BEM modeling of MEMS with thin plates and shells using a total Lagrangian approach

Transcription

1 Boundary Elements XXVII 287 BEM modeling of MEMS with thin plates and shells using a total Lagrangian approach S. Telukunta & S. Mukherjee 2 Sibley School of Mechanical Engineering, Cornell University, Ithaca, NY 4853, U.S.A. 2 Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 4853, U.S.A. Abstract MicroElectroMechanical Systems (MEMS) typically consist of plate shaped conductors that can be very thin with h/l O( ) (in terms of the thickness h and length L of the side of a square pate). Conventional Boundary Element Method (BEM) analysis of the electric field in a region exterior to these thin conductors is difficult to carry out accurately and efficiently especially since MEMS analysis requires computation of charge densities (and then surface tractions) separately on the top and bottom surfaces of such thin plates. A new Boundary Integral Equation (BIE) is derived in this work, and coupled with Finite Element Analysis (FEM) to solve for the deformation of such thin MEMS structures. A fully Lagrangian BEM formulation is developed here. A relaxation scheme is implemented to solve for a selfconsistent state of this coupled problem. Some sample results are presented in this paper. Introduction The field of MicroElectroMechanical Systems (MEMS) is a very broad one that includes fixed or moving microstructures; encompassingmicroelectromechanical, microfluidic, microoptoelectromechanical and microthermomechanical devices and systems. MEMS usually consists of released microstructures that are suspended and anchored, or captured by a hubcap structure and set into motion by mechanical, electrical, thermal, acoustical or photonic energy source(s). Typical MEMS structures consist of arrays of thin beams with crosssections in the order of microns (µm) and lengths in the order of ten to hundreds of microns.

2 288 Boundary Elements XXVII Sometimes, MEMS structural elements are plates. Of interest in this work are small rectangular silicon plates with sides in the order of mm and thicknesses of the order of microns, that deform when subjected to electric fields. Owing to their small size, significant forces and/or deformations can be obtained with the application of low voltages ( 0 volts). Examples of devices that utilize vibrations of such plates are synthetic microjets, microspeakers etc. Space constraints dictate that these plates be densely packed and it is important to accurately carry out the coupled field analysis for such MEMS. Numerical simulation of electrically actuated MEMS devices have been carried out for around a decade or so by using the Boundary Element Method to model the exterior electric field and the Finite Element Method to model deformation of the structure. The commercial software package MEMCAD ], for example, uses the commercial FEM software package ABAQUS for mechanical analysis, together with a BEM code FASTCAP 2] for electric field analysis. There are many other examples of such work for both static and dynamic analysis of MEMS. The coupled BEM/FEM methods employed in the vast literature available perform a mechanical analysis on the undeformed configuration of a structure (Lagrangian approach) and an electrical analysis on the deformed configuration (Eulerian approach). A relaxation method is then used for selfconsistency between the two domains. Therefore, the geometry of the structure must be updated before an electrical analysis is performed during each relaxation iteration. This procedure increases computational effort and introduces additional numerical errors since the deformed geometry must be computed at every stage. Li and Aluru 3] first proposed a Lagrangian approach for the electrical analysis as well, thus obviating the need to carry out calculations based on the deformed shapes of a structure. Additional advantages of the fully Lagrangian approach, for dynamic analysis of MEMS, are described in 4], in which a Newton method has been developed and compared with the relaxation scheme. The reader is referred to Bathe 5] for a comprehensive discussion of the Lagrangian approach in mechanics. The present paper is concerned with 3D coupled quasistatic analysis of MEMS using a fully Lagrangian approach. The geometry under consideration is a rectangular MEMS plate together with the region exterior to the plate. A relaxation scheme is employed for coupling of the BEM and FEM domains. BEM analysis of thin MEMS structures has been recently reported by Mukherjee et al (8]). This new method, along with the use of accurate integration schemes, can simulate densely packed, thin MEMS plates which are of practical interest. Readers are referred to the recent work done by Mukherjee and his group (6, 7, 8, 9]) for more details. This paper is organized as follows. Exterior electrostatic BEM analysis of MEMS plates is presented first. This is followed by the FEM model employed in this work. The relaxation based coupling scheme is presented next. Sample numerical results and discussion complete the paper.

3 Boundary Elements XXVII 289 B b í B b V V Electrostatic Force E lectrostatic Force Figure : A deformable cantilever plate over a fixed ground plane. 2 Electrical problem in the exterior domain Figure shows (as an example of a MEMS device) a deformable cantilever plate over a fixed ground plane. (This schematic figure is very similar to Fig. in 4]). The undeformed configuration is B with boundary B. The plate deforms when a potential V is applied between the two conductors, and the deformed configuration is called b with boundary b. The charge redistributes on the surface of the deformed plate, thereby changing the electrical force on it and this causes the plate to deform further. A selfconsistent final state is reached, and this state is computed in the present work by a relaxation scheme. An alternative is to use a Newton scheme 4]. 2. Boundary integral equations in the deformed configuration The governing Boundary Integral Equation (BIE) for a conductor, in current (deformed) coordinates, is: φ(x) = b G(x, y)σ(y)ds(y) () Here φ(x) is the potential at a surface source point x, y is a surface field point, and σ is the charge density. The Green s function for 3D problems is: G(x, y) = 4πɛr(x, y) (2) where r is the Euclideandistance betweenx and y,and ɛ is the dielectric constant of the medium outside the conductor.

4 290 Boundary Elements XXVII ξ x σ σ V h n plate g x s 2 σ s2 V x n plate 2 σ s 2 s Figure 2: Parallel plate capacitor with two plates. s s ground plane V = Boundary integral equations for a thin plate in the undeformed configuration The BIE, in a Lagrangian framework, is presented next. From Nanson s law: nds = JN F ds (3) where n and N are unit normal vectors to b and B, at generic points x and X, respectively, F = x X is the deformation gradient, J = det(f) and ds is an area element on B. Here, X and x denote coordinates in the undeformed and deformed configurations, respectively. Using (3) one can obtain a lagrangian version of (), shown below for a thin plate. φ(x (X )) = S 2 S Ŝ B(Y)dS(Y) 4πR(X, Y)ɛ B(Y)dS(Y) 4πR(X, Y)ɛ B(Y)dS(Y) Ŝ 4πR(X, Y)ɛ (4) 2 Σ(X ) Σ(X )] = Sˆ B(X ) 4π H = Jσ2 N F 2ɛ S S ˆ = Σ2 N F 2Jɛ N F 2 (5) B(Y)R(X, Y) J(X )(N F )(X ) 4πR 3 (X ds(y), Y) R(X, Y) B(Y)J(X )(N F )(X ) B(X )J(Y)(N F )(Y)] 4πR 3 (X ds(y), Y) 2π 0 cos(ψ(θ))dθ S 2 B(Y)R(X, Y) J(X )(N F )(X ) 4πR 3 (X ds(y), Y) (6)

5 Boundary Elements XXVII 29 where, B =Σ Σ (7) Here Σ refers to the charge density in the undeformed configuration and H is the traction vector per unit undeformed surface area. Solving equation (4), and using (6) as a postprocessing step, one can obtain the charge densities Σ on the top and bottom plates respectively. Next, equation (5) is used to obtain the surface tractions on the plates. 3 Deformation analysis in linear elastic plate Nonlinear deformation of plates, without initial inplane forces, are discussed in this section. The plates are square (side = L), linearly elastic, and are of uniform rectangular crosssection (thickness h). The boundary condition considered here is a plate with all edges clamped. Also, the edges are immovable, i.e. u = v =0 on all edges of the plate. Here u(x, y) and v(x, y) are the inplane and w(x, y) the transverse displacement of the midplane of the plate. The force distribution (per unit area) H(x, y) is applied to the plate. 3. FEM model for plates with immovable edges Each plate element has four corner nodes with 6 degrees of freedom at each node. These are u, v, w, w,x,w,y,w,xy. For each element, one has: u v = w N (I) 0 0 N (O) ] q (I) q (O) ] (8) with: N (I) (x, y)] = N 0 N 2 0 N 3 0 N N 0 N 2 0 N 3 0 N 4 ], N (O) (x, y)] = P,P 2,..., P 6 ] (9) q (I) ]=u,v,..., u 4,v 4 ] T, q (O) ]=w, (w,x ), (w,y ), (w,xy ),..., w 4, (w,x ) 4, (w,y ) 4, (w,xy ) 4 ] T (0) Using the interpolations (8) and minimizing potential energy, results in the element level equations:

6 292 Boundary Elements XXVII K (I) 0 0 K (O) P ] q (I) q (O) ] = (e) ] 0 K (IO) 2K (IO)T N (I) 0 0 N (O) K (NI) ] T H x H y H z ] q (I) q (O) ] ] = P () dxdy (2) where (e) is the area of a finite element. The global version of () is now obtained in the usual way. Note that the inplane and outofplane (bending) matrices K (I) ] and K (O) ] are h and h 3, respectively, the matrix K (IO) ] Ah represents coupling between the inplane and outofplane displacements, and the matrix K (NI) ] A 2 h arises purely from the nonlinear axial strains. It is well known that for the linear theory K (O) ] << K (I) ] as h 0.Itisvery interesting, however, to note that if A/h remains O(), the bending matrix K (O) ], which arises from the linear theory, and the matrix K (NI) ] from the nonlinear theory, remain of the same order as h 0. This fact has important consequences for the modeling of very thin plates 2]. 4 BEM/FEM coupling A simple relaxation scheme is used for the coupling of BEM and FEM solutions. The BEM problem is solved for charge densities. Tractions are obtained and those are transferred to the FEM domain to obtain the deformed configuration. The FEM solves for the displacements and displacement gradients on a thin plate surface and returns them back to the BEM. The BEM problem is solved again for the deformed configuration, and new charge densities are obtained. This process is implemented iteratively until convergence. The deformation gradient is obtained with a membrane assumption as shown below. F = u,x u,y 0 v,x v,y 0 w,x w,y where, u, v and w represent the nodal displacements. 5 Numerical results 5. Material properties (3) Material properties used for Silicon conductors in free space are 0]:

7 Boundary Elements XXVII 293 V/2 silicon insulator V/2 silicon Figure 3: MEMS plates. 5.2 MEMS plates E = 69 GP a, ν =0.22, ɛ = Farad/m (4) 5.2. The problem Deformation of a thin silicon MEMS plate (the silicon is doped so that it is a conductor), subjected to a progressively increasing electrostatic field, is simulated here by the coupled fully Lagrangian BEM/FEM. Each plate is clamped around its edges and two plates are used in order to have a zero voltage ground plane (the plane of symmetry) midway between them (Fig. 3). Each plate is square of side L =3mm and thickness h =0.03 mm, andthegapg between them is 0. mm. Both plates are allowed to deform Discretization The BEM models only the top surface (s k, see Fig (2)) and the FEM the midsurface of each plate. No distinction need to be made between the top and mid surfaces since the plates are very thin. The mesh used here is as follows. The BEM and FEM domains each use 64 Q4 elements. Of course, the BEM has one degree of freedom per node and the FEM has 6 degrees of freedom at each node Results The voltage is applied in steps of 0 volts and tol =for the relaxation algorithm used here. Table provides the normalized central deflection due to the applied voltage. The second column is the value of w 0 /h (where w 0 is the transverse displacement at the center of the top plate, positive downwards) obtained by applying the voltage in several ten volt steps (i.e. 20 volts in 2 steps, 30 volts in 3 steps etc.). These results agree well with 7] where a full plate (i.e not a thin plate) model was used. Figure 4 shows the central deflection of the top plate as a function of the square of the applied voltage.

8 294 Boundary Elements XXVII Table : Central displacement of plate as a function of applied voltage. V Volts w 0 /h from several steps w 0 /h V 2 (volt 2 ) Figure 4: Central displacement of top plate as a function of square of applied voltage for relatively small values of applied voltage. 6 Discussion This work presents a first attempt at a fully Lagrangian approach for the analysis of coupled 3D MEMS problems with thin plates. The Lagrangian approach uses only the (usually simple) undeformed configuration of a plate for both the electrical and mechanical analyses thus obviating the need to discretize any deformed configurations. The hybrid BEM/FEM approach is able to handle thin plates (with h/l = /00) efficiently. Convergence is achieved for relatively large voltage steps. As one can notice from the Figure 4, the nonlinear effects become apparent for an applied voltage higher than about 5 volts 7]. Although equations () includes nonlinear terms, the present numerical implementation of the FEM is a linear one. Work is in progress on extending it to a nonlinear one. The twoplate MEMS exam

9 Boundary Elements XXVII 295 ple solved in this paper has h/l =/00 and g/l =/0. In practice ], one can have h/l =/000 with g h. Research along such lines is currently in progress. Acknowledgements This research has been partially supported by Grant # EEC of the National Science Foundation to Cornell University. We express our sincere gratitude for their support. References ] Senturia SD, Harris RM, Johnson BP, Kim S, Nabors K, Shulman MA, White JK. A computeraided design system for microelectromechanical systems (MEMCAD). Journal of Microelectromechanical Systems 992;:33. 2] Nabors K, White J. FastCap: a multipole accelerated 3D capacitance extraction program. IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems 99;0: ] Li G, Aluru NR. A Lagrangian approach for electrostatic analysis of deformable conductors. Journal of Microelectromechanical Systems 2002;: ] De SK, Aluru NR. FullLagrangian schemes for dynamic analysis of electrostatic MEMS. Journal of Microelectromechanical Systems 2004 (in press). 5] Bathe KJ. Finite element procedures. New Jersey : PrenticeHall, ] Shi F, Ramesh P, Mukherjee S. Dynamic analysis of microelectromechanical systems. International Journal for Numerical Methods in Engineering 996;39: ] Mukherjee S, Bao Z, Roman M, Aubry N. Nonlinear mechanics of MEMS plates with a total Lagrangian approach. Comp. Struc. In press. 8] Bao Z, Mukherjee S. Electrostatic BEM for MEMS with thin conducting plates and shells. Engineering Analysis with Boundary Elements. 2004; 28: ] Telukunta S, Mukherjee S. BEMFEM simulation of MEMS with densely packed thin plates using a total Lagrangian approach. Under Preparation. 0] Petersen KE. Silicon as a Mechanical Material. Proceedings of the IEEE 982;70: ] Roman M, Aubry N. Design and fabrication of electrostatically actuated synthetic microjets. ASME Paper No. IMECE , New York: American Society of Mechanical Engineers, ] Bao Z, Mukherjee S, Roman M, Aubry N. Nonlinear vibrations of beams, strings, plates and membranes without initial tension. ASME Journal of Applied Mechanics. 2004; 7:55559.