An adaptive boundary element for eigenvalue problems with the Helmholtz equation: simplified h-scheme

Advancesin Engineering,Software19 (1994) 143-147 © 1994 Elsevier Science Limited Printed in Great Britain. AH rights reserved 0965-9978/94/$07.00

0965-9978(94)00013-1

ELSEVIER

An adaptive boundary element for eigenvalue problems with the Helmholtz equation: simplified h-scheme N. Kamiya, K. Nogae & S.-Q. Xu Department of Informatics and Natural Science, School of Informatics and Sciences, Nagoya University, Nagoya 464-01, Japan (Received 10 June 1994; accepted 1 July 1994) A simplified h-version of the adaptive boundary elements is proposed for the eigenvalue analysis of the Helmholtz equation. The new scheme considers the effect of each local boundary element refinement, not on the eigenvalue but on the eigenvector, which is devised for possible application of the conventional adaptive mesh construction strategy for boundary value problems. In this paper, for improvement of computational efficiency, the local reanalysis for obtaining the eigenvector is employed. The error indicator of the eigenvector in place of that of the eigenvalue, the global value, decides selectivelythe boundary elements to be refined. Utility of the proposed method is compared, through some examples, with those previously developed. Key words: boundary element method, adaptive mesh, eigenvalue analysis,

Helmholtz equation.

INTRODUCTION

PREVIOUS WORKS

The present authors proposed two different schemes for the adaptive boundary element construction and related analysis for the eigenvalue analysis of the scalar-valued Helmholtz equation, l'z These schemes employed distinct formulations of the integral equation and eigenvalue extractions. They both, however, estimate the effect of the h-version mesh refinement on the variation of the eigenvalue and select the boundary elements with relatively higher effect to be refined. The eigenvalue extractions are based on methods proposed by the authors, 3-5 which are more efficient than those previously developed. One shortcoming is, however, that computational load is heavy because the eigenvalue is a global value and the adaptive scheme using this requires each determination of the eigenvalue for every refinement of the boundary element. We can reduce the load by replacing the effect of refinement on the eigenvector on the boundary. The last makes it possible to use the ordinary adaptive boundary element construction even for the eigenvalue analysis by simply applying the scheme developed for the conventional boundary value problems. We will describe the new scheme with some numerical examples.

The adaptive boundary element construction schemes for the eigenvalue analysis, developed in Refs 1 and 2 is explained below. The problem under consideration is for the following Helmholtz equation in terms of the scalar value u, in the two dimensional bounded domain fL V2U + k2u

= 0

(1)

with the homogeneous boundary conditions u = O (onrl) )

(on)

q -N

=0

(onr2)

i

where k is the wavenumber, unknown for the eigenvalue problem, and n denotes the outward unit normal on the boundary r ( r ; r l + r2). Although the eigenvalue extraction schemes in Refs 1 and 2 are different from each other, the same strategy for the adaptive boundary element construction was employed, which is explained here. Equation (1) is transformed into the corresponding integral equation and is discretized by boundary elements. The resulting system of equations is written in terms of the boundary 143

144

N. Kamiya, K. Nogae, S.-Q. Xu

nonvanishing boundary value vector x,

I I,n',ia'Mesh I

Ax = B0

I_

l

Estimationof EachMesh RefinementEffecton Eigenvalue i_

(7)

where the coefficient matrices A and B remain the polynomial matrices of k composed of the components of H and G. Similarly, for eqn (4), we obtain

I Eigenvalue Solver

ArnXm= BmO.

F.

(8)

The eigenvalues are determined through eqns (7) and (8) as indicated in Refs. 3-5, e.g. in Ref. 5, we employed

I Mesh Re~nementl k.

d d e t A ( k ) = 0.

(9)

Now, provided that the ith eigenvalue/~L is determined for the prescribed initial mesh, then ki satisfies the following equation, = 0. ddetA(/¢i)

I

However, corresponding to eqn. (8), ~det,~m(/~i) deviates somewhat from zero and therefore it can be recognized as a kind of error indicator

Fig. 1. Previous adaptive process. node vectors u and q as,

Hu = Gq

(3)

where H and G are the coefficient matrices, represented as the polynomial matrices with respect to k. 3-5 The above-mentioned formulation simplified the analysis and improved computational efficiency in the eigenvalue analysis by the boundary element method. Equation (3) is revised, when one boundary element m is refined, as (4)

Jim@m = Gmflra

where

"oIraJl}

/'/m-- [/'tm',.,-I

~ = Ld~,m-~

(5)

dm',m' J

UIm~ / um-I/ } t

i,,,

.

(6)

Submatrices with subscript m-1 in eqns (5) and (6) are those in eqn (3) without the ruth element, and the others with subscripts m' and m', m-1 are added by refinement of the element m, getting additional degrees-of-freedom

mp"

d 7,, -----~-~det,~m(/~i)(~ 0). For every boundary element m = 1,2,..., M (M boundary elements), 7i,~ is determined. The element with bigger 7ira relative to the appropriate reference magnitude, say the average of ")'ira,is selected to be refined. 7ira indicates the effect of the refinement of the element m on the eigenvalue /~i, and consequently it was decided to determine the eigenvalue for every boundary element refinement, which is seriously time consuming. The indicated adaptive process is shown in Fig. 1.

SIMPLIFIED SCHEME

/'/m',m',

[ Gin-1 C--lm-l'm]

(10)

By introducing the boundary condition (2) into eqn (3), we obtain the eigenvalue problem in terms of the

Efficiency of the previous adaptive schemes depends mainly on repeated eigenvalue determinations for each mesh refinement. A simplified version shown here will employ the effect of mesh refinement on the eigenvector. While the eigenvalue is a global value, the eigenvector is defined on each boundary element or boundary node and therefore, we can apply to the latter various adaptive methods already developed for boundaryvalue problems. It should be noted, however, that the situation is not intrinsically modified if we need to compute the effect on the eigenvector after determination of the eigenvalue for each mesh refinement. The following simplified assumption is adopted: when one element is refined, its effect is principally confined within the variation of the eigenvector on the related element. This is the concept of the "local reanalysis" employed elsewhere for boundaryvalue problems. 6"7

Simplifiedh-schemefor eigenvalueanalysisof theHelmholtzequation

145

difference between those just obtained and previous values, e ~ and respectively, of u and q on the element m, as

eqm,

I L

[leullm=(Ir e2mdr)U2}

-I Estimation of

Each Mesh Refinement Effect on Eigenvector

Eigenvalue Solver

F._iger~:t~r Solver

Ileqllm

( I r e2qmdF)1/2

(13,

where Fm stands for the boundary corresponding to the boundary element m. (4) Total errors of the eigenvector along the entire boundary are computed by counting every refinement of the element m = 1, 2, ..., M,

I Mesh Refinement J

( )'2/ ( /

Ile.II =

~tisf'y

I

Ile,,ll2m

m=l

(14)

Ileqll= lleqll

Fig. 2. Simplified adaptive process.

m=l

Application of the above-mentioned strategy is shown in Fig. 2. In what follows we explain it more precisely. (1) Compute the eigenvalue and eigenvector for the prescribed mesh (initial rough mesh). (2) Corresponding to the refinments of the element m, we express eqn (4) as follows: a + H,,e,m'0,e = b + (~,,¢,m,~lm,

(5) The relative errors are defined so as to account simultaneously for the values of different dimensions; for the total errors ~.

ile,,l12 -

~ ]/2}

(15)

~" = \llu°l12 + Ile"l12)

(11)

2

where

i/2

% = \llqOll 2 + Ileq[12,]

a -/~'m',m- l U r e

-

1

b - t~',m- lqm- 1

and for the errors on the element m

(12)

f

r/~ = ~lluOl12 + ile.l12 ]

In this equation, we suppose, for the refinement of the element m, that the eigenvector urn-I, qm-l of the remaining elements and the eigenvalue are not influenced and remain invariant, and consequently we employ the value determined in (1). Equation (11) is thought to be the simultaneous equation for the new degrees-of-freedom urn', qm'(m' unknowns), which can be solved easily. (3) Determine the error norm of the eigenvector as the

(16)

( lle, ll: ilq0112+ Ileqll2}

T}q m =

where Ilu°ll, IIq°ll are defined for the prescribed mesh, norms similar to eqn (13). Denominators of eqns (15) and (16) approximate the norms of the solution. (6) If rhm (or in eqn (16) is bigger than the reference criterion, the element m is to be refined.

Oqm)

Table 1. Results of example 1

Mode

1,0 0,1 1,1 2,0 2,1 0,2 1,2 3,0

Analytical solution

Reference (5) Eigenvalue

Final elements 32 32 36 44 60 32 48 28

3.14

3.14

3.93

3.93

5.03 6.28 7.41 7.85 8.46 9.42

5.03 6.29 7.41 7.85 8.47 9.43

Present Refs.

Eigenvalue

2

3.14

2

3.93

2 3 4 2 3 2

5.04 6.28 7.41 7.85 8.46 9.43

Final elements 32 32 34 48 49 32 48 28

Original 16 elements Refs. 2

3.16

2

3.94

2 3 3 2 4 2

5.06 6.32 7.48 7.88 8.52 9.50

146

N. Kamiya, K. Nogae, S.-Q. Xu

D-

_-

:

_-

-C

orriginal mesh (5 elms)

original element q:o

q=O

u=O

q=0

q=O

q:o

q=O

I st refinement (7 elms)

2nd refinement ( 12 elms)

4th refinement (42 elms)

5th refinement (69 elms)

q--O

I /

/ / / .: ......

°,t 0.0

! ! ¢- -7-

A

I st refinement

B

-7 - ,

C

D

A

I

.

.

.

.

.

.

.

.

3rd refinement (23 elms)

i

0°°i/I/I,o,oo/I/If

.

7.7.

2nd refinement

A

B

C

D A

.~7.7. 7.7.~.~

.... f._i

1¢ 6th refinement (75 elms)

....

I

Fig. 4. Iteration process of example 2. A

B C

D A

Fig. 3. Iteration process and error distribution for mode (2,0) of example I. (7) The above process is repeated until the desired eigenvalue converges. The new scheme does not need to compute the eigenvalues for each mesh refinement but employs the variation of the eigenvector as the error indicator under the assumption aimed at simplicity. In general, we often encounter comparison of the values with different dimensions but the process shown above is devised for reducing inefficiency in eigenvalue determination by the boundary element method.

E X A M P L E S AND D I S C U S S I O N A whole set of boundary element computations in the following examples are performed with so-caUed constant elements and the element refinement is carried

out by equal bisection of line segments (h-version refinement). The eigenvalue is searched iteratively through the minimum of the determinant of the coefficient matrix formulated in eqn (9). 5 Terms of the polynomial matrices H and G are taken so as to keep desired accuracy. T h e criterion for convergence of the eigenvalue is up to coincidence of two decimal figures. Example 1: the first example is for a rectangular domain of sides Lx = 1, Ly = 0.8 with the boundary condition q = 0 along a whole boundary. For several eigenmodes within k = 0-10, 16 elements (four elements on each side) are the initial mesh. Table 1 compares various results; the present, the method used in Ref. 5 and the analytical solution for the eigenvalue, final numbers of elements and numbers of adaptive iteration required indicated as 'Refs'. G o o d agreement is recognized and the numbers of iteration are between two and four times. Computation time for the present method is reduced to two thirds of that in Ref. 5. Detailed adaptive refinement from the initial mesh (16 elements) to the third mesh (48 elements) and distribution of error along the boundary for the mode (2, 0) are shown in Fig. 3. This figure explains the reduction

Table 2. Results of example 2

Original

1st

2nd

3rd

4th

5th

6th

7th

Final

Equal 72 elements

1.45

1.46

1.45

1.44

1.46

1.48

1.49

1.49

1.49

1.50

3.84 5.78 6.75 8.84 10.00 ----

3.83 5.69 7.04 9.04 9.91 ----

3.44 4.59 6.40 7.02 8.83 9.85 ---

3.39 4.53 6.26 6.55 8.06 8.18 9.62 9.94

3.39 4.56 6.24 6.54 8.01 8.30 9.56 9.90

3.39 4.57 6.24 6.54 8.01 8.32 9.56 9.89

3.39 4.58 6.24 6.54 8.01 8.33 9.56 9.90

3.39 4.59 6.24 6.54 8.01 8.34 9.56 9.90

3.39 4.59 6.23 6.55 8.01 8.35 9.56 9.90

3.40 4.59 6.25 6.55 8.01 8.33 9.55 9.91

Simplified h-scheme for eigenvalue analysis of the Helmhoitz equation tendency of the error r/urn and its distribution width with the progress of the iterations. Example 2: the second example is considered for the rectangular domain (dimensions as in Example 1) with the combined boundary condition indicated left uppermost in Fig. 4. The validity of the proposed scheme for the eigenvector error norms of different dimensions on the boundary was investigated. The adaptive iteration process, started with only five initial boundary elements for the range k = 0-10, is shown in Fig. 4 and Table 2. Because of the lack of an analytical solution, the solution obtained with sufficiently fine 72 elements is shown for reference. Figure 4 suggests that refined elements concentrate near the points where the indicated boundary values u and q change discontinuously (upper left corner and middle of the upper side). This problem was already treated in Ref. 1 where the last two eigenvalues were not distinguished. The method used in Ref. 1 requires an exchange of the column of the coefficient matrices and was a source of computational instability. On the other hand, this inconvenience was not encountered in the present scheme and a fairly sound result was obtained.

CONCLUSION By considering the effect of each mesh refinement on the variation of the eigenvector, the simplified adaptive boundary element analysis scheme was proposed for the eigenvalue problem with the scalar-valued Helmholtz equation. Reduction of numbers of eigenvalue

147

determination in the process increased computational efficiency with higher stability. Further practical applications will be shown in future studies.

REFERENCES 1. Kamiya, N., Nogae, K. & Andoh, E., Adaptive boundary element for eigenvalueanalysis of the Helmholtz equation. in Proc. 16th Bound. Elms. Conf., ed, C. A. Brebbia, Elsevier Sci. Pub., London Comp. Mech. Pub., Southampton, UK, 1994. 2. Kamiya, N., Andoh, E. & Nogae, K., A new complexvalued formulation and eigenvalue analysis of the Helmholtz equation by boundary element method (in preparation). 3. Kamiya, N. & Andoh, E., Eigenvalue analysis by boundary element method. J. Sound Vib., 160 (1993) 279-87. 4. Kamiya, N., Andoh, E. & Nogae, K. Eigenvalue analysis by boundary element method; new developments. Eng. Anal. Bound. Elms., 12 (1993) 152-63. 5. Kamiya, N,, Andoh, E. & Nogae, K., Iterative local minimum search for eigenvalue determination of the Helmholtz equation by boundary element formulation. Proc. Comp. Acoust. (in preparation). 6. Charafi, A., Wrobel, E. C. & Adey, R., An approach to h-adaptive method using local reanalysis, in Proc. 7th Bound. Elms. Tech. Conf., ed. C. A. Brebbia & M. S. Ingber, Comp. Mech. Pub., Southampton, UK, 1992, pp. 905-918. 7. Charafi, A., Neves, A. C. & Wrobel, L. C., Use of local reanalysis and quadratic h-hierarchical functions in adaptive boundary element models. In Proc. 8th Bound. Elms. Tech. Conf., ed. H. Pina & C. A. Brebbia. Comp. Mech. Pub., Southampton, UK, 1993, pp. 353-368.