MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Computer Aided Design of Micro-Electro-Mechanical Systems From Energy Losses to Dick Tracy Watches D. Bindel Courant Institute for Mathematical Sciences New York University MIT, 3 Nov 2008 The Computational Science Picture MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Application modeling Disk resonator Beam resonator Shear ring resonator, checkerboard, ... Mathematical analysis Physical modeling and finite element technology Structured eigenproblems and reduced-order models Parameter-dependent eigenproblems Software engineering HiQLab SUGAR FEAPMEX / MATFEAP The Computational Science Picture MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Application modeling Disk resonator Beam resonator Shear ring resonator, checkerboard, ... Mathematical analysis Physical modeling and finite element technology Structured eigenproblems and reduced-order models Parameter-dependent eigenproblems Software engineering HiQLab SUGAR FEAPMEX / MATFEAP Outline MIT 08 1 Resonant MEMS and models 2 HiQLab Thermoelastic losses and beam resonators 3 Anchor losses and disk resonators Conclusion 4 Thermoelastic losses and beam resonators 5 Conclusion Resonant MEMS and models HiQLab Anchor losses and disk resonators Backup slides Outline MIT 08 1 Resonant MEMS and models 2 HiQLab Thermoelastic losses and beam resonators 3 Anchor losses and disk resonators Conclusion 4 Thermoelastic losses and beam resonators 5 Conclusion Resonant MEMS and models HiQLab Anchor losses and disk resonators Backup slides What are MEMS? MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides MEMS Basics MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Micro-Electro-Mechanical Systems Chemical, fluid, thermal, optical (MECFTOMS?) Applications: Sensors (inertial, chemical, pressure) Ink jet printers, biolab chips Radio devices: cell phones, inventory tags, pico radio Use integrated circuit (IC) fabrication technology Tiny, but still classical physics Resonant RF MEMS MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Microguitars from Cornell University (1997 and 2003) Conclusion MHz-GHz mechanical resonators Backup slides Favorite application: radio on chip Close second: really high-pitch guitars The Mechanical Cell Phone MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Mixer RF amplifier / preselector Tuning control IF amplifier / filter Local Oscillator Backup slides Your cell phone has many moving parts! What if we replace them with integrated MEMS? ... Ultimate Success MIT 08 “Calling Dick Tracy!” Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Disk Resonator MIT 08 Vin Resonant MEMS and models HiQLab AC Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion DC Backup slides Vout Disk Resonator MIT 08 Vin Resonant MEMS and models HiQLab AC Lx Anchor losses and disk resonators Thermoelastic losses and beam resonators C0 Cx Rx Conclusion Backup slides Vout Electromechanical Model MIT 08 Resonant MEMS and models Assume time-harmonic steady state, no external forces: iωC + G iωB δ Îexternal δ V̂ = −B T K̃ − ω 2 M δ û 0 HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Eliminate the mechanical terms: Y (ω) δ V̂ = δ Îexternal Y (ω) = iωC + G + iωH(ω) H(ω) = B T (K̃ − ω 2 M)−1 B Backup slides Goal: Understand electromechanical piece (iωH(ω)). As a function of geometry and operating point Preferably as a simple circuit Damping and Q MIT 08 Resonant MEMS and models Designers want high quality of resonance (Q) Dimensionless damping in a one-dof system HiQLab d 2u du + u = F (t) + Q −1 2 dt dt Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides For a resonant mode with frequency ω ∈ C: Q := |ω| Stored energy = 2 Im(ω) Energy loss per radian To understand Q, we need damping models! The Designer’s Dream MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides 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. Outline MIT 08 1 Resonant MEMS and models 2 HiQLab Thermoelastic losses and beam resonators 3 Anchor losses and disk resonators Conclusion 4 Thermoelastic losses and beam resonators 5 Conclusion Resonant MEMS and models HiQLab Anchor losses and disk resonators Backup slides Enter HiQLab MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Existing codes do not compute quality factors ... and awkward to prototype new solvers ... and awkward to programmatically define meshes So I wrote a new finite element code: HiQLab Heritage of HiQLab MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides SUGAR: SPICE for the MEMS world System-level simulation using modified nodal analysis Flexible device description language C core with MATLAB interfaces and numerical routines FEAPMEX: MATLAB + a finite element code MATLAB interfaces for steering, testing solvers, running parameter studies Time-tested finite element architecture But old F77, brittle in places Other Ingredients MIT 08 Resonant MEMS and models “Lesser artists borrow. Great artists steal.” – Picasso, Dali, Stravinsky? HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Lua: www.lua.org Evolved from simulator data languages (DEL and SOL) Pascal-like syntax fits on one page; complete language description is 21 pages Fast, freely available, widely used in game design MATLAB: www.mathworks.com “The Language of Technical Computing” O CTAVE also works well Standard numerical libraries: ARPACK, UMFPACK M ATEXPR, MW RAP, and other utilities HiQLab Structure MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators User interfaces (MATLAB, Lua) Solver library (C, C++, Fortran, MATLAB) Core libraries (C++) Problem description (Lua) Element library (C++) Standard finite element structures + some new ideas Conclusion Full scripting language for mesh input Backup slides Callbacks for boundary conditions, material properties MATLAB interface for quick algorithm prototyping Cross-language bindings are automatically generated Outline MIT 08 1 Resonant MEMS and models 2 HiQLab Thermoelastic losses and beam resonators 3 Anchor losses and disk resonators Conclusion 4 Thermoelastic losses and beam resonators 5 Conclusion Resonant MEMS and models HiQLab Anchor losses and disk resonators Backup slides Damping Mechanisms MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Possible loss mechanisms: Fluid damping Material losses Conclusion Backup slides Thermoelastic damping Anchor loss Model substrate as semi-infinite with a Perfectly Matched Layer (PML). Perfectly Matched Layers MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Complex coordinate transformation Generates a “perfectly matched” absorbing layer Idea works with general linear wave equations Electromagnetics (Berengér, 1994) Quantum mechanics – exterior complex scaling (Simon, 1979) Elasticity in standard finite element framework (Basu and Chopra, 2003) Model Problem MIT 08 Resonant MEMS and models Domain: x ∈ [0, ∞) Governing eq: HiQLab 1 ∂2u ∂2u − =0 ∂x 2 c 2 ∂t 2 Anchor losses and disk resonators Thermoelastic losses and beam resonators Fourier transform: d 2 û + k 2 û = 0 dx 2 Conclusion Backup slides Solution: û = cout e−ikx + cin eikx Model with Perfectly Matched Layer MIT 08 σ Resonant MEMS and models HiQLab Regular domain x Transformed domain Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides d x̃ = λ(x) where λ(s) = 1 − iσ(s) dx d 2 û + k 2 û = 0 d x̃ 2 û = cout e−ik x̃ + cin eik x̃ Model with Perfectly Matched Layer MIT 08 σ Resonant MEMS and models HiQLab x Transformed domain Regular domain Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides d x̃ = λ(x) where λ(s) = 1 − iσ(s), dx 1 d 1 d û + k 2 û = 0 λ dx λ dx û = cout e−ikx−k Σ(x) + cin eikx+k Σ(x) Z x Σ(x) = σ(s) ds 0 Model with Perfectly Matched Layer MIT 08 σ Resonant MEMS and models HiQLab Regular domain x Transformed domain Anchor losses and disk resonators Thermoelastic losses and beam resonators If solution clamped at x = L then Conclusion Backup slides cin = O(e−k γ ) where γ = Σ(L) = cout Z L σ(s) ds 0 Model Problem Illustrated MIT 08 Outgoing exp(−ix̃) Resonant MEMS and models HiQLab Anchor losses and disk resonators 1 0.5 0.5 0 0 -0.5 -0.5 -1 0 5 10 15 Thermoelastic losses and beam resonators Conclusion Backup slides Incoming exp(ix̃) 1 20 -1 0 10 5 15 20 Transformed coordinate 0 -2 -4 0 2 4 6 8 10 Re(x̃) 12 14 16 18 Model Problem Illustrated MIT 08 Outgoing exp(−ix̃) Incoming exp(ix̃) 1 Resonant MEMS and models 3 2 0.5 1 0 0 HiQLab Anchor losses and disk resonators -0.5 -1 -1 0 5 10 15 Thermoelastic losses and beam resonators Conclusion Backup slides 20 -2 0 5 15 10 20 Transformed coordinate 0 -2 -4 0 2 4 6 8 10 Re(x̃) 12 14 16 18 Model Problem Illustrated MIT 08 Outgoing exp(−ix̃) Incoming exp(ix̃) 1 Resonant MEMS and models 6 4 0.5 2 0 0 HiQLab Anchor losses and disk resonators -0.5 -2 -1 0 5 10 15 Thermoelastic losses and beam resonators Conclusion Backup slides 20 -4 0 5 15 10 20 Transformed coordinate 0 -2 -4 0 2 4 6 8 10 Re(x̃) 12 14 16 18 Model Problem Illustrated MIT 08 Outgoing exp(−ix̃) Incoming exp(ix̃) 1 Resonant MEMS and models 15 10 0.5 5 0 0 HiQLab Anchor losses and disk resonators -0.5 -5 -1 0 5 10 15 Thermoelastic losses and beam resonators Conclusion Backup slides 20 -10 0 5 15 10 20 Transformed coordinate 0 -2 -4 0 2 4 6 8 10 Re(x̃) 12 14 16 18 Model Problem Illustrated MIT 08 Outgoing exp(−ix̃) Incoming exp(ix̃) 1 Resonant MEMS and models HiQLab Anchor losses and disk resonators 40 0.5 20 0 0 -0.5 -1 0 5 10 15 Thermoelastic losses and beam resonators Conclusion Backup slides 20 -20 0 5 10 20 15 Transformed coordinate 0 -2 -4 0 2 4 6 8 10 Re(x̃) 12 14 16 18 Model Problem Illustrated MIT 08 Outgoing exp(−ix̃) Incoming exp(ix̃) 1 Resonant MEMS and models HiQLab Anchor losses and disk resonators 100 0.5 50 0 0 -0.5 -1 0 5 10 15 Thermoelastic losses and beam resonators Conclusion Backup slides 20 -50 0 5 10 20 15 Transformed coordinate 0 -2 -4 0 2 4 6 8 10 Re(x̃) 12 14 16 18 Finite Element Implementation MIT 08 x2 ξ2 x(ξ) Resonant MEMS and models HiQLab x̃2 ξ1 Ω x̃(x) Ω̃e Ωe Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides x1 Combine PML and isoparametric mappings Z e k = B̃T DB̃J̃ dΩ Ω Z e m = ρNT NJ̃ dΩ Ω Matrices are complex symmetric x̃1 Eigenvalues and Model Reduction MIT 08 Want to know about the transfer function H(ω): Resonant MEMS and models H(ω) = B T (K − ω 2 M)−1 B HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Can either Locate poles of H (eigenvalues of (K , M)) Plot H in a frequency range (Bode plot) Usual tactic: subspace projection Build an Arnoldi basis V for a Krylov subspace Kn Compute with much smaller V ∗ KV and V ∗ MV Can we do better? Variational Principles MIT 08 Variational form for complex symmetric eigenproblems: Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Hermitian (Rayleigh quotient): ρ(v ) = v ∗ Kv v ∗ Mv Complex symmetric (modified Rayleigh quotient): θ(v ) = v T Kv v T Mv First-order accurate eigenvectors =⇒ Second-order accurate eigenvalues. Key: relation between left and right eigenvectors. Accurate Model Reduction MIT 08 Resonant MEMS and models Build new projection basis from V : HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides W = orth[Re(V ), Im(V )] span(W ) contains both Kn and K̄n =⇒ double digits correct vs. projection with V W is a real-valued basis =⇒ projected system is complex symmetric Disk Resonator Simulations MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Disk Resonator Mesh MIT 08 Electrode Resonant MEMS and models Resonating disk HiQLab Wafer (unmodeled) Anchor losses and disk resonators Thermoelastic losses and beam resonators PML region −6 x 10 2 0 Conclusion −2 Backup slides −4 0 1 2 3 4 −5 x 10 Axisymmetric model with bicubic mesh About 10K nodal points in converged calculation Mesh Convergence MIT 08 7000 6000 Resonant MEMS and models Linear Quadratic Cubic 5000 Anchor losses and disk resonators Thermoelastic losses and beam resonators Computed Q HiQLab 4000 3000 2000 Conclusion Backup slides 1000 0 1 2 3 5 4 Mesh density 6 7 8 Cubic elements converge with reasonable mesh density Model Reduction Accuracy MIT 08 Transfer (dB) Resonant MEMS and models 0 -20 -40 -60 HiQLab -80 47.2 Anchor losses and disk resonators Conclusion Backup slides 47.3 200 Phase (degrees) Thermoelastic losses and beam resonators 47.25 Frequency (MHz) 100 0 47.2 47.25 47.3 Frequency (MHz) Results from ROM (solid and dotted lines) nearly indistinguishable from full model (crosses) Model Reduction Accuracy MIT 08 HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion 10−2 |H(ω) − Hreduced(ω)|/H(ω)| Resonant MEMS and models 10−4 10−6 Structure-preserving ROM Arnoldi ROM Backup slides 45 46 47 48 Frequency (MHz) Preserve structure =⇒ get twice the correct digits 49 50 Response of the Disk Resonator MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Variation in Quality of Resonance MIT 08 108 Resonant MEMS and models 106 Anchor losses and disk resonators Q HiQLab 104 Thermoelastic losses and beam resonators Conclusion 102 Backup slides 100 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Film thickness (µm) Simulation and lab measurements vs. disk thickness Explanation of Q Variation MIT 08 0.25 e a HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion d c 0.2 Imaginary frequency (MHz) Resonant MEMS and models b 0.15 0.1 a = 1.51 µm b = 1.52 µm c = 1.53 µm 0.05 d = 1.54 µm Backup slides e = 1.55 µm 0 46 46.5 d c e b a 47 47.5 Real frequency (MHz) Interaction of two nearby eigenmodes 48 Outline MIT 08 1 Resonant MEMS and models 2 HiQLab Thermoelastic losses and beam resonators 3 Anchor losses and disk resonators Conclusion 4 Thermoelastic losses and beam resonators 5 Conclusion Resonant MEMS and models HiQLab Anchor losses and disk resonators Backup slides Thermoelastic Damping (TED) MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Thermoelastic Damping (TED) MIT 08 u is displacement and T = T0 + θ is temperature Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides σ = C − βθ1 ρü = ∇ · σ ˙ ρcv θ̇ = ∇ · (κ∇θ) − βT0 tr() Coupling between temperature and volumetric strain: Compression and expansion =⇒ heating and cooling Heat diffusion =⇒ mechanical damping Not often an important factor at the macro scale Recognized source of damping in microresonators Zener: semi-analytical approximation for TED in beams We consider the fully coupled system Nondimensionalized Equations MIT 08 Continuum equations: Resonant MEMS and models σ = Ĉ − ξθ1 ü = ∇ · σ HiQLab ˙ θ̇ = η∇2 θ − tr() Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Discrete equations: Muu ü + Kuu u = ξKuθ θ + f Cθθ θ̈ + ηKθθ θ = −Cθu u̇ Micron-scale poly-Si devices: ξ and η are ∼ 10−4 . Linearize about ξ = 0 Perturbative Mode Calculation MIT 08 Discretized mode equation: (−ω 2 Muu + Kuu )u = ξKuθ θ Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides (iωCθθ + ηKθθ )θ = −iωCθu u First approximation about ξ = 0: (−ω02 Muu + Kuu )u0 = 0 (iω0 Cθθ + ηKθθ )θ0 = −iω0 Cθu u0 First-order correction in ξ: −δ(ω 2 )Muu u0 + (−ω02 Muu + Kuu )δu = ξKuθ θ0 Multiply by u0T : 2 δ(ω ) = −ξ u0T Kuθ θ0 u0T Muu u0 ! Zener’s Model MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides 1 Clarence Zener investigated TED in late 30s-early 40s. 2 Model for beams common in MEMS literature. 3 “Method of orthogonal thermodynamic potentials” == perturbation method + a variational method. Comparison to Zener’s Model MIT 08 −4 10 HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Thermoelastic Damping Q −1 Resonant MEMS and models −5 10 Zener’s Formula −6 HiQlab Results 10 −7 Conclusion Backup slides 10 5 10 6 10 7 10 8 10 9 10 10 10 Frequency f(Hz) Comparison of fully coupled simulation to Zener approximation over a range of frequencies Real and imaginary parts after first-order correction agree to about three digits with Arnoldi Outline MIT 08 1 Resonant MEMS and models 2 HiQLab Thermoelastic losses and beam resonators 3 Anchor losses and disk resonators Conclusion 4 Thermoelastic losses and beam resonators 5 Conclusion Resonant MEMS and models HiQLab Anchor losses and disk resonators Backup slides Onward! MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators What about: Modeling more geometrically complex devices? Modeling general dependence on geometry? Thermoelastic losses and beam resonators Modeling general dependence on operating point? Conclusion Digesting all this to help designers? Backup slides Computing nonlinear dynamics? Future Work MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Code development Structural elements and elements for different physics Design and implementation of parallelized version Theoretical analysis More damping mechanisms Sensitivity analysis and variational model reduction Application collaborations Use of nonlinear effects (quasi-static and dynamic) New designs (e.g. internal dielectric drives) Continued experimental comparisons Conclusions MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides RF MEMS are a great source of problems Interesting applications Interesting physics (and not altogether understood) Interesting computing challenges http://www.cims.nyu.edu/~dbindel Concluding Thoughts MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides The difference between art and science is that science is what we understand well enough to explain to a computer. Art is everything else. Donald Knuth The purpose of computing is insight, not numbers. Richard Hamming Checkerboard Resonator MIT 08 D− Resonant MEMS and models D+ D+ HiQLab D− S− Anchor losses and disk resonators S+ S+ S− Thermoelastic losses and beam resonators Anchored at outside corners Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Excited at northwest corner Sensed at southeast corner Surfaces move only a few nanometers Checkerboard Model Reduction MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Finite element model: N = 2154 Expensive to solve for every H(ω) evaluation! Build a reduced-order model to approximate behavior Reduced system of 80 to 100 vectors Evaluate H(ω) in milliseconds instead of seconds Without damping: standard Arnoldi projection With damping: Second-Order ARnoldi (SOAR) Checkerboard Simulation MIT 08 x 10 12 HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators −100 −120 10 Amplitude (dB) Resonant MEMS and models 8 −140 −160 −180 6 −200 9 9.2 9.4 9.6 Frequency (Hz) x 10 9.4 9.6 Frequency (Hz) x 10 4 9.8 7 4 2 Phase (rad) 3 Conclusion 0 Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis 0 2 4 6 8 2 1 10 −5 x 10 0 9 9.2 9.8 7 Checkerboard Measurement MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis S. Bhave, MEMS 05 Contributions MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Built predictive model used to design checkerboard Used model reduction to get thousand-fold speedup – fast enough for interactive use General Picture MIT 08 If w ∗ A = 0 and Av = 0 then Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators δ(w ∗ Av ) = w ∗ (δA)v This implies If A = A(λ) and w = w(v ), have w ∗ (v )A(ρ(v ))v = 0. Conclusion ρ stationary when (ρ(v ), v ) is a nonlinear eigenpair. Backup slides If A(λ, ξ) and w0∗ and v0 are null vectors for A(λ0 , ξ0 ), Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis w0∗ (Aλ δλ + Aξ δξ)v0 = 0. Electromechanical Model MIT 08 Kirchoff’s current law and balance of linear momentum: d (C(u)V ) + GV = Iexternal dt 1 ∗ Mutt + Ku − ∇u V C(u)V = Fexternal 2 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Linearize about static equilibium (V0 , u0 ): C(u0 ) δVt + G δV + (∇u C(u0 ) · δut ) V0 = δIexternal Conclusion M δutt + K̃ δu + ∇u (V0∗ C(u0 ) δV ) = δFexternal Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis where K̃ = K − 1 ∂2 (V ∗ C(u0 )V0 ) 2 ∂u 2 0 4 HiQLab’s Hello World MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis 2 0 −2 mesh = Mesh:new(2) mat = −4make_material(’silicon2’, ’planestrain’) mesh:blocks2d( { 0, l }, { -w/2.0, w/2.0 }, mat ) −6 0 2 4 6 8 10 −6 x 10 mesh:set_bc(function(x,y) if x == 0 then return ’uu’, 0, 0; end end) HiQLab’s Hello World MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis >> >> >> >> >> >> mesh = Mesh_load(’beammesh.lua’); [M,K] = Mesh_assemble_mk(mesh); [V,D] = eigs(K,M, 5, ’sm’); opt.axequal = 1; opt.deform = 1; Mesh_scale_u(mesh, V(:,1), 2, 1e-6); plotfield2d(mesh, opt); Continuum 2D model problem MIT 08 Resonant MEMS and models k HiQLab Anchor losses and disk resonators L Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis 1 − iβ|x − L|p , x > L 1 x ≤ L. ∂2u 1 ∂ 1 ∂u + + k 2u = 0 λ ∂x λ ∂x ∂y 2 λ(x) = Continuum 2D model problem MIT 08 Resonant MEMS and models k HiQLab Anchor losses and disk resonators Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis L Thermoelastic losses and beam resonators Conclusion 1 − iβ|x − L|p , x > L 1 x ≤ L. 1 ∂ 1 ∂u − ky2 u + k 2 u = 0 λ ∂x λ ∂x λ(x) = Continuum 2D model problem MIT 08 k Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation L 1 − iβ|x − L|p , x > L 1 x ≤ L. 1 ∂ 1 ∂u + kx2 u = 0 λ ∂x λ ∂x λ(x) = Electromechanical model Hello world! Reflection Analysis 1D problem, reflection of O(e−kx γ ) Discrete 2D model problem MIT 08 Resonant MEMS and models k HiQLab Anchor losses and disk resonators L Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Discrete Fourier transform in y Solve numerically in x Project solution onto infinite space traveling modes Extension of Collino and Monk (1998) Nondimensionalization MIT 08 k Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion L λ(x) = 1 − iβ|x − L|p , x > L 1 x ≤ L. Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Rate of stretching: Elements per wave: Elements through the PML: βhp (kx h)−1 and (ky h)−1 N Nondimensionalization MIT 08 k Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion L λ(x) = 1 − iβ|x − L|p , x > L 1 x ≤ L. Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Rate of stretching: Elements per wave: Elements through the PML: βhp (kx h)−1 and (ky h)−1 N Discrete reflection behavior MIT 08 − log10 (r) at (kh)−1 = 10 1 Resonant MEMS and models 0.5 log10 (βh) 0 -0.5 3 2 4 -1 1 -1.5 Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis 4 4 3 2 Conclusion Backup slides 3 3 1 Thermoelastic losses and beam resonators 2 2 3 Anchor losses and disk resonators 2 2 HiQLab 4 3 5 10 15 4 2 1 20 25 3 30 Number of PML elements Quadratic elements, p = 1, (kx h)−1 = 10 Discrete reflection decomposition MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Model discrete reflection as two parts: Far-end reflection (clamping reflection) Approximated well by continuum calculation Grows as (kx h)−1 grows Interface reflection Discrete effect: mesh does not resolve decay Does not depend on N Grows as (kx h)−1 shrinks Discrete reflection behavior MIT 08 − log10 (r) at (kh)−1 = 10 − log10 (rinterface + rnominal ) at (kh)−1 = 10 1 2 4 2 4 3 10 15 20 25 Number of PML elements 3 30 2 -1 2 3 3 -0.5 3 3 4 2 3 4 1 -1.5 4 2 1 5 2 4 4 1 -1.5 4 4 3 0 1 -1 log10 (βh) 3 3 2 log10 (βh) -0.5 3 1 Conclusion 0.5 2 Thermoelastic losses and beam resonators 0 1 Anchor losses and disk resonators 2 2 3 HiQLab 2 2 Resonant MEMS and models 1 0.5 5 10 15 2 20 3 4 25 30 Number of PML elements Quadratic elements, p = 1, (kx h)−1 = 10 Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Model does well at predicting actual reflection Similar picture for other wavelengths, element types, stretch functions Choosing PML parameters MIT 08 Resonant MEMS and models HiQLab Anchor losses and disk resonators Thermoelastic losses and beam resonators Conclusion Backup slides Checkerboard resonators Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Discrete reflection dominated by Interface reflection when kx large Far-end reflection when kx small Heuristic for PML parameter choice Choose an acceptable reflection level Choose β based on interface reflection at kxmax Choose length based on far-end reflection at kxmin

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