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1 Copyrigh c 5 Tech Science Press CMC, vol., no., pp.39-49, 5 umerical Analysis of Parameers in a Laminaed Beam Model by Radial Basis Funcions Y. C. Hon,L.Ling andk.m.liew 3 Absrac: In his paper we invesigae a hermal driven Micro-Elecrical-Mechanical sysem which was originally designed for inkje priner o precisely deliver small ink droples ono paper. In he model, a iny free-ended beam of meal bends and projecs ink ono paper. The model is solved by using he recenly developed radial basis funcions mehod. We esablish he accuracy of he proposed approach by comparing he numerical resuls wih repored experimenal daa. umerical simulaions indicae ha a ligh (low composie mass) beam is more sable as i does no oscillae much. A sof (low rigidiy) beam resuls in a higher rae of deflecion, when compared o a high rigidiy one. Effecs caused by he values of physical parameers are also sudied. Finally, we give a predicion on he opimal ime for he second curren pulse which resuls in maximum rae of second deflecion of he beam. keyword: Micro-Elecrical-Mechanical sysem, Thermal driven, Radial Basis Funcions mehod Inroducion Micro-Elecro-Mechanical-Sysem (MEMS) is an inegraion of mechanical elemens, sensors, acuaors, and elecronics on a common silicon subsrae hrough he uilizaion of micro-fabricaion echnology. The dimensions of MEMS devices are usually in micromeer scale whose design and manufacuring processes require he use of elecrically-driven moors smaller han he diameer of a human hair. Sudies of he complicaed mechanical, hermal, biological, chemical, opical, and magneic phenomena are paricularly challenging for quaniaive analyses and modelling. During he las decade, Ciy Universiy of Hong Kong, Hong Kong. The work described in his paper was fully suppored by a gran from he Research Grans Council of he Hong Kong Special Adminisraive Region, China (Projec o. CiyU 85/3E). Ciy Universiy of Hong Kong, Hong Kong 3 anyang Technological Universiy, Singapore inensive research sudies in MEMS have enabled many fruiful developmens in he design, consrucion, and fabricaion of he MEMS devices. For insance, he micro-acceleromeer for crash air-bag deploymen sysem and he acive suspension sysem for auomobiles; he micro-valves, micro-pumps and micro-acuaors for fluidic ranspor, mixing and paricle filering; he micromirror swiches and he snapping micro-swiches for radio frequency have increased remendously he porabiliy and applicabiliy of elecronic and elecrical producs; and he human blood pressure moniors and he polymer micro-fluidic chips have been widely adoped for medical diagnosics and drug discovery. The main srucure of many componens like micro swiches and micro mirrors can be simplified as a Dbeam or a D-plae mode which consiss of fourh-order differenial equaions. To furher simplify he model, Huang, Liew, and e al. () used a lumped model composed of mass, spring and damp o sudy he beam srucure (called bridge ) in he micro swich. In heir work hey designed a saic elecromechanical model for he residual sress effec o predic he siffness and pullin volage. They also used a nonlinear dynamic model ha capured he essenial characerisics of he bridge o predic he swiching speed and he Q-facor. Younis, Abdel-Rahman, and ayfeh (3) proposed a macro model for microbeam-based MEMS by discreizing he disribued-parameer sysem using a Galerkin procedure ino a finie-degree-of-freedom sysem, consising of ordinary differenial equaions (ODEs) in ime. Their model considered moderaely large deflecions, dynamics loads and coupling beween mechanical and elecrical forces. The model successfully prediced he pull-in volage, naural frequencies and pull-in ime. Bochobza- Degani and emirovsky () proposed a pull-in model of wo degrees of freedom o model he pull-in parameers of elecrosaic acuaors. Their model is more accurae han he radiional one-degree-of-freedom one. On he opic of sysem-level simulaion, Endemano,

2 4 Copyrigh c 5 Tech Science Press CMC, vol., no., pp.39-49, 5 Desmuillez, and Dunnigan () proposed an analyical orque model o describe he elecrosaic acuaor. The model is included ino sysem level simulaion of a micromoor by hardware descripion language-analogue mixed signal (VHDL-AMS). Mos MEMS models require solving differenial equaion in several coupled energy domains and complicaed acuaion mechanism. Due o he lack of suiable numerical and analyical mehods, mos of he MEMS design processes, however, are sill performed in a primiive rial-and-error fashion. These mehods normally require several modificaions before MEMS devices are finally modeled. This resuls in an inefficien and ineffecive producion cycle for commercial produc developmen. Developmens in advanced numerical echniques and simulaion ools will definiely be beneficial o MEMS design and producion processes. umerical modeling and simulaion for MEMS devices involve knowledge of various disciplines such as mechanical, hermal, fluidic, elecrical, magneic, chemical and opical engineering. Recenly, he developmen of ano-elecro-mechanical sysems (EMS), which are housand imes smaller han MEMS, has given a significan impac on medical, auomobile, aerospace and informaion echnology areas [Kovacs (998)]. Some prospecive applicaions of EMS include random access memory [Rueckes, Kim, Joselevich, Tseng, Cheung, and Lieber ()], super-sensiive sensors [Collins, Bradley, Ishigami, and Zel ()], and nano-weezers for miniaurized roboics [Kim and Lieber (999)]. This paper is organized as follows: In Secion, he physical beam model is saed. The numerical formulaion for solving he model will be given in Secion 3. umerical resuls are hen given in Secion 4 o verify he efficiency and accuracy of he proposed mehod. Physical Model A beam composed of wo maerials, aluminum (Al) and silicon dioxide (SiO ) in a raio of :3, was manufacured a Easman Kodak wih he overall dimensions µm µm 5µm, see Figure. A volage pulse of µs was applied o he beam heaing i up o abou 4K and resuling in a maximum rae of deflecion of abou.ms. The model of his problem is divided ino wo main pars: Hea Transpor Due o he difference in hermal expan- Expand when heaed Figure : Laminaed beam model. Oxide Aluminum Isopar fluid sion coefficien of Al and SiO, he laminaed beam will bend when a curren is supplied. This generaes a cerain amoun of hea. Beam Fluid Ineracion When he beam is heaed, is moion is governed by a beam equaion ha saisfies a oal effecive momen condiion. The moion is linear wih respec o he applied emperaure, a he free end. The model of he laminaed beam was given in [Ross, Biswanger, Bohun, Bridge, Ling, oel, Saujani, Spirn, and Ting ()] as follow: The equaions governing he hea flow are: Isopar Fluid: Silicon Oxide: Aluminum: u ρ f c v f = k u f x, u ρ ox c vox = k u ox x, () u ρ Al c val = k u Al x +Q. Some of he hermal properies of Al, SiO and surrounding Isopar fluid 4 is lised in Table. The boundary condiions are deermined by he empirical fac ha emperaure is coninuous and energy is conserved across he inerface boundaries. These condiions imply θ(inerface ) = θ(inerface + ), k u x (inerface ) = k + u x (inerface+ ), () a any inerface. In addiion, he boundary condiions a infiniy are given o be room emperaure θ(x = ±, )= 3K. For a µs heaing pulse, he resuling emperaure profiles are displayed in Figure. Since he conduciviy 4 Isopar is he brand name for a synheically produced Isoparaffinic fluid. Isopar fluids have excepionally high puriy and uniform composiion. They are available in a wide range of evaporaion raes. Processing is sringenly conrolled o provide producs wih exremely low odor, selecive solvency,excellen sabiliy, and narrow disillaion ranges.

3 umerical Simulaions for MEMS Using RBFs 4 k of aluminum is so high, i can be assumed ha he emperaure variaion across he aluminum is zero and he emperaure of aluminum is spaially uniform. The emperaure of he aluminum layer, θ Al in Figure (b), is of cenral imporance in he second par of his model. ρ c v k Maerial (g cm ) (Jg K ) (Jcm s K ) Isopar Fluid.77. E 3 Silicon Dioxide E Aluminum µ s Table : Thermal properies of Al, SiO and surrounding Isopar fluid. 8 θ 6 4 5µ s where ρ is he weighed densiy, D is he composie flexural rigidiy, P is he exernal pressure, H is he (uniform) hickness of he beam, β and k are he damping coefficiens. µ s The following iniial condiions reflec ha he beam sars a res and will no ben during moion: u(x,)= = u(x,). Isopar Oxide Al Isopar (a) Cross secion of he beam. Since he beam is fixed and clamped a one end, x =, we specify he boundary condiions as: u(,)= = x u(,). θ Al ( ) (µ s) (b) Average emperaure of he Al layer. Figure : Temperaure profile as a µs heaing pulse is applied. In addiion, he following boundary condiion assumes ha he beam does no have shear sress a he free end, x = L: 3 x 3 u(l,)=. Finally, since he beam is laminaed, each of he layers will expand a differen raes when heaed. This imbalance in he srains of he various layers creaes a momen a he free end x = L saisfying he following boundary condiion: x u(l,)=γθ Al(), (4) which is linear wih respec o he applied emperaure. 3 Radial Basis Funcions Mehod The moion of he beam is governed by he beam equaion wih boh he drag and he viscosiy of he fluid aken ino accoun, (β +ρh) u = u D 4 x 4 k u, x [,L], [,T], (3) Finie difference mehod and finie elemen mehod are widely adoped for he numerical modeling and simulaion of he MEMS analysis. One of he major disadvanages of hese radiional compuaional mehods is heir requiremen on he generaion of grids or meshes,

4 4 Copyrigh c 5 Tech Science Press CMC, vol., no., pp.39-49, 5 which hinders heir applicaion o solve high dimensional problems or problems under irregular domains. The rapid developmen of mesh-free mehods during he las decade has recenly overcome his mesh dependen disadvanage. There are basically wo ypes of meshfree mehods: he Galerkin-based ype ha requires a background mesh, for insance, he smooh paricle hydrodynamics mehod [Gingold and Moraghan (977)], he diffuse elemen mehod [ayroles, Touzo, and Villon (99)], he hp-clouds mehod [Liszka, Duare, and Tworzydlo (996)], he Meshless Local Perov-Galerkin [Aluri and Shen ()],[Aluri (4)] he local boundary inegral equaion [Zhu, Zhang, and Aluri (998)]; and he collocaion-based ype ha does no require a background mesh, for insance, he Finie Poin mehod [Onae, Idelsohn, and Zienkiewicz (996)], he radial basis funcions mehod [Chen, Ganesh, Golberg, and Cheng (); Kansa (99a,b); Hon (); Hon and Schaback (); Shu and Yeo (3, 4); Wu and Shu ()], he mehod of paricular soluion [Chen, Muleshkov, and Golberg (999); Golberg, Muleshkov, Chen, and Cheng (3)] and fundamenal soluion [Chan and Chen (996); Chen (995a,b); Chen, Marcozzi, and Choi (999)], he differenial quadraure mehod [Bellman and Casi (97); Shu and Richards (99); Shu (); Wu and Liu ()] and he poin inerpolaion mehods [Liu (3); Liu and Gu ()]. In his secion, a numerical algorihm based on he Radial Basis Funcions (RBFs) is developed for solving he governing equaion (3) in he beam model subjec o he given iniial and boundary condiions. The idea of he RBFs mehod is o inerpolae an unknown mulivariae funcion f (x) R n by a linear combinaion of he radial basis funcions φ( x x j ): f (x) j= α j φ( x x j )+q m n (x), (5) where x j are disinc daa poins in R n and q m n ( ) is a polynomial in R n wih degree up o m. Micchelli (986) proved ha when he daa poins are all disinc, he resulan marix for obaining he undeermined coefficiens α j from a radial basis funcion inerpolaion is always inverible. A direc collocaion is hen performed by assuming ha he represenaion (5) saisfies he given parial differenial equaions (3). From he boundary condiions, a unique se of he undeermined coefficiens α j is obained by solving he resulan sysem of equaions (refer o Franke and Schaback (998) and Wendland (999) for he heoreical foundaion of his mehod). In he compuaion of his paper, we choose m = and φ = x x j 7 o be he smooh spline of order Mehodology The beam model designed in he las Secion consiss of he following parial differenial equaion (PDE): a u(x,)+a 4 u(x,)+ u(x,)=, (6) x4 for (x,) [,] wih iniial condiions (of a resing beam) u(x, )=, u(x,)=, (7) and boundary condiions (a boh ends of he beam) u(,)=, u(,)=, (8) u(,)=θ(), 3 u(,)=, (9) 3 where boh he spaial and emporal variables, x and, have been normalized. The parameers conained in (6) are relaed o hose of he physical model (3) as a = L4 (β +ρh) T, a = L4 k D TD. The source funcion in (9) corresponds o (4) as Θ() := L Γθ Al (). To discreize equaion (6) in ime, we simply ake a cenral difference approximaion o he second parial order ime derivaive and a forward difference approximaion o he firs parial order ime derivaive of he funcion u o obain: ] [(a +a d)u +d 4 x 4 u (x, +d) () =(a +a d)u(x,) a u(x, d), where he fourh order spaial derivaive of he funcion u is evaluaing a ime + d. This higher order spaial derivaive will hen be approximaed by he radial basis funcions as follow.

5 umerical Simulaions for MEMS Using RBFs 43 Given a se of daa cener {x i }, he unknown soluion u(x, +d) of () is approximaed by RBFs as: u(x, +d)= α i φ i (x), () where φ i (x)=φ( x x i ) is a radial basis funcions evaluaed a daa cener x i. Due o he differeniabiliy of he smooh spline φ i used as he RBFs in (), he fourh order derivaive of he unknown soluion can be obained by 4 x 4 u(x, +d)= 4 α i x 4 φ i(x). oe ha here are wo boundary condiions on each side of he beam o be saisfied. Furhermore, hese boundary condiions involve high order derivaives which normally cause numerical oscillaion. The use of smooh spline of order 7 in our proposed mehod, however, does no encouner any oscillaion problem. For he choice of daa ceners {x i }, we use he following funcion given in [Han and Liew (999); Liew and Han (997)] o place he firs 4 daa ceners: ( cos (i )π 5 ), i =,,...,( 4), () x i = in [,]. Following he idea of Liew, Teo, and Han (999), we define he las four daa ceners x 3,...,x o be x ±δ and x ±δ,whereδ = (x x ), a each end of he beam o obain he final se of daa ceners. This choice of daa se allows wo exra degree of freedom o collocae he PDE and he second boundary condiion imposed a boh boundaries. Collocaing (6) a x j for j =,,..., 4 giveshefollowing 4 equaions: ] [(a +a d)φ i (x j )+d 4 x 4 φ i(x j ) α i =(a +a d)u(x j,) a u(x j, d). Furher collocaion a he four boundary condiions resuls in he following 4 equaions: α i φ i ()=, α i 3 3 x φ i(l)=, α i x φ i()=, α i x φ i(l)=θ( +d). Parameer ρh D Γ Value.56E E- Parameer Q β k Value 3.774E7.3776E-3 3 Table : Thermal properies of Al, SiO and he surrounding Isopar fluid. We now have a oal of equaions for he unknown coefficiens α i, i =,,...,. Once hese coefficiens α i have been obained, he beam posiion is given by () a each ime ieraion. The ime profile of he beam posiion is hen obained by ieraing he soluion process unil =. 4 umerical Resuls The soluion of he hea equaion () and boundary condiions () are used o deermine θ Al (), and herefore Θ(). This ime dependen emperaure is hen imposed as a boundary condiion in (9). The parameers values used are lised in Table. The numerical algorihm given in Secion 3 is hen used o compue he coefficiens α i a each ime ieraion. We simply use = in our compuaion. For a µs heaing pulse (see Figure ), he ime profile of he moion of he laminaed beam is displayed in Figure 3. A comparison wih he experimenal daa exraced from [Ross, Biswanger, Bohun, Bridge, Ling, oel, Saujani, Spirn, and Ting ()] is also shown in Figure 4. I can be observed ha he numerical resul agrees exremely well wih he experimenal daa. The maximum rae of deflecion is abou.ms. To perform some parameers analysis, we assume ha all parameers ake he values lised in Table. I is displayed in Figure 5 he dynamic responses of he beam a he free end o differen values of applied currens, from Q = E7oE7, in he hermal equaion (). The numerical resul verifies he direc relaionship beween he impulse and he ampliudes o he inpu currens. oe ha he applied currens do no cause any apparen oscillaion effec o he moion of he beam. This suggess ha his numerical simulaion can deermine he maximum applicable curren for a specify maerial from is fixed meling emperaure.

6 44 Copyrigh c 5 Tech Science Press CMC, vol., no., pp.39-49, 5 x 4 6 x 4 u( x, ) 4. u(, ) Q=E8 Q=8E7 Q=6E7.4 x.6.8 Figure 3 : Profile of he laminaed beam moion for he µs ( =.5) heaing pulse. 3 x umerical soluion Experimenal daa Q=4E7 Q=E7 Figure 5 : Profile of he free end beam moion corresponding o differen volage Q. u(, ) x 4 k = E k = E4 k = E5.5 u(, ) Figure 4 : Profile of he free end beam moion corresponding o Figure 3 and experimenal daa Figure 6 : Profile of he free end beam moion corresponding o differen damping consan k. Figure 6 shows he moion of he beam a he free end under differen value of he damping consan k. As he fricion increases, he deflecion of beam decreases and does no oscillae. The criical damping occurs a k 4E 4. This indicaes ha differen hermal propery of he isopar fluid surrounding he beam should be chosen according o he beam srucure. In oher words, an ideal isopar fluid will cause a close-o-criical damping effec o he beam. ex, we vary he value of he composie mass B = β + ρh. Figure 7 o Figure 9 show he deflecion of he beam for B = 5E 3, 3E 3, and E 3, respecively. The numerical resul indicaes ha a ligh (low composie mass) beam is more sable as i does no oscillae much. The numerical deflecion of he beam under differen value of he composie flexural rigidiy D shown in Figure o Figure suggess ha a sof (low rigidiy) beam resuls in a higher rae of deflecion, when com-

7 umerical Simulaions for MEMS Using RBFs 45 3 x 4 3 x 4.5 B=5E 3 B=3E 3 B=E 3.5 B=5E 3 B=3E 3 B=E 3 u(, ).5 u(, ) Figure 7 : Profile of he free end beam moion corresponding o composie mass B = β +ρh = 5E Figure 9 : Profile of he free end beam moion corresponding o composie mass B = β +ρh = E. u(, ) 3 x B=5E 3 B=3E 3 B=E 3 u(, ) 3 x D= D= D= Figure 8 : Profile of he free end beam moion corresponding o composie mass B = β +ρh = 3E Figure : Profile of he free end beam moion corresponding o composie flexural rigidiy D =. pared o a high rigidiy one. These numerical simulaions conclude ha a ligher and sofer beam is more preferable in conrolling he inkje priner o precisely deliver small ink droples ono paper. Finally, we assume ha a second volage pulse of µs is applied o he beam for anoher delivery of ink. Due o he hermal propery of isopar fluid, he emperaure of he beam canno exceed 4K. The beam has o be cooled down afer he firs heaing pulse. Once he hea supply o he beam has been urned off, he ampliude of he oscillaion a he beam decreases as ime increases. For numerical demonsraion, we le he beam cool down for abou µs (i.e., =.5 in he normalized ime) afer he firs µs heaingpulse. Thesecondpulseishenapplied a =.5,.5,.53, and.54 respecively. The resuling beam end moions are displayed in Figure 3. For beer illusraion, we also display respecively he zoom in figures of he saring and ending phases of he second deflecion in Figure 4. The maximum raes of deflec-

8 46 Copyrigh c 5 Tech Science Press CMC, vol., no., pp.39-49, x 4 3 (b).5 3 x 4 D= D= D= u(, ).5.5 (a) u(, ) Figure : Profile of he free end beam moion corresponding o composie flexural rigidiy D = Figure 3 : Profiles of he free end beam moion for wo µs heaing pulses. ion resuling from hese second heaing pulses are +3%, +5%, +3%, and % respecively of he firs heaing pulse. This indicaes ha =.5 is he bes ime o sar he second pulse. The bes ime for he second curren pulse is herefore he ime when he beam reaches a local minima, which is expeced from he undersanding of he resonance phenomena. u(, ) x D= D= D= Figure : Profile of he free end beam moion corresponding o composie flexural rigidiy D =. 5 Conclusion In his paper we develop a compuaional mehod by using he radial basis funcions, in paricular, he smooh spline, for he numerical soluion of a beam model arising from he sudies of MEMS. In he model a free-end laminaed beam composed o wo maerials, aluminum (Al) and silicon dioxide (SiO ), was supplied wih a volage pulse of µs resuling a a emperaure of abou 4K and a maximum rae of deflecion of abou.ms.umerical simulaions wih respec o differen values of parameers are invesigaed. Simulaed resuls show good mach wih he available experimenal daa. Furhermore, he numerical simulaions indicaed ha an ideal consrucion of he laminaed beam should be ligh in weigh and low in rigidiy. The surrounding isopar fluid (i.e., he ink) should be viscous enough o provide a close-ocriical damping effec o he beam. Finally, we predic ha he opimal ime for a second curren pulse is he ime when he beam reaches a local minima.

9 umerical Simulaions for MEMS Using RBFs 47 u(, ) u(, ).7 x x 4 3 = =.5 =.5 =.53 =.54 (a) Saring phase of deflecion. = =.5 =.5 =.53 = References (b) Ending phase of deflecion. Figure 4 : Zoom in of Figure 3. Aluri, S..; Han, Z. D.; Rajendran, A. M. (4): A new implemenaion of he meshless finie volume mehod, hrough he MLPG mixed approach. CMES: Compuer Modeling in Engineering & Sciences, vol.6, pp Aluri, S..; Shen, S. (): The meshless local Perov-Galerkin (MLPG) mehod: a simple & less-cosly alernaive o he finie elemen and boundary elemen mehods. CMES: Compuer Modeling in Engineering & Sciences, vol. 3, no., pp. 5. Bellman, B., K.-G.; Casi, J. (97): A echnique for he rapid soluion of nonlinear parial differenial equaions. J. Compu. Phys., vol., pp Bochobza-Degani, O.; emirovsky, Y. (): Modelling he pull-in parameers of elecrosaic acuaors wih a novel lumped wo degrees of freedom pull-in model. Sensors and Acuaors A, vol , pp Chan, C. Y.; Chen, C. S. (996): Mehod of fundamenal soluions for muli-dimensional quenching problems. In Proceedings of Dynamic Sysems and Applicaions, Vol. (Alana, GA, 995), pp. 5, Alana, GA. Dynamic. Chen, C. S. (995): The mehod of fundamenal soluions and he quasi-mone Carlo mehod for Poisson s equaion. In Mone Carlo and quasi-mone Carlo mehods in scienific compuing (Las Vegas, V, 994), volume 6 of Lecure oes in Sais., pp Springer, ew York. Chen,C.S.(995): The mehod of fundamenal soluions for non-linear hermal explosions. Comm. umer. Mehods Engrg., vol., no. 8, pp Chen, C. S.; Ganesh, M.; Golberg, M. A.; Cheng, A. H.-D. (): Mulilevel compac radial funcions based compuaional schemes for some ellipic problems. Compu. Mah. Appl., vol. 43, no. 3-5, pp Radial basis funcions and parial differenial equaions. Chen,C.S.;Marcozzi,M.D.;Choi,S.(999): The mehod of fundamenal soluions and compacly suppored radial basis funcions: a meshless approach o 3D problems. In Boundary elemens, XXI (Oxford, 999), volume 6 of In. Ser. Adv. Bound. Elem., pp WIT Press, Souhampon. Chen,C.S.;Muleshkov,A.S.;Golberg,M.A.(999): The numerical evaluaion of paricular soluions for Poisson s equaion a revisi. In Boundary elemens, XXI (Oxford, 999), volume 6 of In. Ser. Adv. Bound. Elem., pp WIT Press, Souhampon. Collins, P.; Bradley, K.; Ishigami, M.; Zel, A. (): Exreme oxygen sensiiviy of elecronic properies of carbon nanoubes. Science, vol. 87, pp

10 48 Copyrigh c 5 Tech Science Press CMC, vol., no., pp.39-49, 5 Endemano, A.; Desmuillez, M.; Dunnigan, M. (): Sysem level simulaion of a double saor wobble elecrosaic micromoor. Sensors and Acuaors A, vol. 99, pp Franke, C.; Schaback, R. (998): Solving parial differenial equaions by collocaion using radial basis funcions. Appl. Mah. Compu., vol. 93, no., pp Gingold, R.; Moraghan, J. (977): Smoohed paricle hydrodynamics: heory and applicaions o non-spherical sars. Man. o. Asrou. Soc., vol. 8, pp Golberg, M. A.; Muleshkov, A. S.; Chen, C. S.; Cheng, A. H.-D. (3): Polynomial paricular soluions for cerain parial differenial operaors. umer. Mehods Parial Differenial Equaions, vol. 9, no., pp. 33. Han, J.-B.; Liew, K. (999): Saic analysis of mindlin plaes: he differenial quadraure elemen mehod (dqem). Compuer Mehods in Applied Mechanics and Engineering, vol. in press. Hon, Y. C. (): A quasi-radial basis funcions mehod for american opions pricing. Compu. Mah. Applic., vol. 43, pp Hon, Y. C.; Schaback, R. (): On unsymmeric collocaion by radial basis funcions. Appl. Mah. Compu., vol. 9, pp Huang, J. M.; Liew, K. M.; e al. (): Mechanical design and opimizaion of capaciive micromachined swich. Sensors and Acuaors A, vol. 93, pp Kansa, E. J. (99): Muliquadrics a scaered daa approximaion scheme wih applicaions o compuaional fluid-dynamics. I. Surface approximaions and parial derivaive esimaes. Compu. Mah. Appl., vol. 9, no. 8-9, pp Kansa, E. J. (99): Muliquadrics a scaered daa approximaion scheme wih applicaions o compuaional fluid-dynamics. II. Soluions o parabolic, hyperbolic and ellipic parial differenial equaions. Compu. Mah. Appl., vol. 9, no. 8-9, pp Kim, P.; Lieber, C. (999): anoube nanoweezers. Science, vol. 86, pp Kovacs, G. (998): Micromachined Tranducers Sourcebook. McGraw-Hill, ew York. Liew, K.; Han, J.-B. (997): A four-node differenial quadraure mehod for sraigh-sided quadrilaeral reissner/mindlin plaes. Communicaions in umerical Mehods in Engineering, vol. 3, no Liew, K.; Teo, T.; Han, J.-B. (999): Comparaive accuracy of da and hdq mehods for hree-dimensional vibraion analysis of recangular plaes. In. J. umer. Meh. Engng., vol. 45, pp Liszka, T.; Duare, C.; Tworzydlo, W. (996): Hpmeshless cloud mehod. Compuer Mehods in Applied Mechanics and Engineering, vol. 39, pp Liu, G. R. (3): Mesh free mehods. CRC Press, Boca Raon, FL. Moving beyond he finie elemen mehod. Liu, G. R.; Gu, Y. T. (): Boundary meshfree mehods based on he boundary poin inerpolaion mehods. In Boundary elemens, XXIV (Sinra, ), volume 3 of In. Ser. Adv. Bound. Elem., pp WIT Press, Souhampon. Micchelli, C. A. (986): Inerpolaion of scaered daa: disance marices and condiionally posiive definie funcions. Consr. Approx., vol., no., pp.. ayroles, B.; Touzo, G.; Villon, P. (99): Generalizing he finie elemen mehod: diffuse approximaion and diffuse elemens. Compuaional Mechanics, vol., pp Onae, E.; Idelsohn, S.; Zienkiewicz, O. (996): A finie poin mehod in compuaional mechanics: applicaions o convecive ranspor and fluid flow. In. J. for umerical Mehods in Engineering, vol. 39, pp Ross, D.; Biswanger, K.; Bohun, C.; Bridge, L.; Ling, L.; oel, D.; Saujani, S.; Spirn, D.; Ting, F. (): Opimal design of a micro-elecrical-mechanical sysems acuaor. In Repor of he Third PIMS Graduae Indusrial Mah Modelling Camp, pp Simon Fraser Universiy, Pacific Insiue for he Mahemaical Sciences (PIMS). Rueckes, T.; Kim, K.; Joselevich, E.; Tseng, G.; Cheung, C.; Lieber, C. (): Carbon nanoube-based nonvolaile random access memory for molecular compuing. Science, vol. 89, pp Shu, C. (): Differenial quadraure and is applicaion in engineering. London, Springer-Verlag. Shu, C.; Richards, B. E. (99): Applicaion of generalized differenial quadraure o solve wo-dimensional incompressible navier-sokes equaions. In. J. umer. Meh. Fluids, vol. 5, pp Shu, C., D.-H.; Yeo, K. S. (3): Local radial basis funcion-based differenial quadraure mehod and

11 umerical Simulaions for MEMS Using RBFs 49 is applicaion o solve wo-dimensional incompressible navier-sokes equaions. Compuer Mehods in Applied Mechanics and Engineering, vol. 9, pp Shu, C., D.-H.; Yeo, K. S. (4): Soluion of parial differenial equaions by a global radial basis funcionbased differenial quadraure mehod. Engineering Analysis wih Boundary Elemens, vol. 8, pp Wendland, H. (999): Meshless Galerkin mehods using radial basis funcions. Mah. Comp., vol. 68, no. 8, pp Wu, T. Y.; Liu, G. R. (): The generalized differenial quadraure rule for iniial-value differenial equaions. J. Sound Vibraion, vol. 33, no., pp Wu, Y. L.; Shu, C. (): Developmen of rbfdq mehod for derivaive approximaion and is applicaion o simulae naural convecion in concenric annuli. Comp. Mech., vol. 9, pp Younis, M. I.; Abdel-Rahman, E. M.; ayfeh, A. (3): A reduced-order model for elecrically acuaed microbeam-based mems. J. MEM Sys., vol., no. 5, pp Zhu, T.; Zhang, J.; Aluri, S. (998): A meshless local boundary inegral equaion (lbie) mehod for solving nonlinear problems. Compuaional Mechanics, vol., pp

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