TOPOLOGY OPTIMIZATION OF COMPLIANT MECHANISM DESIGN WITH STA- TIONARY FLUID-STRUCTURE INTERACTION

Transcription

1 Blucher Mechanical Engineering Proceeding May 2014, vol. 1, num. 1 TOPOLOGY OPTIMIZATION OF COMPLIANT MECHANISM DESIGN WITH STA- TIONARY FLUID-STRUCTURE INTERACTION G. H. Yoon 1 1 Aian Proeor, School o Mechanical Engineering, Hanyang Univeriy, Reublic o Korea (ghy@hanyang.ac.kr/gilho.yoon@gmail.com) Abrac. Thi aer ouline a new rocedure or oology oimizaion in he eady-ae luid-rucure ineracion (FSI) roblem. A review o curren oology oimizaion mehod highligh he diiculie in alernaing beween he wo diinc e o governing equaion or luid and rucure dynamic (hereaer, he luid and rucural equaion, reecively) and in imoing couling boundary condiion beween he earaed luid and olid domain. To overcome hee diiculie, we rooe an alernaive monolihic rocedure emloying a uniied domain raher han earaed domain, which i no comuaionally eicien. In he rooed analyi rocedure, he aial dierenial oeraor o he luid and rucural equaion or a deormed coniguraion i ranormed ino ha or an undeormed coniguraion wih he hel o he deormaion gradien enor. For he couling boundary condiion, he divergence o he reure and he Darcy daming orce are inered o he olid and luid equaion, reecively. The rooed mehod i validaed in everal benchmark analyi roblem. Toology oimizaion in he FSI roblem i hen made oible by inerolaing Young modulu, he luid reure o he modiied olid equaion, and he invere ermeabiliy rom he daming orce wih reec o he deign variable. Keyword: Toology oimizaion, oology oimizaion, luid-rucure ineracion, monolihic aroach 1. INTRODUCTION The numerical imulaion and he oimizaion o luid-rucure ineracion have been imoran ubjec in engineering [1,2,3,4,5,6]. Many innovaive numerical analyi rocedure have been develoed which can be mainly caegorized ino aggered (ariioned) and monolihic analyi rocedure deending on he dierence in he couling mehod. In aggered analyi rocedure, he luid and he rucure equaion are alernaely olved in ime, and he inerace couling boundary condiion in racion and velociy are enorced aynchronouly. On he oher hand, in monolihic analyi rocedure, he couling ineracion along he ineracing boundary beween he luid and he rucure i reaed ynchronouly, a

2 hown in Figure 1. Deending on he roblem characeriic o FSI yem, eiher o he rocedure can be emloyed [2,3,4,5,6]. Solid: u : v, Saggered analyi mehod Monolihic analyi mehod Figure 1. Saggered and monolihic analyi mehod wih earae analyi domain [4]. One o he moivaion o he reen reearch i ha when alying he exiing aggered or monolihic numerical cheme or FSI o one o he rucural oimizaion cheme called oology oimizaion many comlicaion exi: alernaing he diinc governing equaion wih reec o he deniy deign variable and imoing he exlici couling boundary condiion. Thu, in hi reearch, we emloy a new FE-baed monolihic rocedure wih a uniied analyi domain or olving eady-ae luid-rucure ineracion roblem and aly hi monolihic analyi rocedure o oology oimizaion o comlian mechanim deign conidering FSI [2,3,4]. One o he main dierence beween he reen monolihic aroach and reviou aggered and monolihic rocedure lie in ha ac ha a uniied analyi domain having boh he linear elaiciy and he Navier-Soke equaion i emloyed wih he couling boundary condiion. In hi new monolihic aroach, i i oible o alernaively inerolae he wo governing equaion by inerolaing he maerial roerie involved wih he wo equaion. In he reen reearch, we aly our reviou analyi ormulaion and inveigae he diadvanage and advanage o he develoed analyi rocedure in more deail.

3 2. UNIFIED FE FORMULATION FOR STEADY STATE FLUID STRUCTURE IN- TERACTION Becaue he reen udy ocue on he reormulaion o he governing equaion a well a he couling boundary condiion uing he involved maerial roerie o he Navier-Soke equaion and he linear elaiciy equaion wihou exlici ineracion boundary curve, only he eady-ae FSI roblem i conidered. Furhermore, only mall rucural dilacemen are aumed or he rucural engineering rain, i.e., linear rain, which make he ine marix indeenden o he rucural dilacemen. The luid induced orce in he linear elaiciy equaion i deenden on he rucural dilacemen. For more deail decriion, ee our conribuion in [2,3,4]. To deine he uniied ormulaion or rucure and luid, we inroduced he ranormaion o he dierenial oeraor, F. The inie deormaion enor, F, i deined a he arial dierenial o he curren coordinae, x, wih reec o he undeormed coordinae, X. T x X F (1) T u F u, xvf Xv, and ( ) d ( ) d 0 T T T T T ( ) X X v vf v F d F v T F d v v F d n d (2) T ( x v) F d0 (3) X T T T T (, ) S T d d d F S u u F F u F (4) where he luid velociy ield and reure o an incomreible low are decribed by v and, reecively. Noe ha he dierenial oeraor, x, a ime i deined a he conrol volume, ( u ), which imlie ha he conrol volume i deenden on he rucural dilacemen, u. The dynamic vicoiy i or he Newonian low. The Dirichle boundary condiion are imoed on v 0,, and ( u ) or he no-li boundary condiion, inlow/oulow v i boundary condiion, and ineracing boundary condiion, reecively. The Neumann boundary condiion or he alied reure,, i deined a ( ) u wih he normal vecor, n. The linear rain S and aociae re T are deined.

4 3. TOPOLOGY OPTIMIZATION EXAMPLE The ynhee o he comlian mechanim layou maximizing he ouu dilacemen a he dieren locaion o he ring in Figure 2 are conidered uing he reen monolihic analyi cheme. The objecive o he oology oimizaion i o diribue an allowed ma a he deign domain (he righ domain) in order o maximize he rucural x-dilacemen a he ring imulaing a workiece a (5). Max u Subjec o NED e1 v e e V where he objecive uncion,, i he dilacemen or he ring ( u ). The elemen volume and he uer bound o he volume are denoed by v e and i conrained o be le han 10% o he deign domain. (5) V, reecively. The volume limi Comlian mechanim 1 Comlian mechanim 2 (a) -4 ( u -4 = m ) ( u = m ) (b) Figure 2. Problem deiniion o he comlian mechanim and he obained layou [4]. ( = N/m, 3 kg/m, = mkg/m, k 1000 N/m ) C 0.1 MPa, 0.3, V :15%,, max 10 in

5 Uing a uniorm iniial deign ( iniial 0.15 ) aiying he ma conrain, he oimal layou can be obained a Figure 2(b) or he dieren ring locaion. I i likely ha he layou ranmiing he luid orce o he ring are obained. 4. CONCLUSIONS In order o obain oimal comlian mechanim, hi aer develo a monolihic ormulaion baed on our reviou ormulaion or FSI analyi. Common aggering or monolihic analyi mehod have been ued in he analyi and ize/hae oimizaion o FSI yem, bu he analyi rocedure become rohibiively comlicaed in he cae o deniy-baed oological oimizaion. To reolve hi iue, our reviou analyi ormulaion and rocedure have been imlemened and alied; he re redicion are imroved. We olve oology oimizaion or he comlian mechanim conidering luidrucure ineracion o how he validiy o he develoed aroach in oimizaion. The deail ormulaion and examle will be reored in [4]. 5. REFERENCES [1] Bendøe M. P., Sigmund O., Toology Oimizaion Theory Mehod and Alicaion. Sringer-Verlag [2] Yoon G.H., Toology oimizaion or aionary luid-rucure ineracion roblem uing a new monolihic ormulaion. Inernaional Journal or Numerical Mehod in Engineering, 82, , [3] Yoon G. H., Toological layou deign o elecro-hermal-comlian acuaor. Comuer Mehod in Alied Mechanic and Engineering, ,28-44, [4] Yoon G. H., Monolihic luid-rucure ineracion analyi or oological comlian mechanim deign, in review. [5] Andreaen C., Sigmund O., Sauraed oroelaic acuaor generaed by oology oimizaion. Srucural and Mulidicilinary Oimizaion 43, , [6] Kreil S., Pingen G., Evgraov A., Maue K., Toology Oimizaion o Flexible Micro-ic Device. Srucural and Mulidicilinary Oimizaion, 42(4): , 2010.