Numerical and semi-analytical structure-preserving model reduction for MEMS

Transcription

1 Numerical and semi-analytical structure-preserving model reduction for MEMS David Bindel ENUMATH 07, 10 Sep 2007

2 Collaborators Tsuyoshi Koyama Sanjay Govindjee Sunil Bhave Emmanuel Quévy Zhaojun Bai

3 Resonant MEMS Microguitars from Cornell University (1997 and 2003) MHz-GHz mechanical resonators Favorite application: radio on chip Close second: really high-pitch guitars

4 The Mechanical Cell Phone RF amplifier / preselector Mixer IF amplifier / filter... Tuning control Local Oscillator Your cell phone has many moving parts! What if we replace them with integrated MEMS?

5 Ultimate Success Calling Dick Tracy!

6 Example Resonant System V in AC DC V out

7 Example Resonant System V in AC L x C 0 C x R x V out

8 The Designer s Dream Ideally, would like Simple models for behavioral simulation Parameterized for design optimization Including all relevant physics With reasonably fast and accurate set-up We aren t there yet.

9 The Hero of the Hour Major theme: use problem structure for better models ODE structure Complex symmetric structure Perturbative structure Geometric structure

10 SOAR and ODE structure Damped second-order system: Mu + Cu + Ku = Pφ y = V T u. Projection basis Q n with Second Order ARnoldi (SOAR): M n u n + C n u n + K n u n = P n φ y = Vn T u where P n = Q T n P, V n = Q T n V, M n = Q T n MQ n,...

11 Checkerboard Resonator

12 Checkerboard Resonator D+ D D+ D S S+ S+ S Anchored at outside corners Excited at northwest corner Sensed at southeast corner Surfaces move only a few nanometers

13 Performance of SOAR vs Arnoldi 10 5 N = 2154 n = 80 Bode plot Magnitude Phase(degree) Exact SOAR Arnoldi

14 Aside: Next generation

15 Complex Symmetry Model with radiation damping (PML) gives complex problem: (K ω 2 M)u = f, where K = K T, M = M T Forced solution u is a stationary point of I(u) = 1 2 ut (K ω 2 M)u u T f. Eigenvalues of (K, M) are stationary points of ρ(u) = ut Ku u T Mu First-order accurate vectors = second-order accurate eigenvalues.

16 Disk Resonator Simulations

17 Disk Resonator Mesh Electrode Resonating disk Wafer (unmodeled) 2 x PML region x 10 5 Axisymmetric model with bicubic mesh About 10K nodal points in converged calculation

18 Symmetric ROM Accuracy 0 Transfer (db) Frequency (MHz) 200 Phase (degrees) Frequency (MHz) Results from ROM (solid and dotted lines) near indistinguishable from full model (crosses)

19 Symmetric ROM Accuracy 10 2 H(ω) Hreduced(ω) /H(ω) Structure-preserving ROM Arnoldi ROM Frequency (MHz) Preserve structure = get twice the correct digits

20 Perturbative Structure Dimensionless continuum equations for thermoelastic damping: σ = Ĉɛ ξθ1 ü = σ θ = η 2 θ tr( ɛ) Dimensionless coupling ξ and heat diffusivity η are 10 4 = perturubation method (about ξ = 0). Large, non-self-adjoint, first-order coupled problem Smaller, self-adjoint, mechanical eigenproblem + symmetric linear solve.

21 Thermoelastic Damping Example

22 Performance for Beam Example Time (s) Perturbation method First order form Beam length (microns)

23 Aside: Effect of Nondimensionalization 100 µm beam example, first-order form. Before nondimensionalization Time: 180 s nnz(l) = 11M After nondimensionalization Time: 10 s nnz(l) = 380K

24 Semi-Analytical Model Reduction We work with hand-build model reduction all the time! Circuit elements: Maxwell equation + field assumptions Beam theory: Elasticity + kinematic assumptions Axisymmetry: 3D problem + kinematic assumption Idea: Provide global shapes User defines shapes through a callback Mesh serves defines a quadrature rule Reduced equations fit known abstractions

25 Global Shape Functions Normally: u(x) = j N j (X)û j Global shape functions: û = û l + G(û g ) Then constrain values of some components of û l, û g.

26 Hello, World! Which mode shape comes from the reduced model (3 dof)? (Left: 28 MHz; Right: 31 MHz) Student Version of MATLAB Student Version of MATLAB

27 Conclusions Respecting problem structure is a Good Thing! ODE structure Complex symmetric structure Perturbative structure Geometric structure Result: Better accuracy, faster set-up, better understanding.