A new RBF-Trefftz meshless method for partial differential equations

Transcription

1 Hoe Search Collectos Jourals About Cotact us My IOPscece A ew RBF-Trefftz eshless ethod for artal dfferetal equatos Ths artcle has bee dowloaded fro IOPscece. Please scroll dow to see the full text artcle. 00 IOP Cof. Ser.: Mater. Sc. Eg (htt://oscece.o.org/ x/0//07) Vew the table of cotets for ths ssue, or go to the oural hoeage for ore Dowload detals: IP Address: The artcle was dowloaded o 07/07/00 at :5 Please ote that ters ad codtos aly.

2 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 A ew RBF-Trefftz eshless ethod for artal dfferetal equatos Lele Cao,, Qg-Hua Q * ad Ng Zhao School of Mechatrocs, Northwester Polytechcal Uversty, X a, 7007 Cha School of Egeerg, Australa Natoal Uversty, Caberra, ACTON 60, Australa. Eal: qghua.q@au.edu.au Abstract. Based o the radal bass fuctos (RBF) ad T-Trefftz soluto, ths aer resets a ew eshless ethod for uercally solvg varous artal dfferetal equato systes. Frst, the aalog equato ethod (AEM) s used to covert the orgal atal dfferetal equato to a equvalet Posso s equato. The, the radal bass fuctos (RBF) are eloyed to aroxaate the hoogeeous ter, whle the hoogeeous soluto s obtaed by lear cobato of a set of T-Trefftz solutos. The reset schee, aed RBF-Trefftz has the advatage over the fudaetal soluto (MFS) ethod due to the use of osgular T-Trefftz soluto rather tha sgular fudaetal solutos, so t does ot requre the artfcal boudary. The alcato ad effcecy of the roosed ethod are valdated through several exales whch clude dfferet tye of dfferetal equatos, such as Lalace equato, Hellholtz equato, covect-dffuso equato ad te-deedet equato.. Itroducto I the last decade, eshless ethods have bee eergg as ortat schees for uercal soluto of artal dfferetal equatos (PDEs), because the eshless ethods ca avod the colcated eshg roble fte eleet ethods (FEM) ad boudary eleet ethod (BEM) effectvely. The ethods based o the radal bass fucto (RBF) are heretly eshfree due to the deedet of desoalty ad colexty of geoetry, so the research o the RBF for PDEs has becoe very actve. Aog the exstg RBF schees, the so-called Kasa s ethod s a doa-tye collocato techque, whle the BBF-MFS techque whch s evolved for the dual recrocty boudary eleet ethod (DRBEM) [] s a tycal boudary-tye ethodaology. I the RBF-MFS, the ethod of fudaetal solutos (MFS) [] has bee used stead of boudary eleet ethod (BEM) the * Corresodg author: Qghua.q@au.edu.au; tel: ; Fax: c 00 Publshed uder lcece by Ltd

3 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 rocess of the DRBEM. Deste the advatages, the a drawback of the RBF-MFS s the use of the fctous boudares outsde the hyscal doa to avod the sgulartes of the fudaetal soluto. The creato of the fctous boudares s the troublesoe ssue related to the stablty ad accuracy of the ethodology ad there are o effectve rules to follow. I ths study, we troduce a ew RBF-Trefftz eshless ethod based o alyg the radal bass fucto ad T-Trefftz soluto. I cotrast to the RBF-MFS whch s based o MFS, the RBF-Trefftz s based o Trefftz ethod. Trefftz ethod s forulated usg the faly of T-colete fuctos whch are hoogeeous solutos for the goverg equato. The Trefftz ethod was tated 96 [3]. Sce the, t has bee studed by ay researchers (Cheug et al. [4, 5], Zelsk [6], Q [7, 8], Kta [9]). Ulke the ethod of fudaetal soluto whch eeds source ots to be laced outsde the doa order to avod sgularty, T-Trefftz fuctos are o-sgular sde ad o the boudary of the gve rego. Also, the aalog equato ethod (AEM) s develoed to covert the orgal artal dfferetal equato to a equvalet Posso s equato whch has a sler T-Trefftz soluto tha that of the orgal roble requred.. Nuercal ethod ad algorths Cosder a geeral dfferetal equato Lu( X ) = f ( X ), X Ω () wth boudary codtos ux = g X X Γ (), u ux = g( X), X Γq (3) where L s a dfferetal oerator, f ( X ) s a kow forcg fucto, s the ut outward oral to ux d the boudary Γ q ad s the drectoal dervatve drecto. X R, d s the deso of geoetry doa. We start by covertg Equato () to a sle Posso equato through aalog equato ethod, the troduce the RBF-MFS ethod to solve the equvalet Posso equato... The aalog equato ethod (AEM) [0] It s assued here that the oerator L cludes the Lalace oerator, aely, It should be oted out that ths assuto s ot ecessary. Equato () ca be restated as L = + L (4) ux f( X) LuX = (5) that s ux = bx (6) where bx = f( X) LuX (7) The soluto of the above Equato (6) ca be exressed as a suato of a artcular soluto ad a hoogeeous soluto u h, that s: u

4 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 u = u + u (8) h where u satsfes the hoogeous equato u X = b X X Ω (9) but does ot ecessarly satsfy the boudary codtos ()-(3), ad u h satsfes: u X = X Ω (0) h 0 u ( X) = g ( X) u ( X) X Γ u u ( X) h X = g( X) X Γ h u q ().. RBF-Trefftz schee... RBF aroxato for artcular soluto The artcular soluto u ca be evaluated by RBF. To do ths, the rght-had sde ter of Equato (9) s aroxated by RBF [], yeldg N I () = αϕ b X = X X Ω where N I s the uber of terolato ots the doa uder cosderato. Here, ϕ( X) = ϕ( r) = ϕ( X X ) deotes radal bass fuctos wth the referece ot X ad α are terolatg coeffcets to be detered. Sultaeously, the artcular soluto u s slarly exressed as N I = αψ u X = X (3) where ψ rereset corresodg aroxated artcular solutos whch satsfy the followg dfferetal equatos: ψ ϕ = (4) oted the relato betwee the artcular soluto u ad fucto f ( X ) Equato (9). The choce of the RBFs ϕ s very ortat because t affects the effetvess ad accuracy of the terolato. There are several tye of RBF, lke ad hoc bass fucto (+ r ), the olyharoc sles, Th Plate Sle (TPS) ad utquadrcs (MQ). I the atheatcal ad statstcal lterature TPS ad MQs see to be the ost wdely used RBFs []. I ths study, we choose ϕ as a TPS: ϕ = r l r (5) A addtoal olyoal ter P s requred to assure osgularty of the terolato atrx f the RBF s codtoally ostve defte such as TPS [3, 4]. Ad also, to acheve hgher covergece rates for f ( X ), the hgher order sles are cosdered [5]. For exale, 3

5 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 the ϕ = r l r, R (6) N I [ ] αϕ f X = X +P (7) = where [ ] s the order of the TPS, P s a olyoal of total degree ad let { b } l = be a bass for + d P ( l = s the deso of P, ad d= for a deso roble ). The corresodg d boudary codtos are gve by N I αbl( Pl) = 0, l l (8) = However, fro Equato (), t s obvous that the ukow coeffcets α ca ot be straghtly detered sce the hoogeeous ter bx s a ukow fucto deedg o the ukow fucto ux. The roble ca be dealt wth the way descrbed below.... Trefftz fucto for hoogeeous soluto Itroducg olar coordates (, r θ ) wth r = 0 at the cetrod of Ω, t s kow that the set N = { r cos θ} { r s θ} (9) = 0 = are T-Trefftz solutos of the Lalace equato. Hece, the hoogeeous soluto to Equato (0) s aroxated as h = u X = c N X (0) where c are the coeffcets to be detered ad s ts uber of cooets. The ters N ( X) = N( r) = N( X X ) are the T-Trefftz solutos of the Lalace oerator, ad { X } N S = are collocato ots laced o the hyscal boudary of the soluto doa. As a llustrato, the teral fucto N Equato (0) ca be gve the for So, Equato (0) ca be wrtte as N =, N = rcos θ, N = rs θ, L, () 3 k u X r cos θ + r s θ h = 0 = k = () where = k+. Noted that u h Equato (0) ad () autoatcally satsfes the gve dfferetal equato (0), all we eed to do s to eforce u h to satsfy the odfed boudary N codtos (). To do ths, collocato ots { X } S = are laced o the hyscal boudary to ft the boudary codto (). 4

6 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07.3. The costructo of the soluto syste Based o the equatos derved above, the solutos ux to dfferetal equato () ad ()-(3) ca be wrtte as = = (3) αψ u X = X + c N X X Ω ad N I ψ ( X) N ( X) u X = + Ω α c X (4) = = dfferetaltg Equato (3) wth resect to x or y yelds, x = α ( ψ), x +, x = = (5) u c N, y = α ( ψ), y +, y = = (5-) u c N, xx = α ( ψ), xx +, xx = = (5-) u c N, yy = α ( ψ ), yy + ( ), yy = = (5-3) u c N It s coveet for couter rograg to utlze the vector for. Therefore, Equato (3) ca be wrtte as ux = { UX }{ β} (6) where { U( X)} = { ψ, ψ, Lψ, N, N, LN } + T L N c I c Lc + { β} = { α, α, α,,, } Usg Equato(3)-(5), satsfacto of the goverg equato (9) at N I terolato ots sde Ω ad the boudary codto () at N s collocato ots o the hyscal boudary rovdes + Ns equatos to detere ukows α ad c (7) α ( + L) ψ + c( + L) N = f( X) = = αψ + cn = g( X) = = ψ ( X ) N ( X) α + c = g( X) = = (8) 5

7 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 It leads to a syste of lear algebrac equatos atrx for: wth [ M ] { β} = { γ} (9) ( + Ns) ( + ) ( + ) ( + Ns) { β} = { α, α, Lα, c, c, L c}, {} γ = { f, K, fn, g, K gn } (30) If the uber of cooets equals to the uber of collocato ots o the hyscal boudary ( = NS ), ths leads to roerly detered equatos. Alteratvely, case the uber of cooets s saller tha the uber of collocato ots ( < NS ), ths results over-detered equatos. The least square ethod ca be used to solve the over-detered equatos. Oce { β} s obtaed, u ca be couted usg Equato (3) 3. Nuercal leetato I order to evaluate the erforace of the roosed aroach, here we cosder tycal bechark robles whch are take fro the Refereces [, 6] for solvg dfferetal equatos. The geoetry of the test robles s a ellse featured wth aor axs of legth 3 ad or axs of legth uless otherwse secfed. For coarso, the sae odes as Referece [] are eloyed to dscretze the doa. The results are coared wth other uercal ethods ad the aalytcal soluto. 3.. Lalace Equato The D Lalace equato s gve by I u = 0 (3) the boudary codto s aled by the artcular fucto u = x+ y (3) The uercal results are reseted Table together wth those by the BEM ad boudary kot ethod (BKM) for coarso. It ca be see clearly fro Table that the results obtaed by roosed RBF-Trefftz ethods agree well wth the exact soluto ad aear to be ore accurate tha the results obtaed fro BEM. Table. Results for a Lalace roble. x y exact BEM BKM RBF-Trefftz s 6

8 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0// Helholtz equato The D Helholtz equato s gve by u+ u = 0 (33) wth hoogeeous boudary codto u = s x (34) The results are dslayed Table.. It ca be see that the RBF-Trefftz ethod ca acheve hgher accuracy tha other ethods. Table. Results for a Helholtz equato. x y exact DRBEM BKM RBF-Trefftz Varyg -araeter Helholtz roble Cosder the varyg-araeter Helholtz equato u u = 0 (35) x wth the artcular soluto aled as boudary codto u = (36) x The org of the Cartesa coordates syste s dslocated to the ode (3.0) to crcuvet sgularty at x = 0. Fro Table 3 we ca see that the accuracy ad effcecy of the RBF-Trefftz schee are very ecouragg Covecto-Dffuso robles Cosder the equato wth boudary codtos u u u = x y x y u e e (37) = + (38) whch s also a artcular soluto of Equato (37). It ca be see fro Table 4 that the results obtaed by roosed eshless ethods agree well wth the exact soluto ad aear to be ore accurate tha the results obtaed fro other ethods. 7

9 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 Table 3. Absolute errors for varyg araeter Helholtz roble. x y DRBEM BKM RBF-Trefftz e-3.6e-3.0e e- 3.3e-3.5e e-3 4.7e-3 3.0e e-3 4.4e-3 5.4e e-3 9.e-4 8.7e e-4.7e-.4e e-4 3.4e-.0e e-3 5.3e-3.7e e-3 6.3e-3 5.e e-3 5.6e-3 6.8e e-3 3.4e-3 8.7e e-3 8.8e-3.4e-3 Table 4. Results for a Covecto-Dffuso equato. x y exact DRBEM BKM RBF-Trefftz Modfed hoogeeous equato Cosder the equato wth boudary codtos u ku = 4 k( x + y ) (39) u = x + y (40) whch s also a artcular soluto of Equato (39). I our calculato, the soluto doa s a square wth sde legth 3 ad k = 9. The dstrbuto of the uercal soluto s show fgure. ad the corresodg soles are show fgure.. It ca be see that the uercal soluto atches very well wth the aalytcal soluto. 8

10 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 Fgure. dstrbutos of uercal soluto. Fgure. Isoles for the soluto Te-deedet artal dfferetal equato Let us cosder a te-deedet artal dfferetal equato wth boudary codtos ad tal codto uxyt (,, ) uxyt (,, ) = 0 t ( x + y ) uxyt (,, ) = ex( ) 4 π ( t+ 0.) 4( t+ 0.) The soluto doa s a else wth aor axs 3 ad or axs. I the lterature there are dfferet aroaches to hadle te varable, two of whch are: () Lalace trasfor; () te-steg ethod. Sce uercal verso of the Lalace trasfor s ofte ll-osed, here we aly te-steg ethod schee to hadle the te varable. For a tycal te terval [t, t + ] [0,T], u(x, t), ts dervatve wth resect to te varable t + (, ) = θ + ( θ) + u( X, t) u ( X) u ( X) u X t u X u X (43) = t τ It s roved that the Partal dfferetal Equato (PDE) ca be solved accurately usg the lct schee ( θ = ) [7]. Hece, we use θ = ad τ = 0.0 our aalyss. Table 5 shows the soluto at (0,0) Table 5. Maxu absolute errors at varous te stes at ot (0,0) Te Exact Sol. Abs.Errors Te Exact Sol. Abs.Errors e e e e e e e e e e- (4) (4) 9

11 WCCM/APCOM 00 IOP Cof. Seres: Materals Scece ad Egeerg 0 (00) 07 do:0.088/ x/0//07 4. Cocluso A ew RBF-Trefftz eshless aroach s develoed for solvg artal dfferetal equatos. After trasferrg the artal dfferetal equato to equvalet Posso s equato by the aalog equato ethod, the hoogeeous ter s aroxated by lear cobato of radal bass fuctos (RBF) ad the hoogeeous ter by Trefftz bases. RBF-Trefftz s easer to leet sce t s osgular, so t s uecessary to lace source ots outsde the doa for avodg sgularty whch does occur RBF-MFS. The uercal exales show clearly that the ethod reseted s very effectve. Refereces [] Partrdge P, Brebba C ad Wrobel L 99 The Dual Recrocty Boudary Eleet Method (Southato ad Bosto: Coutatoal Mechacs Publcatos) [] Farweather G ad Karageorghs A 998 The ethod of fudaetal solutos for elltc boudary value robles, Adv. Cout. Math , [3] Trefftz E., E G, Proc d It Cog Al Mech, 96, [4] J W, Cheug Y, ad Zekewcz O 990 Alcato of the Trefftz ethod lae elastcty robles, It. J. Nuer. Methods Eg , [5] Cheug Y, J W, ad Zekewcz O 989 Drect soluto rocedure for soluto of haroc robles usg colete, o-sgular, Trefftz fuctos, Coucatos Aled Nuercal Methods , [6] Zelsk A 995 O tral fuctos aled the geeralzed Trefftz ethod, Adv. Eg. Software , [7] Q Q 005 Trefftz Fte Eleet Method ad Its Alcatos, Al. Mech. Rev , [8] Q Q ad Radulescu V 00 Trefftz Fte ad Boudary Eleet Method, Al. Mech. Rev. 54. B99-B00, [9] Kta E, Ikeda Y, ad Kaya N 005 Trefftz soluto for boudary value roble of threedesoal Posso equato, Eg. Aal. Boudary Ele , [0] Katskadels J 994 The aalog equato ethod- A owerful BEM-based soluto techque for solvg lear ad olear egeerg robles, Boudary eleet ethod XVI. 67-8, [] Golberg M 996 Recet develoets the uercal evaluato of artcular solutos the boudary eleet ethod, Al. Math. Cout , [] Golberg M 999 The ethod of fudaetal solutos for otetal, Helholtz ad dffuso robles [3] Kasa E 990 Multquadrcs--A scattered data aroxato schee wth alcatos to coutatoal flud-dyacs--ii solutos to arabolc, hyerbolc ad elltc artal dfferetal equatos, Couters & Matheatcs wth Alcatos , [4] Zerroukat M, Power H, ad Che C 998 A uercal ethod for heat trasfer robles usg collocato ad radal bass fucto, It. J. Nuer. Methods Eg , [5] Muleshkov A, Golberg M, ad Che C 999 Partcular solutos of Helholtz-tye oerators usg hgher order olyhroc sles, Cout. Mech , [6] Che W ad Taaka M 00 A eshless, tegrato-free, ad boudary-oly RBF techque, Couters ad Matheatcs wth Alcatos , [7] Zva R., Vetzal K, ad Forsyth P 000 PDE ethods for rcg barrer otos, Joural of Ecooc Dyacs ad Cotrol , 0