Excerpt from the Proceedings of the COMSOL Conference 2010 Boston

Transcription

1 Ecerp rom he Proceeding o he COMSOL Conerence Boon Compuaional Mehod or Muli-phyic Applicaion wih Fluid-rucure Ineracion Kumni Nong, Eugenio Aulia, Sonia Garcia 3, Edward Swim 4, and Padmanabhan Sehaiyer *,5,5 George Maon Univeriy, ea ech Univeriy, 3 US Naval Academy, 4 Sam Houon Sae Univeriy *Correponding auhor: 44 Univeriy Drive, MS: 3F, Science and ech I, Deparmen o Mahemaical Science, George Maon Univeriy, Faira, VA 3, pehaiy@gmu.edu Abrac: Eicien modeling and compuaion o he nonlinear ineracion o luid wih a olid undergoing nonlinear deormaion ha remained a challenging problem in compuaional cience and engineering. Direc numerical imulaion o he non-linear equaion, governing even he mo impliied luid-rucure ineracion model depend on he convergence o ieraive olver which in urn relie heavily on he properie o he coupled yem. he purpoe o hi work i o model and imulae muli-phyic applicaion ha involve luid-rucure ineracion uing a diribued mulilevel algorihm wih inie elemen. he propoed algorihm i eed uing COMSOL which oer he leibiliy and eiciency o udy coupled problem involving luid-rucure ineracion. Numerical reul or ome benchmark luid-rucure ineracion are preened ha validae he propoed compuaional mehodology or olving coupled problem involving luid-rucure ineracion i reliable and robu. Keyword: Fluid-Srucure Ineracion; Muliphyic; Micro-Air Vehicle; Coupled Syem.. Inroducion he eicien oluion mehodology o comple muli-phyic problem involving luidrucure ineracion (FSI i a challenging problem in compuaional cience. Such problem require udying comple nonlinear ineracion beween he rucural deormaion and he low-ield ha oen arie in applicaion uch a blood-low ineracion wih an arerial wall or compuaional aero-elaiciy o leible micro-air vehicle. In he la wo decade, domain decompoiion echnique have become increaingly popular or obaining a and accurae oluion o problem involving coupled procee [,, 3]. hee viable domain decompoiion echnique have been hown o be able mahemaically and have been ucceully applied o a variey o engineering applicaion [4, 5, 6]. he baic idea i o replace he rong coninuiy condiion a he inerace beween he dieren ub-domain by a weaker one o olve he problem in a coupled ahion. he purpoe o hi paper i o develop a coupled FSI algorihm and implemen he algorihm uing COMSOL o ome benchmark FSI problem. In ecion, we preen he ormulaion o a one-dimenional FSI problem uing an arbirary Lagrangian Eulerian ormulaion and employ he inie elemen mehod via COMSOL or olving he coupled problem. In ecion 3, we preen an opimal conrol ormulaion o he FSI problem or he -D problem preened in ecion. In ecion 4, we conider a hree dimenional FSI problem wih applicaion o micro-air vehicle.. A Coupled One-Dimenional FSI Model Problem For impliciy o preenaion, we ir develop he model or a one-dimenional FSI problem ha involve a rucural domain ineracing wih a luid medium. he model i e up o ha iniially he luid domain occupie he inerval (, and he elaic rucure occupie he inerval (,. A he luid low deorm he adjacen olid, we allow he movemen o he inerace o depend on he velociy o he luid. hi i illuraed in igure below. Figure : Undeormed and deormed compuaional domain We denoe by γ ( he poiion o hi inerace a any poiive ime. For all value o in he inerval (, γ (, we model he luid velociy v and he preure p uing a generalizaion o model employed by [7] uing: v α v ( β v v ε p =

2 In hi model, α = α( ν > i a parameer depending on he kinemaic vicoiy ν o he luid. he conan β [,.5] and ε vary depending on he maerial properie o he luid. Here i he eernal orce on he luid. In hi work we conider he peciic cae o he Burger equaion uing he parameer choice, α = ν, β =.5, ε =. I i well known ha hi choice o value yield abiliy o he numerical mehod employed or he problem whenever no preure erm i included. he luid model i coupled wih an elaic model ha repreen he rucure. In paricular, we conider he wave equaion ha model he olid diplacemen u o any poin in he adjacen rucure rom i iniial poiion given by: v α v (.5v w v = For he boundary condiion, we le he luid velociy v(, =in(.π and he rucural diplacemen u(, = or all ime. For iniial condiion, boh he luid and he rucural domain are a re iniially. he FSI coupled yem decribed wa implemened in COMSOL and he reul o he inie elemen implemenaion o he model i ummarized in igure ha diplay he plo o he luid velociy v and he rucural velociy u over he ime period rom o. u u = Here µ > and i he eernal orce on he rucure. Alo, he poiion o he inerace beween he wo ub-domain mu aiy he movemen a he inerace or all ime. A he inerace beween he luid and rucure, we enorce coninuiy o he luid velociie and he acion-reacion principle: v( γ (, = u (, α v ( γ (, = µ u (, In order o accoun or he changing naure o he luid, we conider he arbirary Lagrangian- Eulerian (ALE ormulaion [8]. hi will allow or a dynamic compuaional gird ha avoid ereme meh diorion near he inerace. o do hi one can move he numerical grid independenly o he luid velociy on he luid domain. Deining he grid velociy a: w= & γ ( γ ( Noe ha w( γ (, = & γ (, w(, =. hu he grid velociy i conien wih he velociy o he luid a he endpoin o he luid domain. Addiionally, we aume γ ( [,] or all ime. he ALE orm o he luid equaion ha we hen olve i: Figure : FSI imulaion uing COMSOL or ime = o. 3. Coupled FSI problem wih Conrol A relaed apec ha we conider in hi work i inveigaing diribued conrol or FSI problem. In paricular, one can udy he coupled FSI problem uing an opimal conrol ormulaion o predic he diribued conrol ha correpond o a precribed velociy and diplacemen daa û ha aiie he boundary condiion. Speciically, we wan develop a model ha predic he orce ha reul in he minimizaion o he error in he luid velociy v and he olid diplacemen u. For impliciy, le u conider he onedimenional model problem preened in ecion and eend i o include conrol apec. oward hi end, le u conider he relaed co uncional or he aociaed non-linear FSI problem (noe ha one can imilarly conider he linear FSI problem by dropping he correponding linear erm which i no decribed here given by:

3 M = ( v ( u ( l( ρ v v.5vv ( g( ρ u µ u where l and g are he Lagrange muliplier correponding o he luid velociy and he olid diplacemen repecively. Moreover, we alo impoe he coninuiy o he velociie over he coupled domain. Proceeding uing he andard opimal conrol approach o minimizing he co uncion by aking he variaion yield he ollowing auiliary yem o governing equaion: α α Figure 3: Diplacemen proile comparing he precribed olid diplacemen wih linear and non-linear conrol model. In he luid domain : ρ v v ρ v l.5vv.5vl l α In he olid domain : ρ g ρ v u g u = = v = g = α For numerical eperimen, we conider a imple olid diplacemen proile and he luid velociy proile given by: =.5( = ( he problem wa implemened in COMSOL o yield he ollowing velociy and diplacemen proile in he repecive domain. hi i illuraed in Figure 3 and 4 where he precribed oluion i ploed again boh he oluion o he boh he linear and non-linear conrol problem. Figure 4: Fluid-velociy proile comparing he precribed velociy wih linear and non-linear conrol model 4. A Coupled muli-dimenional FSI problem wih applicaion o MAV A Micro Air Vehicle (MAV i a ype o radioconrolled miniaure aircra ha can ly a very low peed. Due o he compleiie o he wing rucure o a MAV, a compuaional model o he aircra wing require a combinaion o many rucural elemen ineracing wih eernal luid. he wing ypically coni o a leible membrane maerial braced wih a leading edge par and chordwie baen (ee Figure 5. he rucural model mu combine he model o he membrane maerial ogeher wih he model o he rigid baen. Mo curren model rea he baen a large-deniy membrane elemen. Modeling hi coupled wo-dimenional rucural model ineracing wih a hreedimenional luid make he problem very challenging.

4 φ= in Ω3 r φ n= on Γ N Figure 5: A imple model o a leible MAV wing In hi work, we aemp o come up wih a mahemaical model ha can provide inigh ino he dynamic o MAV. he model preened herein i impliic; however, one may eend hi o accommodae oher eaure. In order o ge an inigh ino he modeling and dynamic o a MAV, le u conider a cylindrical compuaional domain. In hi domain we will aume ha he luid aiie a poenial equaion and ha a wo-dimenional rucural model (ha will repreen he MAV i a par o one o he circular urace. hi laer urace will repreen he oulow boundary o he compuaional domain which i illuraed below: Inlow Figure 6: Compuaional domain or MAV Oulow Le he compuaional domain be pariioned ino hree ub-domain Ω i, i=,,3. Le Ω 3 repreen he cylinder in he compuaional domain where he ollowing governing equaion hold: N Here Γ correpond o he laeral urace o he cylinder where Neumann boundary condiion are precribed. he oulow par o he compuaional domain coni o he ollowing ub-domain: Γ correpond o he oulow region ha i no O a par o he rucural domain; Ω correpond o he rucural domain ha involve he hree baen; Ω correpond o he rucure (haded grey ha doe no involve he hree baen. We will aume he ollowing adorpion condiion: O φ n r = aφ on Γ I where a i a conan. Alo, Γ correpond o he inlow urace where we precribe: I φ n r =..5in(π on Γ For he rucural model o he MAV ha i modeled via he ub-domain Ω and Ω we conider he ollowing governing membrane equaion or he delecion o he membrane w: ( ρ ρ w E ρ w v= w E w E v yy w= yy ε v = ρ φ in in Ω Ω in Ω Ω where E, E are he conan correponding o he elaic modulu and econd momen o area or each o he ub-domain Ω and Ω ρ,ρ repecively. Alo, are he repecive deniie o he membrane and he baen. I mu alo be poined ou ha hal o he MAV edge wa kep rigid o relec he leading edge par. he wo yem are alo coupled hrough he coninuiy o he velociie: r φ n= w on Ω he ully coupled yem decribed herein wa modeled and olved in COMSOL and he reul or he membrane delecion are hown in Figure 7 and 8.

5 6. Reerence Figure 7: hree-dimenional MAV membrane wing delecion 5. Concluion Figure 8: wo-dimenional MAV membrane wing delecion In hi work, a coupled compuaional mehodology o olve problem ha involve luid-rucure ineracion ha been preened or variou benchmark problem. he problem conidered in hi work included a one dimenion problem coupling luid and rucure wih and wihou conrol and an applicaion problem in hree dimenion involving MAV. he one dimenional problem provide a grea inigh ino he naure o he coupled behavior o he ineracion beween he luid velociy and he rucural diplacemen. he imporance o he non-linear erm in he luid equaion wa illuraed in he conrol problem ha helped decreae he error beween he precribed and compued oluion.. Sehaiyer, P. and Suri, M., hp ubmehing via non-conorming inie elemen mehod, Compuer Mehod in Applied Mechanic and Engineering, 89, 3, (.. Ben Belgacem, F., Chilon, L.K. and Sehaiyer, P., he hp-morar Finie Elemen Mehod or Mied elaiciy and Soke Problem, Compuer and Mahemaic wih Applicaion, 46, 35 55, (3. 3. Swim, E.W. and Sehaiyer, P., A nonconorming inie elemen mehod or luid-rucure ineracion problem, Compuer Mehod in Applied Mechanic and Engineering, 95, 88 99, (6. 4. Aulia, E., Manervii, S. and Sehaiyer, P., A compuaional mulilevel approach or olving D Navier-Soke equaion over non-maching grid, Compuer Mehod in Applied Mechanic and Engineering, 95, , (6. 5. Aulia, E., Manervii, S. & Sehaiyer, P., A non-conorming compuaional mehodology or modeling coupled problem, Nonlinear Analyi, 6, (5. 6. Aulia, E., Manervii, S. and Sehaiyer, P., A mulilevel domain decompoiion approach o olving coupled applicaion in compuaional luid dynamic, Inernaional Journal or Numerical Mehod in Fluid, 56, (8. 7. C. Grandmon, V. Guime and Y. Maday. Numerical analyi o ome decoupling echnique or he approimaion o he uneady luid rucure ineracion. Mahemaical Model and Mehod in Applied Science, : , (. 8. J. Donea, S. Giuliani and J. Halleu. An arbirary Lagrangian-Eulerian inie elemen mehod or ranien luid-rucure ineracion. Compuer Mehod in Applied Mechanic and Engineering, 33:689-73, ( Acknowledgemen hi work wa uppored in par by gran rom he NSF and NIH o he correponding auhor.