Reparameterization and Adaptive Quadrature for the Isogeometric Discontinuous Galerkin Method

Transcription

1 Rparamtrization and Adaptiv Quadratur for th Isogomtric Discontinuous Galrkin Mthod Agns Silr, Brt Jüttlr 2 Doctoral Program Computational Mathmatics 2 Institut of Applid Gomtry Johanns Kplr Univrsity Linz, Altnbrgr Str. 69, 4040 Linz, Austria agns.silr@dk-compmath.jku.at, brt.juttlr@jku.at Abstract. W us th Poisson problm with Dirichlt boundary conditions to illustrat th complications that aris from using non-matching intrfac paramtrizations within th framwork of Isogomtric Analysis on a multi-patch domain, using discontinuous Galrkin (dg) tchniqus to coupl trms across th intrfacs. Th dg-basd discrtization of a partial diffrntial quation is basd on a modifid variational form, whr on introducs additional trms that masur th discontinuity of th valus and normal drivativs across th intrfacs btwn patchs. Without matching intrfac paramtrizations, firstly, on nds to idntify pairs of associatd points on th common intrfac of th two patchs for corrctly valuating th additional trms. W will us rparamtrizations to prform this task. Scondly, suitabl tchniqus for numrical intgration ar ndd to valuat th quantitis that occur in th discrtization with th rquird lvl of accuracy. W xplor two possibl approachs, which ar basd on subdivision and adaptiv rfinmnt, rspctivly. Introduction Isogomtric Analysis (IgA) [5,6] uss th sam spacs of splin functions for rprsnting th gomtry of a physical domain and for prforming a discrtization in th contxt of a PDE-basd numrical simulation. This mthod is basd on a paramtrization of th physical domain, i.., on a gomtry map that rlats th physical domain and th paramtr domain. Many approachs rly on tnsor product paramtrizations, whr th domain is a unit squar or a unit cub. Consquntly, mor complx domains hav to b dividd into svral singl patchs, forming a multi-patch rprsntation. Thr xist svral mthods for coupling th discrt discontinuous Galrkin IgA patch wis solution across th intrfacs btwn singl patchs and nhancing global continuity of th solution. Ths includ Nitsch s [6] mthod and mortar tchniqus [2] as wll as th discontinuous Galrkin (dg) approach, which is th focus of th prsnt papr.

2 DG mthods discrtiz th variational form of a partial diffrntial quation taking into account th discontinuity of th finit dimnsional discrtization spacs. Th publications [4,7] provid a gnral dscription of dg tchniqus in th contxt of finit lmnts, which hav bn transfrrd to th isogomtric stting in [3,2,3,4]. So far, only matching intrfac paramtrizations hav bn studid in th contxt of dg-iga mthods. Mor prcisly, whnvr two patchs mt in an intrfac, thn th paramtrizations rstrictd to ths intrfacs ar assumd to b idntical (possibly aftr affin transformations of th paramtr domains), s [2,3,4,8]. On th on hand, this limitation provids th advantag that th lmnts of th patchs on both sids of th intrfac ar prfctly matching, which significantly simplifis th implmntation of such mthods. On th othr hand, it complicats substantially th cration of multi-patch paramtrizations. As notabl xcptions w mntion th rcnt publications [0,], whr th authors study gaps and ovrlaps at th intrfacs. Whil th thory prsntd in ths paprs dos not rquir any assumptions rgarding matching intrfacs, such conditions ar assumd to b satisfid in all th computational xampls. Mor prcisly, th mshs of th considrd domains fulfill rstrictiv corrspondnc conditions, which ar quit similar to th matching cas. This is du to th lack of an implmntation for th non-matching cas [9]. This rcnt work has motivatd us to invstigat th ffct of non-matching intrfac paramtrizations in th contxt of dg-iga in th prsnt papr. W aim to giv a complt dscription of th ncssary computational stps for applying th thortical rsults of [0,,2,3,4,8] to th cas of non-matching paramtrizations at th intrfacs. In ordr to kp th prsntation simpl, w rstrict ourslvs to planar two-patch domains and w assum that th intrfacs ar gomtrically matching, thus thy hav nithr ovrlaps nor gaps. Howvr, it is clar that th rsults from [0,] apply to th non-matching cas also, as th thory prsntd thr is sufficintly gnral. Mor prcisly, th assmbly of th local stiffnss matrics drivd from th dg bilinar form rquirs th computation of intgrals of th typ Db k i(x)d b l j(x)dx, () whr is an intrfac btwn Ω k and Ω l in th physical domain, x is a point on th intrfac, b k i, bl j ar isogomtric basis functions dfind on patchs Ω k, Ω l Ω of th multi-patch domain Ω R, and D,D ar diffrntial oprators. As w shall s, non-matching intrfac paramtrizations giv ris to two problms that nd to b tratd sparatly. Th first on concrns th valuation of b k i (x) and bl j (x) at th sam position x on th intrfac. Du to th us of non-matching paramtrizations, a point x will hav two possibly diffrnt primags in th paramtr domains of th two patchs joind by th intrfac rspctivly. To idntify pairs of corrsponding primags w us rparamtrizations of th primags of th intrfac. W also invstigat th influnc of th quality of th rparamtrization on th accuracy of th ovrall rsult. 2

3 Th scond problm is rlatd to th us of numrical intgration mthods. W nd to find a quadratur mthod whos xactnss dos not dpnd on th smoothnss of th intgrands. W prsnt diffrnt approachs, on rsulting from dividing th lmnt on which quadratur is prformd and anothr on making us of automatizd lmnt splitting. Th prformanc of both approachs is xplord in numrical xprimnts. Th rmaindr of this papr is structurd as follows: W stablish th notation and dscrib th xampl problm w will focus on in th nxt sction. W thn stat th two issus of valuation and numrical intgration, as dscribd abov. Sction 3 trats th first problm of finding suitabl rparamtrizations, whil Sction 4 is dvotd to th diffrnt quadratur tchniqus. Rsults of numrical xprimnts ar prsntd in Sction 5. Finally w conclud th papr. 2 Prliminaris W rcall th discontinuous Galrkin isogomtric (dg-iga) discrtization of a givn modl problm and discuss th computation of th stiffnss matrix lmnts in th cas of non-matching intrfac paramtrizations. Hrby, w rstrict ourslvs to th two-patch cas shown in Figur du to bttr radability. All obsrvations gnraliz dirctly to domains with mor than two patchs. ˆΩ ˆΩ2 0 0 G L R G 2 Ω Ω 2 Ω Fig.. Multi-patch domain with two patchs Ω,Ω 2, on intrfac and gomtry mappings G, G 2. Th mappings L and R will b introducd latr. 3

4 2. Th Modl Problm and th Multi-patch Discrtization Givn a domain Ω R 2, w considr th Poisson problm { (α u) = f on Ω Find u : u = 0 on Ω, (2) whr f is givn and α > 0 is th known diffusion cofficint. W allow α to b picwis constant, i.. α may tak diffrnt valus on vry singl patch (s blow). Mor prcisly, w considr a multi-patch domain Ω R 2 that consists of 2 non-ovrlapping singl patchs Ω, Ω 2 such that Ω Ω 2 = Ω. W us uppr indics to rfr to th numbr of th patch, and thus α k dnots th valu of th diffusion cofficint on th k-th patch, k =,2. An intrfac btwn th two patchs is th intrsction = Ω Ω 2. W considr intrfacs that ar curv sgmnts only and ignor th rmaining ons. Each physical patch Ω k is paramtrizd by an associatd gomtry mapping G k with paramtr domain ˆΩ k = [0,] 2, k =,2. Ths mappings ar tnsor product splin functions G k (ξ) = i R k P k i β k i (ξ), ξ ˆΩ k, (3) which ar dfind by control points Pi k R 2 and tnsor product B-splins βi k, whr R k is th indx st of th k th patch. Th lowr indx i idntifis th i-th dgr of frdom of th k-th patch. W do not assum that th knot vctors of th patchs ar idntical. Ths knot vctors split ach paramtr domain into lmnts. W will us opn knot vctors which implis that th boundaris of th patchs ar B-splin curvs. An isogomtric basis function b k i on th physical patch Ω k is th pushforward of a B-splin β k dfind on th paramtr domain ˆΩ k, {( b k βi k i(x) = ( G k) ) (x) if x Ω k (4) 0 othrwis. Ths functions span th spac which is usd to driv th dg-iga discrtization. For latr rfrnc w dfin th st of all dgs Γ = 2 {G k ([0,],0), G k ([0,],), G k (0,[0,]), G k (,[0,])} (5) k= of th multi-patch domain. It is th disjoint union of th st of th intrfac dgs Γ C = { Γ : Ω Ω 2 } (6) and th st of boundary dgs Γ D = { Γ : Ω k Ω, k =,2}. (7) 4

5 2.2 DG-IgA Discrtization Th discontinuous Galrkin isogomtric(dg-iga) discrtization spac considrs th subspac V h = span{b k i : i R k, k =,...,n} 2 H ( Ω k), (8) of th brokn Sobolv spac, s [7]. Functions in V h ar continuously diffrntiabl on th intrior of th singl patchs but not ncssarily smooth across intrfac dgs. This smoothnss of th solution is achivd approximatly by introducing a pnalty trm that considrs th jump of th solution across th intrfac. Bfor stating th final variational formulation w nd to dfin avrags and jumps, s [4,7]. For ach patch indx k, any function v 2 k= H( Ω k) has a wll-dfind trac along any dg Ω k. Thus, any such function v dfins two tracs on th intrfac Γ C, which w dnot as v Ω and v Ω 2, rspctivly. W us thm to dfin th avrag and th jump k= {v} = 2 (v Ω +v Ω2) (9) [v] = v Ω v Ω 2 (0) across th intrfac Γ C. Ths dfinitions ar furthr xtndd to boundary dgs Γ D, Th dg-iga discrtization {v} = v Ω k and [v] = v Ω k, k =,2. () Find u V h : a(u,v) = F(v) v V h (2) of th Poisson problm (2) uss th bilinar form a(u,v) = with 2 a k (u,v) 2 k= Γ C Γ D ( a 2, (u,v)+a 2,2(u,v) ) + Γ C Γ D a 3(u,v) (3) a k (u,v) = α k u vdω, (4) Ω k a 2,(u,v) = { u n} [v] d, a 2,2(u,v) = { v n} [u] d, (5) a δ 3(u,v) = h [u] [v] d (6) 5

6 and th linar form F(v) = Ω fvdω. (7) Th scond group of trms a 2, and a 2,2 considrs normal vctors n = n of th intrfac, which nd to comply with th chosn orintation of th dgs (dtrmind by th patch numbring). Th last trms a 3 in th bilinar form ar th pnalty trms mntiond bfor, which involv th sufficintly larg paramtr δ. Thy dpnd on th lmnt siz h, i.. on th lngth of th knot spans 3. A dtaild drivation of th dg discrtization is givn in [7]. Th adaptation to th isogomtric stting is discussd in th thsis [3], which also commnts on th choic of th δ, and in th rcnt articl [2]. Th discrtization (2) dfins th associatd dg norm u 2 dg = 2 a k (u,u)+ k= Γ C Γ D a 3(u,u), (8) whr in a (u,u) th gradint of u is rstrictd to Ω k, s again [2]. Th cofficints u k i of th approximat solution u h = 2 k= ar found by solving th linar systm Su = b with S = ( ) s (i,k),(j,l) (i,k),(j,l) b = ( ) b (j,l), (j,l) u = ( u k ) i whr (i,k), i R k u k ib k i (9) s (i,k),(j,l) = a ( b k i,b l j), i R k, j R l, k,l =,2, and b (j,l) = F ( b l j), j R l, l =,2., 2.3 Intgrals along Intrfacs Evaluating th forms in (3) involvs intgrals along intrfacs, which pos considrabl difficultis. W discuss th valuation of ths quantitis in mor dtail, considring again th domain shown in Figur. As a rprsntativ xampl w shall focus on a 2,. All obsrvations gnraliz dirctly to othr trms. 3 For simplicity w considr uniform knots only. If this is not th cas thn on may considr quasi-uniform knots instad, choosing a paramtr that controls th siz of all knot spans. 6

7 In this situation w obtain a 2,(u,v) = ( u Ω n)v Ω +( u Ω 2 n)v Ω ( u Ω n)v Ω 2 ( u Ω 2 n)v Ω 2d. Th stiffnss matrix is a combination of svral matrics, ach of which is contributd by on of th four forms in (3) dfining it. In particular w focus on th contribution of a 2,. Taking into account that w find that only th xprssions b 2 i Ω = 0, b 2 i Ω = 0 i R2, b i Ω 2 = 0, b i Ω 2 = 0 i R, a 2,( b k i,b l ) j = ( ) l+ ( b k i Ω k n ) b l j Ω ld (20) contribut to th lmnt s (i,k),(j,l) of th stiffnss matrix. In ordr to comput ths valus w us an appropriat numrical quadratur rul, which mans that w hav to valuat ths products on th intrfac. This is no major problm if k = l sinc th intgral involvs only on trac in this cas. Howvr, th situation is mor complicatd if k l sinc th (possibly diffrnt) paramtrizations of th intrfac nd to b takn into account. In th rmaindr of this sction w discuss th valuation of a 2,( b i,b 2 j) in mor dtail. Th intrfac = G ([0,] 2 ) G 2 ([0,] 2 ) = G (,[0,]) = G 2 (0,[0,]) (2) is paramtrizd by th rstrictions L = G (G ) () and R = G 2 (G2 ) (), (22) s Figur. Ths two diffrnt rprsntations of th sam intrfac ar rlatd by th rparamtrizations and via λ : [0,] {} [0,] (23) : [0,] {0} [0,] (24) L λ = R. (25) Th construction of suitabl rparamtrizations λ and is th first major problm rlatd to th valuation of this trm. W will discuss it in th nxt sction. 7

8 Ths paramtrizations will b usd to rprsnt th dg as = (L λ)([0,]) = (G λ)([0,]) = (G 2 )([0,]) = (R )([0,]). (26) Finally w dfin P = L λ = R and arriv at a 2,( b i,b 2 ) ( j = b i (x) Ω n(x) ) b 2 j(x) Ω 2dx [ ( G = (x) ) ( β i (G ) (x) ) Ω n(x)] βj 2 ( (G 2 ) (x) ) Ω 2dx = = 0 0 [ ( G (P(t)) ) ( ) n(p(t))] β i L (P(t)) βj 2 ( R (P(t)) ) P(t) dt [ ( G (P(t)) ) β i (λ(t)) n(p(t))] β 2 j ( (t)) P(t) dt. Th intgral in th last lin is valuatd by a quadratur rul. Howvr, th choic of th quadratur rul, which is th scond major problm rlatd to th valuation of this trm, is nontrivial and will b discussd furthr in Sction 4. In fact, th choic of th ruls nds to tak th diffrnt knots of th functions βi, β2 j, λ and into account. Whil on will gnrally choos th sam knots for λ and, th knots of βi and β2 j ar subjct to a non-linar transformation and cannot b assumd to b idntical. 3 Finding th Rparamtrizations It is quit common in th litratur to assum matching paramtrizations or almost matching ons, s [5, p. 448], [6, p. 87], [2,3,4,8]. In this situation, th choic of th rparamtrizations λ and is trivial, as thy ar simply linar paramtrizations (possibly rvrsing th orintation) of th primags of th intrfac in th paramtr domains. Howvr, th rstriction to matching paramtrizations poss constraints on th construction of multi-patch paramtrizations, making it ssntially impossibl to paramtriz th individual patchs sparatly. This fact motivats us to study th non-matching cas. Mor prcisly, w considr situations whr th condition (25) cannot b satisfid by considring linar rparamtrizations λ and. Clarly, th condition dos not dtrmin λ and uniquly. W fix on of th mappings, say λ, and comput th rmaining on,. Figur 2 visualizs th rlations btwn th mappings. Th unknown mapping satisfis = R L λ. W comput it by lastsquars approximation of point sampls, as follows:. For a givn numbr N of sampls, w valuat ( ) i i = R L λ N (27) 8

9 0 λ t ˆΩ ˆΩ2 0 0 G L R G 2 Ω Ω 2 Ω Fig.2. Multi-patch domain with gomtry maps G and G 2, thir rstrictions L and R to th primags of th intrfac and rparamtrizations λ and by prforming th closst point computations i = argmin ξ {0} [0,] ( ) i L λ R(ξ) N, i = 0,...,N, (28) whr is th Euclidan norm. This formulation also applis to th cas of gomtrically inxact intrfacs (cf. [0,]). 2. W choos a suitabl splin spac (.g. linar, quadratic or cubic splins with a fw uniformly distributd innr knots) and find th control points c j {0} [0,] of th associatd B-splins N j, j =,...,m, by solving th linar last-squars problm N m ( ) i c j N j i N i= j= 2 min, (29) cf. [7]. Th influnc of th choic of th splin spac for will b discussd latr in Sction 5. Th givn rparamtrization λ is chosn as a linar polynomial. W will rfr to th cas whr at last on of th mappings λ and is diffrnt from th idntity as non-matching paramtrizations at th intrfac. 9

10 4 Numrical Intgration Th valuation of a 2,( b i,b 2 ) j = 0 [ ( G (P(t)) ) β i (λ(t)) n(p(t))] β 2 j ( (t)) P(t) dt. (30) rquirs intgration with rspct to th paramtr t, which varis in th paramtr domain [0, ]. This is don by applying numrical quadratur and w prsnt svral stratgis for doing so. 4. Gauss Quadratur with Exact Splitting Gauss quadratur can b applid to sgmnts of analytic functions. Consquntly, w split th paramtr domain [0, ] into sgmnts (sparatd by junctions) whr th intgrand satisfis this rquirmnt. Thr typs of junctions aris: th invrs imags λ (κ i ) of th knots κ i that wr usd to dfin th B-splins β j, th invrs imags (κ 2 i ) that wr usd to dfin th B-splins β2 j, and th knots τ i that wr usd to dfin th B-splins N j in (29). Ths typs ar visualizd in Figur 3. Consquntly w prform Gauss quadratur with xact splitting by applying th following algorithm: Comput all junction points (all thr typs) in [0, ], sort th junction points, subdivid th domain into sgmnts accordingly, subdivid th rsulting sgmnts if thy ar too long, apply a Gauss quadratur rul to ach sgmnt and sum up th contributions. As a disadvantag, th invrsion of λ and is costly and has to b don with high accuracy, as th sorting dpnds on it. Furthrmor, th mthod may rsult in many sgmnts of varying lngths. W us Gauss quadratur with p+ nods pr lmnt (which xactly intgrats polynomials of dgr 2p+), whr p is th dgr usd for dfining th dg-iga discrtization, cf. [5]. 4.2 Gauss Quadratur with Uniform Splitting A computationally simplr approach is to us uniform subdivision, as follows: Split th domain [0,] uniformly into M sgmnts, whr M is a multipl of th numbr of knot spans usd to dfin th B-splins N j in (29), apply a Gauss quadratur rul to ach sgmnt and sum up th contributions. 0

11 0 0.5 λ t L R Ω Fig.3. Exact splitting of a knot span and application of a quadratur rul to ach subsgmnt As w shall s latr, it is mandatory to us larg valus of M in ordr to rach th dsird lvl of accuracy. This is du to th fact that th junctions of th first two typs listd in th prvious sction may still b locatd within th sgmnts obtaind by uniform splitting. On th othr hand, th us of uniform rfinmnt also crats many small sgmnts that could b mrgd into largr ons without compromising th accuracy. This can b xploitd by using adaptiv quadratur. 4.3 Adaptiv Gauss Quadratur W rcall th main ida of adaptiv quadratur, cf. [8]. In ordr to valuat th intgral I = b a f(x)dx (3) of an intgrabl function f ovr an intrval [a, b] adaptivly on computs two diffrnt stimats I and I 2 of I by using two diffrnt intgration mthods. On assums that on of ths stimats, say I, is mor accurat than th othr. Nxt, on computs th rlativ distanc btwn I and I 2 taking into account a givn (or chosn) tolranc tol,.g. machin prcision. If th diffrnc is small nough, on chooss I as th valu of th intgral b f(x)dx. If this is a not th cas on splits th intrval [a,b] into two subintrvals, [a,b] = [a,m] [m,b], whr m = a+b 2,

12 and valus I by summing up th two contributions. This mans that on applis th procdur of computing two diffrnt stimats and chcking thir rlativ diffrnc to both subintrvals. Adaptiv quadratur is thrfor a rcursiv procdur, which is summarizd in Algorithm. Algorithm Adaptiv Quadratur: Basic routin. adaptivquadratur(f, a, b, tol) : Input: f, a, b, tol whr f is an intgrabl function, a and b ar th intrval boundaris and tol is a givn tolranc 2: Choos knots u i and wights w i, i =,...,n. 3: Comput I = n i= wif(ui). 4: Choos knots ũ i and wights w i, i =,...,m. 5: Comput I 2 = m i= wif(ũi). 6: if I I 2 tol I thn 7: Rturn I 8: ls 9: Rturn adaptivquadratur 0: nd if ( f,a, a+b ) ( 2,tol + adaptivquadratur f, a+b ) 2,b,tol. Not that th stopping critrion has to b chosn with car and in fact lin 6 in th algorithm is a slight ovrsimplification of it. S[8] for furthr information. W apply this procdur to th knot spans that wr usd to dfin th B-splins N j in (29). Thrfor w choos I as a Gauss quadratur rul with p + quadratur knots, whr again p is th dgr of th basis functions in th dg-iga discrtization spac. For th computation of I 2 w split th intrval manually into two halvs, apply a a Gauss quadratur rul of th sam ordr on both halvs, and sum up. Th tolranc tol is st to machin prcision. As an advantag, adaptiv quadratur can b prformd without invrting th rparamtrizations. Morovr, it avoids th ovrsgmntation problm that was prsnt for th prvious approach. W obsrvd xprimntally that th adaptiv procdur accuratly dtcts th junction points and subdivids th domain accordingly. Clarly, th implmntation is mor costly and rquirs a rcursiv algorithm. 5 Numrical Rsults W xamin th prformanc of th quadratur mthods prsntd in Sction 4 as wll as th influnc of th accuracy of th rparamtrization. All xprimnts wrprformdusingg+smo 4,anobjct-orintdC++IgAlibrarynamd Gomtry + Simulation Moduls. 4 G+Smo: gs.jku.at 2

13 Fig. 4. Patch and its control nt. Lft: matching paramtrizations at th intrfac. Right: non-matching paramtrizations at th intrfac. 5. Rfrnc Rsults As a rfrnc w will first show th convrgnc plot of th solution of th Poisson quation in th cas of matching paramtrizations, i.. for λ = = id. In this cas w can rstrict ourslvs to a simpl quadratur rul. Thr is no nd for using mor laborat vrsions of numrical intgration. Furthrmor, sinc λ = = id, w do not nd to considr th influnc of th quality of th rparamtrization. Mor prcisly, w considr th two-patch domain with biquadratic matching intrfac paramtrizations shown in Figur 4, lft. Figur 5 dmonstrats th convrgnc bhaviour of th numrical solutions that wr obtaind for various valus of th lmnt siz h that was usd to dfin th dg-iga discrtization. W considr a problm with a known solution and masur th rror as th diffrnc to it. Th quadratur mthod w usd is Gauss quadratur with thr quadratur knots. A convrgnc rat of thr for th L2 rror and of two for th dg rror is clarly visibl. This is in accordanc with th thortical prdictions, s [,6]. 5.2 Influnc of th Quadratur Rul W now considr a diffrnt paramtrization of th sam computational domain, with non-matching paramtrizations of th intrfac, s Figur 4, right. Again w us biquadratic patchs. Now w nd to us a mor complicatd intgration tchniqu, and w considr th thr approachs that wr dscribd in Sction 4. Figur 6, lft and right, visualizs th convrgnc bhaviour masurd in th L2 and dg norms rspctivly. Each plot contains four curvs, corrsponding to four diffrnt numrical quadratur tchniqus. Mor prcisly, w considr Gauss quadratur with xact splitting (yllow), Gauss quadratur with uniform splitting into 0 (blu) and into 30 (rd) sgmnts, and adaptiv Gauss quadratur (purpl). W obsrv that th first and th last mthod prform bttr than th rsults basd on uniform splitting and thy achiv th optimal convrgnc rats (compar with Figur 5). In particular w not that using uniform quadratur lads to a rducd ordr of convrgnc for smallr msh sizs h. 3

14 Fig. 5. Matching paramtrizations at th intrfac, convrgnc bhaviour of rror in diffrnt norms: L2 norm (blu curv), dg norm (rd curv). Evn th us of a vry fin but uniform sgmntation (30 (rd) instad of 0 (blu) sgmnts) dos not improv this significantly. Basd on ths obsrvations w dcidd to us solly adaptiv and xact Gauss quadratur in th rmaining xampl. 5.3 Influnc of th Rparamtrization Nxt w analys th influnc of th quality of th rprsntation of th rparamtrization. Considr again th paramtrization of th domain in Figur 4, right, with non-matching paramtrizations of th intrfac. W compar thr diffrnt choics of th rparamtrizations λ and. For th first rparamtrization, which gnrats th rsults rprsntd by th blu curv in Figur 7, w choos polynomials λ and such that th quation L λ = R is xactly satisfid. In this cas it was possibl to find such polynomials, du to th particular construction of th xampl. Howvr, this would b impossibl in gnral and it is usd hr to gnrat a rfrnc rsult. Th scond and third rparamtrizations (rd and yllow curvs) wr obtaind using th Algorithm from Sction 3 to find, whil λ was chosn as a linar polynomial. Th scond rparamtrization uss a linar splin with 8 sgmnts and has an L2 rror of.3 0 2, and th third rparamtrization uss a cubic splin with 4 sgmnts and has an L2 rror of Figur 7 compars th rrors in th L2 (lft) and dg norms (right) for th thr rparamtrizations. W obsrv that using a high quality rparamtrization is ssntial for th convrgnc of th mthod. Dpnding on th accuracy of 4

15 Fig. 6. Influnc of th quadratur rul. Lft: Convrgnc bhavior of th rror in L2 norm. Right: Convrgnc bhavior of th rror in dg norm. Blu and rd curvs: 0 and 30 uniform sgmnts pr t-knot span. Yllow curvs: xact splitting of th knot spans. Purpl curvs: adaptiv quadratur. Not that th yllow curv coincids with th purpl on for smallr valus of h. Exact rprsntation of th rparamtrizations λ and. 5

16 th rparamtrization, h-rfinmnt only works until it rachs a critical msh siz, whr furthr rfinmnt dos not hav any ffct. Th plots show th rsults obtaind by using adaptiv Gauss quadratur. Th xact mthod givs virtually idntical rsults. Fig. 7. Influnc of th rparamtrization. Adaptiv quadratur on intrfac intgrals. Lft: Convrgnc bhaviour of th rror in L2 norm. Right: Convrgnc bhaviour of th rror in dg norm. Blu curvs: Exact rprsntation of λ and. Rd curvs: Approximation rror of Yllow curvs: Approximation rror of Comparison of Exact and Adaptiv Quadratur W prform an xprimntal comparison of th computational complxity of xact and adaptiv quadratur for th domain in Figur 4, right. First w dmonstrat th ffct of using adaptiv quadratur, by showing th automatically cratd splitting points in Figur 8. W usd an accuracy of 6

17 0 6 instad of machin prcision for this pictur to obtain a clarr imag. Both patchs wr uniformly rfind into 4 4 lmnts by knot insrtion. Th mappings λ and ar cubic splins on [0,] with four knot spans of qual lngth. Thir knots τ i coincid with th invrs imags λ (κ i ), as th first mapping is simply th idntity. Th adaptiv quadratur, which is applid to th knot spans [τ i,τ i+ ], thus crats additional splitting points around th invrs imags (κ 2 i ), as shown in th Figur. In this particular cas, only on splitting point (at ) is cratd nar (κ 2 2) = sinc this suffics to rach th dsird accuracy. Fig. 8. Splitting points cratd by adaptiv quadratur - s txt for dtails. Ths rsults indicat that, unlik uniform Gauss quadratur, adaptiv quadratur avoids ovr-sgmntation of th intgration domains. Still, it splits th knot spans mor oftn than xact Gauss quadratur, which also rsults in a highr numbr of quadratur knots and thus valuations. In ordr to analyz this ffct, Figur 9 compars th numbr of valuations (i.., quadratur knots) usd by xact and adaptiv Gauss quadratur for incrasing numbrs of lmnts. In addition, w also show th numbr of root finding oprations (which ar mor xpnsiv than valuations) ndd to comput th splitting points of xact Gauss quadratur. Clarly, adaptiv quadratur rquirs mor valuations than xact splitting. Howvr, for sufficintly fin discrtizations, th numbr of valuations in th intrior of th patchs dominats th total ffort. 6 Conclusion W usd a simpl modl problm to invstigat th complications that aris from using non-matching intrfac paramtrizations within th framwork of Isogomtric Analysis on a multi-patch domain, using discontinuous Galrkin tchniqus to coupl trms across th intrfacs. Mor prcisly, w studid two particular problms. Firstly, w xplord th us of rparamtrizations to idntify pairs of associatd points on th common intrfac. This was found to b usful for corrctly valuating crtain trms in th dg discrtization. Scondly, w addrssd th construction of a suitabl procdur for numrical intgration. As dmonstratd in our numrical xprimnts, both problms ar important for nsuring th optimal rat of convrgnc for th numrical simulation basd on th isogomtric dg discrtization. 7

18 Fig. 9. Numbr of quadratur knots and root finding oprations ndd by xact and adaptiv quadratur for incrasingly finr discrtizations. Futur work may b dvotd to th xtnsion of th adaptiv quadraturbasd approach to th thr-dimnsional cas. Morovr, w ar currntly studying dg-typ tchniqus for prforming multi-patch splin surfac fitting with approximat gomtric smoothnss across patch intrfacs. Rfrncs. Bazilvs, Y., d Viga, L.B., Cottrll, J.A., Hughs, T.J., Sangalli, G.: Isogomtric analysis: approximation, stability and rror stimats for h-rfind mshs. Mathmatical Modls and Mthods in Applid Scincs 6, (2006) 2. Brivadis, E., Buffa, A., Wohlmuth, B., Wundrlich, L.: Isogomtric mortar mthods. Computr Mthods in Applid Mchanics and Enginring 284, (205) 3. Brunro, F.: Discontinuous Galrkin Mthods for Isogomtric Analysis. Mastr s thsis, Univrsità dgli Studi di Milano (202) 4. Cockburn, B.: Discontinuous Galrkin mthods. Journal of Applid Mathmatics and Mchanics 83, (2003) 5. Cottrll, J.A., Hughs, T.J.R., Bazilvs, Y.: Isogomtric analysis: CAD, finit lmnts, NURBS, xact gomtry and msh rfinmnt. Computr Mthods in Applid Mchanics and Enginring 94, (2005) 6. Cottrll, J.A., Hughs, T.J.R., Bazilvs, Y.: Isogomtric Analysis. Toward Intgration of CAD and FEA. John Wily and Sons, Chichstr, England (2009) 7. Dirckx, P.: Curv and Surfac Fitting with Splins. Monographs on Numrical Analysis, Oxford Scinc Publications (995) 8. Gandr, W., Gautschi, W.: Adaptiv quadratur - rvisitd. BIT Numrical Mathmatics 40(), 84 0 (2000) 9. Hofr, C.: prsonal communication 8

19 0. Hofr, C., Langr, U., Toulopoulos, I.: Discontinuous Galrkin isogomtric analysis of lliptic diffusion problms on sgmntations with gaps. SIAM Journal on Scintific Computing (206), accptd manuscript, also availabl at arxiv: Hofr, C., Toulopoulos, I.: Discontinuous Galrkin isogomtric analysis of lliptic problms on sgmntations with non-matching intrfacs. Computrs and Mathmatics with Applications 72, (206) 2. Langr, U., Mantzaflaris, A., Moor, S.E., Toulopoulos, I.: Multipatch discontinuous Galrkin isogomtric analysis. In: Jüttlr, B., Simon, B. (ds.) Isogomtric Analysis and Applications. pp. 32. Springr, Hidlbrg (204), also availabl as NFN Tchnical Rport No. 8 at 3. Langr, U., Moor, S.E.: Discontinuous Galrkin isogomtric analysis of lliptic PDEs on surfacs (204) 4. Langr, U., Toulopoulos, I.: Analysis of multipatch discontinuous Galrkin IgA approximations to lliptic boundary valu problms. Computing and Visualization in Scinc 7(5), (206) 5. Mantzaflaris, A., Jüttlr, B.: Intgration by intrpolation and look-up for Galrkinbasd isogomtric analysis. Computr Mthods in Applid Mchanics and Enginring 284, (205) 6. Nguyn, V.P., Krfridn, P., Brino, M., Bordas, S.P., Bonisoli, E.: Nitsch s mthod for two and thr dimnsional NURBS patch coupling. Compuational Mchanics 53(6), (204) 7. Rivièr, B.: Discontinuous Galrkin Mthods for Solving Elliptic and Parabolic Equations: Thory and Implmntation. SIAM (2008) 8. Zhang, F., Xu, Y., Chn, F.: Discontinuous Galrkin mthods for isogomtric analysis for lliptic quations on surfacs. Communications in Mathmatics and Statistics 2(3), (204) 9