be con- sidered.

Example 5 - - D R M results f o r a right isosceles triangle

Consider a fight isosceles triangle of hypotenuse length

x /~ with N e u m a n n boundary condi t ion (q = 0). This could, for example, describe the seiching of water waves in a fight tr iangular basin of constant depth.

In the case of a right isosceles triangle, the exact values

for /~ are known to be Mn, M x , / ~ , for M = 1, 2, 3 . . . . (This is one of the very few nonseparable geometries for which a solution is known. Its eigenvalues and eigenmodes form a subgroup of those of a square for which these values are known analytically).

The boundary condit ions are q = 0. Moreover , a small

arbi t rary value qo (i.e. q = 0.1), was imposed at one boundary node.

In order to test sensitivity to the number of boundary elements and also to the number of internal nodes, different discretizations were tested.

B E M - D R M results for /A are given in Table 5, while Fig. 13 depicts the mode shapes of the fiypotenuse of the fight isosceles triangle.

Analysing the results it may be seen that approxi- mately three boundary nodes for each quarter wave- length for the highest mode to be obtained must be used. Provided that the above rule for the boundary nodes was applied, the results show that it is impor tant to include at least L = N /2 uniformly distributed internal nodes.

C O N C L U S I O N S

It has been shown that using the Dual Reciprocity Method an accurate solution to the Helmhol tz problems can be obtained without the need to define internal cells. When internal nodes are defined, each one is indepen- dently placed, and they do not form part of any element or cell.

By incrementing/z and repeated solution, the vibration modes may be obtained if an adequate discretization is used, wi thout encounter ing complex eigenvalues.

In order to ensure an adequate discretization, two rules must be followed:

i) approximately three boundary nodes for each quar ter wavelength for the highest mode to be obtained in this direction

ii) approximately three internal nodes for each half wavelength for the highest mode to be obtained in this direction.

A fine boundary discretization is not necessary for problems with fixed boundaries, such as membranes. However, for these problems the presence of internal nodes is indispensable.

A C K N O W L E D G M E N T

The first au thor would like to thank Dr. C. A. Brebbia, Director of the Computa t iona l Mechanics Institute, for his encouragement .

Table 4

CIRCULAR MEMBRANE (R = 1.0 m)

(n, m) EXACT

L = 17 L = 3 7 L=41

N = I 6

L = 4 9 L=65 L=81 L = 93

(0. l) 2.41 2.44 2.41 2.42 2.41 2.41 2.41 2.41 ( I, 1 ) 3.83 4.04 3.90 3.95 3.88 3.87 3.86 3.85 (2, 1) 5.14 5.80 5.38 5.58 5.34 5.29 5.26 5.24 (0, 2) 5.52 7.19 5.76 5.85 5.73 5.66 5.62 5.60

L = 1 7 L=37 L=41

N = 30

L = 49 L = 65 L=81 L = 93

2.44 2.41 2.42 2.41 2.41 2.41 2.41 4.05 3.91 3.96 3.89 3.87 3.87 3.86 5.83 5.40 5.61 5.36 5.30 5.27 5.24 7.21 5.78 5.88 5.75 5.68 5.64 5.61

L = 17: L = 37: L =41: L = 49: L = 65: L=81: L = 93:

node at centre, 16 nodes on r = 0.5 node at centre, 8 on r = 0.25, 12 on r = 0.5, 16 on r = 0.75 node at centre, 8 nodes on r = !/6, 2/6, 3/6, 4/6, 5/6 node at centre, 16 on r = 0.25, 0.5, 0.75 node at centre, 16 on r = 0.2, 0.4, 0.6, 0.8 node at centre, 16 nodes on r = 1/6, 2/6, 3/6, 4/6, 5/6 node at centre, (8 on r = 0.125, 0.25), (12 on r = 0.375, 0.5), (16 on r = 0.625, 0.75), 20 on r = 0.875

14 Adv. Eng. Software, 1991, VoL 13, No. 1

THE HOOE SHAPES 0 r A CIRCULA~ HENBRANE FOR N - 0 , M - )

~ N

O9O

0,10

O.~e

0 5 0

O ~ _ - - .

8 I~ ~ 2 B g g 11' 8 2 8 2

[ e x a c t ) R - I . S - - - >

IHE MODE SHAPES OF A CIRCULAR NEMBRANE FOR N - i . 14-|

o , 1 0 . . . . , . . , . . . . . . i , , i

0,*e

0,30

o. to

o .oo , 4 - - - - -* . . . . . ~ . . . . . . .

.-0A0

..O2O

-~ . !0

-0/,O

[ _ _ e x a c t ) R - I . S - - - >

THE 140DE SHAPES OF A CIRCULAR MEMBRANE FOR N=2 . I~-~ THE HODE SHAPES OF A CIRCULAR 14EI4BRANE FOR N - O , 1 4 - 2

O.~

O.II

O ~

O ~

0 .~

O,M

0.~0

~' oA6

0. t~

0 . N ,

0 . ~

0 , N

C _ _ e x a c t ) R - l , S - - - >

I . N

I . B

-Q.~0

-O.aR

• Q • • 8 £ Q • • •

( e x e c t ] R - I . S - - - >

e.M

0,40

0.~

o

Fig. 12. Mode shapes o f a circular membrane ( N = 30, ooo L = 37, x x x L = 93)

Table 5

RIGHT ISOSCELES TRIANGLE

N = 30 (11 per side) N = 42 (15 per side) N --- 60 (21 per side) ~i EXACT

L = 1 L = 10 L = 3 6 L = 1 L = 2 1 L = 7 8 L = 1 L = 3 6

1 3.14 3.17 3.15 3.14 3.17 3.15 3.14 3.17 3.14 2 4.44 4.49 4.47 4.45 4.49 4.45 4.44 4.49 4.45 3 6.28 6.49 6.40 6.32 6.45 6.34 6.30 6.43 6.31 4 8.89 9.49 9.25 9.02 9.38 9.07 8.93 9.32 8.97

Adr. Eng. Software, 1991, Vol. 13, No. 1 15

0 . 0 0 .

0011.

0.,14

0 . l ~ .

0 ~

-# .~1 .

- 0 . ~ .

- O . M ,

• " 0 . M .

- t t ~ .

FIRST EIGENNOOE ON HYPOTENUSE 0F A RIGHT ISOSCELES TRIANGLE

8

( e x s c t ] t~y~ote~use - - - >

0.10

0.00 I . . J

SECONO EIGENHOOE ON HYPOTENUSE OF a ~IGHT ISOSCELES T~ANGLE

0.70

0.S0

• 2 2 S • • 2

( e x a c t ) hyt)otef~use - - - >

THIRO EIGENMOOE ON HYPOTENUSE OF a RIGHT ISOSCELES TRIANGLE FOURTH EIGENNOOE ON HYPOTENUSE OF A PlOHT ISOSCELES TRIANGLE

t . ~ . .

I , m I

0 . g o ,

0 , 1 M .

0 , d 0 .

o l . 0 .

0 0 0

- 0 , N

- t a m .

- t .~,o .

8 ~g • I I 8 8 • l 8

0 . M

0 . ~

0 .70

0 .60

0 ~

0 . ~ 0

o. 10

[ - - - e~act 1 hyDoten~s~ - - - > ~ - - e x a c : t ) hyDOter~tJ~e - - - >

Fio. 13. E ioenmodes on hypotenuse o f a right isosceles trianole ( N = 42, o o o L = 21, x x x L = 78)

R E F E R E N C E S

1 Brebbia, C. A., Telles, J. F. C. and Wrob¢l, L C. Boundary Element Techniques, Springer-Verla$, Berlin and New York, 1984

2 Brebbia, C. A. and Dominguez, J. Boundary Elements: An Intro- ductory Course, Computational Mechanics Publications, South- ampton and McGraw Hill, New York, 1989

3 Partridge, P. W., Brcbbia, C. A. and Wrob¢l, L. C. The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, 1990

4 Partridge, P. W. and Brebbia, C. A. The dual reciprocity boundary element method for the Helmholtz equation, Proceedings of the International Boundary Element Symposium, Nice, France, 1990

5 Nardini, D. and Brebbia, C. A. A new approach to free vibration analysis using boundary elements, Boundary Element Methods in Engineering, Brebbia, C. A., (ed.) Springer-Verlag, 1982

6 Nardini, D. and Brebbia, C. A. Boundary Integral Formulation of mass matrices for dynamic analysis, Topics in Boundary Element Research, Vol. 2, Springer-Verlag, Berlin and New York, 1985

7 Wrob¢l, L. C., Brebbia, C. A. and Nardini, D. The dual reciprocity boundary element formulation for transient heat conduction, VI International Conference on Finite Elements in Water Resources. Lisbon, Portugal, 1986

8 Partridge, P. W. and Brebbia, C. A. The BEM dual reciprocity method for diffusion problems, VIII International Conference on Computational Methods in Water Resources, Venice, Italy, 1990

9 Craig,, R. R. Structural Dynamics, John Wile.y, 1981 10 Graft K. F. Wave Motion in Elastic Solids, Ohio State University

Press, 1975 11 Connor, J. J. and Brebbia, C. A. Finite Element Techniques jbr

Fluid Flow, Newnes-Butterworths, 1976 12 Loettler, C. F. An Alternative BEM Formulation Applied to Scalar

Field Problems, DSc Thesis, COPPE/UFRJ, Brazil, 1988, (in Portuguese)

13 Niku, S. M. and Brebbia, C. A. Dual reciprocity boundary element formulation for potential problems with arbitrary distributed sources, J. Eng. Anal. with Boundary Elements, 1988, 5, 46

16 Adv. Eng. Sr~[tware, 1991, Vol. 13, No. 1