The numerical tool for a posteriori detection of boundary layers in hp-adaptive analysis

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1 The umercal tool for a posteror detecto of boudary layers hp-adaptve aalyss ukasz Mazo ad Grzegorz Zbosk, Faculty of Techcal Sceces, Uversty of Warma ad Mazury ul.oczapowskego ; -79 Olszty, Polad e-mal: lukasz.mazo@uwm.edu.pl Isttute of Flud Flow Machery, Polsh Academy of Sceces ul. Fszera 4, 8-95 Gdask, Polad e-mal: zbo@mp.gda.pl Abstract The preseted paper cocers a posteror detecto of the boudary layers th or thck elastc structures aalyzed by meas of the hp-adaptve fte elemet methods (FEM). I the paper we recall some aalytcal cosderatos cocerg boudary layers theory. The we preset the dea of our detecto tool based o the elemet resdual method (ERM) of error estmato. The method of detecto s based o soluto of three local (elemet) problems defed aalogously to ERM local problems. The frst problem s solved for a boudary couple of two udvded tragular-prsm elemets. I the secod ad thrd problems the two-elemet mesh s dvded by two uformly ad expoetally towards the boudary. Ths way two four-elemet local problems are formed. The correspodg solutos are compared wth the soluto of the frst local problem. Fally, we check umercally effectveess of our detecto method for the meshes of varyg desty appled to the model plate problem. Keywords: plates, fte elemet methods, adaptvty, error estmato, umercal aalyss. Itroducto.. Geeral remarks The boudary layer effect appears the case of the th or thck-walled structures. The aalytcal explaato of ths effect s preseted the umerous papers ad text books (see [, ], for example). I most cases of such structures, the soluto s composed of the smooth part the teral zoe of the structure ad the part of hgh soluto gradets the boudary zoes. I the umercal mplemetatos, the problem soluto eeds applcato of the soluto spaces rch eough to reflect hgh soluto gradets the vcty of the boudares. The most commo method of space erchmet les applcato of the mesh refed expoetally the drecto ormal to the boudary. Numerous papers address the ssue (see [3, 4], for example). Note that such space erchmet s both umercally complex ad costly due to applcato of the o-uform mesh subdvsos ad dfferet treatmet of the teral ad boudary zoes of the structure. However, wthout the space erchmet the soluto covergece ca be affected serously... Objectve of our research The preseted paper s a part of the geeral research o hpadaptve statc ad dyamc aalyss of complex structures composed of th-walled, thck-walled ad sold parts. We search for a qute geeral umercal tool for adaptve modelg ad aalyss of wde varety of problems wth sold mechacs. I ths approach we apply: herarchcal modelg, herarchcal hp-approxmatos, elemet resdual equlbrato method of error estmato, ad error-cotrolled hp-adaptvty. I ths paper we address the problem of automatc umercal resoluto of the boudary layers, sutable for the stadard hpadaptve approach. I partcular, we search for a umercal tool for detecto, ad possbly rage ad testy assessmet, of the boudary layers. Wth ths tool at had, we do hope to be able to h-refe our tal mesh towards the boudares expoetally so as to obta the modfed mesh capable of resoluto of the boudary layers va stadard two-step hpapproach. The startg pot for the research are the results of our prevous efforts descrbed [6, 8]. The results of these efforts have bee qute promsg, yet ot totally satsfactory, especally terms of geeral applcablty of the prevously proposed approach. The ma objectve of ths paper s to preset the ehaced geeral umercal tool for detecto of the boudary layers a posteror ad to assess ts effectveess. Note that the authors of ths papers are ot aware of ay alteratve geeral tools for such detecto.. A posteror boudary layer detecto.. The boudary layer pheomeo It should be stated frst that I the case of Krchhoff plates the boudary layer pheomea do ot appear [6]. It results from the fact that the problem s descrbed wth a bharmoc equlbrum equato. The soluto to such a equato s always smooth, regardless of the appled boudary codtos. I our further cosderatos, however, we wll address the case of Reser-Mdl plates, the equlbrum of whch s descrbed wth the relatos derved work [5] D, u3,, u3 u3 q x x () ()

2 where:, =, rotato agles of the shell md-surface, u 3 plate deflecto,,, 3 the dces deotg two logtudal ad thrd trasverse drectos of the global Cartesa system of coordates, D a auxlary varable, defed as the case of the frst order plate theory, 3.e. D, q = q(x,x ) stads for the trasverse (ormal) tracto of the plate. It s worth otcg that dferetato of two relatos () wth respect to two drectos ( =, ) ad ther substtuto to the left-had sde sum of (), leads to the bharmoc equatos of the form (compare [6]): u u3 u3 3 q D q q. (3) It ca be cocluded the, that the soluto defg the deflecto of the Resser-Mdl plate s smooth, smlarly to the Krchhoff plate soluto [6]. The smooth character of the fucto u 3 seems to be clear. O the cotrary, the solutos for rotatos, =, do ot ecessarly possess ths feature. I vew of ths, we wll assume u G x, x x x 3 u3,, (4) x x x, x x, x,,, G, (5) where: G oto of the smooth part of the soluto, oto of the boudary part of the soluto. The smooth soluto for the trasverse dsplacemet u G 3 x, x ca ow be obtaed drectly, substtutg (4) to (3). Ths soluto ca ow be assocated wth the equato (), G whch we assume x, x x, x,,. Ths allows to determe the smooth part of the rotatos. Determato of the boudary part of the trasverse dsplacemets s ot ecessary, as u 3 x, x. Takg ths to accout, oe may wrte equatos () ad () the smplfed form D,,, (6), (7) where (7) has bee substtuted to (6). The above relatos, after assumg that x, x x x, allow for, determato of the boudary part of the rotatos, x, x. asg o the above cosderatos oe ca fd the soluto for plates, subject to boudary layer pheomeo, of a fte legth the drecto of x ad the wdth a the drecto x. Ths s doe the works [4, 8, 4]. The approach preseted these works s based o the perturbato method ad the assumpto of a soluto the form of asymptotc seres u 3 3,, G x, x t u x x, (8),, G x x t x, x x, x,,,, (9) where the cosecutve terms represetg the boudary part of the soluto ca be wrtte as x,, a t x, x C x e,, () ad where: t the plate thckess. I the above relato, the terms C, x x a t ad e descrbe a smooth fucto the drecto tagetal to the plate boudary ad a expoetally decreasg fucto the drecto ormal to ths boudary, wth stadg for a costat depedat o the order of regularty of the soluto. The boudary solutos for rotatos clude a dfferet umber of terms () the seres (8) ad (9). The lower the order of the frst term of the seres, the greater the effect o the boudary soluto [7]. It s the expoetal fucto of () that makes t ecessary to use fe meshes the case of boudary layer pheomea... The dea of the detecto tool The proposed method of detecto of a boudary layer s related the method of our work [6]. The method takes advatage of the local (elemet) solutos, cotrbutg to the upper boud of the exact soluto, ad obtaed wth the equlbrated resdual method. I order to detect the pheomeo we solve the local problem thrce wth the doma V c of a chose couple of prsmatc elemets adjacet to the boudary S of a structure. The stresses r o the teral boudary S c of the chose doma are kow from the global soluto obtaed from the tal mesh. They equlbrate the exteral loadg due to body ad surface tractos, actg V c ad o the supported Q ad loaded P parts of the structure boudary. Frstly, we obta the referece local soluto o the two-elemet mesh based o the dvso patter from the global problem. We choose two elemets adjacet to the boudary ad call them paret oes. The correspodg dvso umbers the drectos ormal ad taget to the boudary are M = ad M s =, respectvely. Next we solve two four-elemet problems, applyg algebrac ad expoetal dvsos the drecto ormal to the boudary (M = ad M s = ). The stuato s llustrated Fg.. The correspodg dvso ratos are deoted as. Ths way we form two sub-domas f =, wth two prsmatc elemets each sub-doma ( =,). All the problems to be solved ca be defed as follows M HPQ HPQ HPQ f u, f u f L f u f f HPQ HPQ u T f r, f u f ds f ScS PQ f () where: stads for the blear form represetg vrtual stra eergy, L s the vrtual work of exteral forces, whle u represets dsplacemet vector. I the above equato H, P ad Q represet elemet sze, ad ts horzotal ad vertcal orders of approxmato, all appled the local problem. Whle solvg the secod local problem (), oe has to determe local dscretzato parameters H, P ad Q. Ther values take advatage of the correspodg values of h, p, ad q from the global problem. The local values of the parameters

3 have to correspod to boudary layer appearace, wthout umercal lockg preset. Takg ths to accout, we assume H H, H H s, H h s, H H, P P 8 art Pmax, Q Q q. [N/m ] ad the Posso s rato equals.3. The plate s loaded wth uform ormal tracto p = -4, 6 [N/m ] ad clamped aroud ts lateral boudares. We apply two 3D-based mechacal models of the plate: the herarchcal shell model of the trasverse order q = ad the Resser-Mdl plate model (q = ). I our umercal tests we utlze regular 3x3, 4x4 ad 8x8 meshes dsplayed Fgs, 3 ad 4, respectvely. The respectve umberg of elemets s also preseted these fgures. Fgure : A boudary par of paret elemets (a) ad four elemets obtaed by subdvso (b) I the thrd local problem the mesh dmesos are set so as the boudary layer effect does ot appear. Ths meas that the H H H, H H h followg relatos hold: exp, s s s, P Pmax Q Q Q. H H, P P 8, q Above, the szes H ad exp art H correspod to expoetal ad arthmetc subdvso by two of a paret par of elemets the drecto ormal to the boudary,.e. M M M exp. I ths stuato we have: h/ art, H art exp H 9h / wth f correspodg to the boudaryadjacet par of elemets. Let us troduce ow a codto based o estmates U 4 of the stra eergy correspodg to two above four-elemet local problems [6] M f M H PQ HPQ HPQ HPQ u f, uf f u f, uf f. () f Fulfllmet or ot of the above codto leads to the cocluso that the boudary layer pheomeo occurs or ot, respectvely. Our elemet method of detecto of the edge effects ca be appled to each fte elemet par, adjacet to the cocered boudary segmet. I practce, t appears eough, however, to check the codto () for the selected pars of fte elemets alog the boudary segmet..3. The detecto crtero The proposed crtero takes advatage of the codto (). The crtero s based o comparso of the relatve values of the dffereces betwee the stra eerges, U ad U 4, obtaed from the two- ad four-elemet local problems, respectvely. We calculate two such dffereces. Oe s based o the four-elemet mesh refed arthmetcally ad the other oe o the expoetally dvded mesh. If the soluto from the expoetal mesh s better,.e. the value of the correspodg stra eergy s hgher, the the boudary effect may play a sgfcat role. I the opposte case the boudary layer effect s eglgble. 3. Numercal expermets Our calculatos are performed for a model plate problem. Due to the symmetry we aalyze a quarter of the square plate. The plate legth s l =, [m], whle ts thckess s equal to t =,3 -. The Youg modulus s E=, Fgure : The global umberg of elemets (3x3 regular Fgure 3: The global umberg of elemets (4x4 regular Fgure 4: The global umberg of elemets of the 8x8 mesh I order to detect the boudary layer ad to assess ts fluece o the soluto we perform a kd of sestvty aalyss. We chage the dvso rato the four-elemet local problem ad refer the correspodg value of the stra eergy to ts value obtaed from the two-elemet local problem,.e. U 4 U % %. (3) U The results of our aalyss are show Fgs. 5 to, where the values of % are dsplayed as a fucto of the dvso rato. The value of the rato equal to correspods to the uform (algebrac) dvso, whle the value of says that two elemets adjacet to the boudary are te tmes arrower tha the other par of elemets resultg from the dvso. Each of the preseted curves s marked wth the umbers of two correspodg paret elemets chose out of the elemets dsplayed Fgs, 3 ad 4.

4 % Fgure 5: Local problem results (Resser-Mdl plate model, 3x3 % Fgure 6: Local problem results (herarchcal plate model, 3x3 % /... Fgure 7: Local problem results (Resser-Mdl plate model, 4x4 % /... Fgure 8: Local problem results (herarchcal plate model, 4x4 % /5 8/8... Fgure 9: Local problem results (Resser-Mdl plate model, 8x8 % /5 8/8... Fgure : Local problem results (herarchcal plate model, 8x8 Aalyzg the obtaed results oe ca observe the qualtatve dfferece betwee the course of the curves obtaed for the Resser-Mdl (Fgs 5, 7 ad 9) ad herarchcal shell models (Fgs 6, 8 ad ). Comparg the values of % correspodg to values equal to (arthmetc subdvso by ) ad (expoetal subdvso), oe ca observe the decrease for the former model ad the crease for the latter oe. Ths suggests the cosderable fluece of the boudary layer pheomeo o the soluto results the case of the herarchcal shell model ad lack of such a fluece the case of the Resser-Mdl oe. Moreover, comparg the curves for three dfferet meshes t ca be otced that for each of them, the results are good qualtatve agreemet. Ths s true for both mechacal models of the plate. I spte of ths, some local effects (the crease or decrease % depeds o the elemet par locato alog the boudary) appear the case of 8x8 mesh ad the Resser- Mdl model. I order to assess effectveess of our tool we compare the above results wth the cotrol results obtaed from the global problems whch the global mesh s dvded as the local four-elemet problems. The results are show Fgs to 6. They correspod to the local results preseted Fgs 5-, respectvely. Comparg the results of the global ad local aalyses oe ca see the qualtatve agreemet for both models ad each of three regular meshes. The relato betwee values of % ad, correspodg to global results ca be characterzed exactly the same way as for the local results. I most cases the qualtatve agreemet of the local ad global results s qute astoshg.

5 % Fgure : Global problem results (Resser-Mdl plate, 3x3 % Fgure : Global problem results (herarchcal plate model, 3x3 mesh ) % /... Fgure 3: Global problem results (Resser-Mdl model of the plate, 4x4 % /... Fgure 4: Global problem results 4x4 (herarchcal model of the plate, 4x4 % /5 8/8... Fgure 5: Global problem results (Resser-Mdl plate model, 8x8 % /5 8/8... Fgure 6: Global problem results (herarchcal plate model, 8x8 Edg our dscusso o the obtaed results, we would lke to address codtog of the four-elemet problems for the extreme values of. We have ot observed ay loss of soluto stablty for such values. Ths s cofrmed by the values of % for =. ad =. Ths ca be terpreted as detty of the stra eerges obtaed from two- ad fourelemet problems for whch the codtog problem does ot exst ad mght have appeared, respectvely. Ths observato was also cofrmed wth the equvalece of the umber of commo decmal dgts the values of the mmzed stra eergy ad potetal eergy four-elemet problems. For stll lower or hgher values of oe may checked the codtog of the local problems wth the stadard method, appled by us prevously [9] to global problems. 4. Coclusos I the case of the Resser-Mdl plate the results from the expoetally dvded mesh are worse tha those from the uform mesh (compare Fg. 5, 7 ad 9). I ths case the boudary layer does ot fluece the results very much. Ths result s cosstet wth the theory. I the case of the herarchcal shell model of the plate the stuato s opposte (compare Fg. 6, 8 ad ). The fluece of the boudary layer pheomea s vsble. The global results cofrm the effectveess of the proposed detecto tool (compare Fgs. 5 ad, 7 ad 3, 9 ad 5, as well as 6 ad, 8 ad 4, ad 6). It ca be otced that the detecto of the boudary layer ca be based o the results for oly oe chose par of elemets from the boudary, provded that our choce s reasoable.

6 Refereces [] Arold D. N., R.S. Falk., Asymptotc aalyss of the boudary layer for the Resser-Mdl plate model, SIAM J. Math. Aal., 7, pp , 996. [] Dauge M., Gruas I., Rossle A., The fluece of lateral boudary codtos o the asymptotcs the th elastc plates, SIAM J. Math. Aal., 3, pp ,. [3] Cho J. R., Ode J. T., Lockg ad boudary layer herarchcal models for th elastc structures, Comput. Methods Appl. Mech. Egrg, 49, pp , 997. [4] Schwab C., Sur M., Lockg ad boudary layer effects the fte elemet approxmato of Resser-Mdl plate model, Proc. Symp. Appl. Math., 48, pp , 994. [5] Wemper G. Mechacs of solds, wth applcatos to th bodes. McGraw-Hll, New York, 973. [6] Zbosk, G., Herarchcal Modellg ad Fte Elemet Method for Adaptve Aalyss of Complex Structures ( Polsh), Sc. Istal. of IFFM, No. 5/479, IFFM, Polsh Academy of Sceces, Gdask,. [7] Zbosk G., Ostachowcz W., Krawczuk M. Modfcatosof the adaptve procedur es for complex structures the case of the mproper soluto lmt, lockg ad boudary layers (I Polsh), Report o. 4/, IFFM, Polsh Academy of Sceces, Gdask,. [8] Mazo,., The umercal tools for a posteror detecto of boudary layers ad ther effectveess ( Polsh), Master s Thess (supervsor G. Zbosk), Faculty of Techcal Sceces, Uversty of Warma ad Mazury, Olszty, 8. [9] Zbosk, G., 3D-ased hp-adaptve frst order shell fte elemet for modellg ad aalyss of Complex Structures. Part. Applcato to structural aalyss. Iteratoal Joural for Numercal Methods Egeerg, 7, pp , 7.