based Devices - evgenii.rudnyi.ru

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


E.B. Rud

Introduction

Re

♦

E

ATb

2

Tu

♦

JSM

ints

.im de/simulation/

c

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


E.B. Rud

Introduction

Co

♦

I

♦

S

♦

S

♦

Is

♦

L

♦

N round picture © Ponderosa Forge

Forging a smaller System

Backg


ntentsntroduction to the idea

tatement of the problem

mall linear systems

ntroduction to Krylov ubspaces

arge linear systems

onlinear systems


E.B. Rud

Introduction

Ge

Reduced Order System of ODEs

4 5

Discrete System of ODEs
3

neration Paths

Device Description

Direct Translation Lumped Model

Discretization MeshPDEs & Geometry

1

2


E.B. Rud

Introduction

W

♦

Cs

♦

♦

diagram:

ca

uced

FEM +model red.

stem levelulation to

velop inteligentcuit

el

se

tion

Red

Sysimdecir

mod


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


E.B. Rud

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


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


E.B. Rud

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

) =


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

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


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------ =


of the Problem

PiRe

. .

.

Befo

Afte

S Output

Σ =

Σ =

C

C

=

=

ystem


Statement

ctorial presentation

.

. .

re:

r:

User Input

A BC

A B

C

A

A

B

B

=

=

+

+


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


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


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 =


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⋅


Linear Systems

De lues

Antoulos♦

2(


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+ + )


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– )


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


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 to
Ti

60

70

43


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


Small Pause

Su♦ H

g

♦ Stsl

♦ Sts


e the problem

li systems

u to Krylov ac

li systems

ne stems

ction

nt of

near

ctiones

near

ar sy


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


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 subs
ompally g err

d in 1 cent


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(


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 =


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


Small Pause

Su♦ T

n♦

♦ T♦

♦ Tn♦


e the problem

li systems

u to Krylov ac

li systems

ne stems

ction

nt of

near

ctiones

near

ar sy


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


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


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 =


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⋅


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 =


Linear Systems

Pr

ℜ

1k

v

vO1

⊥ 2

liq rojectionOrt

Arn Lanczos

SimMore natural:

ControllabilityObservability

v

ue P


Large

ojection Idea

1k

ℜ 2k

ℜ

v

v⊥ 1

Obhogonal Projection

oldi

pler


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=
ogon
fast tfast m

Solvndit

emeniply: elp h

Q R⋅


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 =


Linear Systems

Ex♦ 6

♦ C

EM Circuit: PVL

: Z. Bai, R. Freund, A Partial Padé-via-LanzcosMethod for Reduced-Order Modelling

Source


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


Linear Systems

Ra♦ E

a

♦ T♦♦

♦ CEQ

Kry n

PartialRealization

Padé

ShiftedPadé

RationalInterpolation

True

True

Red Thesis
lov Projectiouction, PhD

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


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


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( )∼


Small Pause

Su♦ P

♦ Aga

♦ Ru

♦ F♦♦


e the problem

li systems

u to Krylov ac

li systems

n stems

ction

nt of

near

ctiones

near

ear sy


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


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(=


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,(


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(


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


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


Non

D Example: MEMS

15 V,

7.4 V, Step input


Small Pause

Su♦ A

s♦

♦

♦ P♦♦

♦ N


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.


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


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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