Meshfree Boundary Particle Method Applied to Helmholtz Problems

Engineering Analysis with Boundary Elements 26 (2002) 577–581 www.elsevier.com/locate/enganabound

Meshfree boundary particle method applied to Helmholtz problems

W. Chen

Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway

Received 27 September 2001; accepted 17 April 2002

Abstract This paper is concerned with the boundary particle method (BPM), a new boundary-only radial basis function collocation schemes. The method is developed based on the multiple reciprocity principle and applying either high-order nonsingular general solutions or singular fundamental solutions as the radial basis function. Like the multiple reciprocity BEM (MR-BEM), the BPM does not require any inner nodes for inhomogeneous problems and therefore is a truly boundary-only technique. On the other hand, unlike the MR-BEM, the BPM is meshfree, integration-free, symmetric, and mathematically simple technique. In particular, the method requires much less computational effort for the discretization than the MR-BEM. In this study, the accuracy and efﬁciency of the BPM are numerically demonstrated in some 2D inhomogeneous Helmholtz problems under complicated geometries. q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Multiple reciprocity principle; Boundary particle method; Radial basis function; Meshfree; Method of fundamental solution; Boundary knot method; Multiple reciprocity BEM

1. Introduction BPM is an easy-to-program, inherently meshfree, integration-free, and mathematically simple approach. The In last decade the dual reciprocity BEM (DR-BEM) and computational effort is also relatively reduced significantly. multiple reciprocity BEM (MR-BEM) have been emerging In particular, inspired by Fasshauer’s Hermite RBF as two most promising BEM techniques to handle interpolation [7], we give the symmetric BPM scheme. inhomogeneous problems [1,2], where the dual reciprocity This paper testifies this method to some Helmholtz principle and multiple reciprocity principle are, respect- problems under complicated geometry. ively, used. It is well known that unlike the DR-BEM, the Developing the BPM was mainly motivated by some MR-BEM does not require inner nodes for general recent substantial advances on the method of fundamental inhomogeneous problems and therefore is a truly bound- solution (MFS), boundary knot method (BKM), and RBF. ary-only technique, especially attractive for high-dimension Golberg [8] combines the dual reciprocity principle and free surface and unbounded domain problems. The draw- RBF with the MFS to solve some typical Poisson problems backs of the MR-BEM, however, are that compared with the in a simple and efficient manner. The strategy is essentially DR-BEM, the method in general requires more computational meshfree and RBF-only method, where the fundamental efforts [3] and is not easily applied to nonlinear problems [2]. solution can be regarded as the RBF for the evaluation of Despite these disadvantages, in recent years the MR-BEM has homogeneous solution. Very recently Chen and Tanaka [9, been successfully used for a broad variety of problems [1,4]. 10] further developed a boundary knot method, which In this study, we introduce the boundary particle method applies the nonsingular general solution, RBF and dual (BPM) [5,6], a new exact boundary-only discretization reciprocity principle and therefore effectively eliminates the technique based on applying the radial basis function (RBF) need of controversial fictitious boundary outside physical and multiple reciprocity principle. The method is regarded domain in the MFS. Both the MFS and BKM, however, as a boundary-type RBF collocation scheme since we use need to use the inner nodes for inhomogeneous problems to either high-order nonsingular general solutions or singular guarantee the stability and accuracy of the solution. For fundamental solutions as the RBFs to evaluate high-order more details on the MFS and BKM see Refs. [11,12]. On the homogeneous solutions, and then their sum approximates other hand, since Kansa’s pioneer work [13] in 1990, the the particular solutions. Compared with the MR-BEM, the research on the RBF method for PDEs has become very active. The methods based on radial basis function are E-mail address: wenc@ifi.uio.no (W. Chen). inherently meshfree due to the fact that the RBF method 578 W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 577–581 uses the one-dimensional distance variable irrespective of using inner nodes. In contrast, the multiple reciprocity dimensionality of problems [14]. Therefore, the RBF methods method also involves the higher order homogeneous are independent of dimensionality and complexity of solution. For the first-order homogeneous solution, we have geometry. These recent developments on the MFS, BKM no R1 u1 ¼ 0; x [ V; ð8Þ and RBF inspire the present author to present the boundary 8 noh no particle method which uses the multiple reciprocity principle, > R0 1ð Þ ¼ ð Þ R0 1ð Þ <> uh xi f xi 2 up xi ; x [ Su; nonsingular general solution and fundamental solution. nono ð9Þ In what follows, the BPM is introduced first in Section 2, > ›R0 u1ðx Þ › f ðx Þ 2 R0 u1ðx Þ :> h j j p j followed by numerical validations in terms of some 2D ¼ ; x [ ST ; inhomogeneous Helmholtz problems under complicated ›n ›n geometries in Section 3, and finally, Section 4 concludes where R1{}¼ R{R{}}; R0{}¼ R{}: Note that here we some remarks based on the reported results. employed the external forcing function f ðxÞ and its normal derivative, respectively, as the corresponding Dirichlet and Neumann boundary conditions. Through a similar incre- 2. Boundary particle methods mental differentiation operation over Eq. (9) via operator R{}; we have successive higher order boundary differential To clearly illustrate our idea, consider the following equations example without loss of generality n n R fguh ¼ 0; x [ V; ð10Þ R{u} ¼ f ðxÞ; x [ V; ð1Þ 8 no > Rn21 n Rn22 Rn21 n <> fguhðxiÞ ¼ {f ðxiÞ} 2 upðxiÞ ; x [ Su; uðxÞ¼RðxÞ; x [ Su; ð2aÞ nono > Rn21 n Rn22 Rn21 n ›uðxÞ > › uhðxjÞ › {f ðxjÞ} 2 upðxjÞ ¼ NðxÞ; x [ S ; ð2bÞ : ¼ ; x [ S ; ›n T ›n ›n T where R is a differential operator, x means multi- n ¼ 2; 3; …; ð11Þ dimensional independent variable, and n is the unit outward normal. The solution of Eq. (1) can be expressed as where Rn{ } denotes the nth order operator of R{}; say R2 ¼ R R1 u ¼ u0 þ u0; ð3Þ {} { {}}: It is stressed that the inhomogeneous h p term f ðxÞ was repeatedly differentiated as the boundary 0 0 where uh and up are the zero-order homogeneous and conditions for higher-order operator equations. particular solutions, respectively. The multiple reciprocity In terms of the multiple reciprocity method, the nth order method evaluates the particular solution in Eq. (3) by a sum n particular solution up is approximated by of higher-order homogeneous solution, namely X1 X1 un ¼ um: ð12Þ u0 ¼ um; ð4Þ p h p h m¼nþ1 m¼1 where superscript m is the order index of the homogeneous The mth-order homogeneous solution can be analogized by solution. Eq. (4) is also essential in the MR-BEM to evaluate XL m m # the particular solution [1]. Thus, the solution of inhomo- uh ðxÞ¼ bk umðrkÞ; ð13Þ geneous equation (Eq. (1)) can be expressed as k¼1 X1 where L is the number of boundary nodes, k is the index of u ¼ u0 þ u0 ¼ um: ð5Þ # h p h source points on boundary; um is the nonsingular general m¼0 solution of operator Rm{}: b are the desired coefficients, It is noted that albeit the widespread use of the multiple k and rk ¼ kx 2 xkk represents the Euclidean distance norm. reciprocity method, a general mathematical proof is not p Note that we can use the fundamental solution u instead of found in the literature. m general solution u# in Eq. (13). However, the use of The zero-order homogeneous solution has to satisfy both m fundamental solution requires a fictitious boundary outside the governing equation and the boundary conditions, i.e. no physical domain as encountered in the MFS. 0 Collocating Eqs. (7), (9) and (11) at all boundary knots in R uh ¼ 0; x [ V; ð6Þ 8 terms of the representation (13), we have the BPM boundary > u0ðx Þ¼Rðx Þ 2 u0ðx Þ x [ S ; discretization equations: < h i i p i u 9 0 0 ð7Þ XL > :> ›uhðxjÞ ›upðxjÞ 0 # 0 > ¼ Nðx Þ 2 x [ S ; bku0ðrikÞ¼RðxiÞ 2 upðxiÞ > ›n j ›n T = k¼1 0 > ¼ b ; ð14Þ where i and j indicate, respectively, the response knots XL ›u#ðr Þ ›u0ðx Þ > located on the Dirichlet and Neumann boundary S , S .Eq. b0 0 jk ¼ Nðx Þ 2 p j > u G k ›n j ›n ; (7) is in fact the dual reciprocity method [1] formula without k¼1 W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 577–581 579 9 XL no no > The system matrix of the above equations is symmetric if 1R0 # R0 1 > bk u1ðrikÞ ¼ f ðxiÞ 2 upðxiÞ > operator R{ } is self-adjoint. Note that i and j are reciprocal k¼1 = nono 1 indices of Dirichlet (Su) and Neumann boundary (SG) nodes. > ¼ b ; ð15Þ XL ›R0 u#ðr Þ › f ðx Þ 2 R0 u1ðx Þ > In terms of the multiple reciprocity, the successive 1 1 jk j p j > bk ¼ ; process is truncated at some order M, namely, let k¼1 ›n ›n no 9 RM21 uM ¼ 0: ð21Þ XL no no > p nRn21 # Rn22 Rn21 n > bk unðrikÞ ¼ {f ðxiÞ} 2 upðxiÞ > k¼1 = It is expected that as in the MR-BEM [1,3], the truncated no ÈÉ > order M in the BPM may not be large in a variety of practical XL n21 # Rn22 Rn21 n > ›R u ðr Þ › {f ðxjÞ} 2 upðxjÞ > uses if the problem has been properly scaled. However, a n n ik ¼ > bk ; general strategy deciding M a prior is not available now. k¼1 ›n ›n The practical solution procedure of the BPM is a reversal ¼ bn; n ¼ 2; 3; … ð16Þ recursive process: bM ! bM21 ! ···! b0: ð22Þ It is noted that the use of the naı¨ve representation (13) will simply lead to the unsymmetric BPM interpolation matrix It is noted that due to (14)–(16) due to the mixed boundary conditions. In order to no n21 # # get symmetric BPM scheme for self-adjoint operators, by R unðrÞ ¼ u0ðrÞ; ð23Þ analogy with Fasshauer’s Hermite RBF interpolation [7], we presented the following RBF approximate expression to the coefficient matrices of all successive equations (Eqs. homogeneous solution instead of expression (13) (14)–(16) or Eqs. (7), (9) and (11)) are the same, i.e. n n XLD LDXþLN ›u# ðr Þ Qb ¼ b ; n ¼ M; M 2 1; …; 1; 0: ð24Þ umðxÞ¼ b u# ðr Þ 2 b m s ; ð17Þ h s m s s ›n s¼1 s¼LDþ1 Thus, the LU factorization algorithm is suitable for this task. where n is the unit outward normal as in boundary condition Finally, after the solution of the above recursive algebraic (2b), and LD and LN are, respectively, the numbers of knots equations (Eqs. (7), (9), (11) or (14)–(16)), we can employ at the Dirichlet and Neumann boundary surfaces. The minus the obtained expansion coefficients b to calculate the BPM sign associated with the second term is due to the fact that solution at any knots. The unsymmetric and symmetric the Neumann condition of the first-order derivative is not BPM solutions are given, respectively, by self-adjoint. In terms of expression (17), the collocation XM XL analogue equations (Eqs. (14)–(16)) are rewritten as n # 9 uðxiÞ¼ bk unðrikÞð25Þ XLD LDXþLN ›u#ðr Þ > n¼0 k¼1 b u#ðr Þ 2 b 0 is ¼ Rðx Þ 2 u0ðx Þ > s 0 is s ›n i p i => and s¼1 s¼LDþ1 0 1 XL # L XþL 2 # 0 > XM XLD LDXþLN # D ›u ðr Þ D N › u ðr Þ ›u ðx Þ > @ n # n ›unðrisÞ A 0 js 0 js ¼ ð Þ p j > uðx Þ¼ b u ðr Þ 2 b : ð26Þ bs 2 bs 2 N xj 2 ; i s n is s ›n ›n ›n n¼0 s¼1 s¼L þ1 ›n s¼1 s¼LDþ1 D 0 ¼ b ð18Þ The BPM can use either singular fundamental solution or nonsingular general solution, relative to the MFS and BKM, ÈÉ 9 L noL þL no > XD DX N ›R0 u#ðr Þ > where the former requires an artificial boundary outside R0 #ð Þ 1 is ¼ ð Þ R0 1ð Þ > bs u1 ris 2 bs f xi 2 up xi > physical domain while the latter not. It is noted that the s¼1 s¼L þ1 ›n = noD nono BPM with M ¼ 1 degenerates into the BKM or MFS 0 # 2 0 # 0 1 > XLD R LDXþLN R R > › u1ðrjsÞ › u1ðrjsÞ › f ðxjÞ 2 upðxjÞ > without using the inner nodes. The only difference between b 2 b ¼ > s ›n s ›n2 ›n ; s¼1 s¼LDþ1 the BKM (MFS) and BPM lies in how to evaluate the particular solution. The former applies the dual reciprocity ¼ b1 ð19Þ principle, while the latter employs the multiple reciprocity

ÈÉ 9 XLD noLDXþLN n21 # no > ›R unðrisÞ > b Rn21 u#ðr Þ 2 b ¼ Rn22{f ðx Þ} 2 Rn21 unðx Þ > s n is s ›n i p i => s¼1 s¼LDþ1 no nono¼ bn; n ¼ 2; 3; … ð20Þ n21 # 2 n21 # n22 n21 n > XLD R LDXþLN R R R > › unðrjsÞ › unðrjsÞ › {f ðxjÞ} 2 upðxjÞ > b 2 b ¼ > s ›n s ›n2 ›n ; s¼1 s¼LDþ1 580 W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 577–581 principle. The advantage of the BPM over the BKM and Table 1 MFS is that it dose not require interior nodes which may be L2 norm of relative errors for 2D inhomogeneous Helmholtz problems by especially attractive in such problems as moving boundary, the unsymmetric and symmetric BPMs inverse problems, and exterior problems. However, the UBPM (26) UBPM (40) SBPM (26) SBPM (40) BPM may be more mathematically complicated and pffiffi computationally costly due to the iterative use of higher- g ¼ 2 2.8 £ 1023 4.5 £ 1024 1.3 £ 1023 2.9 £ 1024 order fundamental or general solutions. pffiffi UBPM (77) UBPM (91) SBPM (91) SBPM (119) 23 24 23 24 The present form of the BPM uses the expansion g ¼ 5 2 1.5 £ 10 4.1 £ 10 6.7 £ 10 7.8 £ 10 coefficients rather than the direct physical variables in the approximation of boundary value. Therefore, such scheme is called as the indirect BPM. Chen [5] also gave the direct corresponding high-order general solutions of Helmholtz is BPM with physical variable as basic variable. straightforward according to [15]

# 2n=2þ1þm umðrÞ¼AmðgrÞ Jn=221þmðgrÞ; ð29Þ 3. Numerical experiments 2 where Am ¼ Am21=ð2mg Þ; A0 ¼ 1; n is the dimension of The 2D irregular geometry tested is illustrated in Fig. 1, the problem; m denotes the order of general solution; J which includes a triangle cut-out. Except for Neumann represents the Bessel function of the ﬁrst kind. boundary conditions shown in Fig. 1, the boundary The experimental results are displayed in Table 1 where conditions are all of the Dirichlet type. Equally spaced the UBPM and SBPM denote, respectively, the unsym- knots were applied on the boundary. The L2 norms of metric and symmetric BPM schemes. It is found that both relative errors are calculated based on the BPM BPM schemes produce very accurate solutions with a small numerical solutions at 364 inner and boundary nodes. number of boundary nodes for inhomogeneous Helmholtz The absolute error is taken as the relative error if the problems. We also observed that the maximum relative absolute value of the solution is less than 0.001. Note that errors occurred in random locations in difference cases. different nodes are used for BPM coefﬁcients and for L2 Note that we did not use any inner nodes here. Ref. [16] norm of relative errors. applied the BPM successfully to the steady convection– Helmholtz system equation is given by diffusion problems. 72u þ g2u ¼ 2 sinðdxÞ cosðdxÞþ4x cosðdxÞ cosðdyÞ: ð27Þ

The accurate solution is 4. Concluding remarks

2 u ¼ x sinðdxÞ cosðdyÞð28Þ The BPM circumvents the troublesome singular integral pffiffi for 2D inhomogeneous Helmholtz problem ðg ¼ d 2Þ: The inherent in the BEM and are very easy to learn and program. The method has advantage not requiring inner nodes for inhomogeneous problems, and therefore is very suitable for problems whose higher-order homogeneous solution quickly tends to zero. It is noted that unlike the MR-BEM, the BPM does not need to generate more than one interpolation matrix, which tremendously reduces computing effort and memory requirements. In addition, the BPM is essentially meshfree, spectral convergence and symmetric technique. It is expected that compared with other numerical techniques, the BPM may become more efficient for higher- dimensional complex-shape geometry problems since the general solutions of high-dimensional operators are simpler and radial basis function is independent of dimensionality and geometry complexity. Similar to the BEM and MFS, the drawbacks of the method are the poor condition of the global interpolation and costly computing effort for full matrix. Some fast solvers based on the multipole [17] and wavelets may be promising to cure these inefficiencies. More investigations on the method are still under way and will be reported in Fig. 1. Configuration of a square with a triangle cutout. subsequent papers. W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 577–581 581

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