Model Library and Tool Support for MEMS Simulationpublications.eas.iis.fraunhofer.de/papers/2001/014/slides.pdf · Model Library and Tool Support for MEMS Simulation Peter Schwarz,

Fraunhofer InstitutIntegrierte Schaltungen

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Model Library and Tool Support for MEMS Simulation

Peter Schwarz, Peter Schneider

Fraunhofer Institute for Integrated CircuitsDesign Automation Division EAS DresdenModelling and Simulation Department

International Symposium on Microelectronic and MEMS Technologies, Edinburgh, Scotland, UK, 30 May - 1 June 2001


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Outline

1. Introduction

2. Model generation and tool support

3. Modelling with generalized KIRCHHOFFian networks

4. Modeling by order reduction

5. Approximation

6. Optimization

7. Simulator coupling

8. Summary


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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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IntroductionMicrosystems: complex, heterogeneous

First integrated design environments exist, but tool support for model generation and coupled simulation is not sufficient yet.

In this paper:- a general modelling approach - a unified model description proposal- some tools for model generation - libraries

will be presented shortly.


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MEMS design: different physical domains and levels of abstraction

System

Subsystem

Component


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System

Subsystem

Component

Modelling byabstraction


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System

Subsystem

Component

Modelling bytransformation



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System

Subsystem

Component

Si

mulator

Coupling

Simulator coupling Modelling bytransformation



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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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Different ways to model a microsystemMicrosystem

Modelling (manually):

Generalized KIRCHHOFFian networks

Mathematical description of

Parameterizable analytical element models

Geometrical structure Mathematical description: PDE

consisting of basic elements

basic elements (analytically)

decomposition into multipoles


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Modelling (manually): Discretization (FEM, FDM, ...)

Generalized KIRCHHOFFian networks DAE, ODE, algebraic equations

Mathematical description of Order reduction


Reduced system matrices

Numerically generated




behavioural models



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Mathematical description of Order reduction Simulation in time or

Black-box model generation


Reduced system matricesSimulation results





behavioural models

frequency domain



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behavioural models

frequency domain



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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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Modelling with Generalized Networks

Many physical quantities may be considered as flow or difference quantities.

There exist conservation laws for flow and difference quantities: generalized KIRCHHOFF’s laws.

Partioning of a large system may be interpreted as decom-position into network elements.

The network concept is valid in the electrical as well as in many non-electrical domains:

fluidics, translational and rotational mechanics,magnetism, ... .

Observation


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Network elements

Two-pole:f

d

Generalized KIRCHHOFF’s laws:

Multipole:f1f2f3

f4

f5

d1d2d3

d4d5


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Network elements and conservation laws

Two-pole:f

d


d1

d3d2

Multipole:f1f2f3

f4

f5

d1d2d3

d4d5

d4


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Two-pole:f

d


Multipole:f1f2f3

f4

f5

d1d2d3

d4d5


Σ di = 0

d1

d3d2

d4

mesh law


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Two-pole:f

d

Multipole:f1f2f3

f4

f5

d1d2d3

d4d5

Generalized KIRCHHOFF’s laws:Generalized KIRCHHOFF’s laws:

d1

d3d2

d4

f2

f1 f3

Σ di = 0

mesh law


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Two-pole:f

d

Σ fj = 0

Multipole:f1f2f3

f4

f5

d1d2d3

d4d5

Generalized KIRCHHOFF’s laws:Generalized KIRCHHOFF’s laws:

d1

d3d2

d4

f2

f1 f3

Σ di = 0

node lawmesh law


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Modeling of an accelerometer

suspensionseismic mass

arrangement of fingers(comb structure)

Typical questions: Effects of parameter variations, models for system simulation, ...

fastening

By courtesy of Robert Bosch GmbH, Germany


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l

m

n

1 2

L


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l

m

n

1 2

L

Fl2Fm2

Fn2Tl2Tm2

Tn2

wl2wm2

wn2ϕl2ϕm2

ϕn2

FT

M wϕ

⋅ D wϕ

⋅ S wϕ

⋅+ +=

Fl1Fm1Fn1Tl1Tm1

Tn1

wl1wm1

wn1ϕl1ϕm1

ϕn1

..

...

.

multi-terminal beam model


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S

EAL

-------- 0 0 0 0 0 EAL

--------– 0 0 0 0 0

012EIn

L3--------------- 0 0 0

6EIn

L2------------ 0

12EIn

L3---------------– 0 0 0

6EIn

L2------------

0 012EIm

L3----------------- 0

6EIm

L2--------------– 0 0 0

12EIm

L3-----------------– 0

6EIm

L2--------------– 0

0 0 0GItL

-------- 0 0 0 0 0GItL

--------– 0 0

0 06EIm

L2--------------– 0

4EImL

-------------- 0 0 06EIm

L2-------------- 0

2EImL

-------------- 0

06EIn

L2------------ 0 0 0

4EInL

------------ 06EIn

L2------------– 0 0 0

2EInL

------------

EAL

--------– 0 0 0 0 0 EAL

-------- 0 0 0 0 0

012EIn

L3---------------– 0 0 0

6EIn

L2------------– 0

12EIn

L3--------------- 0 0 0

6EIn

L2------------–

0 012EIm

L3-----------------– 0

6EIm

L2-------------- 0 0 0

12EIm

L3----------------- 0

6EIm

L2-------------- 0

0 0 0GItL

--------– 0 0 0 0 0GItL

-------- 0 0

0 06EIm

L2--------------– 0

2EImL

-------------- 0 0 06EIm

L2-------------- 0

4EImL

-------------- 0

06EIn

L2------------ 0 0 0

2EInL

------------ 06EIn

L2------------– 0 0 0

4EInL

------------

=


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M

ρV2

------ 0 0 0 0 0 0 0 0 0 0 0

0 ρV2

------ 0 0 0 0 0 0 0 0 0 0

0 0 ρV2

------ 0 0 0 0 0 0 0 0 0

0 0 0ρLIp

2------------ 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 ρV2

------ 0 0 0 0 0

0 0 0 0 0 0 0 ρV2

------ 0 0 0 0

0 0 0 0 0 0 0 0 ρV2

------ 0 0 0

0 0 0 0 0 0 0 0 0ρLIp

2------------ 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

= D

d1 0 0 0 0 0 0 0 0 0 0 0

0 d2 0 0 0 0 0 0 0 0 0 0

0 0 d3 0 0 0 0 0 0 0 0 0

0 0 0 d4 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 d1 0 0 0 0 0

0 0 0 0 0 0 0 d2 0 0 0 0

0 0 0 0 0 0 0 0 d3 0 0 0

0 0 0 0 0 0 0 0 0 d4 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

=


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Network modeling of a valveless micropump

240 µm fluid inlet

b=120 µm

h=60 µm

piezo

Si-substrateoutlet2.4 mm

200 µm


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pump inletinlet diffuser/outletchannel

inlet reservoir

piezo-electrical membrane

outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir


outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics


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inlet reservoir

piezo

outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronics

mechanicsand membrane


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inlet reservoir

piezo

outletnozzle

nozzle region

control

outlet

chamber channel

reservoir

electronicspiezoelectronics



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inlet reservoir

piezo

outletnozzle

nozzle region

control

outlet

chamber channel

reservoir



em


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inlet reservoir

piezo

outletnozzle

nozzle region

control

outlet

chamber channel

reservoir



me

mf


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Library elements

Micromechanics Mechatronics Fluidics Magnetics

• beam

- local coordinates- reference coordinates- reference coordinates (consideration of geo- metric nonlinearities)

• link (reference coordinates)

• plate (local coordinates)

• shell (reference coordi-nates)

• coordinate transformer

• sinusoidal excitation

• gear: spur, planet

• motors

• P-controller

• PI-controller

• PID-controller

• limit

• nonlinear friction

• hysteresis

• measuring devices

• diffusor, nozzle

• fluidic inertia

• fluidic resistance

• outlet - free surface

• channels

• fluidic capacitance

• reservoir

• transducer:membranepiezoelement

• ferromagnetic sections( nonlinear / linear )

• air gap with force effectand variable width

• air gap: constant width

all models with several shapes of cross section: rectangular, circular, pipe, elliptic


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Unified multipole descriptionfa

db

fb

da

network

fa F1 da da· fb f·b ain a· in s s· bin p t, , , , , , , , , ,( )=

db F2 da da· fb f·b ain a· in s s· bin p t, , , , , , , , , ,( )=

0 F3 da da· fb f·b ain a· in s s· bin p t, , , , , , , , , ,( )=

controlled flow quantities

controlled difference quantities

p = parameter vectors = internal states


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db

fb

da

ain aout

network

block diagram(signal flow)

K

- f(.)-



aout F4 da da· fb f·b ain a· in s s· bin p t, , , , , , , , , ,( )=




non-conservative quantities

p = parameter vectors = internal states


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db

fb

da

ain aout

bin bout

network

block diagram(signal flow)

logical block

K

-

RS

f(.)-



aout F4 da da· fb f·b ain a· in s s· bin p t, , , , , , , , , ,( )=


bout F5 da da· fb f·b ain a· in s s· bin p t, , , , , , , , , ,( )=



non-conservative quantities

digital / discrete quantitiesp = parameter vectors = internal states


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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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Different ways to model a microsystem: order reduction (1)Microsystem



Mathematical description of Order reduction


Reduced system matrices





behavioural models



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Model export from FEM simulators

Partial differential equations

Large system of ordinary differential equations

F

F

Real system

FEM simulator(e.g. ANSYS)

Model export

xx..xx................................................xxxx....................xx..xx................................................xxxx......................xx....................xx..xx..............................................xx..xx....................xx..xx..............................................xxxx..xxxx..............................................xxxxxx..................xx..xxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxx.xx..............................................xxxxxx....................xx.xxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx......................xxxxxx..............................................xxxxxx....................xxxxxx..............................................xxxxxx..xx..................xxxx..xx............................................xxxx..xx..................xxxx..xx............................................xxxx..........................xxxx..xx..........................................xx..........................xxxx..xx..........................................xx..xx....................xxxxxx..xx..........................................xx..xx....................xxxxxx..xx..........................................xx..............................xx....................xxxxxx..................................................xx....................xxxxxx..............................................xxxx..xxxx......................................xxxx..........................xxxx..xxxx......................................xxxx................................xxxxxx..................................xxxxxx................................xxxxxx..................................xxxxxx..................................xxxxxx..............................xxxxxx....................................xxxxxx..............................xxxxxx......................................xxxxxx..........................xxxxxx........................................xxxxxx..........................xxxxxx..........................................xxxxxx......................xxxxxx............................................xxxxxx......................xxxxxx..............................................xxxxxx..................xxxxxx................................................xxxxxx..................xxxxxx..................................................xxxxxx..............xxxxxx....................................................xxxxxx..............xxxxxx......................................................xxxxxx..........xxxxxx........................................................xxxxxx..........xxxxxx..........................................................xxxxxx......xxxxxx............................................................xxxxxx......xxxxxx..............................................................xxxxxx..xxxxxx................................................................xxxxxx..xxxxxx..............................................xx..................xxxxxxxxxx................................................xx..................xxxxxxxxxx..................xx..xx........................xx....................xxxxxx....................xx..xx........................xx....................xxxxxx....................xx..xxxx......................xx..................xxxxxxxxxx..................xx..xxxx......................xx..................xxxxxxxxxx......................xxxxxx......................................xxxxxx..xxxxxx....................xxxxxx......................................xxxxxx..xxxxxx......................xxxxxx..................................xxxxxx......xxxxxx....................xxxxxx..................................xxxxxx......xxxxxx......................xxxxxx..............................xxxxxx..........xxxxxx....................xxxxxx..............................xxxxxx..........xxxxxx......................xxxxxx..........................xxxxxx..............xxxxxx....................xxxxxx..........................xxxxxx..............xxxxxx......................xxxxxx......................xxxxxx..................xxxxxx....................xxxxxx......................xxxxxx..................xxxxxx......................xxxxxx..................xxxxxx......................xxxxxx....................xxxxxx..................xxxxxx......................xxxxxx......................xxxxxx..............xxxxxx..........................xxxxxx....................xxxxxx..............xxxxxx..........................xxxxxx......................xxxxxx..........xxxxxx..............................xxxxxx....................xxxxxx..........xxxxxx..............................xxxxxx......................xxxxxx......xxxxxx..................................xxxxxx....................xxxxxx......xxxxxx..................................xxxxxx..xx..................xxxxxxxx..xxxx......................................xxxx..xx..................xxxxxxxx..xxxx......................................xxxx

Entire system Reduced systems (with adjustable order)

x... x ..x ... x.. x. x ....x . xx. x. x ..x .x . x

FF

...

x. . .x .. . ...x . .. x. . .... x ..... x.... x. .. . .xx. . .x .x . ...x . .. x. x .... . .x .x . x... . .. x. x .x.. x ...x . x.... x. .. x .x


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Reduced systems

x ... x.. x .. .x.. x. x....x .xx . x. x.. x .x .x

Description of interface

ENTITY balken ISGENERIC( para_1 REAL;

...para_n: REAL);

PIN( ux1,...uy2: MECHANICAL);

END ENTITY balken;

ARCHITECTURE balken_4 OF balken ISSTATE s_d1,s_d2,s_d3: ANALOG;...VARIABLE var1: ANALOG;...

BEGINRELATION...PROCEDURAL FOR AC,DC,TRANSIENT...END RELATION

END ARCHITECTURE balken_4;

x ...x ...... x... x ......x ..... x .... x..... xx ...x . x.... x... x .x ......x . x. x ...... x .x . x..x ... x. x .... x...x . x

ARCHITECTURE balken_2 OF balken ISSTATE s_d: ANALOG;...VARIABLE var1: ANALOG;...

BEGINRELATION...PROCEDURAL FOR AC,DC,TRANSIENT...END RELATION

END ARCHITECTURE balken_2;

Description ofbehaviour

Description ofbehaviour

Behavioural descriptions with different levels of precision

Model export from FEM simulators: model description


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Geometrical description

Order reduction


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Meshing in the FEM simulator

Order reduction


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Component definition

Order reduction


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Definition of interface nodes−> master degree of freedomterminal and internal nodes

Order reduction


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Definition off flow quantitiesd difference quantitiesxi internal states

Order reduction

f1

d1

f2

d2


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Order reduction

f

d xi





Definition off flow quantitiesd difference quantitiesxi internal states

Export of reduced model equations−> behavioural model

f1

d1

f2

d2


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Order reduction: an ANSYS postprocessor

FEM-Description

Substructuring

Generation of

MAST HDL-A VHDL-AMS

(internal reduction of the order)

M D S, ,

behavioral models

furtherlanguages

system matrices(component description)

M̃ D̃ S̃, , ANSYS-output file *.out containingthe reduced system matrices

Selection of boundary nodes(terminals of the model)

parameter file *.datwith internal representation

Selection of interface nodes


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Electro - thermal interaction

Electrical network

temperature

Thermal model

power dissipation, heat flow chip

header


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Circuit simulator

Electrical network

temperature

power dissipation, heat flow

qT

Thermal model(multipole)


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TSMG - Thermal Simulator and Model Generator

Thermal ModelSPICE-FormatsMAST, HDL-AVHDL-AMS

TSMG

Chipsize, HeadersizePosition and size of devices

Package Model


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Transient simulationT

t

Isotherms

TSMG

Chipsize, HeadersizePosition and size of devicesPower dissipation

Package Model


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t

Isotherms

TSMG

ODE


Package Model

PDE

(large system)

FDM


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t

Isotherms

Order reduction

ODE

TSMG

ODE


Package Model

PDE

(large system)

(small system)

FDM


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Different ways to model a microsystem: order reduction (2)Microsystem











behavioural models

frequency domain



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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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with M = m : interpolation M < m : approximation

• weighted sum of radial basis functions • linear polynomial extension

• smooth and differentiable approximation function• no oszillation between data points• location of given points arbitrary (no equidistant data

required)• independent of the dimension of x

F x( ) aiϕ x xi–( ) bkxk b0+k 1=

d∑+

i 1=

M∑=

Approximation with radial basis functions

Approximation function

Properties


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Radial basis functions

Table of functions

Radial basis function Typelinear function

multiquadric

inverse multiquadric

cubic function

thin plate spline

ϕ r( ) r=

ϕ r( ) r2 c2+=

ϕ r( ) 1 r2 c2+⁄=

ϕ r( ) r3=

ϕ r( ) r2 rlog=

multiquadric inverse multiquadric

thin plate splinecubic

Graphic plot


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MARABU

• Read-in data points (dat, axf, cou, crv)

• Determine RBF aproximation function

• Print calculated parameters

• Generate behavioural models and separate C- program

Program for Multivariate Approximation with Radial Basis Functions

MASTVHDL-AMSHDL-A

RBF approximation

Model generation

Iinput data

Parameters

C-program


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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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Simulation and Optimization


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simulationsystem

optimizationalgorithm

new set ofparameters

error value


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optimizationsystem

simulationalgorithm

error value


simulationsystem

optimizationalgorithm



error value


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Flexible integration of optimization algorithms, simulators, and problem-specific tools

• Simple exchange of algorithms(optimization methods, simulators, ...)

• Change of abstraction level

Focus on industrial design systems (simulators!)

For combination with designer’s experiences and decisions

• Problem analysis• Display of parameter values, progress of optimization,

and optimization results• Support of unexpected behaviour of the optimization

algorithms

Open system

Modular system

Interfaces

Interactive GUI

Visualization

Requirements


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MOSCITOmodular system for constraint nonlinear micro system optimization

simulationmodelgeneration

optimizationerror

calculation


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front end

File Edit View HelpRun

optimizationerror

calculation


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initial

initial values

values

constraints, bounds


optimizationerror

calculation

front end


values


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front end


initial values,

generic

constraints, bounds

modelp1

p2p3


optimizationerror

calculation

model, ready for simulation

initialvalues


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front end


actual

simulator

initial values

generic

constraints, bounds

modelp1

p2p3


control

behavior

optimizationerror

calculation


initialvalues


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front end


actual

objective function weights

specification

simulator

initial values

generic

constraints, bounds

modelp1

p2p3


control

behavior

(desired behavior)

optimizationerror

calculation


initialvalues


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front end


actual

objective function

actual

weights

specification

simulator

initial values

algorithm

generic

parameters

constraints, bounds

modelp1

p2p3


control

control

behavior

(desired behavior)

optimizationerror

calculation



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• Conjugate Gradients• Nelder Mead Simplex• Powell• BFGS (Broyden Fletcher Goldfarb Shanno)• L-BFGS-B • Simulated Annealing• FSQP (Feasible Sequential Quadratic Programming)

• DiRect (Dividing rectangles)• BTRK (Boender Timmer Rinnooy Kan)• N2FB (Nonlinear Mean Square Approximation)• ...

Algorithms integrated

Algorithms planned

Open for other optimization algorithms !

Optimization Algorithms


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• Saber• ELDO• Spice• KOSIM

• Matlab/SIMULINK

• ANSYS• CAPA

• Mathematica• C- or Java-Routines

Circuit simulation

Control systems

FEM

Math-Codes

Open for other simulators !

Simulators


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Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


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Solution of coupled, large systems of nonlinear differential equations. Restricted data exchange and simulator control via the simulator interfaces.

Choice of different basic approaches:

- PVM Parallel Virtual Machine

- CORBA Common Object Request Broker Architecture

- MPI Message Passing Interface

- Java RMI Remote Method Invocation

- COM / DCOM Distributed Common Object Model

- ...

- TCP/IP direct application ( own programs )

Algorithmic problems

Software problems

Simulator coupling


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Simulator coupling

Controlled acceleration sensor

C1

C2

acceleration

F1

F2

displacement

SM


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Controlled acceleration sensor

C1

C2

acceleration

F1

F2

displacement

SM

Simulator coupling


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Simulator coupling: results

1.0 1.2 1.4 1.6 1.8 2.00.8

0

-1e-8

displacement [m]

with control

without control

time [ms]


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Electrical network

temperature

Thermal model

power dissipation, heat flow chip

header


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Electro - thermal interaction: simulator coupling

Circuit simulator FEM/FDM Simulator

Electrical network

temperature

Thermal model

power dissipation, heat flow

qT

chip

header


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Coupling of two simulators

y1 y1out T1


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y1 y1out

y2out y2

T1

T2


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y1

y2

T1

T2


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y2

y2 = T1(y1)

y1

y1 = T2(y2)

Solution


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Relaxation method( y1

(0),y2(0))

y2

y2 = T1(y1)

y1

y1 = T2(y2)

initial solution


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(0),y2(0))

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

initial solution Simulator 1y2

(1) = T1(y1(0))

i = 0, y1(0), y2

(0)


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(0),y2(0))

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

Simulator 1y2

(1) = T1(y1(0))

Simulator 2y1

(1) = T2(y2(1))

i = 0, y1(0), y2

(0)

y1(1)


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(0),y2(0))

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

Simulator 1y2

(1) = T1(y1(0))

Simulator 2y1

(1) = T2(y2(1))

|| y1(1) - y1

(0) || < ε || y2

(1) - y2(0) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(1)


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Relaxation method

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

Simulator 1y2

(2) = T1(y1(1))

Simulator 2y1

(2) = T2(y2(2))

|| y1(2) - y1

(1) || < ε || y2

(2) - y2(1) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(1)


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Relaxation method

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

Simulator 1y2

(2) = T1(y1(1))

Simulator 2y1

(2) = T2(y2(1))

|| y1(2) - y1

(1) || < ε || y2

(2) - y2(1) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(1) y1

(2)


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Relaxation method

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

Simulator 1y2

(2) = T1(y1(1))

Simulator 2y1

(2) = T2(y2(1))

|| y1(2) - y1

(1) || < ε || y2

(2) - y2(1) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(1) y1

(2)


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Relaxation method

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

y2(3)

Simulator 1y2

(3) = T1(y1(2))

Simulator 2y1

(3) = T2(y2(2))

|| y1(3) - y1

(2) || < ε || y2

(3) - y2(2) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(2)


IIS

Relaxation method

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

y2(3)

Simulator 1y2

(3) = T1(y1(2))

Simulator 2y1

(3) = T2(y2(2))

|| y1(3) - y1

(2) || < ε || y2

(3) - y2(2) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(3) y1

(2)


IIS

Relaxation method

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

y2(3)

Simulator 1y2

(3) = T1(y1(2))

Simulator 2y1

(3) = T2(y2(2))

|| y1(3) - y1

(2) || < ε || y2

(3) - y2(2) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(3) y1

(2)


IIS

Relaxation method

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(4)

y2(3)

Simulator 1y2

(4) = T1(y1(3))

Simulator 2y1

(4) = T2(y2(4))

|| y1(4) - y1

(3) || < ε || y2

(4) - y2(3) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(3)


IIS

Relaxation method

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(4)

y2(3)

Simulator 1y2

(4) = T1(y1(3))

Simulator 2y1

(4) = T2(y2(4))

|| y1(4) - y1

(3) || < ε || y2

(4) - y2(3) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(3) y1

(4)


IIS

Relaxation method

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(4)

y2(3)

Simulator 1y2

(4) = T1(y1(3))

Simulator 2y1

(4) = T2(y2(4))

|| y1(4) - y1

(3) || < ε || y2

(4) - y2(3) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(3) y1

(4)

numerical solution


IIS


(0),y2(0))

y2(1)

y2

y2 = T1(y1)

y1

y1 = T2(y2)

y2(2)

y2(3)

initial solution Simulator 1y2

(i+1) = T1(y1(i))

Simulator 2y1

(i+1) = T2(y2(i+1))

|| y1(i+1) - y1

(i) || < ε || y2

(i+1) - y2(i) || < ε

next time interval

i=i+1no

yes

i = 0, y1(0), y2

(0)

y1(1) y1

(2)

numerical solution


IIS

Newton’s method

y2

y1

y1 = T2(y2)

y2 = T1(y1)

initial solution


IIS

Newton’s method

y2

y1

y1 = T2(y2)

y2 = T1(y1)y2

(0)

y1(0)


IIS

Newton’s method

y2

y1

y1 = T2(y2)

y2 = T1(y1)

y2(1)

y1(1)


IIS

Newton’s method

y2

y1

y1 = T2(y2)

y2 = T1(y1)

y2(2)

y1(2)

numerical solution


IIS

Outline

1. Introduction




5. Approximation

6. Optimization


8. Summary


IIS

Summary

Generalized KIRCHHOFFian networks can be used to model multi-domain problems.

Modelling tools (order reduction, black-box approximation, opti-mization, ... ) and first multi-domain libraries exist: prototypes, very incomplete.

Standardized modelling languages / Hardware Description Languages like VHDL-AMS and Modelica will be supported by many system simulators.

Simulator coupling has been proven as a powerful alternative to model generation of heterogeneous systems.

A widely accepted modelling methodology.

Order reduction techniques for very large nonlinear dynamic systems.

Automatic modelling of coupled field problems.

State-of-the art

Open problems

Documents

Model Library and Tool Support for MEMS Simulationpublications.eas.iis.fraunhofer.de/papers/2001/014/slides.pdf · Model Library and Tool Support for MEMS Simulation Peter Schwarz,