Electro-Thermal-Mechanical MEMS Devices Multi-Physics Problems

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1 Electro-Thermal-Mechanical MEMS Devices Multi-Physics Problems Presentation for ASEN 5519 Joseph Pajot Department of Aerospace Engineering CU Boulder November 15, 2004 Typeset by FoilTEX 1

2 The ETM Problems Background Electro-thermal-mechanical coupling Joule heating Thermal expansion Application MEMS Typeset by FoilTEX 2

3 The ETM Problem: Electrostatic Governing Physics Poisson Equation ( ) V σ xx x x + y ( σ yy V y ) + z ( σ zz V z ) = z Isotopic conduction (σ = σ xx = σ yy = σ zz ) σ 2 V = z Electrostatic conduction finite element discretization R E = K E V Z = 0 V = ˆV on Ω V Z = Ẑ on Ω Z Typeset by FoilTEX 3

4 The ETM Problem: Electrostatic Governing Physics Poisson Equation x ( ) V σ xx x + y ( σ yy V y ) + z ( σ zz V z ) = z Isotopic conduction (σ = σ xx = σ yy = σ zz ) σ 2 V = z Electrostatic conduction finite element discretization R E = K E V = 0 V = ˆV on Ω V Typeset by FoilTEX 4

5 The ETM Problem: Coupling Electrothermal Joule Heating q = J E J = σe E = V Element expression Q J = Ω σe T EdΩ Typeset by FoilTEX 5

6 The ETM Problem: Thermal Governing Physics Poisson Equation ( ) T k xx x x + y ( ) T k yy y + z ( ) T k zz z = q Isotopic conduction (k = k xx = k yy = k zz ) k 2 T = q Conduction finite element discretization R Q = K Q T Q ext = 0 T = ˆT T = ˆf on Ω T on Ω F Typeset by FoilTEX 6

7 The ETM Problem: Thermal Governing Physics R Q = K Q T Q ext = 0 T = ˆT T = ˆf on Ω T on Ω F Q ext contributions 1. Joule heating 2. Convection 3. Radiation Typeset by FoilTEX 7

8 The ETM Problem: Thermal Convection Convective bc s proportional to temperature Q c h(t T) For shell elements convecting through one lateral side Q c = K c (T T) Typeset by FoilTEX 8

9 The ETM Problem: Thermal Radiation Blackbody radiation proportional to temperature to the fourth Q r h r (T 4 r T 4 ) Element radiation from one end will produce load vector Q r = N (e) h r (T 4 r T 4 )dω Ω Linearization of residual ( K Q + K c Q ) r T T + R Q (T ) = 0 K Q T + R Q (T ) = 0 Typeset by FoilTEX 9

10 The ETM Problem: Coupling Thermal strain (in plane) Structural-Thermal e t ij = { αij (T T 0 ) if i = j = 1, 2 0 else Elemental external force f ext = Ω C ijkl e t kldω Element orientation independent force in local coord Typeset by FoilTEX 10

11 The ETM Problem: Structure Governing Physics Continuum mechanics σ ij + p i = 0 u i = ū i σ ij n j = ˆt i on Ω u on Ω t TPE functional Π[u] = U W = Ω σ ij e ij dω σ ij = C ijkl (e kl e t kl) Ω p i u i dω Ω t ˆt i n i dω t Typeset by FoilTEX 11

12 The ETM Problem Governing Equations Residual finite element equations in each domain Structural: R S = f int (u) f ext (u, T) = 0 Thermal: R Q = Q int (u, T) Q ext (u, V) = 0 Electrostatic: R E = Z int (u, V) Z ext = 0 K T R Q ext f T 0 u K Q Qext V R E u 0 K E u (n) T (n) V (n) u (n+1) = u (n) + u (n) T (n+1) = T (n) + T (n) V (n+1) = V (n) + V (n) = R (n) S R (n) Q R (n) E Typeset by FoilTEX 12

13 The ETM Problem Governing Equations Residual finite element equations in each domain Structural: R S = f int (u) f ext (u, T) = 0 K T K Q K E Thermal: R Q = Q int (u, T) Q ext (u, V) = 0 Electrostatic: R E = Z int (u, V) Z ext = 0 u (n) T (n) = V (n) u (n+1) = u (n) + u (n) T (n+1) = T (n) + T (n) V (n+1) = V (n) + V (n) R S (u (n), T (n) ) R Q (u (n), T (n), V (n) ) R E (u (n), V (n) ) Typeset by FoilTEX 13

14 The ETM Problem: Coupling One Way Coupling The Linearized operator Φ = K T R Q ext f T 0 u K Q Qext V R E u 0 K E Assume thermal and electrostatic independent of structural response R Q u = 0 R E u = 0 Typeset by FoilTEX 14

15 The ETM Problem: Coupling One Way Coupling The one way linearized operator Φ = ext f T 0 0 K Q Qext V 0 0 K E K T Eliminate costly computations Upper diagonal one solve per domain Effect on solution/accuracy? Typeset by FoilTEX 15

16 The ETM Problem: Verification Energy Balance Check that power balance is correct Electrical Power in equals thermal dissipation Electrical Power: P = IV = I 2 R = V/R 2 I = J ˆndA Thermal Power: P = ˆf ˆndA ˆf = k T Typeset by FoilTEX 16

17 The ETM Problem: Solution How does one run these problems Modularity Parallel computation One-to-one interface Body is the interface Solution Procedure Typeset by FoilTEX 17

18 The ETM Problem: Solution Modularity Which values should be sent from code to code? Codes should be kept dumb to each other Thermal codes at the middle of coupling Master-slave relationship Typeset by FoilTEX 18

19 Modular requirements The ETM Problem: Solution Parallel computation Split problem into manageable sizes Φ = ext f T 0 0 K Q Qext V 0 0 K E K T If domains have 10k nodes, full system is larger than 30k by 30k! Typeset by FoilTEX 19

20 The ETM Problem: Solution Parallel computation Handling communications between domains (e.g. MPI) Nontrivial issue Pass minimum amount of information (vectors vs matrix) Computations carried out within each domain Resultants sent back Typeset by FoilTEX 20

21 The ETM Problem: Parallel Computation Optimization 1. Problem definition min s z(s) g i (s) 0 h i (s) = 0 s L s s U 2. Analysis of physical fields (FEM) - Optimization criteria q = q(s, u(s), T(s), V(s)) - State variables u = u(s), T = T(s), V = V(s) Typeset by FoilTEX 21

22 ETM Optimization Background Linking optimization variables to physical problem Size optimization Shape optimization Typeset by FoilTEX 22

23 ETM Optimization Background Topology optimization is a material distribution problem Indicator function χ χ(r) = { 1 r Ω1 0 r Ω 0 Ω = Ω 0 Ω 1 Typeset by FoilTEX 23

24 ETM Optimization Sensitivity Analysis Require criteria gradients dq j = q j + q j du + q j dt + q j dv s i u T V Differentiate Finite Element Residual Equations with respect to s i K T R Q ext f T 0 u K Q Qext V R E u 0 K E du dt dv = Φ du dt dv = R S s i R Q s i R E s i Typeset by FoilTEX 24

25 ETM Optimization Sensitivity Analysis Rewrite criteria sensitivity expression dq j = q j s i [ qj u q j T q j V ] Φ 1 R S s i R Q s i R E s i Matrix-vector solves required Minimize this expensive procedure Direct vs Adjoint Typeset by FoilTEX 25

26 K T R Q ETM Optimization Staggered Solution ext f T 0 u K Q Qext V R E u 0 K E du dt dv = Gauss-Seidel solve on submatricies R S s i R Q s i R E s i K T K Q K E du dt dv = R Q s i R S s i + R Q u R E s i f ext dt T du + Qext V R E u du dv Effect of one way coupling? Typeset by FoilTEX 26

27 ETM Optimization Examples Maximize output force Uses less than 30% of available mass (d) Current Density (e) Temperature (f) Deformation Typeset by FoilTEX 27

28 ETM Optimization Examples Maximize output force Uses less than 30% of available mass (a) Current Density (b) Temperature (c) Deformation Typeset by FoilTEX 28

29 ETM Optimization Examples Maximize output force Uses less than 30% of available mass (a) Current Density (b) Temperature (c) Deformation Typeset by FoilTEX 29

Fundamentals of Linear Elasticity

Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy