A COMPLEX VARIABLE BOUNDARY ELEMENT METHOD FOR AN ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS

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1 A COMPLEX VARIABLE BOUNDARY ELEMENT METHOD FOR AN ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS Y. S. PARK a adw.t.ang b Computatioal ad Mathematical Scieces Group Faculty of Iformatio Techology Uiversiti Malaysia Sarawak Kota Samaraha, Malaysia Abstract A boudary elemet method based o the Cauchy itegral formulae is proposed for the umerical solutio of a boudary value problem govered by a secod order elliptic partial differetial equatio with variable coefficiets. The boudary value problem has applicatios i egieerig problems ivolvig ohomogeeous media. The method reduces the boudary value problem to the task of solvig a system of liear algebraic equatios. It ca be easily implemeted o the computer. Keywords: complex variable boudary elemet method, elliptic partial differetial equatio with variable coefficiets. This is a preprit of the article i Commuicatios i Numerical Methods i Egieerig 16 (2000) The URL of the publisher is: 1

2 1 INTRODUCTION The secod order elliptic partial differetial equatio (PDE) µ K(x) φ + µ K(x) φ =0, φ = φ(x, y), (1) x x y y where K(x) is a give fuctio of the Cartesia coordiate x such that K(x) > 0 i the domai of iterest, has extesive egieerig applicatios i structural ceramics, modelig of mechaics of soils ad other particulate materials (see e.g. Brea [1] ad Gibso [2]). A boudary value problem (BVP) of iterest is to solve (1) i a fiite regio R bouded by a simple closed curve C (o the 0xy plae) subject to the coditios φ = p(x, y) for (x, y) D, (2) φ = q(x, y) for (x, y) E, (3) where p ad q are suitably prescribed fuctios, D ad E are o-itersectig curves such that C = D E, φ/ = φ ad =[ 1, 2 ] is the uit ormal outward vector to E. Clemets [3] had derived a geeral solutio of (1) i the form ( ) 1 X φ(x, y) = p Re f (x)φ (z), (4) K(x) where i = 1adz = x + iy. The fuctios f are particular solutios of the recurrece equatios 2 df +1 dx with f 0 (x) =1ad =0 = d2 f dx 2 + Λ(x)f for =0, 1, 2, 3,, (5) Λ(x) = d dx " µdk 2 /(4K)# / dx µ dk. (6) dx 2

3 The complex fuctios Φ (z) are holomorphic i the domai of iterest ad satisfy dφ dz = Φ 1(z) for =1, 2, 3,. (7) I the preset article, the solutio (4) together with the Cauchy itegral fromulae is applied to derive a umerical method for solvig approximately the BVP defied by (1), (2) ad (3). The method requires oly the boudary C of the domai R to be discretized ad is kow as the complex variable boudary elemet method (CVBEM). It reduces the BVP to solvig a system of liear algebraic equatios. The coefficiets of the liear algebraic equatios are easy to compute. Hece the method ca be easily implemeted o the computer. The proposed method should offer itself as a iterestig alterative to earlier BEM approaches (see e.g. Clemets [3] ad Ragogi [4]) for solvig certai BVPs govered by (1). The CVBEM was origially itroduced by Hromadka ad Lai [5] for BVPs govered by the two-dimesioal Laplace equatio, i.e. for the case where K(x) is a costat fuctio i (1). Recetly, Ag ad Park [6] developed adifferet versio of the method for a more geeral system of secod order elliptic PDEs with costat coefficiets. The approach of [6] which differs from that of [5] i the approximatio of the relevat complex fuctios ad i the treatmet of the boudary coditios is adopted here to solve the BVP uder cosideratio. 2 CVBEM Firstly, the boudary C is discretized as follows. Place M well-spaced out poits (x (1),y (1) ), (x (2),y (2) ),, (x (M 1),y (M 1) )ad(x (M),y (M) )iaaticlockwise order o C. Let C (k) be the straight lie segmet from (x (k),y (k) ) to (x (k+1),y (k+1) )(k =1, 2,,M), where (x (M+1),y (M+1) )=(x (1),y (1) ). We the make the approximatio C C (1) C (2) C (M 1) C (M). (8) 3

4 Repeated applicatio of the recurrece relatio (7) gives Z z 1 Φ (z) = (z t) 1 Φ 0 (t)dt for =1, 2,, (9) ( 1) a where a is a suitably chose complex umber. Let z (k) = x (k) + iy (k) (k =1, 2,,M +1), bz (p) = ξ (p) + iη (p), ξ (p) = (x (p) + x (p+1) )/2 adη (p) =(y (p) + y (p+1) )/2 (p =1, 2,,M). I (9), if we take a = z (1),z= bz (p) ad do the itegratio from a to z alog the path C (1) C (2) C (p 1) C b(p) (where C b(p) is the straight lie segmet from z (p) to bz (p) ), the Φ (bz (p) )= ( 1 Xp 1 ( 1) Z z (k+1) z (k) Z bz (p) + z (p) (bz (p) t) 1 Φ 0 (t)dt (bz (p) t) 1 Φ 0 (t)dt for p =1, 2,,M ad =1, 2,, (10) For t C (k), if we expad Φ 0 (t) as a Taylor-Maclauri series about t = bz (k), i.e. if we write Φ 0 (t) =Φ 0 (bz (k) )+(t bz (k) )Φ 0 0 (bz(k) )+ 1 2 (t bz(k) ) 2 Φ 00 0 (bz(k) )+, (11) the, after igorig terms whose magitudes are O( z (k+1) z (k) 2 )orhigher order, we fid that (10) gives (approximately) ) Φ (bz (p) )= px Γ (pk) Φ 0 (bz (k) ) for p =1, 2,,M ad =0, 1, 2,, (12) where Γ (pk) 0 = δ pk ad Ã Γ (pk) 1 = (1 δ pk ) ( 1) Z z (k+1) z (k) (bz (p) t) 1 dt + δ pk Z bz (p) z (p) (bz (p) t) 1 dt for =1, 2, 3,. (13) 4

5 Notice that Γ (pk) ca be easily evaluated. If we write Φ 0 (bz (p) )=u (p) + iv (p) where u (p) ad v (p) are real costats the use of (12) together with (4) i the coditio (2) yields ( ) where α (pk) px = u (k) X =0 f (ξ (p) )α (pk) v (k) X =0 f (ξ (p) )β (pk) q K(ξ (p) )p(ξ (p), η (p) ) if φ is specified over C (p), (14) ad β (pk) Now, from (4) ad (7), we fid that à ( ) 1 X p Re f (x)φ (z) K(x) =Re =0 ( ( 1 + i 2 ) p K(x) Φ 0 0(z)+ are real parameters give by Γ (pk) =0 = α (pk) + iβ (pk). à #) X f +1 (x) Φ (z) " 1 p + F (x) + i 2f +1 (x) p, K(x) K(x) (15) ³ where F (x) =d f (x)/ p K(x) /dx for =0, 1, 2,. Thus, to deal with the coditio i (3), we are required to evaluate Φ 0 0(z) at z o the boudary. We shall apply the Cauchy itegral formula M 2πiΦ 0 0 (z) = X Z Φ 0 (t)dt for z R. (16) (t z) 2 C (k) If we substitute (11) ito (16) ad igore terms whose magitudes are O( z (k+1) z (k) 2 ) or higher order, we obtai the approximatio M 2πiΦ 0 0 (z) = X Φ 0 (bz (k) ) q(z (k),z (k+1),z)+ir(z (k),z (k+1),z) for a fixed z R, where q ad r are real parameters defied by (17) q(a, b, c)+ir(a, b, c) = 1 b c + 1 a c. (18) 5

6 However, if we repeat the same exercise but with z bz (p) (the midpoit of the p-th straight lie segmet), we fid that πiφ 0 0 (bz(p) )= Φ 0 (bz (k) ) q(z (k),z (k+1), bz (p) )+ir(z (k),z (k+1), bz (p) ). (19) Notice the differece i the factors that is multiplied to the derivative Φ 0 0 i (17) ad (19). With (12), (15) ad (19), the coditio i (4) gives à u (k) (p) 2 q(z (k),z (k+1), bz (p) )+ (p) 1 r(z (k),z (k+1), bz (p) ) π p K(ξ (p) ) + + px where [ (p) 1, (p) σ (p) u (k) X =0 =0 α (pk) µ (p) β(pk) σ (p) à v (k) (p) 1 q(z (k),z (k+1), bz (p) ) (p) 2 r(z (k),z (k+1), bz (p) ) π p K(ξ (p) ) px X v (k) α (pk) σ (p) + β(pk) µ (p) = q(ξ (p), η (p) ) if φ is specified over C(p), (20) 2 ] is the uit ormal (outward) vector to C (p) ad µ (p) ad are real parameters defied by " à # µ (p) + iσ(p) = (p) f +1 (ξ (p) ) 1 p K(ξ (p) ) + F (ξ (p) ) + i(p) 2 f +1 (ξ (p) ) p. (21) K(ξ (p) ) The equatios (14) ad (20) give M equatios i 2M ukows u (k) ad v (k) (k =1, 2,,M). To obtai aother M equatios, we apply the Cauchy itegral formula 2πiΦ 0 (z) = Z C (k) Φ 0 (t)dt (t z) for z R. (22) 6

7 Proceedig as before, i.e. substitutig (11) ito (22) ad igorig terms whose magitudes are O( z (k+1) z (k) 2 )orhigherorder,weobtai 2πiΦ 0 (z) = (u (k) + iv (k) ) γ(z (k),z (k+1),z)+iθ(z (k),z (k+1),z) for z R, (23) where γ ad θ are real parameters defied by γ(a, b, c) =l b c l a c Θ(a, b, c) if Θ(a, b, c) [ π, π] θ(a, b, c) = Θ(a, b, c)+2π if Θ(a, b, c) [ 2π, π) Θ(a, b, c) 2π if Θ(a, b, c) (π, 2π] Θ(a, b, c) =Arg(b c) Arg(a c), (24) where Arg(z) deotes the pricipal argumet of the complex umber z. If thesolutiodomaiiscovexishape, θ(a, b, c) ca be calculated directly from à θ(a, b, c) =cos 1 b c 2 + a c 2 b a 2. (25) 2 b c a c We may take the real part of equatio (23) ad let z bz (p) to obtai 2πv (p) = u (k) γ(z (k),z (k+1), bz (p) ) v (k) θ(z (k),z (k+1), bz (p) ) ª for p =1, 2,,M. (26) The equatios (14), (20) ad (26) costitute 2M equatios from which the ukows u (k) ad v (k) (k =1, 2,,M) ca be determied. Oce the ukows are determied, we ca calculate Φ (z) ( =1, 2, )atthepoit z = bz (k) (k =1, 2,,M) usig (12) ad the at the poit z = c i the iterior of R via 2πiΦ (c) = Φ (bz (k) ) γ(z (k),z (k+1),c)+iθ(z (k),z (k+1),c). (27) 7

8 Notice that (27) is also valid for =0. Thus, all the complex fuctios eeded for the solutio of the BVP ca be determied umerically at all poits i R C. 3 A TEST PROBLEM For a test problem, we shall apply the CVBEM proposed above to solve µ (x +2) 4 φ +(x +2) 4 2 φ =0i0<x<1, 0 <y<π, (28) x x y2 subject to φ =0 for0<x<1, (29) y=0 φ =0 for0<x<1, (30) y=π φ(0,y)=0 for0<y<π, (31) φ(1,y)=cos(y) for 0 <y<π. (32) The exact solutio of the BVP ca be obtaied by the method of separatio of variables. It is give by φ(x, y) = 27 ( 3x 3+x exp( 2x)+3exp( 2x)) exp(x +2) 2e (3e 2 2) (x +2) 3 cos(y). (33) For the PDE i (28), K(x) =(x +2) 4, f 0 (x) =1, f 1 (x) = 1 (x +2) f (x) =0 for 2, (34) ad F 0 (x) = 2 (x +2) 3, F 1(x) = 3 (x +2) 4, F (x) = 0 for 2. (35) 8

9 The CVBEM described i the previous sectio is applied to solve (28) subject to (29)-(32). The umerical values of φ are the calculated at selected iterior poits ad compared with the exact values from (33). A compariso is made i Table 1 for umerical results obtaied by puttig 10 elemets o each side of the rectagular boudary ad also 40 elemets per side (i.e. for M =40adM = 120) with the exact solutio. For M =40, the largest legth of the elemets is approximately 0.31 uits, while for M = 120 the largest legth is about uits. It is obvious from Table 1 that there is sigificat improvemet i the accuracy of the umerical results whe the umber of elemets is icreased from 40 to 120. The accuracy is i geeral good except for iterior poits extremely close to the boudary. For iterior poits very close to the boudary, it is ecessary to employ a larger umber of elemets for better accuracy. For example, at the poit (0.10, 3.10) which is extremely close to the boudary y = π, the percetage error i the calculatio of φ is about 47% if the computatio is carried out usig M =40. However, usig M = 120, we are able to reduce the percetage error to 6.1%. Further calculatio usig M = 400 brigs the percetage error dow to well uder 1%. Table 1. A compariso of the CVBEM umerical results with the exact solutio at selected iterior poits. Iterior poit CVBEM CVBEM (x, y) M =40 M =120 Exact ( ) (0.40, 0.60) (0.80, 0.70) (0.50, 0.50) (0.99, 2.10) (0.10, 3.10) SUMMARY A CVBEM is proposed for the umerical solutio of BVPs govered by a secod order elliptic PDE with variable coefficiets. A geeral solutio of 9

10 the PDE, expressible i terms of a arbitrary complex fuctio that is aalytic i the domai of iterest, is available i the literature (Clemets [3]). The proposed method makes use of the Cauchy itegral formulae to costruct approximately the relevat complex fuctio that gives the required solutio of the BVP. The task of costructig the complex fuctio requires maipulatio of the ukow data o oly the boudary of the solutio domai ad ca be evetually reduced to solvig a system of liear algebraic equatios. The method ca be efficietly implemeted o the computer, as oly the boudary eeds to be discretized ad the coefficiets of the algebraic equatios are easy to compute. It is used to solve a test problem which has a explicit exact solutio. The umerical results obtaied idicates that the proposed CVBEM works. Covergece of the umerical solutio to the exact oe is also observed whe the calculatio is refied by icreasig the umber of boudary elemets. Ackowledgemet. The first author (YSP) would like to ackowledge the assistace provided by Uiversiti Malaysia Sarawak i the form of a research grat for carryig out the work reported. Refereces [1] Brea, JJ. Iterfacial studies of refractory glass-ceramic matrix/advaced SiC fiber reiforced composite. Aual Report, ONR Cotract N C-0699, 1991, Uited Techologies Research Ceter. [2] Gibso, RE. Some results cocerig displacemets ad stresses i a ohomogeeous elastic half-space. Geotechique 1967; 17: [3] Clemets, DL. A boudary itegral equatio method for the umerical solutio of a secod order elliptic partial differetial equatio with variable coefficiets. Joural of the Australia Mathematical Society (Series B) 1980; 22:

11 [4] Ragogi, R. A solutio of Darcy s flow with variable permeability by meas of BEM ad perturbatio techique. Proceedigs of the 9th World Coferece o the Boudary Elemet Method 1987 (Edited by C. A. Brebbia) Spriger-Verlag, Berli. [5] Hromadka II, TV ad Lai, C. The Complex Variable Boudary Elemet Method i Egieerig Aalysis 1987, Spriger-Verlag, Berli. [6] Ag WT ad Park, YS. CVBEM for a system of secod-order elliptic partial differetial equatios. Egieerig Aalysis with Boudary Elemets 1998 ; 21: