Green s function based finite element formulations for isotropic seepage analysis with free surface

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1 1991 Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac Abtract A olution procdur uing th Grn function bad finit lmnt mthod (FEM) i prntd for two-dimnional nonlinar tadytat pag analyi with th prnc of fr urfac in iotropic dam. In th prnt algorithm, an itration tratgy i dignd to convrt th ovr-pcifid fr urfac problm to a rgular partial diffrntial quation problm. Thn, at ach itration tp, th Grn function for iotropic linar pag partial diffrntial quation i mployd to contruct th lmnt intrior watr had fild, whil th convntional hap function ar ud for th indpndnt lmnt fram watr had fild. Thn th two indpndnt fild ar connctd by a doubl-variabl hybrid functional to produc th final olving quation ytm. By man of th phyical dfinition of Grn function, all two-dimnional lmnt domain intgral in th prnt algorithm can rduc to on-dimnional lmnt boundary intgral, o that vratil multinod lmnt i contructd to implify mh rcontruction during itration. Finally, numrical rult from th prnt Grn function bad FEM with iotropic Grn function krnl ar compard with othr numrical rult to vrify and dmontrat th prformanc of th prnt mthod. Hui Wang a* Ya-Ting Gao b Qing-Hua Qin c* a,b Dpartmnt of Enginring Mchanic, Hnan Univrity of Tchnology, Zhngzhou, , China. a,c Rarch School of Enginring, Autralian National Univrity, Acton, ACT 601, Autralia. Corrponding author: a* huiwang@haut.du.cn c* qinghua.qin@anu.du.au Rcivd In rvid form Accptd Kyword Spag; fr urfac; iotropic dam; hybrid finit lmnt mthod; Grn function. 1 INTRODUCTION During th tratmnt of watr hazard, on of practical maur i to contruct dam. So th drawdown pag analyi i fatal for th curity of dam ubjctd to watr lvl chang. On of th main difficulti i that th prdiction of th fr urfac, which i uually ovr-pcifid. Du to th complx boundary condition and gomtrical domain, numrical mthod including FEM (Numan, 1973; Bath and Khohgoftaar, 1979; Bath, 006; Luo t al., 008) and boundary

2 199 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac lmnt mthod (BEM) (Brbbia and Chang, 1979; Chn t al., 1994; Tay t al., 1997) ar oftn ud to analyz pag problm of th dam with fr urfac. For xampl, Bath and Khohgoftaar dvlopd a FE olution procdur for nonlinar fr urfac pag analyi without mh itration (Bath and Khohgoftaar, 1979). Luo t al. applid th FEM to tablih thr dimnional computational modl of tranint pag for dp foundation pit dwatring in th Yangtz Rivr Dlta (Luo t al., 008). Brbbia and Chang tablihd boundary lmnt formulation for pag problm in aniotropic oil (Brbbia and Chang, 1979). Chn t al. contructd boundary lmnt formulation for pag problm by introducing a dual intgral quation with hypringular intgral (Chn t al., 1994). In addition to th FEM and BEM mntiond abov, mhl/mhfr mthod including th mthod of fundamntal olution (MFS) (Young t al., 006; Chaiyo t al., 011), th natural lmnt mthod (NEM) (Ji t al., 013), th hybrid boundary nod mthod (Tan t al., 010), th moothd fixd grid finit lmnt mthod (Kazmzadh-Pari and Danhmand, 01) and th cald boundary finit-lmnt mthod (Bazyar and Talbi, 014) wr rcntly dvlopd for th numrical dtrmination of th fr urfac in pag problm. Gnrally, th itration procdur i ncary for th fr urfac problm. Howvr, th mh rcontruction during itration incra xtrmly th difficulty of olution. To bypa thi difficulty, dvlopmnt of upr lmnt (or multi-nod larg lmnt) i dird. In thi tudy, anothr numrical mthod diffrnt to th convntional FEM and BEM (Bath, 006; Qin, 1994; 1995; 003; Qin and Mai, 00), calld th fundamntal olution-bad (or Grn function bad) hybrid FEM (HFS-FEM), i formulatd for olving uch problm in two-dimnional iotropic dam and a multi-nod lmnt i dvlopd to modl th rgion clo to th fr urfac for implifying th mh rdfinition. Th HFS-FEM wa firtly prntd by Wang and Qin for hat tranfr analyi (Wang and Qin, 009) and thn wa xtndd to analyz latic tr fild (Wang and Qin, 010b; Wang and Qin, 011a; Wang and Qin, 01b), thrmal proprti of advancd functional/compoit matrial (Wang and Qin, 011b; Wang t al., 01; Wang t al., 013) and biohat tranfr in biological tiu (Wang and Qin, 010a; Wang and Qin, 01a) with gnral/pcial lmnt to achiv th purpo of high accuracy and mh rduction. It hould b mntiond that convrgnc of th HFS-FEM wa fully dicud in th work. Diffrnt to th convntional FEM, th HFS-FEM involv lmnt boundary intgral only, and allow for contructing arbitrary n-idd polygonal lmnt and accuratly calculating fild vrywhr in th domain, du to th u of fundamntal olution. Bid, both th HFS-FEM and th BEM ar dpndnt of fundamntal olution, howvr, th BEM m mor complicatd for olving multi-matrial problm, bcau it rquir parat boundary intgral quation for ach matrial domain and additional connctiv quation for atifying th continuou condition btwn adjacnt matrial domain. In thi tudy, th HFS-FEM wa xtndd for olving two-dimnional nonlinar pag problm with fr urfac. In th prnt hybrid FE algorithm, th linar combination of Grn function of iotropic pag govrning quation i mployd to contruct th lmnt intrior watr had fild by placing ourc point outid th phyical boundary of th domain, a don in th claical MFS (Chn t al., 008), whil th convntional hap function ar ud for th lmnt fram watr had fild, which i indpndnt of th lmnt intrior fild. Thn th two fild ar connctd by a doubl-variabl hybrid functional. Du to phyical dfinition of th Grn function, th lmnt intrior fild can naturally atify th iotropic pag govrning quation and thu, all lmnt domain intgral in th functional ar convrtd

3 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 1993 into noningular lmnt boundary intgral, which hav lowr dimnion than domain intgral. Mot importantly, on can flxibly contruct larg lmnt with mor nod and dg than th convntional plan lmnt for practical u. In th papr, a larg multi-nod lmnt i dignd to modl th rgion narby th fr urfac to allviat difficulty of unavoidabl mh rcontruction during itration procdur. Th outlin of th prnt work i a follow: Dtail of th pag boundary valu problm ar providd in Sction, and th prnt Grn function bad FEM i formulatd in Sction 3. Subquntly, numrical xprimnt ar conductd by th prnt mthod in Sction 4 and om concluion ar finally prntd in Sction 5. MATHEMATICAL FORMULATIONS.1 Govrning quation and boundary condition Gnrally, th govrning partial diffrntial quation of two-dimnional incompribl tadytat flow through iotropic dam can b dcribd a: k H k x x H H , x1 x (1) whr i a boundd domain in two-dimnional pac rfring to th rctangular coordinat x1, x. H i th unknown watr had in th domain, and k 0 0 dnot iotropic hydraulic conductivity or prmability cofficint of matrial. According to th Darcy law, th pag vlocity componnt qi ( i 1, ) can b writtn in trm of th hydraulic had H, that i, H q k, q k H x1 x () Bid, gnralizd boundary condition ar of th typ H H on q qn 1 1 qn q on H q (3) whr H and q ar rpctivly th known boundary data on th boundary H and q. q i th vlocity in th dirction normal to th boundary, and ni ( i 1, ) ar th dirction coin of th normal with rpct to th global coordinat ytm. For th convninc of following drivation, th normal vlocity q i rwrittn in matrix form a q 1 q n1 n (4) q with

4 1994 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac H q k 0 x x q 0 k 0H (5) Figur 1 how a two-dimnional pag problm with fr urfac in a dam, in which th matrial i aumd to b iotropic. AE and DG ar th boundari with pcifid contant watr had H 1 and H, rpctivly. Th bottom urfac AD of th dam i aumd to b imprmabl, that i q 0. Th curv EF i th fr urfac and F i th paration point. On th fr urfac EF, two boundary condition hould b atifid imultanouly: H x and q 0, o EF i ovrpcifid. On th pag urfac FG, thr i a givn watr had in trm of th coordinat variabl x, i.. H x. Sinc th watr outflow th dam on thi boundary, it hould alo atify th condition of q 0 on FG. In ummary, th boundary ction AE, EF, FG, GD, and AD conit of th complt th boundary of th olution domain. For th ak of clarity, all boundary condition rlatd to th pag hown in Figur 1 ar givn a H H1 on AE H H on DG q 0 on AD q 0 on EF H x on EF H x on FG (6) Figur 1: Two-dimnional pag problm with fr urfac in an iotropic dam.. Itration of th fr urfac Sinc th fr urfac problm i a nonlinar problm, it i invitabl to prform itration in dtrmining th location of th fr urfac diplayd in Figur 1.

5 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 1995 (1) Firtly, an initial gu of th location of fr urfac i aumd to tart th computation. For xampl, a traight lin of EF can b initially givn by tting th paration point F a th midpoint of th lin CG to dtrmin th olution domain AEFDA ( Figur ). () During itration, only th condition q 0 i pcifid on th fr urfac EF and th linar watr condition H x i givn on th pag urfac FG. (3) Aftr ach tp of itration, th watr had at th nod on th fr urfac can b obtaind uing th propod numrical mthod. (4) Examining th rror norm dfind by x H (7) i i whr x i ar th nodal vrtical coordinat on th fr urfac and H i th nodal watr had on th fr urfac. (5) Updating th nodal vrtical coordinat x i on th fr urfac by tting i 1i i x H x, x (8) (6) Th polynomial intrpolation of th vrtical coordinat of nodal point on th fr urfac i ud to dtrmin th nw location of th paration point F. (7) Updating th nodal coordinat and linar watr had boundary condition on th pag urfac. ( n) th (8) Th itration proc can b trminatd if th rror εε or th diffrnc btwn at th n itration and ( n 1) at th ( n 1) th itration i qual to or mallr than a givn tolranc. Figur : Initial gu of th fr urfac. 3 HYBRID FINITE ELEMENT FORMULATION In th prnt Grn function bad FE formulation, two indpndnt lmnt fild ar paratly dfind and thn ar linkd by a hybrid functional, from which th tiffn quation in trm of unknown nodal potntial and th rlationhip of th two indpndnt lmnt fild can b obtaind.

6 1996 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 3.1 Hydraulic had within th lmnt In th intrior of an lmnt, ay lmnt, th hydraulic had fild can b approximatd by a linar combination of Grn function a H( P) N ch PQ, 1 * i Nc (9) whr N i th numbr of ourc point locating outid th lmnt domain. c ( 1 N ) prnt unknown intrpolation cofficint and H * ( PQ, ) tand for th iotropic Grn function rlatd to th fild point P and th ourc point Q, which locat outid th lmnt to avoid th ingularity of th Grn function, a don in th MFS (Chn t al. and Karagorghi t al., 008; Chaiyo t al. and Rattanadcho t al., 011). Compard to th MFS, which mploy th boundary dicrtization of th ntir domain, th prnt mthod divid th ntir domain into vral mall lmnt and in ach lmnt, th linar combination of Grn function i mployd to rprnt th intrior hydraulic had. Thu, th prnt mthod ha bttr intrpolation tability than th MFS and th location of th ourc point can b flxibly arrangd outid th lmnt domain (Wang and Qin, 009; Wang and Qin, 010a; Wang t al., 013). In Eq. (9), th vctor N and c ar rpctivly 1 H * PQ H * PQ H * N PQ (10),,, N and T 1 N c c c c (11) Diffrntiating Eq. (9) yild th following normal pag flow qp ( ) Qc (1) whr with Q n n N k 0 x 0 k N x 0 (13) * * * (, (, ) 1) (, ) H PQ N H PQ H PQ N ( i 1, ) (14) x i xi xi x i Th Grn function for a two-dimnional iotropic pag problm i dfind a k,, H PQ H PQ PQ, 0 * * 0 x1 x (15)

7 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 1997 who olution i whr * 1 H PQ, lnr (16) k r x x x x (17) i an Euclidian ditanc btwn th fild point ( 1, ) Q x1, x, and i th Dirac dlta function. Furthrmor, th drivativ of th Grn function in trm of coordinat variabl can b writtn a Px x and th ourc point * H 1 x x k r 1 0 * H 1 x x k r 0 1 x1 x (18) Obviouly, th aumd intrior watr had fild (9) analytically atifi th govrning quation (1), and thi fatur will b ud to implify th hybrid functional. 3. Hydraulic had along th lmnt boundary In ordr to nforc conformity on intr-lmnt boundary btwn two adjacnt lmnt, th fram watr fild H i indpndntly prcribd in trm of lmnt nodal hydraulic had d by HP ( ) Nd (19) whr N i an intrpolation hap function vctor, in which th intrpolation function ar chon to b th convntional quadratic hap function commonly ud in th ioparamtric quadratic lin lmnt in th FEM (Bath, 006) and th BEM (Brbbia, 198). 3.3 Doubl-variabl hybrid functional For th particular lmnt, which occupying a ub-domain with boundary, th hybrid functional in trm of H and H i dfind a (Qin, 000) 1 H H k 0 k 0 d qhd q H H d x q 1 x (0) Applying th Gau thorm, on ha 1 qhd qhd qh d (1) q

8 1998 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac in which only lmnt boundary intgral ar involvd. Th ubtitution of th intrior fild (9) and th fram fild (19) into Eq. (1) yild 1 T T T c Hc d gc Gd () Subquntly, th minimization of th functional with rpct to th unknown vctor c and d rpctivly giv Hc Gd 0 c T T G c g 0 d T (3) from which th optional rlationhip btwn c and d 1 c =H Gd (4) and th olving ytm of quation in trm of th nodal hydraulic had vctor d Kd =g (5) can b producd. In Eq. (5), th ymmtric and par cofficint matrix i writtn by T 1 K =G H G (6) 4 NUMERICAL EXPERIMENTS 4.1 Vrification of th prnt mthod To vrify th accuracy of th prnt approach, lt conidr a linar iotropic pag problm in a rctangular dam by fixing th fr urfac. Th rctangular dam i 4 m in hight and 16 m in width. Th uptram (lft) watr had i 4 m, and th downtram (right) watr had i 4 m, a hown in Figur 3. Th prmability cofficint i k 0 1 m/ d. Th fr urfac i aumd to b an inclind traight lin conncting th point (0, 4) and (16, 14), which i th initial gu of th fr urfac in th nxt xampl. In th computation, th FE mh hown in Figur 3 i ud to modl th rctangular dam. In th mhing chm, thr 8-nod gnral lmnt i ud to modl th bottom rgion of th dam and on -nod upr lmnt i mployd in th rgion clo to th fr boundary to provid mor fficint rmhing proc than th convntional FEM during itration by updating only th location of nod on th fr urfac and th pag urfac. Bid, th u of larg lmnt i alo bnficial to dcra th computation tim, in contrat to th convntional FEM.

9 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 1999 Figur 3: Rctangular dam with traight fr urfac and Mh tratgy in th HFS-FEM. To dmontrat modling fficincy and olution accuracy of th prnt mthod, numrical rult from th convntional FEM implmntd by ABAQUS ar alo providd. Figur 4 and 5 rpctivly how th variation of th watr had on th imprmabl bottom urfac and th traight-lin fr urfac. It i obrvd that thr i a good agrmnt btwn th prnt HFS- FEM and th FEM although much l lmnt ar ud in th propod lmnt modl. It can b alo n from Figur 5 that thr i an uddn jump of th watr had at tho point clo to th pag urfac, du to th rtriction of natural boundary condition at th paration point, o that curv fitting tchnology hould b ud to obtain a mooth fr urfac and to updat th location of th paration point. Finally, th contour plot from th convntional FEM and th prnt mthod for th watr had ar prntd in Figur 6, and imilar ditribution can b found for both mthod. Thrfor, th prnt mthod i fficint and accurat to imulat th ditribution of th watr had in th domain. Figur 4: Ditribution of watr had along th bottom urfac of th dam.

10 000 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac Figur 5: Ditribution of watr had along th fr urfac of th dam. Figur 6: Contour plot for th watr had in th dam. 4. Iotropic pag in a rctangular dam with iz 4 m x 4 m In thi xampl, th rctangular dam in th firt tt i conidrd again to dtrmin th final location of th fr urfac by man of itration approach. In ordr to tart th itration, th inclind traight lin givn in Figur 3 i lctd a an initial gu. Th rror tolranc in th itration proc i takn a Th itration rult ar hown in Figur 7. Aftr 1 itration, th itration convrg. Figur 8 how th convrgd configuration of th fr urfac from th prnt mthod. Alo, th rult from th BEM (Chn t al., 007) and th MFS (Chaiyo t al., 011) ar providd in Figur 8 for comparion. W can that thr i no ignificant diffrnc on th computation rult btwn th prnt mthod and othr mthod. Th mall dviation at th paration point i mainly caud by diffrnt curv fitting tchniqu. In th dvlopd algorithm, th cubic polynomial intrpolation i mployd. For th ak of clarity, th hight of th paration point rpctivly prdictd by

11 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 001 th prnt HFS-FEM, th BEM, th MFS, and th FDM (Aitchion, 197) ar litd in Tabl 1, and it found that thy agr vry wll. Finally, th convrgd hap of th dam and th corrponding contour plot for th watr had ar litd in Figur 9. Figur 7: Th chang of location of th fr urfac during itration. Figur 8: Th convrgnt location of th fr urfac. Rfrnc Hight of th paration point (m) Prnt HFS-FEM MFS 1.88 BEM 1.68 FDM 1.79 Tabl 1: Th hight of th paration point.

12 00 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac Figur 9: Mh diviion and contour plot for th watr had in th dam aftr itration convrgnc. 4.3 Iotropic pag in a rctangular dam with iz 6 m x 4 m A anothr xampl in th litratur (Ji t al., 013), a rctangular dam with 4 m in width and 6 m in hight i conidrd and it gomtrical iz i hown in Figur 10. Th watr lvl uptram (lft) and downtram (right) ar 6 m and 1 m, rpctivly. Th prmability cofficint i k m/ d. Th rror tolranc in th itration proc i takn a Th xampl i analyzd by uing th prnt HFS-FEM. Figur 11 diplay th initial computational mh, in which thr gnral 8-nod hybrid lmnt and on -nod hybrid lmnt ar mployd to modl th domain. Aftr 11 itration, th mh of th computing domain i updatd to a convrgnt configuration, which i diplayd in Figur 11 too. Figur 1 plot th final location of th fr urfac prdictd from th prnt mthod and th tt (Ji t al., 013), and it can b found that a good agrmnt i achivd btwn thm. Thi dam ha alo bn analyzd by Ji t.al. uing th natural lmnt mthod (NEM) (Ji t al., 013). For th purpo of comparion, th hight of th paration point from diffrnt mthod ar tabulatd in Tabl, from which it can b n that th prnt mthod ha mallr drivation to th tt valu. Figur 10: Rctangular dam with a hight of 6 m and a width of 4 m.

13 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 003 Figur 11: Initial (lft) and convrgnt (right) mh configuration in th HFS-FEM. Figur 1: Final location of th fr urfac. Rfrnc Hight of th paration point (m) Prnt HFS-FEM 3.58 NEM 3.64 Tt valu 3.50 Tabl : Th hight of th paration point.

14 004 H. Wang t al. / Grn function bad finit lmnt formulation for iotropic pag analyi with fr urfac 5 CONCLUSIONS A a kind of domain-typ numrical mthod, th prnt hybrid FEM with Grn function krnl nd only lmnt boundary intgration and vratil lmnt contruction can b fulfilld in th HFS-FEM for your practical nd. Thu, th prnt HFS-FEM i mor uitabl for daling with problm with moving boundari than th convntional FEM. In thi tudy, th HFS-FEM i formulatd for th dtrmination of th fr urfac and th hight of th paration point in iotropic dam, and a larg multi-nod lmnt i mployd to modl th rgion clo to th fr urfac for th purpo of implifying mh rcontruction during itration proc, o that only nod on th fr urfac and th pag urfac ar updatd during itration. Th numrical rult dmontrat that th dvlopd Grn function-bad hybrid FEM i capabl to calculat fr urfac with good accuracy and fficincy. Acknowldgmnt Th upport from th Natural Scinc Foundation of China (Grant no ; ) and th Crativ Talnt Program of Univriti in Hnan Provinc (Grant no. 13HASTIT019) ar fully acknowldgd. Rfrnc Aitchion, J., (197). Numrical tratmnt of a ingularity in a fr boundary problm. Procding of th Royal Socity of London. Sri A 330: Bath, K.J., (006). Finit lmnt procdur, Klau-Jurgn Bath. Bath, K.J., Khohgoftaar, M.R., (1979). Finit lmnt fr urfac pag analyi without mh itration. Intrnational Journal for Numrical and Analytical Mthod in Gomchanic 3: 13-. Bazyar, M.H., Talbi, A., (014). Tranint pag analyi in zond aniotropic oil bad on th cald boundary finit-lmnt mthod. Intrnational Journal for Numrical and Analytical Mthod in Gomchanic, in pr. DOI: /nag.91. Brbbia, C.A., (198). Boundary lmnt mthod in nginring, Springr, Nw York. Brbbia, C.A., Chang, O.V., (1979). Boundary lmnt applid to pag problm in zond aniotropic oil. Advanc in Enginring Softwar 1: Chaiyo, K., Rattanadcho, P., Chantairiwan, S., (011). Th mthod of fundamntal olution for olving fr boundary aturatd pag problm. Intrnational Communication in Hat and Ma Tranfr 38: Chn, C.S., Karagorghi, A., Smyrli, Y.S., (008). Th Mthod of Fundamntal Solution: A Mhl Mthod. Dynamic Publihr, Atlanta. Chn, J.T., Hong, H.K., Chyuan, S.W., (1994). Boundary lmnt analyi and dign in pag problm uing dual intgral formulation. Finit Elmnt in Analyi and Dign 17: 1-0. Chn, J.T., Hiao, C.C., Chiu, Y.P., L, Y.T., (007). Study of fr-urfac pag problm uing hypringular quation. Communication in numrical mthod in nginring 3: Ji, Y.X., Liu, L.Z., Xu, W.J., Li, G.X., (013). Application of NEM in pag analyi with a fr urfac. Mathmatic and Computr in Simulation 89: Kazmzadh Pari, M.J., Danhmand, F., (01). Unconfind pag analyi in arth dam uing moothd fixd grid finit lmnt mthod. Intrnational Journal for Numrical and Analytical Mthod in Gomchanic 36:

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