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
The ETM Problems
– Background –
• Electro-thermal-mechanical coupling
– Joule heating
– Thermal expansion
• Application
– MEMS
– Typeset by FoilTEX –
2
The ETM Problem: Electrostatic
– Governing Physics –
Poisson Equation
∂
∂
∂
∂V
∂V
∂V
σxx
+
σyy
+
σzz
= −z
∂x
∂x
∂y
∂y
∂z
∂z
Isotopic conduction (σ = σxx = σyy = σzz )
σ∇2V = −z
Electrostatic conduction finite element discretization
RE = K˜EV − Z = 0
V = V̂ on ΩV
Z = Ẑ on ΩZ
– Typeset by FoilTEX –
3
The ETM Problem: Electrostatic
– Governing Physics –
Poisson Equation
∂
∂
∂
∂V
∂V
∂V
σxx
+
σyy
+
σzz
= −z
∂x
∂x
∂y
∂y
∂z
∂z
Isotopic conduction (σ = σxx = σyy = σzz )
σ∇2V = −z
Electrostatic conduction finite element discretization
RE = K˜EV = 0
V = V̂ on ΩV
– Typeset by FoilTEX –
4
The ETM Problem: Coupling
– Electrothermal –
Joule Heating
= J·E
q
J = σE
E = ∇V
Element expression
Z
QJ =
σETEdΩ
Ω
– Typeset by FoilTEX –
5
The ETM Problem: Thermal
– Governing Physics –
Poisson Equation
∂
∂T
∂
∂T
∂
∂T
kxx
+
kyy
+
kzz
= −q
∂x
∂x
∂y
∂y
∂z
∂z
Isotopic conduction (k = kxx = kyy = kzz )
k∇2T = −q
Conduction finite element discretization
RQ = K˜QT − Qext = 0
– Typeset by FoilTEX –
T = T̂
on ΩT
∇T = f̂
on ΩF
6
The ETM Problem: Thermal
– Governing Physics –
RQ = K˜QT − Qext = 0
T = T̂
on ΩT
∇T = f̂
on ΩF
• Qext contributions
1. Joule heating
2. Convection
3. Radiation
– Typeset by FoilTEX –
7
The ETM Problem: Thermal
– Convection –
Convective bc’s proportional to temperature
Qc ∝ h(T∞ − T)
For shell elements convecting through one lateral side
Qc = K̃c(T∞ − T)
– Typeset by FoilTEX –
8
The ETM Problem: Thermal
– Radiation –
Blackbody radiation proportional to temperature to the fourth
Qr ∝ hr(T4r − T4)
Element radiation from one end will produce load vector
Z
Qr =
N(e)hr(T4r − T4)dΩ
Ω
Linearization of residual
∂Qr
K˜Q + K̃c −
∆T + RQ(T∗) = 0
∂T
KQ∆T + RQ(T∗) = 0
– Typeset by FoilTEX –
9
The ETM Problem: Coupling
– Structural-Thermal –
Thermal strain (in plane)
etij =
αij (T − T0) if i = j = 1, 2
0
else
Elemental external force
f ext =
Z
CijkletkldΩ
Ω
• Element orientation independent force in local coord
– Typeset by FoilTEX –
10
The ETM Problem: Structure
– Governing Physics –
Continuum mechanics
∇σij + pi = 0
ui = ūi
on Ωu
σij nj = t̂i
on Ωt
TPE functional
Z
Π[u] = U − W =
Z
σij eij dΩ −
Ω
Z
piuidΩ −
Ω
t̂inidΩt
Ωt
σij = Cijkl(ekl − etkl)
– Typeset by FoilTEX –
11
The ETM Problem
– Governing Equations –
Residual finite element equations in each domain
Structural:
Thermal:
Electrostatic:




KT
∂RQ
∂u
∂RE
∂u
ext
− ∂∂f T
KQ
0
RS = |f int(u) − f ext(u, T)| = 0
RQ = |Qint(u, T) − Qext(u, V)| = 0
RE = |Zint(u, V) − Zext| = 0
0
∂ Qext
− ∂V
KE


(n)
∆u

  ∆T(n)  = 


∆V(n)


(n)
−RS
(n)
−RQ
(n)
−RE



u(n+1) = u(n) + ∆u(n)
T(n+1) = T(n) + ∆T(n)
V(n+1) = V(n) + ∆V(n)
– Typeset by FoilTEX –
12
The ETM Problem
– Governing Equations –
Residual finite element equations in each domain
Structural:
Thermal:
Electrostatic:


KT 0
0


0
K
0


Q
0
0 KE
RS = |f int(u) − f ext(u, T)| = 0
RQ = |Qint(u, T) − Qext(u, V)| = 0
RE = |Zint(u, V) − Zext| = 0
(n)


(n)
(n)

∆u
−RS (u , T )
∆T(n)  =  −RQ(u(n), T(n), V(n)) 
∆V(n)
−RE (u(n), V(n))
u(n+1) = u(n) + ∆u(n)
T(n+1) = T(n) + ∆T(n)
V(n+1) = V(n) + ∆V(n)
– Typeset by FoilTEX –
13
The ETM Problem: Coupling
– One Way Coupling –
The Linearized operator


Φ = 

KT
∂RQ
∂u
∂RE
∂u
ext
− ∂∂f T
KQ
0
0
∂ Qext
− ∂V
KE




Assume thermal and electrostatic independent of structural response
∂RQ
=0
∂u
∂RE
=0
∂u
– Typeset by FoilTEX –
14
The ETM Problem: Coupling
– One Way Coupling –
The ”one way” linearized operator

KT

Φ =  0
0
ext
− ∂∂f T
KQ
0
0

∂ Qext
− ∂V


KE
• Eliminate costly computations
• Upper diagonal → one solve per domain
• Effect on solution/accuracy?
– Typeset by FoilTEX –
15
The ETM Problem: Verification
– Energy Balance –
• Check that power balance is correct
• Electrical Power in equals thermal dissipation
• Electrical Power: P = IV = I 2R = V /R2
R
I = J · n̂dA
• Thermal Power: P =
R
f̂ · n̂dA
f̂ = k∇T
– Typeset by FoilTEX –
16
The ETM Problem: Solution
– Solution Procedure –
• How does one run these problems
– Modularity
– Parallel computation
• One-to-one interface
• Body is the interface
– Typeset by FoilTEX –
17
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
The ETM Problem: Solution
– Parallel computation –
• Modular requirements
• Split problem into manageable sizes

KT

Φ =  0
0
∂ f ext
− ∂T
KQ
0
0

∂ Qext
− ∂V


KE
If domains have 10k nodes, full system is larger than 30k by 30k!
– Typeset by FoilTEX –
19
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
The ETM Problem: Parallel Computation
– Optimization –
1. Problem definition
min z(s)
s
gi(s)
≥
0
hi(s)
=
0
sL
≤
s ≤ sU
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
ETM Optimization
– Background –
• Linking optimization variables to physical problem
– Size optimization
– Shape optimization
– Typeset by FoilTEX –
22
ETM Optimization
– Background –
• Topology optimization is a material distribution problem
– Indicator function χ
χ(r) =
1 r ∈ Ω1
0 r ∈ Ω0
Ω = Ω0 ∪ Ω1
– Typeset by FoilTEX –
23
ETM Optimization
– Sensitivity Analysis –
Require criteria gradients
∂qj ∂qj du ∂qj dT ∂qj dV
dqj
=
+
+
+
dsi
∂si ∂u dsi ∂T dsi ∂V dsi
Differentiate Finite Element Residual Equations with respect to si




KT
∂RQ
∂u
∂RE
∂u
ext
− ∂∂f T
KQ
0
– Typeset by FoilTEX –
0
ext
− ∂ ∂QV
KE




du
dsi
dT
dsi
dV
dsi




 = Φ
du
dsi
dT
dsi
dV
dsi


 
=

S
− ∂R
∂si
∂R
− ∂sQi
E
− ∂R
∂si




24
ETM Optimization
– Sensitivity Analysis –
⇒ Rewrite criteria sensitivity expression

∂qj h
dqj
=
−
dsi
∂si
∂qj
∂u
∂qj
∂T
∂qj
∂V
i
−1 
Φ


∂RS
∂si
∂RQ
∂si
∂RE
∂si




• Matrix-vector solves required
• Minimize this expensive procedure
– Direct vs Adjoint
– Typeset by FoilTEX –
25
ETM Optimization
– Staggered Solution –




KT
∂RQ
∂u
∂RE
∂u
ext
− ∂∂f T
KQ
0
0
∂ Qext
− ∂V
KE




du
dsi
dT
dsi
dV
dsi


S
− ∂R
∂si

 
∂RQ 

 =  − ∂si 

∂RE
− ∂si
Gauss-Seidel solve on submatricies


KT
0
0


0
K
0


Q
0
0 KE
du
dsi
dT
dsi
dV
dsi


 
=

∂RS
∂ f ext dT
− ∂si + ∂ T dsi
∂RQ
∂RQ du
∂ Qext dV
− ∂si − ∂ u dsi + ∂ V dsi
∂RE du
E
−
− ∂R
∂si
∂ u dsi




⇒ Effect of one way coupling? ⇐
– Typeset by FoilTEX –
26
ETM Optimization
– Examples –
• Maximize output force
• Uses less than 30% of available mass
(d) Current Density
– Typeset by FoilTEX –
(e) Temperature
(f) Deformation
27
ETM Optimization
– Examples –
• Maximize output force
• Uses less than 30% of available mass
(a) Current Density
– Typeset by FoilTEX –
(b) Temperature
(c) Deformation
28
ETM Optimization
– Examples –
• Maximize output force
• Uses less than 30% of available mass
(a) Current Density
– Typeset by FoilTEX –
(b) Temperature
(c) Deformation
29