Modeling of Thermoelastic Damping in MEMS Resonators

Transcription

1 Modeling of Thermoelastic Damping in MEMS Resonators T. Koyama a, D. Bindel b, S. Govindjee a a Dept. of Civil Engineering b Computer Science Division 1 University of California, Berkeley

2 MEMS Resonators Micron size devices Mechanical Filters in IC 2

3 MEMS Resonators Micron size devices. Mechanical Filters in IC 3

4 MEMS Resonators Micron size devices. Mechanical Filters in IC Electrostatically actuated 4

5 MEMS Resonators Micron size devices. Mechanical Filters in IC Electrostatically actuated 5

6 MEMS Resonators Micron size devices. Mechanical Filters in IC Electrostatically actuated The Quality factor (Q) Index of how narrow and sharp the peak is. 6

7 MEMS Resonators Micron size devices. Mechanical Filters in IC Electrostatically actuated The Quality factor (Q) Index of how narrow and sharp the peak is. 7

8 MEMS Resonators Micron size devices. Mechanical Filters in IC Electrostatically actuated The Quality factor (Q) Index of how narrow and sharp the peak is. 8

9 Zener s Model(1937) Obtained closed form algebraic relation for Q ted of a beam. Experimentally verified. Based on Euler-Bernoulli beam theory Applicable geometry is restricted. 9 C.Zener, Physical Review, 1937,pp

10 Outline Governing equations (Non-dimensionalized form) Finite Element Discretization Time-harmonic response Forced response Modal response Closure 10

11 Governing Equations(Non-dimensionalized form) Equations obtained from thermodynamical principles Polysilicon is linear elastic, MEMS deformations are small. -Assume linear elasticity. -Temperature fluctuations are small (Removes non-linearity in energy balance) Balance of linear momentum Weak coupling Energy Balance Strong coupling (PolySilicon values have been used for ξ.)

12 Finite Element Discretization Weak form Discretized system of equations 1. M and C are singular, K is unsymmetric. 2. Diagonal blocks of M,C, and K are s.p.d Coupling term has relation,

13 Time-harmonic Response Would like to evaluate: the Quality factor(q ) Assume time-harmonic response. Forced response Evaluate transfer function Modal response 13

14 First-Order Form(GEP) By introducing an auxiliary variable we obtain a generalized eigenvalue problem(gep). Formulation where we obtain two Symmetric but Indefinite matrices Quadratic eigenvalue problem Increases size GEP 14 N=N u +N θ 2N u +N θ 67%(2D), 75%(3D) increase

15 Perturbation Method Exploit weak coupling in balance of linear momentum 0. Assume solution is equal to the mechanical problem plus a small perturbation. 1.Solve for initial guess. 2.Compute corresponding thermal vector N u GEP N θ linear solve 3. Solve for update by adding constraint N u +1 linear solve 15 2N u +N θ GEP Reduces size N u GEP N θ, N u +1 linear solve Decreases to 27%(2D), 40%(3D) Use LU from GEP

16 Numerical example:beam Structure 2D Plane Stress assumptions 16 W.-T.Hsu, J.R.Clark,C.T.-C.Nguyen,Transducers 01,pp

17 ROM of the Forced Response Compute transfer function from forced response. Must solve size N system for each data point. Computationally expensive. Compute a reduced order model, accurate around a center frequency, based on the Second Order Arnoldi method. 1. Generate sequence of vectors(spans 2 nd Order Krylov subspace) which can describe the response. 2. Construct ROM by a Galerkin projection of the system onto the generated smaller subspace of solutions. 17

18 SOAR for the Thermoelastic problem SOAR procedure(bai and Su 2004) Choice of basis for projection SOAR basis (n) (2n) M/T split basis Preserves matrix structure: 1. h.p.d of diagonal submatrices 2. zero structure 18 Z. Bai and Y. Su. SIAM J. Sci. Comp., 2004.

19 Numerical Example:Ring resonator 2D Plane Stress NDOF=38989 ROM = 21 -Iterations: 20(SOAR Basis) 10(M/T Split Basis) 19

20 Numerical Example:Ring resonator 2D Plane Stress NDOF=38989 ROM = 21 -Iterations: 100 points Hours. Seconds. 20(SOAR Basis) 10(M/T Split Basis) 20 Forced at 705.5[MHz]

21 Closure Modal Response Perturbative structure 2N u +N θ GEP N u GEP N θ, N u +1 Linear solve Forced Response SOAR N M/T split basis Extends to incorporate anchor loss with Perfectly Matched Layers(PML) n Preserves structure Higher-order accuracy Complex symmetric matrices 21

22 22 Reference Slides

23 Zener s Model Proposed by Zener in Evaluated Q of a beam in flexural mode Linear coeffecient of thermal expansion Specific heat (const. volume) Density Young s modulus 23 Thermal conductivity Reference temperature Beam width C.Zener, Physical Review, 1937,pp

24 Thermoelastic Damping Energy dissipation mechanism due to coupling between the mechanical and thermal domain. Apply external forces Mechanical Domain Time varying local stresses Time varying local strains Thermal Domain Local temperature gradients Irreversible heat flow Resulting in energy dissipation 24

25 Governing Equations Equations obtained from thermodynamical principles Polysilicon is linear elastic, MEMS deformations are small. -Assume linear elasticity. -Temperature fluctuations are small (Removes non-linearity in energy balance) Balance of linear momentum Energy Balance 25

26 Non-dimensionalization Expression for coefficients

27 First-Order Form(GEP) By introducing an auxiliary variable we obtain a generalized eigenvalue problem(gep) RHS matrix is s.p.d 2. --Symmetric but indefinite matrices 27 N u +N θ quadratic eigenvalue problem Increases size 2N u +N θ GEP 67%(2D), 75%(3D) increase

28 Example with PML:Michigan FF Beam TED+PML PML 28

29 Jacobi-Davidson Type of interation The residuals of the eigenvalue problem are, Then, By adding a regularizing constraint on u, 29 If we now drop the coupling term we obtain the Perturbation Method equations. This iteration may be repeated to further refine the solution.

30 Example:Ring Resonator with PML 2D Plane Stress PML parameter f0=20 NDOF=38989 ROM = 21 Normal n=20 M/T, Real n=10 M/T+Real n=5 30

31 TED vs. TED+PML(Ring Resonator) TED TED+Anchor Loss 31

32 Numerical Example:Ring resonator 2D Plane Stress NDOF=38989 ROM = 41 Normal n=20 M/T, Real n=10 M/T+Real n=5 32

33 33 Numerical Example:Ring resonator

34 34 Numerical Example:Ring resonator

35 Choosing the Galerkin Projection Basis Choice of basis complex basis real basis [diagonal submatrices remain s.p.d] nonsplit basis (n) (2n) 2 nd order accurate for pure mech. case m/t split basis [matrix structure (zeros) preserved] (2n) Preserves 1. s.p.d.(h.p.d.) of diagonal submatrices 2. zeros in matrices (4n) Preserves 1. complex symmetry of diagonal submatrices 2. zeros in matrices 35 Only TED TED + Perfectly Matched Layer