SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT SMA-HPC ?2002 MIT 3-D Laplace’s Equation First Kind Issues z Spike Function=1 on a disk 22 0 1 xyRσ = +≤ 2 00 0 1 2) 1 ( i R disk c dS rd R xx d r x r π θ πσ = ′ ? ′′ = ∫∫∫ 0 2 ()1 disk dS Rx πσ ′′ = ∫ Singular Kernel Case Smooth Kernel Case Smooth kernel 0 faster as 0, more singularR→→ 22 0 0 xyRσ = +> The singular Kernel Saves the day SMA-HPC ?2002 MIT Convergence Analysis 3-D Laplace’s Equation Quick review of FEM Convergence for Laplace 2 ufin?= ? Partial Differential Equation form 0uon=Γ is the volume domain? is the problem surfaceΓ “Nearly” Equivalent weak form Introduced an abstract notation for the equation u must satisfy ( ) 1 for (,) ) ll ( a uvdx fvdx v H auv lv ?? ?? = ∈ ? ∫∫ 1424314243 (,) ()auv lv= ( ) 1 for all vH∈ ? SMA-HPC ?2002 MIT Convergence Analysis 3-D Laplace’s Equation Quick review of FEM Convergence for Laplace is a weighted sum of basis funct o ins n u? The basis functions define a space 1 Introduce an approximate soluti n o n n ii i u α? = = ∑ 1 ? 2 ? 3 ? 5 ? 4 ? 6 ? “Hat” basis functions Piecewise linear Space Example 1 |for sme 's n nnii i i XvXvβ? β = ?? =∈ = ?? ?? ∑ SMA-HPC ?2002 MIT Convergence Analysis 3-D Laplace’s Equation Quick review of FEM Convergence for Laplace Using the norm properties, it is possible to show Key Idea (,) () n ii au lIf ??= min Pr nn nn wX uu uw ojectionSolution Error E Then rror ∈ ?= ? 1424314243 U is restricted to be 0 at 0 and1!! {} 12 ,,.for all ., in ? ?? ?∈ () 1 0 defines a norm on(, ( ) ),au au uHuu? ≡ SMA-HPC ?2002 MIT Convergence Analysis 3-D Laplace’s Equation Quick review of FEM Convergence for Laplace n u u The question is only How well can you fit u with a member of X n But you must measure the error in the ||| ||| norm 1 na n error uu u u O n ?? ?≤?Π= ?? ?? 14243 For piecewise linear: SMA-HPC ?2002 MIT Convergence Analysis 3-D Laplace’s Equation () () 1 Must exclude ' where 0xs xdS xx σσ ′ ′ = ′ ? ∫ ( ) for all vH∈ Γ ( ) The difficulty is defining with right propertiesH Γ () () () () () () 1 , vx xdSdS vx xdS xx lvav σ σ ΓΓ ′′ =Ψ ′ ? ∫∫ ∫ 1442443144442444443 “Weak” Form for the integral equation . ( ) is a fractional Sobolev SpaceH Γ We won’t say more about this Applying FEM approach to first kind integral equations SMA-HPC ?2002 MIT Convergence Analysis 3-D Laplace’s Equation Using the norm properties, it is possible to show Use FEM key Idea (,) () n ii alIf σ? ?= min Pr nn nn wX w ojection Solution Error E Then rror σσ σ ∈ ?= ? 1424314243 {} 12 ,,.for all ., in ? ?? ?∈ ( ) defines a norm on (,) (,)aaHσσ σσ σΓ = Applying FEM approach to first kind integral equations () { 1 Basis Functions n n ii i xσα? = = ∑ 1 |for sme 's n nnii i i XvXvβ? β = ? ? =∈ = ? ? ?? ∑ SMA-HPC ?2002 MIT MEMS Performance Depends on Air Damping of Complicated 3-D Structures Bosch angular rate sensor ADXL76 accelerometer TI 3x3 mirror array Resonator Lucent micromirror SMA-HPC ?2002 MIT Velocity integral equation for Stokes flow dsxxGxfxu o ij s i o j )()( 8 1 )( ??= ∫ πμ where ; ?? 3 R xx R G jiij ij += δ ioii o xxxxxR ?=?= ? ; Drag In MEMS is Incompressible Stokes SMA-HPC ?2002 MIT Null Space of the Stokes Equation ? ? ? =?? ?+??= 0 0 2 u uP r r μ 1 () () ( ) 8 oo jiij s ux fxGxxds πμ =? ? ∫ r rrr Constant pressure a singular mode, generates zero velocity. If constant, 0; ()()0 ii i o ij i s PufPn Gxxnxds = ==? ?? = ∫ rr r Differential Form of Stokes, independent of absolute pressure Integral Form of Stokes, constant pressure must not change velocity SMA-HPC ?2002 MIT Null Space of the Singular BEM Operators ? Stokes Integral Operator has a null space – The solution is not uniquely defined. – A pressure boundary condition is needed. ? Null space must be removed – so as to avoid numerical error. ? Two-step method: 1. Modify GMRES to calculate a null-space-free solution. 2. Use pressure condition to adjust solution ε+Χ+= NFF solutioncorrect SMA-HPC ?2002 MIT Krylov Subspace Iterative methods Determine the Krylov Subspace 00 rbAx=? Select Solution from the Krylov Subspace {} 10 0 0 0 , , ,..., kkk k xxyyspan r Ar A r + =+ ∈ k GMRES picks a residual-minimi ing yz. Start with Ax b= Linear System {} 00 0 Krylov Subspace , ,..., k span r Ar A r≡ SMA-HPC ?2002 MIT Modify Krylov-Subspace Method to Calculate Null-Space-Free Solution ? The discretized Stokes equation ? The Krylov subspace is ? If then Remove Null(G) from every Krylov subspace vector { } 234 ,, , , ,spanUGUGUGUGUκ = LL ()F FNulG ⊥ =⊥()Null Gκ ⊥ GF U= SMA-HPC ?2002 MIT Computation finished in 10 minutes FastStokes Simulation Result Drag Force (nN) Q Total Bottom Top Inter- finger End and others Couette Model 123.7 108.9 14.8 50.1 1-D Stokes 137.1 108.9 13.5 14.8 45.20 FastStokes 223.7 123.0 (55%) 26.8 (12%) 73.8 (33%) 27.7 Measurement 27 In-plane motion, 3-D steady incompressible Stokes, 16k panels SMA-HPC ?2002 MIT Micromirror Q-factor Q Mode Simulated Measured Error (%) Mirror+Gimbal 2.36 2.31 2.16 Mirror 1 Mirror 3.14 3.45 8.99 Mirror+Gimbal 4.69 4.27 9.84Mirrror 2 Mirror 10.16 10.63 4.42 Mirror Gimbal Rotate Summary Reminder about 2 nd Kind theory Convergence Theory Fredholm Alternative for 2 nd Kind Finite Dimensional Null Space First Kind Convergence Theory, sort of Connection to the FEM results MEMS Drag Example