Finite-element modeling of the transient response of MEMS sensors

Transcription

1 Smart Mater. Struct. 6 (1997) Printed in the UK PII: S (97) Finite-element modeling of the transient response of MEMS sensors Young-Hun Lim, Vasundara V Varadan and Vijay K Varadan Research Center for the Engineering of Electronic and Acoustic Materials, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA Received 3 June 1996, accepted for publication 16 October 1996 Abstract. The finite-element method of analysis is unrestricted by size considerations and is well suited for the study of very small structures such as MEMS devices embedded in structures. This paper presents the numerical approach and results for a silicon-based micro-flow sensor for pulsed-flow sensing. The new approach presented here is the treatment of transient problems. A finite-element formulation is presented for modeling the dynamic response of piezoelectric ceramic sensors embedded in a micro-cantilever subjected to mechanical loading resulting from fluid flow. An unconditionally stable method (called the α-method) is used for the direct integration of the equations of motion and implicit explicit procedures are used for the transient analysis of the linear system of equations. For verification of the code, the device is tested for step, rectangular pulse and sinusoidal loading. For the case of the micro-fluid flow sensor, numerical results are in good agreement with the available experimental data based on a piezoresistive sensor. The numerical approach presented here may be used in CAE models for microsensors under more realistic, transient excitation. 1. Introduction Smart structures with integrated self-sensing using embedded MEMS (micro-electromechanical system) sensors for diagnosis and control capabilities could lead to a new design dimension for the next generation of smart structures. In order to optimize the design of complicated structures that are subjected to critical loading, dynamic simulation of such smart structures is necessary. Robust numerical simulations of advanced smart structures will result in the development of designs that perform better than their passive counterparts while costing less. This new technology can then be applied to the design of large-scale space structures, aircraft structures and satellites, as well as automobiles and manufacturing systems 1. The finite-element method is very attractive since it can be applied to any geometry for any set of material properties and loading conditions as long as the appropriate constitutive relationships and equilibrium conditions are met. Since the method is not restricted by size, one can use the so called zoom feature in finite-element meshing to use different-size elements to describe a miniature MEMS device relative to the very large structure in which it is embedded. Thus one can realize computational economy without sacrificing accuracy. Many researchers have used the finite-element method for modeling piezoelectric sensors and actuators since the 1970s. The first finiteelement formulation was proposed by Allik and Hughes 2. However, the derived isoparametric hexahedron and tetrahedral elements were too thick for thin continuum applications. For more advanced applications, Kim et al 3 have proposed a hybrid element method using 3D elements for the piezoelectric device region to obtain sufficient detail, plate elements for the structure to reduce numerical stiffening and developed transition elements at the interface of 3D piezoelectric elements and plate elements. Kim et al 4 have also conducted numerous studies on the numerical aspects of modeling electrical boundary conditions, the treatment of electrodes as equipotential surfaces, the optimal placement of sensors and actuators on a structure and other aspects using finite-element modeling. A comprehensive paper was written by Lerch 5 on the simulation of piezoelectric devices that included time domain modeling. When structures are subjected to disturbances that are applied suddenly or that undergo sudden changes in the environment, time explicitly enters the system equations of motion. The objective of this investigation is to develop finite-element analysis for modeling the electromechanical response of composite structures containing MEMS devices under dynamic mechanical loading. In this paper, a simple micro-cantilever beam under transient loading due to fluid flow at its tip is used to numerically simulate the response of a miniature pulsed flow sensor. Although finite-element methods have been developed for piezoelectric structures, the use of finite-element techniques for predicting the response of MEMS sensors is still in its infancy. Until now, a considerable amount /97/ $19.50 c 1997 IOP Publishing Ltd 53

2 Y-H Lim et al of work for MEMS structures has focused heavily on experimentation, fabrication and processing. Only a few static analytical and numerical models have been developed. There is an increasing need for CAD/CAE modeling to assist designers of devices because of the geometric complexity of MEMS devices as well as the coupling of different physical effects. This paper is concerned with a microfluidic component, particularly a micro-flow sensor for measuring drag force. A preliminary understanding of the performance characteristics of a micro-flow sensor has been established through experimentation. An accurate analysis of the electromechanical performance of this device is needed for design optimization. Therefore, a 3D finite-element method is proposed to accurately predict the transient response of a silicon based microsensor subject to a known drag force. The dynamic matrix equations derived for linear piezoelectricity are found to be reducible, in form, to the ordinary matrix equation encountered in structural dynamics which is calculated by a direct integration algorithm (the so called α-method) 6. The sensor response measured as an electrical voltage can be solved from the derived system of equations. Several cases are studied, and the dynamic responses of the system (such as displacement, and electric potential) are presented as a function of time. The calculations are in good agreement with experimental results. 2. Modeling and formulation 2.1. Formulation of piezoelectric elements Piezoelectric materials are anisotropic and the elastic field in such materials is coupled with the electric field. Finiteelement equations for piezoelectric materials have already been formulated in many papers 2 5. The linear constitutive relations expressing the coupling between the elastic field and the electric field can be written as T = C E S h T E D = hs + b S E where T is the stress tensor, D is the electric displacements, S is the strain tensor, h is the piezoelectric coupling constant, E is the electric field, b S is the dielectric constant at constant strain, C E is the elastic stiffness tensor evaluated at constant E field and h T is the transpose of h. The electric field E is related to the electric potential φ by E = φ. The displacement and potential for each element can be expressed, respectively, as u = N u û φ = N φ ˆ where u is the displacement vector, N u and N φ the interpolation functions for the variables of u and φ and ˆ denotes the nodal point values. Putting the strain displacement relation in terms of the nodal displacement yields S = B u û where B u is the product of the differential operating matrix relating S to (1) (2) the shape function matrix N u. Similarly let E = φ= N φˆ = B φˆ. The equation of motion for a piezoelectric body can be derived from the principle of minimum potential energy by means of a variational functional. The resultant equations can be represented in matrix form from the assembly of all the individual finite-element equations 2. Muu ü + Kuu u + Kuφ ={F} K T uφ u + Kφφ ={Q} (3) where M uu = ρnu TN u dv is the kinematically constant mass matrix, K uu = Bu TCE B u dv the elastic stiffness matrix, K uφ = Bu ThT B φ dv the piezoelectric coupling matrix, K φφ = Bφ TbS B φ dv the dielectric stiffness matrix, {F }= V NT b f bdv+ S 1 NS T 1 f s ds 1 +Nu Tf c the mechanical force, {Q} = S 2 NS T 2 q s ds 2 Nφ Tq c the electrical charge, f b the body force, f s the surface force, f c the concentrated force, q s the surface charge and q c the point charge. For convenience, the ˆ sign is omitted. Generally, all structures are slightly damped due to structural damping. Since it is difficult to quantify the structural damping matrix, artificial linear viscous damping is added, such that (3) can be modified to ü Muu + Cuu u + Kuu u + Kuφ ={F} (4) K T uφ u + Kφφ ={Q} where C uu = ηm uu + λk uu where η and λ are called Rayleigh coefficients. Damping constants η and λ are determined empirically by examining critical damping at two different frequencies 6. The electric field boundary condition requires that the electrode surface is an equipotential one and the summation of the nodal electric charges on it should be zero. ˆ i = ˆ i+1 =...=constant Qi = Condensation of system matrices The basic concept of matrix condensation Guyan s reduction 8, 9 scheme is based on Gaussian elimination solution of equations for unknowns but stopping before the stiffness matrix has been fully reduced. A congruent transformation matrix T c can be calculated using the static system equation, Tc = I, K uφ Kφφ 1. After assembling all element matrices, and performing the condensation of the degrees of freedom, the system dynamic equation is written as Muu ü + Cuu u + K u = { F } (5) where K = K uu Kuφ Kφφ 1 { F } ={F} K uφ Kφφ 1{Q} K T uφ and electrical potential can be recovered by the relationship { } = 1 K φφ {Q} Kuφ T u. 54

3 Transient response of MEMS sensors 2.3. Time-history analysis: the direct integration method In direct integration methods or step-by-step methods, a finite-difference approximation is used to replace the time derivatives appearing in (4) (i.e., ü and u) by differences of displacements u at various instants of time. The α- method 7 which is implicit and unconditionally stable was proposed by Hilber et al. It has numerical damping which cannot be introduced in the Newmark method, and retains second-order accuracy with appropriate choice of parameters. It provides numerical dissipation without introducing excessive algorithmic damping in the important low-frequency modes. It also reduces spurious, nonphysical oscillations that may occur in the high-frequency response due to excitation of spatially unresolved modes. The Wilson θ method and Houbolt s method, which are also implicit and unconditionally stable, are too dissipative in the lower modes, requiring a time step to be taken that is smaller than that needed for accuracy. A disadvantage of Newmark methods is that algorithmic dissipation can only be obtained at the expense of reduced accuracy. The α-method, which is a one-parameter family of algorithms, does not have this weakness. The parameter α defines the amount of numerical dissipation. The term implicit means that the displacement vector is a function of both previous (known) and current (unknown) displacements, velocities and accelerations. The term unconditionally stable means that the solution of a linear system will never diverge, no matter how large the integration time step ( t) is. The method uses finitedifference formulas of the Newmark method with a modified equation of motion as given below: Muu ün+1 + (1 + α) C uu un+1 α C uu un +(1 + α) K u n+1 α K u n = { F ( )} t n+α where t n+α = (1 + α)t n+1 αt n = t n+1 + α t. The parameters are selected such that α 1 3, 0, γ = (1 2α)/2, and β = (1 α) 2 /4 from results of a stability and accuracy analysis of the α-method 7. For each value of α in the admissible range, the algorithm is unconditionally stable and of second-order accuracy. When these guidelines are used, with α = 0, the method reduces to the Newmark method (trapezoidal rule), which has no numerical damping. Decreasing α increases the amount of numerical damping. Thus α = 0.1is appropriately chosen to give adequate dissipation in higher modes and at the same time guarantee that the lower modes are not affected too strongly. Hughes and Liu 10 endeavored to design improved implicit explicit algorithms which were amenable to stability and accuracy analysis at the same time. The basic ingredients in implicit explicit procedure are the three main steps. The first step is the definition of an implicit integrator. The α-method is selected for this purpose. The second step is the development of an explicit predictor corrector method, which is constructed to be naturally associated with the chosen implicit integrator. Table 1. A computational flow chart for direct integration by the α-method. The third step and final step is the synthesis of the implicit and explicit schemes by way of a modified time-discrete equation of motion. In table 1, a flow chart for direct integration by the α-method is presented. Numerical dissipation inherent in the α-method is very effective when solving structural dynamics problems, such as wave propagation and impact problems which are characterized by large gradients or discontinuities in the response due to a propagating wave front. 3. Cantilever plate example for code verification A good example of the applications of MEMS in a typical smart structure is a silicon cantilever plate containing a piezoelectric sensor with one end fixed to the structure. This is a relatively simple problem and is presented in this work for code verification. A true MEMS example of a fluid micro-flow sensor will be presented in the next section. Figure 1 depicts the cantilever beam example. Three 20- node solid elements are used to model the piezoelectric device regions and nine-node flat-shell elements are used in the remaining part of the plate. Transition elements connect the solid elements to shell elements 3 (figure 2). It is assumed that the piezoelectric sensor is bonded perfectly to the plate. Before proceeding with the dynamic analysis of the micro-sensor, the algorithm is tested by comparing the natural frequencies and response for pure elastic vibrations of the structure (no piezoelectric sensing). Analytical 55

4 Y-H Lim et al Figure 1. An aluminium cantilever plate with a piezoelectric sensor for FEM modeling. Figure 2. Different types of finite element: (a) 20-node solid element; (b) 13-node transition element; (c) nine-node shell element. Figure 3. The uniform cantilever beam subjected to a step force (area, m 2 ; moment of inertia, m 4 ). Figure 4. Types of load history function used in the numerical models: (a) unit step; (b) rectangular pulse; (c) triangular pulse; (d) sinusoidal force. solutions of the continuous system to external excitation can be obtained conveniently by using normal modes and modal superposition. The clamped free uniform beam of figure 3 is subjected to a step force of 100 N at its tip. Table 2 shows the comparison between exact and FEM results. The numerical solution provides reasonable accuracy compared to the analytical solution. As might be expected, the finiteelement model is a little stiffer than the analytical model. From these results, the use of combination models of solid, transition and shell elements shows good accuracy even though only four elements are used. The discretization error in the finite-element solution can be reduced by using a sufficiently refined mesh. The properties of aluminum and PZT-5H are listed in the appendix and obtained from 11. The accuracy of the transient solution including a piezoelectric sensor also increases with decreasing time step. The smaller the time step, however, the larger the number of iterations required for solution. Therefore, the time step chosen should be small enough for an accurate solution but large 56

5 Transient response of MEMS sensors Figure 5. Tip displacement and piezoelectric sensor response for a cantilever plate for a step excitation. Figure 6. Tip displacement and piezoelectric sensor response for a cantilever plate for a rectangular pulse excitation. enough to minimize iteration. At least ten time steps per period must be taken for accuracy. The time step used is t = 2/ω max = sec where ω max is the highest natural frequency of detk ω 2 M = 0. Structural damping is assumed as η = 7.5 and λ = and α = 0.1. Four loading cases (figure 4) are studied in this section: (i) a step function with a rise time of t 0 = 0 s and a peak magnitude of F 0 = 100 N; (ii) a rectangular pulse of height F 0 = 100 N and a duration t 0 = 0.17 s; (iii) a triangular pulse of duration t 0 = 0.17 s and a peak magnitude of 100 N; (iv) a harmonic force of 57

6 Y-H Lim et al Figure 7. Tip displacement and piezoelectric sensor response for a cantilever plate for triangular pulse excitation. Figure 8. Tip displacement and piezoelectric sensor response for a cantilever plate for sine wave excitation. amplitude 100 N and τ = 0.5 s with initial displacement u 0 = m. Figure 5 presents a time history plot of sensor output and tip displacement for a step function force. A suddenly applied force produces a peak response (displacement, electric potential) that is twice that of a slowly applied force of the same magnitude. The result shows that the peak response approaches the static value due to structural damping effects. In figures 6 8, the tip displacement and sensor voltage response are plotted for rectangular 58

7 Transient response of MEMS sensors Figure 9. A schematic view of the micro-flow sensor. Figure 10. Geometry and dimensions for the cantilevered micro-flow sensor. Figure 11. Drag force at the tip of the silicon cantilever plate. and triangular pulse excitation and a sinusoidal pulse, respectively. In all three cases, we can notice an initial transient response phase, then damped vibration at the natural frequency of the system and approach to the static 59

8 Y-H Lim et al Figure 12. Sensor response corresponding to the micropump pulsating at 1 Hz. Table 2. A solution comparison for the cantilever beam. Natural frequency Natural frequency Maximum deflection ω 1 (rad s 1 ) ω 2 (rad s 1 ) at tip of plate (m) Analytical solution FEM solution Error percentage (%) steady state value. The sensor response faithfully follows the structural response. 4. Numerical simulation of a MEMS micro-flow sensor A schematic description of the micro-flow sensor is presented in figure 9. These devices have been fabricated by micromachining of two silicon or Pyrex wafers which are assembled to form inlets and outlets for the fluid. The flow sensor is a silicon cantilever with a piezoelectric ceramic deposited on the cantilever plate as shown in figure 10 (plate dimensions, 2260 µm 250 µm 9.65 µm; sensor dimensions 200 µm 250 µm 2.5 µm) located 440 µm from the fixed end. Cubic crystal silicon material properties are obtained from 12. The calibration of the flow sensor is achieved by applying a known drag force exerted by the fluid on the tip of the plate. The calculation of the drag force is done by using the Navier Stokes law with the experimentally measured flow rate for a silicon micropump pulsating at a frequency of 1 Hz 13. Figure 11 represents the typical characteristics of the drag force in which the negative values are probably due to a degradation of the valve seat resulting in a back flow. Figure 12 presents the sensor response curves, calculated by the 3D finite-element analysis described in the previous sections for a given pumping condition. The piezoelectric sensor has a response fast enough for measuring the pulsating flow. The dark area in figure 12 corresponds to the characteristic transient behavior, which will disappear after a long time, i.e. it approaches the steady state. It can be also seen that a small volume of liquid flows back through the pump when the valve closes 13. The shape and the magnitude of the sensor output are in good agreement with the experimental results that are obtained by using a piezoresistive sensor of slightly different dimensions (3000 µm 1000 µm 30 µm) for the cantilever. The main difference between the numerical result and experimental result may be due to the inaccurate estimate of structural damping and in part due to the indirect estimation of drag force in numerical analysis. The sensing mechanism for the piezoresistive sensor used in the experimental study is based on change in electrical resistance due to mechanical strain. It appears that the piezoresistive sensor has higher sensitivity than the piezoelectric sensor used in the finite-element analysis especially at peak values of the flow. 60

9 Transient response of MEMS sensors 5. Discussion and conclusions This paper is concerned with the finite-element modeling of the dynamic response of a MEMS device. Guyan s reduction scheme was employed to condense the elastodynamic and electric degrees of freedom. The time-history response was calculated by a direct integration algorithm (αmethod) to accommodate piezoelectric constitutive equations. To numerically simulate the dynamic response of MEMS, we used the example of a micro-flow sensor subjected to pulse flow. In order to verify the proposed analysis, finite-element results were compared with known analytical results for an ordinary cantilever structure subjected to various dynamic loads. The agreement is good. The influence of dynamic loading on the transient response of smart structures is demonstrated. The results indicate that a system would generally oscillate during excitation with varying amplitudes, and at a frequency that corresponds to the natural frequency of the system, but oscillations (voltage, displacement) would gradually reach a steady state value because of structural damping. The transient analysis of a silicon-based micro-flow sensor was studied next. Its operation is based on a piezoelectric sensor that measures fluid flow by measuring the induced strain that it causes in a microcantilever. Agreement with available experimental data is good. The numerical model captures the transient response very well. The method is unconditionally stable. It is believed that this computer code can be further developed into a useful tool for designing MEMS devices embedded in structures. The use of other active materials, such as optical fibers, electrorheological fluids, piezoresistive materials and shape memory alloys, in conjunction with MEMS devices, needs to be explored further. Finite-element formulations incorporating the constitutive equations for such materials are available and can be included in the formulation used here. Appendix. Material properties (i) Aluminum. Young s modulus = Nm 2, Poisson s ratio = 0.32, density = 2800 kg m 3. (ii) PZT-5H. Density = 7500 kg m 3. The elastic property matrix with constant electric fields is C E = sym Nm The piezoelectric strain matrix is h T = Cm 2. The dielectric property matrix with constant strain is b = Fm (iii) Crystal silicon. Density = 2329 kg m 3. The elastic property matrix is C E = sym Nm 2. References 1 Varadan V K and Varadan V V MEMS electronics and applications for smart structures Micromachining and Microfabrication 95; SPIE Proc Allik H and Hughes T J R 1970 Finite element method for piezoelectric vibration Int. J. Numer. Methods Eng Kim J, Varadan V V and Varadan V K 1996 Finite element modeling of structures including piezoelectric active devices Int. J. Numer. Methods Eng. at press 4 Kim J W, Varadan V V and Varadan V K 1995 Finite element optimization methods for the active control of radiated sound from a plate structure J. Smart Mater. Struct Lerch R 1990 Simulation of piezoelectric devices by twoand three-dimensional finite elements IEEE Trans. Ultrason. Freq. Control UFC Hughes T J R 1987 The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Englewood Cliffs, NJ: Prentice-Hall) 7 Hilber H M, Hughes T J R and Taylor R L 1977 Improved numerical dissipation for time integration algorithms in structural dynamics Earthquake Engineering and Structural Dynamics pp Guyan R J 1965 Reduction of stiffness and mass matrices AIAA Cook R D and Shah V N 1978 A cost comparison of two static condensation-stress recovery algorithms Int. J. Numer. Methods Eng Hughes T J R and Liu W K 1978 Implicit explicit finite elements in transient analysis: implementation and numerical examples J. Appl. Mech Auld B A 1973 Acoustic Fields and Waves in Solids vols 1 and 2 (New York: Wiley) 12 Hjort K and Soderkvist J 1994 Gallium arsenide as a mechanical material J. Micromech. Microeng Gass V, van der Schoot B H and de Rooij N F 1993 Nanofluid handling by micro-flow-sensor based on drag force measurements Proc. IEEE MEMS Workshop (Fort Lauderdale, FL, 1993) pp