Avalable onlne at www.scencedrect.com Proceda Proceda Computer Computer Scence Scence 1 (01) 00 (009) 589 597 000 000 Proceda Computer Scence www.elsever.com/locate/proceda Internatonal Conference on Computatonal Scence (ICCS 010) The Sngular Functon Boundary Integral ethod for Laplacan problems wth boundary sngulartes n two and three dmensons Chrstos Xenophontos* Evgena Chrstodoulou Georgos Georgou Department of athematcs and Statstcs Unversty of Cyprus PO BOX 057 cosa 1678 CYPRUS Abstract We present a Sngular Functon Boundary Integral ethod (SFBI) for solvng ellptc problems wth a boundary sngularty. In ths method the soluton s approxmated by the leadng terms of the asymptotc soluton expanson whch exsts near the sngular pont and s known for many benchmark problems. The unknowns to be calculated are the sngular coeffcents.e. the coeffcents n the asymptotc expanson also called (generalzed) stress ntensty factors. The dscretzed Galerkn equatons are reduced to boundary ntegrals by means of Green s theorem and the Drchlet boundary condtons are weakly enforced by means of Lagrange multplers the values of whch are ntroduced as addtonal unknowns n the resultng lnear system. The method s descrbed for two dmensonal Laplacan problems for whch the analyss establshes exponental rates of convergence as the number of terms n the asymptotc expanson s ncreased. We also dscuss the extenson of the method to three dmensonal Laplacan problems wth exhbts edge sngulartes. c 01 Publshed by Elsever Ltd. Open access under CC BY-C-D lcense. Keywords: boundary sngulartes; Lagrange multplers; stress ntensty factors; boundary approxmaton methods 1. Introducton In the past few decades several methods for treatng ellptc boundary value problems wth boundary sngulartes have been proposed. Among them one fnds the so-called hybrd methods [1] whch ncorporate drectly or ndrectly the form of the local asymptotc expanson for the soluton u n the approxmaton scheme. Ths expanson s known n many occasons and has the followng form: ur ( ) W r (1) 1 where (r ) are polar coordnates centered at the sngular pont are the sngular coeffcents and W r r f ( ) () * Correspondng author. Tel.: +57-8-9610; fax: +57-8-9601. E-mal address: xenophontos@ucy.ac.cy. 1877-0509 c 01 Publshed by Elsever Ltd. do:10.1016/.procs.010.04.9 Open access under CC BY-C-D lcense.
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 590 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 are the sngular functons wth f beng the egenvalues and egenfunctons of the problem whch are unquely determned by the geometry and the boundary condtons along the boundares sharng the sngularty. It s mportant to note that the sngular functons W satsfy the governng partal dfferental equaton (PDE) and the boundary condtons (BCs) along the boundary parts sharng the sngularty. Knowledge of the sngular coeffcents s of great mportance n many engneerng felds such as fracture mechancs. any methods have been proposed n the lterature for ther effectve and effcent approxmaton ncludng hgh order p/hp fnte element methods wth post-processng [ ] and Trefftz methods [1]. In the former the soluton s frst approxmated on a refned grd desgned especally to capture the sngularty and the coeffcents are obtaned by an extracton formula whch uses the computed soluton. ethods that do not requre any postprocessng and/or nclude nformaton about the exact soluton n the approxmaton scheme such as Trefftz methods are more attractve f the approxmaton of the coeffcents s the man obectve. The SFBI whch was developed by Georgou and co-workers [4] for Laplacan problems wth a boundary sngularty also has ths trat. In the SFBI the soluton s approxmated by the leadng terms of the expanson (1).e. 1 u W r () where are the approxmate sngular coeffcents. The method has been successfully appled to a number of benchmark problems [5 6]. In [7] t was shown theoretcally that ts convergence rate s exponental as. The method has also been extended to bharmonc problems n two-dmensons arsng from sold and flud mechancs [8 9]. The man advantages of the SFBI are: The dmenson of the problem s reduced by one leadng to consderable computatonal savngs The sngular coeffcents are calculated drectly hence avodng the need for post-processng In ths work we present the method and ts propertes for a model two dmensonal Laplacan problem wth a boundary sngularty ncludng numercal experments for llustraton purposes. We then dscuss ts extensons to three-dmensonal Laplacan problems wth an edge sngularty.. The SFBI.1. Descrpton of the method For concreteness we consder the followng Laplacan problem (see also Fgure 1 below): Fnd u such that u n u n u 0 n 0 on S1 u 0 on S u f on S g on S4. (4) Fg. 1. A planar Laplacan problem wth a boundary sngularty at O.
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 591 It s assumed that the data of the problem f and g are smooth and such that no other sngulartes arse. ow by applyng Galerkn s prncple we have W u 0 1.... A double applcaton of Green s second dentty reduces the above ntegral to u W W u u W 0 1... n n. Takng nto account the boundary condtons of the problem as well as the fact that the sngular functons satsfy the governng PDE we further obtan W W u u u W Wg 1... S n S4 n S n. S4 (5) and assumng that Lettng u / n S 1 (6) where 1... are pecewse polynomals of degree p defned on a subdvson of S n elements of mesh wdth h; s then proportonal to 1/h. In the case when the s are the typcal (Lagrange based) bass functons used n the Fnte Element ethod we have that the constants n (6) are the nodal values of. We then mpose the boundary condton on S weakly by means of Lagrange multplers.e. we requre that ( u f) 0 1... S. (7) Equatons (5) and (7) yeld the followng block system of ( + ) ( + ) equatons K1 K F 1 T K 0 (8) F K X b T n whch...... 1 1 S T W W K1 1... W W S n S n 4 K W 1... 1... S F 1 Wg 1... S4 F f 1... Solvng the system gven by eq. (8) wll produce the approxmate sngular coeffcents as well as the dscrete Lagrange multplers. We emphasze that all ntegrals appearng above are one dmensonal (thus the dmenson of the problem s reduced by one) and are carred out on portons of the doman away from the sngularty; therefore they can be evaluated by standard technques such as Gaussan quadrature. We should also menton that n the above system the coeffcent matrx K becomes sngular when > so the number of (dscrete) Lagrange multplers has to be smaller than the number of sngular functons. (9)
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 59 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597.. Error Analyss The SFBI has been analyzed n the context of Laplacan problems and n ths subsecton we summarze the k man results from [7]. Let H ( ) denote the usual Sobolev space of functons on whose (generalzed) partal dervatves of order 0 1 k are square ntegrable and let k denote the assocated norms. We then defne and 1 1 H w H w S 1/ 1 0 * : 0 H ( ) w H : w H ( ) H 1/ ( ) as the closure of H 0 ( ) wth respect to the norm w 1/ wv sup. 1/ vh ( ) v 1/ ow for any functon w such that w W (10) 1 we can always wrte w w r where 1 1. (11) w W r W Under the assumpton that there exst postve constants C 1 C and (0 1) such that r r C1 C (1) n t was shown n [7] that u u uu nf nf 1 C u v 1 n vv (1) 1 V 1/ S n 1 where 1 such that V span W and the space V s defned as follows: Let S be dvded nto sectons = 1 n 1 n S h h maxh. 1 1 n Then wth ( I ) the set of polynomals of degree p on I we set p V : p 1... n. (14) (We have dm( V1 ) ( p n) dm( V ).) Usng (1) t was also shown n [7] that f u/ nh k ( ) for some k 1 then there exsts a postve constant C ndependent of and such that u uu C h p 1 m k n 1/ S where m = mn{k p + 1}. oreover snce a a u u t follows that 0 (15) a a C (16) whch shows that the method approxmates the sngular coeffcents at an exponental rate as.
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 59. umercal results We now present the results of numercal computatons for the test problem depcted graphcally n Fgure below. u 0 S u 0 S u f( ) O S 1 u 0 0 Fg.. Test Laplacan problem on a sector. The functon f s taken as / gven by f wth beng the angle of the crcular sector. The local soluton s 1 1 u r sn. (17) Snce ths problem can be solved analytcally we have that the exact sngular coeffcents are gven by 16 R 1 (18) where R s the radus of the sector. The numercal results that follow correspond to = R = 1 and our goal s to llustrate the method and ts convergence rate. In fgure we show the percentage relatve error n the soluton u versus n a sem-log scale for dfferent values of the polynomal degree p used to approxmate the Lagrange multplers. We see that ndependently of p the method converges at an exponental rate as predcted by equaton (1). 100 u - u SFBI 1 / u 1 10 1 10 0 10-1 10 - Error n the approxmaton of the soluton u p = 1 p = p = 10-6 8 10 1 14 16 18 0 Fg.. Effect of the order p on the convergence of the soluton u.
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 594 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 In fgure 4 we show the percentage relatve error n the frst four sngular coeffcents versus n a sem-log scale for the case when p = 1 (other values of p gave smlar results). The exponental convergence as predcted by equaton (14) s agan clearly vsble. 10 10 0 Error n th coeffcent p = 1 = 1 = = = 4 100 - SFBI / 10-10 -4 10-6 10-8 10-10 10 15 0 5 0 5 Fg. 4. Error n the frst four sngular coeffcents for p = 1. Fnally n fgure 5 we show the error n u/ n versus n a log-log scale when u/ n on the curved sde of our doman s dscretzed by pece-wse polynomals of (fxed) degree p on a mesh wth wdth h.e. ~ p/h. The error estmate (1) states that the convergence rate s algebrac of order p and ndeed ths s what fgure 5 shows. 10 1 Error n the approxmaton of the Lagrange multplers p = 1 p = p = 100 max ( k ) - h ( k ) / ( k ) 10 0 10-1 slope - 1 slope - slope - 10-10 0 10 1 10 Fg. 5. Error n u/ n for dfferent values of p.
4. Extenson to three-dmensons C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 595 In ths secton we dscuss the extenson of the SFBI to three-dmensonal Laplacan problems an area that to our knowledge stll possesses several mportant open questons. In partcular we consder the followng: Fnd u such that u 0 n u g on S 1 1 u g on S u g on S u n 0 on S4 S5 (19) where the functons g = 1 are gven and the doman s shown n fgure 6 below. Fg. 6. Doman for the -D Laplace equaton gven by (19). A sngularty wll arse along the edge AB and we assume that the gven data s smooth enough such that no other sngulartes are present. The dfference between two and three-dmensons s substantal snce now the local soluton s gven by 1 r u r f x x 1 1 x n1n( n) 4 (0) where f and are the egenfunctons and egenvalues respectvely of the two-dmensonal problem (posed on the face S 1 ). oreover the sngular coeffcents are no longer constants but now they are functons of the thrd coordnate x ; for ths reason they are called Edge Stress Intensty Functons (ESIFs) [10 11]. evertheless they are known to be smooth functons [11] hence they can be approxmated by polynomals of say degree as where the coeffcents 1 k1 k k 1 x a x (1) a k must be determned for each. Once agan the soluton u s approxmated by the leadng
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 596 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 terms of the asymptotc expanson (0) as n the two-dmensonal case: 1 r u r f x x 1 1 x n1n( n) 4. () We note that the nfnte sum n () wll termnate after a fnte number of terms snce x s a polynomal. Consequently the number of unknowns n the above expresson s ( + 1). To determne them we wegh the governng PDE by l where the functons x 1 r 4 ˆ l l l W ( r x) r f x x () 1 x n1n( n) are at our dsposal we may choose them as polynomals. It s easy to show that ˆ l W satsfes the governng PDE and the boundary condtons on ether sde of the sngular edge ndependently of the l choce for x. As a result we have ˆ l W u 0 l 1.... (4) Usng Green s second dentty twce we get ˆ l ˆ l ˆ l u W u W W u n n 0 l 1.... (5) Snce the functons ˆ l W satsfy the PDE and the boundary condtons along S 4 and S 5 we have ˆ l u W ˆ l W u 0 l 1... S1SS n n (6) whch s the analog of our -D method (cf. eq. (5)). The next step s to represent u / nby a Lagrange multpler functon on each face S = 1 (expanded as a sum of polynomal bass functons ) and to mpose the Drchlet condtons weakly vz. 1 1 (7) g u 0 1.... (8) S Then eq. (6) becomes 1 S ˆ l W ˆ l W u 0 l 1... n
C. Xenophontos et. al./ Proceda Computer Scence 00 (010) 000 000 C. Xenophontos et al. / Proceda Computer Scence 1 (01) 589 597 597 1 whch along wth (7) yeld a lnear system analogous to (8) wth unknown vector [ a1 a... a ] T where a1 [ a11 a1... a1 1] a [ a1 a... a 1 ]... a [ a1 a... a 1] 1 1 1 1 [ 1... ] [ 1... ] [ 1... ]. Solvng the system wll produce the coeffcents for the functons x multplers (cf. (7)). l Currently we are nvestgatng possble choces for the polynomals x (cf. (1)) as well as for the Lagrange and. The results of ths study and the mplementaton of the method wll be reported n a future communcaton. Acknowledgements EC was partally supported by the Cyprus Research Foundaton //008/014. References 1. Z. C. L Combned ethods for Ellptc Equatons wth Sngulartes Kluwer The etherlands 1998.. I. Babuška and A. ller Int. J. umer. eth. Engg. 0 (1984) 1111.. B. Szabó and Z. Yosbash Int. J. umer. eth. Engg. 9 (1996) 409. 4. G. Georgou L. Olson and Y. S. Smyrls Comm. umer. eth. Engg. 1 (1996) 17. 5.. Ellots G. Georgou and C. Xenophontos Comm. umer. eth. Engg. 18 (00) 1. 6.. Ellots G. Georgou and C. Xenophontos Appl. ath. Comput. 169 (005) 485. 7. C. Xenophontos. Ellots and G. Georgou SIA J. Sc. Comp. 8 (006) 517. 8.. Ellots G. Georgou and C. Xenophontos Eng. Anal. Bound. Elem. 0 (006) 100. 9.. Ellots G. Georgou and C. Xenophontos Int. J. umer. eth. Fluds 48 (005) 1001. 10. Z. Yosbash R. Acts and B. Szabó Int. J. umer. eth. Fluds 5 (00) 5. 11. Z. Yosbash. Omer. Costabel and. Dauge Int. J. Fracture 16 (005) 7.