ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
Au uction for Tra of MEMS-
es
orvinkTechnologyeiburg
tomatic Model Rednsient Simulation
based DevicEvgenii B. Rudnyi and Jan G. K
IMTEK–Institute for Microsystem Albert Ludwig University Fr
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
E.B. Rud Page 2
Introduction
Re
♦
E
ATb
2
Tu
♦
JSM
ints
.im de/simulation/
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i@ k.de
tek.
timte
nyi – AISEM tutorial, Trento 2003
view. B. Rudnyi, J. G. Korvink, utomatic Model Reduction for ransient Simulation of MEMS-ased Devices, Sensors Update, 002, 11, 3-33.
torial. G. Korvink, IEEE Sensors hort Course on Compact odelling, 2002.
Prepr♦ www
Conta♦ rudny
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
E.B. Rud Page 3
Introduction
Co
♦
I
♦
S
♦
S
♦
Is
♦
L
♦
N round picture © Ponderosa Forge
Forging a smaller System
Backg
nyi – AISEM tutorial, Trento 2003
ntentsntroduction to the idea
tatement of the problem
mall linear systems
ntroduction to Krylov ubspaces
arge linear systems
onlinear systems
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
E.B. Rud Page 4
Introduction
Ge
Reduced Order System of ODEs
4 5
Discrete System of ODEs
3nyi – AISEM tutorial, Trento 2003
neration Paths
Device Description
Direct Translation Lumped Model
Discretization MeshPDEs & Geometry
1
2
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
E.B. Rud Page 5
Introduction
W
♦
Cs
♦
♦
diagram:
ca
uced
FEM +model red.
stem levelulation to
velop inteligentcuit
el
se
tion
Red
Sysimdecir
mod
nyi – AISEM tutorial, Trento 2003
hy is it useful ?ompact model for system level
imulation:Reduced model fits naturally in system level simulators.The generatation of the reduced model can be almost automatic.
♦ Block
Devicespecifi
Changein devic
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
E.B. Rud Page 6
Small Pause
Su
♦
Mtt
♦
Ibp
s
♦
M
ue
’s Next?
u to the idea
e the problem
li systems
u to Krylov ac
li systems
ne stems
ction
nt of
near
ctiones
near
ar sy
nyi – AISEM tutorial, Trento 2003
mmaryany computational nodes in a
ypical FEM (...) model appear o be “redundant”.
t appears that much of the ehaviour of a system takes lace in a low dimensional ubspace.
OR can greatly improve the se of simulation tools during ngineering design.
What♦ Introd
♦ Statem
♦ Small
♦ Introdsubsp
♦ Large
♦ Nonli
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG
E.B. Rud Page 7
ent of Problem
De
♦
D
♦
I
♦
S
it Form:
F such that
for
s “small”:
E
E
M
z
A b
– A ℜ n ℜ n×∈1– b ℜ n∈
⋅
z k k n«
∈t( min x t( ) X z t( )⋅–
ind
i
x⋅ +
1 F⋅
f⋅
Xz εεεε+
ℜ∈
ℜ n
) =
nyi – AISEM tutorial, Trento 2003
Statem
livered ODEsiscussion Limited to 1st Order
mplicit Form (MNA, T-psi):
econd Order (mechanics):
:
♦ Explic
♦ Goal:
where
and
dxdt------⋅ F x⋅ f+=
F, ℜ n ℜ n×∈ f x t( ), ℜ n∈
d2y
dt2---------⋅ D dy
dt------⋅ K y⋅+ + f=
dydt------= M 0
0 I
ddt----- z
y⋅ D K
I– 0zy
⋅– f0
+=
dxdt------ =
A E=
b E=
x X=
εεεεmin εεεε
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 8
ent of Problem
Sy♦ O
o
♦
♦
♦
♦♦
♦
-scale dynamic system
e tem
en
B u⋅+=
y x⋅
A B u⋅+
y z
in y t( ) y t( )–
d sys
ce
A x⋅
C=
ˆ z⋅
C ⋅=
m
nyi – AISEM tutorial, Trento 2003
Statem
stem Theory Versionften solution is not needed ver entire domain, i.e., with:
Inputs
Outputs
Scatter Matrix
Gather Matrix Multiple input–multiple output MIMOSingle input–single output SISO
♦ Large
♦ Reduc
♦ Differ
u ℜ m∈
y ℜ p∈
B ℜ n ℜ m×∈
C ℜ p ℜ n×∈
dxdt------
dzdt------ =
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 9
of the Problem
PiRe
. .
.
Befo
Afte
S Output
Σ =
Σ =
C
C
=
=
ystem
nyi – AISEM tutorial, Trento 2003
Statement
ctorial presentation
.
. .
re:
r:
User Input
A BC
A B
C
A
A
B
B
=
=
+
+
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 10
Small Pause
Su♦ S
npr
♦ Mms
’s Next?u to the idea
e the problem
li systems
u to Krylov ac
li systems
ne stems
ction
nt of
near
ctiones
near
ar sy
nyi – AISEM tutorial, Trento 2003
mmaryystem theory provides a atural language to describe a roblem of model order eduction.
ost results comes from athematicians working for the
ystem theory.
What♦ Introd
♦ Statem
♦ Small
♦ Introdsubsp
♦ Large
♦ Nonli
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 11
Linear Systems
HaVa
♦ S
♦ I
♦ I
♦ S
mians:
n uations:
B BT eATt⋅ ⋅ ) td0
∞
CT C eAt⋅ ⋅ ⋅ ) td0
∞
T B BT⋅+ + 0=
A CT C⋅+ + 0=
σ λ i P Q⋅( )
ov eq
eAt ⋅(
eATt(
P A⋅
Q ⋅
i =
nyi – AISEM tutorial, Trento 2003
Small
nkel Singular lues (HSV)
ystem:
mpulse response:
nput-to-state:
tate-to-output:
♦ Gram
♦ Lyapu
♦ HSV:
Σ A BC
=
h t( ) C eAt B⋅ ⋅=
ξ t( ) eAt B⋅=
η t( ) C eAt⋅=
P ∫=
Q ∫=
A P⋅
AT Q⋅
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 12
Linear Systems
De lues
Antoulos♦
2(
nyi – AISEM tutorial, Trento 2003
Small
cay of Hankel Singular VaTheory
Error bound: If the system is reduced to one with k largest HSV then
G G– ∞<
σk 1+ … σn+ + )
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 13
Linear Systems
Tr♦ I
Di♦ S
♦
♦ H♦
♦ B♦
ter than HNA. not preserve the tio state.la turbation Appr.se he stationary state.en eighted model tio
G
G( W ∞
naryr Perrve tcy-wn
G– )
nyi – AISEM tutorial, Trento 2003
Small
ansfer Functionn Laplace Domain:
fferent methodstable systems
In unstable systems something should be done with unstable poles.
ankel Norm Appr.Produces an optimal solution
alanced Truncation Appr.Most often used.
♦ Fas♦ Do
sta♦ Singu
♦ Pre♦ Frequ
reduc
♦
Σ s( ) C sI A–( ) 1– B⋅ ⋅=
V
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 14
Linear Systems
SL♦ F
fw♦
♦ I♦♦♦♦
♦ H♦ M
e computational lexity is .i “small” systems:
me SerialTime Parallel (4 processor)
25
3 130
46 666
2668
O N3( )
ted toTi
60
70
43
nyi – AISEM tutorial, Trento 2003
Small
ICOT LibraryORTRAN Code + Examples ound at:ww.win.tue.nl/niconet
European Community BRITE-EURAM III Thematic Networks Programme.
mplements all methods:Balanced Truncation Appr.Singular Perturbation Appr.Hankel Norm Appr.Frequency-weighted MOR.
as a parallel version.atlab has licensed SLICOT.
♦ Yet, thcomp♦ Lim
Order
600
1332
2450
3906
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 15
Small Pause
Su♦ H
g
♦ Stsl
♦ Sts
’s Next?u to the idea
e the problem
li systems
u to Krylov ac
li systems
ne stems
ction
nt of
near
ctiones
near
ar sy
nyi – AISEM tutorial, Trento 2003
mmaryankel singular value theorem
ives error bound.
ystem theory has mature heory for MOR of linear ystems. We can find optimal ow dimensional subspace.
LICOT library implements heory and is “easy to use” for mall linear systems.
What♦ Introd
♦ Statem
♦ Small
♦ Introdsubsp
♦ Large
♦ Nonli
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 16
ylov Subspaces
De♦ A
♦ Ts
o
♦ Ao
s the left Krylov subspace of and of order .
es ow-dimensional of paces of order .
c utation is ric unstable because of in ors.
e 0 top algorithms of th ury.
{
{
l, ) A l k
k
the l subsompally g err
d in 1 cent
nyi – AISEM tutorial, Trento 2003
Introduction to Kr
finitionction of matrix on vector :
his is the right Krylov ubspace of and
f order .
ction of transposed matrix n vector :
♦ This i
♦ Definbasis
♦ Directnumeround
♦ Includthe 20
A r
r A r⋅ … Ak 1– r⋅, , , }
KR A r,( ) A b
k
Al
l AT l⋅ … ATk 1–l⋅, , , }
KL A(
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 17
ylov Subspaces
Ar♦ M♦ P
m
♦
♦♦♦ A
op
os Algorithmos ors:
bi-orthogonal:
on :
tr onal matrix.n ast for large k.
V
V
H
s r A r … Ak 1– r⋅, ,⋅, }
sp AT l … AT k 1–( ) l⋅, ,⋅ }W
g δ1 δ2 … δk, , ,( )
⋅ HL
vect
are
to
i-diagcy: F
pan{
an l,{
dia=
A
W =
nyi – AISEM tutorial, Trento 2003
Introduction to Kr
noldi Processodified Gram-Schmidt.
roduces basis and small atrix
is orthonormal:
is upper-Hessenberg matrix new vector must be rthogonalized to all the revious vectors.
Lancz♦ Lancz
♦ and
♦ Relati
♦ is Efficie
VHA
VT V⋅ I=T A V⋅ ⋅ HA=
A
V =
W =V
VT W⋅
VT A⋅HL
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 18
Small Pause
Su♦ T
n♦
♦ T♦
♦ Tn♦
’s Next?u to the idea
e the problem
li systems
u to Krylov ac
li systems
ne stems
ction
nt of
near
ctiones
near
ar sy
nyi – AISEM tutorial, Trento 2003
mmarywo algorithms can form umerically stable basis:
Both are amenable to large, sparse systems due to matrix-vector product.
he Lanczos algorithm is faster.Basises are biorthogonal.
he Arnoldi algorithm is more umerically stable.
Basis is orthogonal.
What♦ Introd
♦ Statem
♦ Small
♦ Introdsubsp
♦ Large
♦ Nonli
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 19
Linear Systems
Pa♦ C
f
♦♦ P
♦ M
ed func. is small rational.
no :é oximant: match
oments.é approximant: li match less
mit hing is numerically
blE mptotic wavefrom lu does not work.
G
s
m
a s z1–( )… s zk 1––( )p1– )… s pk–( )
--------------------------------------------=
2
logy appr
m-typecitlyents. matce: - asyation
s(--------
k
nyi – AISEM tutorial, Trento 2003
Large
dé Approximantsan always express transfer
unction matrix elements using:
are the zeroes, poles.
adé matches moments about
:
oment matching
for
♦ Reduc
♦ Termi♦ Pad
♦ Padimpmo
♦ Explicunsta♦ AW
eva
ij s( )a s z1–( )… s zn 1––( )
s p1–( )… s pn–( )-----------------------------------------------------=
zi pi,k
0 Gij s( ) mi s s0–( )p
p 0=
k n<
∑=
i miˆ= i 0 … q, ,=
Gij s( )
q =
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 20
Linear Systems
ImM♦ A
p
♦
♦
♦
os: Also left subspace
ce , and such
di licitly matches nos licitly matches n
K
N
T, MT CT⋅=
X Y
I s0HL+( )⋅=
YT M⋅
k
2k
,
s
impts. impts
L) L
HL
HL1–
HL1– ⋅
C X⋅
nyi – AISEM tutorial, Trento 2003
Large
plicit Moment atchingrnoldi: Right subspace
, and
roduces and such that:
♦ Lancz
produthat:
♦
♦
♦♦ Arnol
mome♦ Lancz
mome
kr M N,( ) M A s0I–( ) 1–
=
M B⋅=HA X
A HA1– I s0HA+( )⋅=
B HA1– XT M⋅ ⋅=
C C X⋅=
Kkl M(
A
B =
C =
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 21
Linear Systems
Pr
ℜ
1k
v
vO1
⊥ 2
liq rojectionOrt
Arn Lanczos
SimMore natural:
ControllabilityObservability
v
ue P
nyi – AISEM tutorial, Trento 2003
Large
ojection Idea
1k
ℜ 2k
ℜ
v
v⊥ 1
Obhogonal Projection
oldi
pler
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 22
Linear Systems
Co♦ T
♦♦
♦ B
♦
QR Decomposition
h ality e riangle solvee atrix multiply
ve ers:co ioner from Appl.
pl t fast matrix lt Again, application h ere
F
Q 1– QT=
ogonfast tfast m
Solvndit
emeniply: elp h
Q R⋅
nyi – AISEM tutorial, Trento 2003
Large
mputing Inverses ypical case:
Do not compute : Bad ideaFind such that
y LU Decomposition:
Two fast triangle solves
♦ By
♦ Ort♦ On♦ On
♦ Iterati♦ Pre♦ Im
mucan
F 1– w⋅
F 1–
x F x⋅ w=
L U⋅=
L U x⋅( )⋅ w= L y⋅ w=⇔U x⋅ y=
F =
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 23
Linear Systems
Ex♦ 6
♦ C
EM Circuit: PVL
: Z. Bai, R. Freund, A Partial Padé-via-LanzcosMethod for Reduced-Order Modelling
Source
nyi – AISEM tutorial, Trento 2003
Large
amples from EE4 Pin RF IC: Padé via Lanczos:
D Player: Comparison
♦ PEEC
Pin 1 Error
Source: Z. Bai, R. Freund, A Partial Padé-via-LanzcosMethod for Reduced-Order Modelling
Source: Antoulas, Sorenson, Gugercin, A Survey ofModel Reduction Methods for Large Systems
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 24
Linear Systems
Ra♦ E
a
♦ T♦♦
♦ CEQ
Kry n
PartialRealization
Padé
ShiftedPadé
RationalInterpolation
True
True
Red Thesis
lov Projectiouction, PhDnyi – AISEM tutorial, Trento 2003
Large
tional Krylovxpansion usually accurate bout .
his suggests:Multiple expansion points Matching transfer function moments at all points
hallenge: Where to place xpensive solves (Many LU or R decompositions)
s0
si
si
Source: E. Grimme:Methods for Model
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 25
Linear Systems
SoEq♦ P
gK
♦ S♦
♦
♦ V
al remedy: Low rank ximation of grammians.
ns trix rs trix o ov-based, also for an .
A K (Matlab based)e .org/lyapack
O n2( )∼ O n( )∼
e mae maKrylcing
PACtlib
nyi – AISEM tutorial, Trento 2003
Large
lving Lyapunov uationsadé approximants do not have lobal error estimates ... SVD-rylov.
teps:Solve Lyapunov equations for Grammians and .Eigen-decompose .
ery expensive:
♦ Generappro
♦ De♦ Spa♦ Als
bal
♦ See LYwww.n
P QPQ
O n3( )∼
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 26
Small Pause
Su♦ P
♦ Aga
♦ Ru
♦ F♦♦
’s Next?u to the idea
e the problem
li systems
u to Krylov ac
li systems
n stems
ction
nt of
near
ctiones
near
ear sy
nyi – AISEM tutorial, Trento 2003
mmaryadé and Krylov are related.
rnoldi and Lanczos can enerate implicitly Padé pproximants (PVL).
ational Krylov improves sing many expansions.
uture holds promise for:Large Lyapunov solvers.Large matrix exponential approximants.
What♦ Introd
♦ Statem
♦ Small
♦ Introdsubsp
♦ Large
♦ Nonli
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 27
linear Systems
Sp♦ B
♦ Sp
♦ R
nonlinear by Taylor sion:
xA' x⋅+
Tx …+
Li f NLi ∂xj⁄
jk NLi ∂xj∂xk⁄
) x) x⋅ C u⋅+
f NL0
A''⋅ ⋅
j ∂=
∂ f=
A(=
nyi – AISEM tutorial, Trento 2003
Non
ecial Casesasic Problem:
plitting linear and nonlinear arts:
educe linear part as usual
♦ Treat expan
♦
f x u,( )= y g x( )=
f f L f NL+=
ALij f L0 ∂ f Li ∂xj⁄+=
f NL =
12---x+
A'N
A''NLi
f x u,(
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 28
linear Systems
PO♦ S
♦ Ass
♦ P
the truncated snapshots pping the smallest
lar es:
du system, form:
va es:l n near solve
w mpute ?e ition
x
S
S
Sk UΣkVT
=
T ˆ x⋅ u, )
x⋅ )=
UTA x( )U
valu
ced
ntagonli
to co intu
f U(⋅
g U(
nyi – AISEM tutorial, Trento 2003
Non
D Ideaolve the full nonlinear system
t appropriate times, take napshots , and collect the napshots in a matrix:
, m is many!
erform SVD of :
♦ Form by dro
singu
♦ For re
♦ Disad♦ Ful
♦ Ho♦ Som
f x u,( )= y g x( )=
si
s1 s2 … sm, , ,{ }=
S
UΣVT σiui vi⊗i 1=m∑= =
x U=
y
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 29
linear Systems
PO
e: J. Chen, S. Kangiaues for Coupled Circuit and MEMS Simulation
R itch:FD nonlinear beamin ing squeeze film& rodynamic forceK en-Loève + Galerkin
20
SourcTechn
F Sw for
clud electarhun
kHz
nyi – AISEM tutorial, Trento 2003
Non
D Example: MEMS
15 V,
7.4 V, Step input
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 30
Small Pause
Su♦ A
s♦
♦
♦ P♦♦
♦ N
nyi – AISEM tutorial, Trento 2003
mmarypplication-oriented
implifications exist, but:May need symbolic manipulations.May need expensive evaluations.
OD is general, and works, but:Computationally expensive.Requires user interaction.
onlinear MOR is tough.
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud Page 31
Conclusions
Sm♦ E
k♦ F
g
La♦ R♦ A♦ L
m♦ W♦ P
r
e may yield:punov for large systems.
pr ation of matrix o al for large te
n s l application-d chniques.e tion or Splitting.
O tw y snapshots?
oximnentims.
earpeciaent tearizaD, bu man
nyi – AISEM tutorial, Trento 2003
all Linearxcellent state: complete nowledge.or accuracy goal, automatic uaranteed reduced model.
rge Lineareasonably good choices.rnoldi more stable.anczos matches more oments.hen to stop reducing?
adé local, nonoptimal for wide ange. Remedy: rational Krylov.
♦ Futur♦ Lya♦ Ap
expsys
Nonli♦ Either
depen♦ Or Lin♦ Else P
♦ Ho
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