Boudary Elemets ad Other Mesh Reducto Methods XXIX 3 No-overlappg doma decomposto scheme for the symmetrc radal bass fucto meshless approach wth double collocato at the sub-doma terfaces H. Power, A. Heradez & A. La Rocca The Uversty of Nottgham, School of Mechacal, Materal ad Maufacturg Egeerg, UK Abstract I the partcular case of solvg large-scale boudary value problems, the computatoal cost derved as a result of the applcato of ay umercal scheme represets a determat factor the determato of ts computatoal effcecy. The preset work studes the fluece of the o-overlappg doma decomposto o the symmetrc radal bass collocato method, as a way to mprove ts effcecy uder hgh demadg umercal codtos. Due to the Hermta character of the symmetrc scheme at each of the collocatos pots of the sub-doma terfaces t s possble to mpose smultaeously all the correspodg matchg codtos. A mult-zoe problem s cosdered as a test example, comparso betwee the umercal result ad the aalytcal soluto for two set of dfferet physcal parameters are preseted. Keywords: symmetrc RBF meshless approach, doma decomposto ad double collocato. Itroducto The use of a mesh s a basc characterstc of tradtoal umercal approaches for the soluto of partal dfferetal equatos, as s the case of the fte dfferece, elemet, volume ad the boudary elemet methods. I the frst cases, assumptos are made for the local approxmato, whch requre teral mesh to support them. O the other had, the case o boudary methods, a WIT Trasactos o Modellg ad Smulato, Vol 44, 007 WIT Press www.wtpress.com, ISSN 743-355X (o-le) do:0.495/be0700
4 Boudary Elemets ad Other Mesh Reducto Methods XXIX boudary mesh s requred to obta a umercal approxmato of the resultg boudary tegrals. Durg recet years, cosderable effort has bee gve to the developmet of the so-called free-mesh methods (meshless approach). The am of ths type of approach s to elmate at least the structure of the mesh ad approxmate the soluto etrely usg odes values sde ad/or the boudary quas radom dstrbuted the doma. Recetly, some sgfcat developmets meshless methods for solvg boudary value problems of partal dfferetal equatos have bee reported the lterature. Kasa [, ] troduced the cocept of solvg PDEs usg radal bass fuctos (RBFs) (Usymmetrc scheme). Ths type of approach, whch approxmates the whole soluto of the PDE drectly usg RBFs, s very attractve due to the fact that ths s truly a mesh free techque. The Kasa s method has bee appled successfully several cases (see for example [3 5]). However, o exstece of soluto ad covergece aalyss s avalable the lterature ad for some cases, t has bee reported that the resultg matrx was extremely ll-codtoed ad eve sgular for some dstrbuto of the odal pots (see [6]). Several techques have bee proposed to mprove the codtog of the coeffcet matrx ad the soluto accuracy, as are: the use of hgh order terpolato fuctos, replacemet of global solvers by block parttog, LU decomposto schemes, matrx precodtoers, overlappg ad ooverlappg doma decomposto etc (see [7]). Fedoseyev et al. [8] proposed the use of a set of addtoal odes at the boudary ad beyod the boudary (at the exteror) where the goverg equato s requred to be satsfed. It was foud that the suggested approach yelds to more accurate results tha oly mposg the goverg equato at teral odes. Fasshauer [9] suggested a alteratve approach to the Usymmetrc scheme based o the Hermte terpolato property of the radal bass fuctos, whch states that the RBFs ot oly are able to terpolate a gve fucto but also ts dervatves. The covergece proof for RBF Hermte-Brkhoff terpolato was gve by Wu [0] who also proved the covergece of ths approach whe solvg PDEs (see Wu [] ad Schaback ad Frake []). Aother advatage of the Hermte based approach s that the matrx resultg from the scheme s symmetrc, as opposed to the completely ustructured matrx of the same sze resultg from Usymmetrc schemes. The ma obectve of ths work s to study ad test some of the above metoed techques prevously used to mprove the effcecy of the Usymmetrc approach order to crease the computatoal effcecy of the radal bass fucto symmetrc approach. I partcular, we wll study the ooverlappg doma decomposto wth a double collocato at the sub-doma terfaces. The doma decomposto approach s tself a very powerful ad popular scheme umercal aalyss, whch have recetly creased ts popularty due to ts use parallel computg algorthms. WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
Symmetrc radal bass fucto meshless approach Let us cosder a boudary value problem defed by: [ C]( x) f ( x) [ C]( x) g ( x) L = (-a) B =. (-b) where the operators L ad B are lear partal dfferetal operators o the doma Ω ad at the cotour Γ respectvely. A symmetrc RBF collocato method (Fasshauer [9]), represets the soluto of the above boudary value problem by the terpolato fucto: C ( x) = λkb Ψ( x ξk ) + λk Lξ Ψ( x ξk ) + Pm ( x) k = Boudary Elemets ad Other Mesh Reducto Methods XXIX 5 N ξ () k = + wth as the umber of odes o the boudary of Ω ad N the umber of teral odes. Here, Ψ ( x x ) s a codtoally postve defte RBF of order m ad P a polyomal term of order m. I the above expresso L ξ ad B ξ are the dfferetal operators used (-a,b), but actg o Ψ vewed as a fucto of the secod argumet ξ (see Fasshauer [9]). Ths expaso for C leads to a collocato matrx A, whch s of the form BxBξ [ Ψ] BxLξ [ Ψ] BxPm A = LxBξ [ Ψ] LxLξ [ Ψ] LxPm (3) T T BxPm LxPm 0 where the followg ortogoalty codtos s requred to complete the system: k = k N T T xpm + λk Pm = 0 k = + λ B (4) The matrx (3) s of the same type as the scattered Hermte terpolato matrces ad thus o-sgular as log as Ψ s chose approprately (see Wu [6]). A maor pot favour of the Hermte based approach s that the matrx resultg from the scheme s symmetrc, as opposed to the completely ustructured matrx of the same sze resultg from Usymmetrc schemes. 3 Covecto-dffuso problem The steady state dfferetal equato cosdered ths work s of the form: d C d C D u + k C = 0 (5) dx dx The partal dfferetal operators o the matrx represetato (3) of the symmetrc collocato umercal soluto of equato (5), whe satsfyg WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
6 Boudary Elemets ad Other Mesh Reducto Methods XXIX boudary codtos of the frst ad secod kd (Drchlet ad Neuma), are defed by the followg expressos: d d d d Lx = D u,, + k Lξ = D u + k dx dx dξ dξ (6) D N d D D d Bx =, Bx = ( x), Bξ =, Bξ = ( ξ ) dx dξ I the above relatos the super dex D ad N the operator B represet the type of boudary codtos mplemeted,.e. Drchlet ad Neuma. I ths work we wll use the geeralzed TPS. Furthermore to avod sgularty at r = 0 o the resultg dfferetal operators of the matrx A, we use the represetato formula () the geeralzed TPS ψ = r 6 log r (7) together wth the correspodg cubc polyomal. 3 3 P3 ( x) = λn + x + x + 3x x + 4xx + 5x + λn + 6 x + 7 xx + 8x + 9 x + 0 (8) 4 Doma decomposto approach Doma decomposto methods are frequetly used two cotexts. Frst: the dvso of problems to smaller problems usually through artfcal subdvsos of the doma, as a way to mprove the performace of a umercal techque. Secod, may problems volve more tha oe mathematcal model each posed o a dfferet doma, so that doma decomposto occurs aturally. Whe dealg wth the umercal smulato of large problems, t s usual to use the method of doma decomposto, whch the orgal doma s dvded to sub-regos, ad o each of them the orgal goverg equatos are mposed. The ma obectve of the doma decomposto method s to decompose oe large global problem to smaller sub-doma problems. I the mplemetato of the doma decomposto approach, two dfferet alteratves are possble to use: overlappg ad o-overlappg schemes. I the o-overlappg techque, the doma s dvded to o-overlappg sub-domas havg commo terface surfaces. I each sub-doma the orgal umercal scheme s mplemeted. Owg to the lack of the boudary codto o the terface betwee sub-domas, addtoal surface ukows eed to be determed,.e. the preset case the value of the cocetrato ad the surface flux. For each terface boudary pot, the umber of ukows s more tha the umber of the equatos ad therefore the resultg system s uderdetermed. However, oce the matchg codtos are mposed ad the sub-doma assembled, the s possble to obta a close system. WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
Boudary Elemets ad Other Mesh Reducto Methods XXIX 7 I the overlappg approach, the problem ca be solved by a teratve scheme terms of oe of the Schwarz methods,.e. by solvg recursvely each of the sub-domas, or stead the complete close system ca be solved drectly, after mposg the terface matchg codtos betwee subdomas, wthout the used of a teratve scheme. It s mportat to observe that the o-overlappg doma decomposto approach s aturally suted for the umercal soluto of mult-zoe problems, where the goverg equatos have dfferet values of the problem parameters at dfferet regos of the problem doma. Several applcato of the doma decomposto approach has bee reported the lterature whe solvg partal dfferetal equatos wth the use of the Usymmetrc radal bass fucto collocato approach. I the work by Kasa ad Carlso [3] they coclude that oe of the most effcet techque whe solvg dese system of lear equato s to use precodtog ad to make use of doma decomposto techques. I ths work wll be mplemet the o-overlappg o-teratve doma decomposto approach for the umercal soluto of boudary values problems based o the symmetrc radal bass fucto collocato approach, wth applcato to mult-zoe problems. 5 Mult-zoe problems Cosderg a problem that cotas dfferet regos, whch the coeffcets of the goverg equato are costat but dfferet each of them. I the mplemetato of the o-overlappg doma decomposto approach for the soluto of mult-zoe problems, the problem s doma s dvded to a fte umber of o-overlappg zoes accordg wth the behavour of the goverg equato. I order to mplemet the symmetrc approach to solve ths type of mult-zoe doma problem, the soluto at each zoe s represeted by ts correspodg symmetrc terpolato usg a set of collocato pots wth each of the zoes ad the pots at the terface betwee them. At the terfaces pots that cocde wth the physcal boudary of the problem, the correspodg boudary codtos are mposed, whle at the teral pots, t s requred that the goverg equato, wth correspodg value of the parameters at each zoe, should be satsfy. To solve ths type of problem t s ecessary to mpose the cotuty of flux at the terfaces betwee the zoes,.e. the flux leavg oe sub-zoe has to be equal to the flux eterg the other. Therefore, t s ecessary that the followg flux matchg codtos hold at the m th terface of the sub-zoes ad +: C C D + + + C u = D C u + (9) m m Besdes the above codtos, the cocetrato at each terface eeds to be cotuous,.e.: WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
8 Boudary Elemets ad Other Mesh Reducto Methods XXIX + [ C ] m [ C ] m = (0) The Hermta terpolato property of the symmetrc approach (whch takes to accout the fucto ad t s dervate) makes ths method a atural techque to deal wth the above matchg codtos. There are two dfferet alteratves to mpose the terface matchg codtos. Frst two dfferet set of pots at each sub-domas terface are defed. I each set of pots a dfferet matchg codto s mposed,.e. of the = + terface pots are requred to satsfy the cotuty of the cocetrato ad o the remag pots the flux matchg codto s satsfed. O the other had, due to the depedece of the Hermte terpolato o the partal dfferetal operators, t s possble to mpose smultaeously both codtos at each terface pot. I ths last case, as we are usg a Hermte terpolato scheme, the resultg matrx system s o-sgular as log as the partal dfferetal operators appled to each pot are learly depedet, eve f a sgle ode we mpose two dfferet dfferetal codtos (see Wu []). I ths case, at each terface pot both matchg codtos,.e. cocetrato ad flux, are requred to be satsfed. Therefore, the Hermte terpolato fucto s represeted by: C b ( x) = λk Bξ Ψ( x ξk ) + λkψ( x ξk ) k = k = b + + b + λ k k = b + + () N D ( x ) k kl ( x k ) Pm ( x) Ψ ξ + Ψ + λ ξ ξ ξ k = b + wth as the umber of odes o the boudary of a subdoma that cocde b wth the physcal boudary,.e. at a teral subdoma b 0, the umber of odes o the terfaces commo wth other subdomas ad N ( b + ) the umber of teral odes at the subdoma. As before, the above expresso L ξ ad B ξ are the dfferetal operators correspodg to the partal dfferetal equato at the subdoma ad the boudary dfferetal operator. I the above terpolato fucto, the flux matchg codto (9) at the terface m s reduced to + C + C D = D () m m sce we are mposg smultaeously at each terface pot the cotuty of cocetrato ad flux, besdes the covectve velocty feld eeds to be cotuous across the sub-domas. It s mportat to pot out that the above double collocato strategy at the terface pots ca also be used the stadard Usymmetrc approach (Kasa method). However due to lack of depedece o the dfferetal operators of the WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
Boudary Elemets ad Other Mesh Reducto Methods XXIX 9 correspodg terpolato fucto, ths alteratve wll results a overdetermed system of algebrac equatos. 5. Numercal examples Let us cosder the steady state heat trasfer problem a crcular cylder wth a crcular hollow. At the org a costat value of temperature s gve as well as at the exteror wall. The cylder cossts of three rgs of costat but dfferet parameters (see fgure ), uder these codtos, cyldrcal co-ordates, the problem s descrbed by the followg equato: K d d T r + α = 0 =,,3 (3) r dr dr where K ad α are the thermal coductvty ad heat geerato rate at the zoe (rg), respectvely, ad r s the radal dstace. The matchg codtos at the cotact rego betwee rgs,.e. cotuty of temperature ad flux are gve by: + K d T = T + K d + ad ( rt ) = ( rt ) at r = rm, m =,, 3 (4) r d r r d r The aalytcal soluto of the problem s gve by Carslaw ad Jaeger [3]. By expadg the cyldrcal Laplaca operator equato (), we obta the followg expresso: d T K d T K + + α = 0 (5) dr r dr whch ca be terpreted as a oe-dmesoal covecto dffuso equato wth varable egatve covectve velocty feld, u = K r. r / r3 r r Fgure : Cyldrcal doma cosstg of three rgs wth dfferet costat coeffcets at each rg. WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
0 Boudary Elemets ad Other Mesh Reducto Methods XXIX The oe-dmesoal problem defed by equato (5), the matchg codtos gve by equato (4) ad the correspodg boudary codtos at r = r 0 ad r = r3, wll be solve here as two dmesoal covecto dffuso problem the rectagular doma x r3 = 7dm ad 0 y dm wth zero lateral flux, were at each zoe (rg) the followg goverg equato s satsfed: T T K u + α = 0 where u = K / x (6) x x Two cases are cosdered, wth dfferet parameters each zoe ad the same boudary codtos; T(0, y ) =, T(7, y ) = ad T / = 0 at y = 0 ad y =. I fgures ad 3, t s possble to apprecate the excellet agreemet foud betwee the umercal results usg the above symmetrc meshless collocato method ad the aalytcal soluto. I the frst example (fgure ), a total of 84 collocato pots uformly dstrbuted were used order to acheve the obtaed accuracy. The secod case s more computatoal demadg due to the drastc chages the heat producto term, α ( =,,3), betwee the dfferet zoes, gve by α = 5, α = 3, α 3 = 0 (temperature/sec). I ths case, we compare how the umercal result s affect by creasg the total umber of collocato pots. I fgure 3, the results for two dfferet set of uform dstrbuto of collocato pots (84 ad 987) are preseted, showg the covergece of the umercal scheme. 0 9 8 7 C (mol/l) 6 5 4 3 0 3 4 5 6 7 x (mm) Fgure : Comparso betwee the aalytcal soluto ad the umercal results for the followg parameter: α =, α = 0, α 3 = 4 (temperature/sec) ad 3 K = K = K = ( dm /sec). WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
Boudary Elemets ad Other Mesh Reducto Methods XXIX C (mol/l) 6 4 0 8 6 4 0 3 4 5 6 7 x (mm) Fgure 3: Comparso betwee the aalytcal soluto ad the umercal results obtaed wth a total of 84, ( ), ad 987, ( ), collocato pots, for the followg parameters: α = 5, α = 3, α 3 = 0 (temperature/sec) ad 3 K = K = K = ( dm /sec). 6 Coclusos The use of symmetrc radal bass fucto collocato method to solve partal dfferetal equatos provdes a smply accurate ad truly meshes free techque. It s mportat to pot out that the case whe ths scheme s appled to solve large scale problems wth a large umber of data pots, the codtoal umber of the resultg collocato matrx could be very large ad the computatoal performace poor. As a way to overcome these problems, a doma decomposto scheme wth double collocato at the terfaces og eghbourg sub-domas s proposed. The proposed doma decomposto techque makes possble to mprove the ll-codtog problem through the reducto of the sze of the full coeffcet matrx to be solved a global maer. Ackowledgemets Ths research was bee partally sposored by the GABARDINE proect (Cotract umber 588) part of the FP6-006-TTC-TU Europea Commsso Programme. Refereces [] E.J.Kasa, Multquadrcs- A scattered data approxmato scheme wth applcatos to computato flud-dyamcs-i: Surface approxmatos ad partal dervatves estmates; Computers Math. Applc. 9, pp 7-45, (990) WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press
Boudary Elemets ad Other Mesh Reducto Methods XXIX [] E.J.Kasa, Multquadrcs- A scattered data approxmato scheme wth applcatos to computato flud-dyamcs-ii: Soluto to parabolc, hyperbolc ad ellptc partal dfferetal equatos; Computers Math. Applc. 9, pp 47-6, (990) [3] Dubal M.R. Doma decomposto ad local refemet for multquadrc approxmatos. I: secod-order equatos oe-dmeso, Joural of Appled Scece,, No., 46-7 (994). [4] Y. C. Ho ad X. Z. Mao A effcet umercal scheme for Burgers' equatos, Appl. Math. ad Comp. 95, 37-50 (998). [5] Z Zerroukat M., Power H. ad Che C.S., A umercal method for heat trasfer problems usg collocato ad radal bass fuctos, It. J. Numer. Meth. Egg, 4, 63-79, (998). [6] Dubal M.R., Olvera S.R. ad Matzer R.A. I Approaches to Numercal Relatvty, Edtors: R.d Ivero, Cambrdge Uversty Press, Cambrdge UK, (993). [7] Kasa E.J. ad Ho Y.C., Crcumvetg the ll codtog problem wth multquadrc radal bass fuctos: applcatos to ellptc partal dfferetal equatos, 39, 3-37, (000). [8] Fedoseyev AI, Fredma MJ, Kasa EJ. Improved multquadratc method for ellptc partal dfferetal equato va PDE collocato o the boudary. Comput. Math. Appl. 00, 43, 439-455 [9] Fasshauer G.E. Solvg Partal Dfferetal Equatos by Collocato wth Radal Bass fuctos, Proceedgs of Chamox, Edtors: A. Le Méchauté, C. Rabut ad L.L. Schumaker, -8, Vaderblt Uversty Press, Nashvlle, TN (996). [0] Wu Z., Hermte-Brkhoff terpolato of scattered data by radal bass fuctos; Approx. Theory, 8:, - (99). [] Wu Z., Solvg PDE wth radal bass fucto ad the error estmato; Advaces Computatoal Mathematcs, Lecture Notes o Pure ad Appled Mathematcs, 0, Edtors: Z. Che, Y. L, C.A. Mcchell, Y. Xu ad M. Dekker, GuagZhou (998). [] Schaback R ad Frake C., `Covergece order estmates of meshless collocato methods usg radal bass fuctos', Advaces Computatoal Mathematcs, 8, Issue 4, 38-399, (998). [3] Kasa E.J. & Carlso 99, Improved accuracy of multquadrc terpolato usg varable shape parameters, Computers & Mathematc wth Applcato, vol. 4, 99-0. [4] Carslaw H.S. ad Jaeger J.C., Coducto of heat solds, Oxford at the Claredo press, Oxford, (959). WIT Trasactos o Modellg ad Smulato, Vol 44, www.wtpress.com, ISSN 743-355X (o-le) 007 WIT Press