Uncertainty Quantification in MEMS

Transcription

1 Uncertainty Quantification in MEMS N. Agarwal and N. R. Aluru Department of Mechanical Science and Engineering for Advanced Science and Technology

2 Introduction Capacitive RF MEMS switch Comb drive Various sources of uncertainties in MEMS: Material properties such as Young s modulus, Poisson s ratio etc. Geometrical features such as dimensions, gap between electrodes etc. Operating environment and Boundary conditions Objectives: In order to design reliable and efficient MEMS devices Quantify the effect of uncertainties on various performance parameters Use UQ information to identify critical design parameters Treatment of uncertainties in MEMS: Safety factors: over conservative designs Monte Carlo (MC) simulations: computationally expensive

3 Coupled Electro-Mechanical Problem Electromechanical coupling Mechanical Analysis Electrostatic Analysis

4 Mathematical Representation of Uncertainty Probability space Random variable Elementary event uncertain parameters can be modeled as random variables Random field or process uncertain spatial or temporal functions can be represented as random fields Generalized Polynomial Chaos Stochastic input Total number of terms Stochastic process Deterministic functions Orthogonal polynomials From Askey scheme n: dimension; p:order [N. Wiener, 1938], [Ghanem and Spanos, 1991], [Xiu and Karniadakis, SIAM, 2002]

5 UQ for Coupled Electro-Mechanical Problem In order to model stochastic electro-mechanical interaction, we need two components Stochastic Electrostatic Analysis Input: uncertain geometry or deformation Output: uncertain electrostatic pressure Stochastic Mechanical Analysis Input: uncertain material properties, geometry and/or loading Output: uncertain deformation Challenges Geometrical uncertainty: random computational domain Multiphysics: uncertainty propagation across various physical fields

6 Geometrical Variations in Electrostatics Objective: To quantify the uncertainty in surface charge density and output parameters such as capacitance and electrostatic pressure Uncertain geometry Lagrangian g Electrostatics dω dω Conductor 1 Conductor 2 Boundary integral equations (BIE) Undeformed Deformed Lagrangian BIE [Li and Aluru, 2002]

7 Stochastic Lagrangian Electrostatics Expand uncertain displacement in terms of GPC basis functions: Uncertain geometry Mean geometry Stochastic Green s function Uncertain Displacement Position of a point on the uncertain boundary: Stochastic deformation gradient Stochastic Lagrangian BIE [N. Agarwal and Aluru, JCP, 2007]

8 Electrostatics: Stochastic Discretization Step 1: Unknown variables Step 2: Expanding the nonlinear functions in terms of GPC basis functions Step 3: Using Galerkin projection in the space of PC basic functions [Debusschere et al., SIAM, 2004] Coupled Intergal Eqs. Step 4: Spatial discretization using Boundary Element Method (BEM) [N. Agarwal and Aluru, JCP, 2007]

9 Numerical Examples: Interconnect Circuits Effect of uncertain gap on capacitance Random translational displacement Mean and SD For bottom surface

10 Interconnect Circuits (contd.) Empirical i lvs LBIE Empirical i lformula for capacitance: PDF of capacitance for PC, Emp and MC Mean and standard deviation for capacitance

11 Stochastic Mechanics Deterministic Case : Stiffness matrix Residual vector Incremental nodal displacement vector After linearization and spatial discretization Stochastic Case : Stochastic stiffness matrix Random residual vector Random incremental nodal displacement vector [Acharjee and Zabaras, CMAME, 2006]

12 Mechanics: Stochastic Discretization Step 1: Expand the unknown using GPC : Step 2: Express the random stiffness matrix and residual vector in terms of fgpc basis functions : Step 3: Galerkin projection in the space of GPC basis functions :

13 Numerical Example: Fixed Beam Effect of uncertain Young s modulus Uniformly distributed ibtdbetween limits it where, PDF of midpoint vertical displacement Mean vertical displacement with error bars

14 Coupled Stochastic Algorithm [Agarwal and Aluru, CMAME, 2008]

15 Numerical Example: MEMS Switch Effect of uncertain Young s modulus and gap, assumed uniformly distributed ib td 80 [E 1 0 ΔE,E 0 + ΔE], E 0 =169 GPa, ΔE = 0.1E 0 V g [g 0 Δg,g 0 + Δg], g 0 =11 μm, Δg = 0.1 μm PDF of tip displacement using GPC and MC, V=7.0 V Vertical displacement with error bars, V=7.0 V

16 MEMS Switch: Verification using MC Mean and std for peak displacement, V=7.0 V MC simulations using 10,000 realizations PDF of tip displacement using GPC for different applied voltage V=5.0 V V=7.4 V V=7.8 V

17 MEMS Switch: Sensitivity Analysis Sensitivity of tip displacement to variations in Young s modulus and gap Sensitivity w.r.t. Young s modulus Sensitivity w.r.t. gap Observations Tip displacement is more sensitive to variations in gap than Young s modulus Sensitivity information can be used to identify critical design parameters

18 Comb Drive Effect of uncertain Young s modulus [E 0 ΔE,EE 0 + ΔE], E 0 = 200 GPa, ΔE = 0.1E 0 PDF of capacitance, V=4.6 V Mean and standard deviation of capacitance, V=4.6 V Mean and standard deviation of displacement, V=4.6 V

19 Summary Stochastic Lagrangian framework based on generalized polynomial chaos (GPC) to handle uncertainties es in mechanical ca properties es and geometrical parameters for static analysis of electrostatic MEMS Propagation of uncertainties from one energy domain to other energy domains Effect of uncertainties on various design parameters can be easily computed Verification using Monte Carlo (MC) simulation