University of Hertfordshire Department of Mathematics. Some comments on the approximation of the radial basis functions in the dual reciprocity method

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1 Univesit o Hetodshie Depatment o Mathematics Some comments on the appoimation o the adial basis unctions in the dual ecipocit method Wattana Toutip Technical Repot 4 August 999 Peace The dual ecipocit method (DRM) is a technique b which the domain integal is tanseed to an equivalent bounda integal b using suitable appoimation unctions. The appoimation can be constucted in seveal was. In the engineeing communit, the basis unction, is widel used o this pupose, whee is the usual Euclidean distance unction. Such unctions ae called the adial basis unctions (RBF) and ae used in the contet o intepolation. In the past, the choice o appoimation unctions in DRM has been lagel empiical and the lack o a sound theoetical basis has been a mao citicism. Golbeg and Chen (994) pa attention use the theo o RBF to povide a mathematical oundation o DRM. Among seveal available choices o RBF, the equentl used distance unction, and the thin plate splines log( ) ae studied in this wok. We eamine the oot mean squae (RMS) eo o both unctions. Finall we test the DRBEM pogam using the two adial basis unctions with the Tosion poblem (Patidge, Bebbia and Wobel, 99) b seveal unctions on the ight hand sides o Poisson equations.

2 . The DRM appoach We conside Poisson s equation u b () whee b b,, that is, b is consideed to be a known unction o position. The solution to equation () can be epessed as the sum o the solution o a homogenous and a paticula solution, e.g. u uh uˆ whee u h is the solution o the homogeneous equation and û is a paticula solution o the Poisson equation such that uˆ b () I thee ae bounda nodes and intenal nodes, as shown in Figue : Bounda node : Intenal node Figue : Bounda and intenal nodes thee will be om values o û and the appoimation o b can be witten in the b (3)

3 whee the :,,..., is a set o coeicients and the ae appoimating unctions called basis unctions. The paticula solutions û, and the appoimating, ae linked though the elation uˆ (4) The unction in (3) can be compaed with the usual intepolation unction i in epansions such as which ae used on the bounda elements themselves. Substituting equation (4) into (3) gives b u iu i (5) uˆ (6) Equation (6) can be substituted into the oiginal equation () to give the ollowing epession u Multipling b the undamental solution we obtain uˆ (7) * u and integating b pats ove the domain, * u u d * ( u) u d ˆ (8) ote that the same esult ma be obtained om * * ( u) u d bu d (9) Integating the aplacian tems b pats in (8), poduces the ollowing integal equation o each souce node i, c u i i * * * q ud u qd c u q uˆ i i d * ˆ u qˆ d (0) The tem qˆ in equation (0) is deined as nomal to, and can be epanded to u qˆ ˆ, whee n is the unit outwad n 3

4 uˆ uˆ qˆ () n n ote that equation (0) involves no domain integals. The net step is to wite equation (0) in discetized om, with summations ove the bounda elements eplacing the integals. This gives o a souce node i the epession ciui q * * * * ud u qd ciuˆ i q uˆ d u q ˆ d () k k k k k k k k Ate intoducing the intepolation unction and integating ove each bounda element, the above equation can be witten in tems o nodal values as c u i i k H ik u k k G ik q k ciuˆ i H ikuˆ k Gik qˆ k (3) k k The inde k is used o the bounda nodes which ae the ield points. Ate application to all bounda nodes using a collocation technique, equation (3) can be epessed in mati om as Hu Gq Hu Gˆ ˆ (4) q whee the matices H and G ae deined in equation (3). I each o the vectos û and qˆ is consideed to be one column o the matices Û and Qˆ espectivel, then equation (4) ma be witten without the summation to poduce Hu Gq Huˆ Gˆ (5) q Equation (5) is the basis o the application o the Dual Recipocit Bounda Element Method and involves discetization o the bounda onl. The ight-hand side o the equation is a known vecto and equation (5) becomes Hu Gq d (6) whee d HUˆ GQˆ (7) Appling bounda conditions to (6), this equation educes to the om A (8) whee contains unknown bounda values o u and q. 4

5 . Radial basis unctions The main pocess o DRM is to avoid the domain integal and to use an equivalent bounda integal instead. This is achieved b epanding the unction b in equation (3) as b (9) in tems o adial basis unctions at some chosen numbe o bounda () and intenal () nodes in the domain. In this stud, two adial basis unctions and log( ) ae condideed o use in (9). Both o them ae alsoused in the DRBEM pogam, whee esult ae discussed in a late section. We istl discuss the mathematical back gound o the both adial basis unctions. The distance unction unction was adopted ist b adini and Bebbia and then b most eseaches as the simplest altenative. The deinition o is that (0) whee and ae the components o in the diection o the and aes. Fo the linea unction () using the elation (4) and double integation we obtain 3 uˆ () 4 9 and using elation () togethe with patial deivative we have qˆ (3) n n 3 On the othe hand, o the thin plate splines log (4) 5

6 6 Appling the theoem o Duchon (Golbeg, 996) we appoimate the ight hand side unction using the epession c b a b ) log( ), ( (5) whee 0 To compute we need to add 3 equations and use adial basis unction as k k k log o k =,, 3,, + (6) (7) (8) 3 (9) B elation (4) and double integation we have 3 6 log ˆ 4 4 u k o k =,, 3,, + (30) 4 ˆ u (3) 6 ˆ 3 u (3) 6 ˆ 3 3 u (33) Theeoe, b elation () togethe with patial deivative we obtain n n q k 6 4 log ˆ o k =,, 3,, + (34) n q ˆ (35) n q ˆ (36)

7 qˆ 3 (37) n Thee ae vaieties o comments about these two adial basis unctions. Golbeg(995) suggested that thin plate splines might be a bette and moe mathematicall deensible choice because o its convegence and accuac. Howeve, seveal authos indicated that in pactice TPS showed little o no impovement ove the taditional one. To make some comments about this subect we ae going to test these two adial basis unction in the net section. 7

8 3. umeical esults Results will be pesented o a poblem o tosion o an elliptical bounda. Conside the Poisson equation on the elliptical domain as shown in Figue. u b(, (38) (0,) (-,0) 9 (,0) (0,-) 6 7 : Bounda node : Intenal point : Multiple node Figue : Discetization o the bounda into 6 elements 7 intenal points and 4 multiple nodes in elliptical domain The elliptical section shown in Figue has a semi-mao ais a = and a semimino ais b =. The equation o the ellipse is a b (39) The bounda condition is the Diichlet condition with u = 0 on the bounda. To investigate behaviou o the adial basis unction, the bounda is patitioned into 6, 3, 64, and 8 elements. The souce tem b (, is consideed in 3 cases as ollows: () b (, () b(, (3) (4) b(, b(, e 8

9 RMS eo An oveall estimato o the intepolation eo is the oot mean squae (RMS) eo. The RMS eo is deined as, i RMS eo = whee e i is the eact value o unction at node i and unction at the node i. e i ap i (40) ap i is the appoimation o The DRBEM pogam uses both and the thin plate spline to appoimate unction. The RMS eos b appoimation o the unction o the souce tem b (, with the numbe o bounda nodes ae 8, 6, 3, 64, 8 togethe with 7 intenal nodes ae shown in Table. Table : The RMS eos o the souce tem b (, odes Method = + = TPS 8 3.7E E E E E-06 0 We see om Table that the RMS eos using the thin plate splines ae absolutel zeo. The ecellent esult is oced b adding the linea tems in the TPS unction. Consequentl, in equation (5), the vecto is zeo, b c 0,and hence a. On the othe hand the RMS eos using the linea unction ae quite dieent. Similal, o the souce tem b(, with the same condition, the RMS eos act as the pevious one as shown in Table. Table : The RMS eos o the souce tem b(, odes Method = + = TPS 8.08E E E E E

10 We obseve that o the unction with is linea unction o and, the thin plate spline woks bette than the othe one. We have done with this esult but it is not pesented hee. Howeve, o quadatic unction, such as b(,, the RMS eo using the linea unction ae less than those o using the thin plate splines. This esult is shown is Table 3. Table 3: The RMS eos o the souce tem b(, odes Method = + = TPS E-07.6E E E E-05.4E E-05.8E E E-03 In the same manne, o the eponential unction b(, e, the linea unction woks bette than the thin plate spline. This esult is shown in Table 4. Table 4: The RMS eos o the souce tem b(, e odes Method = + = TPS 8.58E E E-05.6E E-04.98E E-04.3E E E-0 Although the appoimation using the thin plate splines is less than that using the linea unction in some situation, the convegence o the intenal solution appling its appoimation in the DRBEM pogam is still moe accuac as the numbe o nodes incease. We ae going to discuss the intenal solution and nomal deivative o the poblem using these appoimation in the DRBEM pogam in the est o this section. 3. The case b (, The eact solution is 0

11 The nomal deivative is 0 u.8 (4) q 0.( 8 ) (4) The intenal solutions with 6 bounda nodes and 7 intenal points ae shown in Table 5. Table 5: Intenal solution o b (, with = 6 and = 7 Point Radial basis unction Eact = + =TPS solution To investigate the convegence o intenal solution using these two adial basis unctions. We patition the bounda into 8, 6, 3, 64, 8 nodes with the same 7 intenal points. The elative eos o the intenal solutions using the linea unction ae shown in Table 6. Table 6: Relative eos o the intenal solution using Point umbe o elements (.5,0.0) (.,-0.35) (0.6,-0.45) (0.0,-0.45) (0.0,0.0) (0.3,0.0) (0.9,0.0) The elative eos o the intenal solutions using thin plate spline with a numbe o patitions ae shown in Table 7.

12 Relative eo RBF-DRBEM 03/08/09 Table 7: Relative eos o the intenal solution using thin plate splines Point umbe o elements (.5,0.0) (.,-0.35) (0.6,-0.45) (0.0,-0.45) (0.0,0.0) (0.3,0.0) (0.9,0.0) We see om Table 6 and Table 7 that the intenal solutions using thin plate splines convege moe apidl that those o using the linea unction, paticulal, at the point (.5,0) as shown in Figue 3. Convegence o intenal solution at (.5,0) umbe o elements = + = TPS Figue 3: Convegence o intenal solution at the point (.5,0)

13 3. The case b(, The govening equation is The eact solution is given b u (43) u (44) 7 4 which satisies the bounda condition u 0 on and poduces 3 q (45) 4 The intenal solutions with 3 bounda nodes and 7 intenal points ae shown in Table 8. Table 8: Intenal solution o b(, with = 3 and = 7 Point Radial basis unction Eact = + =TPS solution The elative eos o the intenal solutions using the linea unction shown in Table 9. ae Table 9: Relative eos o the intenal solution using Point umbe o elements (.5,0.0) (.,-0.35) (0.6,-0.45) (0.3,0.0) (0.9,0.0)

14 Relative eo RBF-DRBEM 03/08/09 The elative eos o the intenal solutions using thin plate splines with a numbe o patitions ae shown in Table 0. Table 0: Relative eos o the intenal solution using thin plate splines Point umbe o elements (.5,0.0) (.,-0.35) (0.6,-0.45) (0.3,0.0) (0.9,0.0) We see om Table 9 and Table 0 that the intenal solutions using thin plate splines convege moe apidl that those o using the linea unction, paticulal, at the point (0.9,0) as shown in Figue 4. Convegence o intenal solution at (0.9,0) umbe o elements = + = TPS Figue 4: Convegence o the intenal solution at (0.9,0) 3.3 The case b(, In this case the govening equation is u (46) 4

15 The eact solution is given b u (47) 46 which again satisies the bounda condition u 0 on and poduces q (48) The intenal solutions with 8 bounda nodes and 7 intenal points ae shown in Table. Table : Intenal solution o b(, with = 8 and = 7 Point Radial basis unction Eact = + =TPS solution The elative eos o the intenal solutions using the linea unction shown in Table. ae Table : Relative eos o the intenal solution using Point umbe o elements (.5,0.0) (.,-0.35) (0.6,-0.45) (0.0,-0.45) (0.0,0.0) (0.3,0.0) (0.9,0.0)

16 Relative eo RBF-DRBEM 03/08/09 The elative eos o the intenal solutions using thin plate splines with a numbe o patitions ae shown in Table 3. Table 3: Relative eos o the intenal solution using thin plate splines Point umbe o elements (.5,0.0) (.,-0.35) (0.6,-0.45) (0.0,-0.45) (0.0,0.0) (0.3,0.0) (0.9,0.0) We see om Table and Table 3 that the intenal solutions using thin plate splines still convege moe apidl that those o using the linea unction, paticulal, at the point (0.6,-0.45) as shown in Figue 5. Convegence o intenal solution at (0.6,-0.45) umbe o elements = + = TPS Figue 5: Convegence o intenal solution at (0.6, -0.45) 6

17 4. Conclusion We have eamined behavio o the two equentl used adial basis unctions namel the linea unction and the thin plate spline unction log( ). The appoimation in DRM using these unctions woks well in dieent kind o unctions to intepolate. Thee is still lack o theoems to indicate that which one is the best. We ound that the thin plate spline is absolutel peect to appoimate unctions which compise one o components in linea tem o unction a b c. On the othe, o a quadatic unction and an eponential unction e, the RMS eos using the linea unction ae quite less than those o using the thin plate spline. Howeve, the convegence o intenal solutions using the thin plate spline, especiall in the Tosion poblem, is moe apidl than that o using the linea one. This is because the Duchon theoem (Golbeg and Chen, 994) poved this act. 7

18 RBF-DRBEM 3/8/09 Reeences Golbeg, M.A. (994) The theo o adial basis unctions applied to the BEM o inhomogneous patial dieential equations. Bounda Elements Communications 5, Golbeg, M.A. (996) The method o undamental solutions o Poisson's equation. Engineeing Analsis with Bounda Elements 6, Golbeg, M.A., Chen, C.S., Bowman, H. and Powe, H. (998) Some comments on the use o adial basis unctions in the dual ecipocit method. Computational Mechanics, Kau, S.R. and Ramachandan, P.A. (994) Radial basis unction appoimation in the dual ecipocit method. Mathematical and Compute Modelling 0, Patidge, P.W. and Bebbia, C.A. (989) Compute implementation o the BEM dual ecipocit method o the solution o Poisson tpe equations. Sotwae o Engineeing Wokstations 5, Patidge, P.W. and Bebbia, C.A. (990) Compute implementation o the BEM dual ecipocit method o the solution o geneal ield equations. Communications in Applied umeical Methods 6, Patidge, P.W., Bebbia, C.A. and Wobel,.C. (99) The dual ecipocit bounda element method, Southampton: Computational Mechanics Publications. Powell, M.J.D. (994) The uniom covegence o thin plate spline intepolation in two dimension. umeical Mathematics 68, ======================================== 8