Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Engineering design issues arising when applying MOR to MEMS: Three case studies

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Outline

Microelectromechanical RF Resonator IRST RF MEMS Switch

Model Order Reduction of a Nonlinear ODE System IBM Millipede memory device

Automatic parametric MOR for MEMS Design IMEGO Butterfly Gyroscope

Small Demo

2

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

What is design?

A registered design is a monopoly right for the appearance of the whole or a part of a product resulting from the features of, in particular, the: lines, contours, colors, shape, texture and materials of the product or its ornamentationUK design office

Design is finding the right connections or simply making things rightRalph Kaplan

Cenk Acar: Distributed-Mass Gyroscope, 1st place 2003 MEMS Design Challenge

Butterfly Gyro: www.monolitsystem.se

3

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Specifics of MEMS Design

Restrictions by technology Materials Processes Size 2.5 D-Layout (Manhattan Design)

Scaling effects Unusual physical effects

(volume versus surface,high gradients)

Novel actuation concepts Reduced design freedom Often in contradiction to everday

experience Many repeated parts

www.zdf.de

www.rootsweb.com

4

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

MEMS Simulation Overview

Fluidic Thermal Chemical Electrical Structural

Transient ParametricFrequency

limitNonlinear

Physical Domain

Equations

Simulation Tasks

Model Order Reduction

Compact Modelling

Transient HarmonicSteady

StateSystem Export

Error

Estimation

5

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

IRST RF MEMS Switch

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

MEMS RF Capacitive Switch

7

Gold membrane, suspended over electrode

Squeeze film damping, modelled using Rayleigh model

CMOS compatible process, directly on top of the electronic circuits

Membrane has ca. 30 MPa tensile stress ➔ Tunes resonance frequency

Deflection measured using an interferometric profilometer

FEM model has ca. 45000 DoF Needed is small accurate circuit

model

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

MEMS RF Capacitive Switch

7

Gold membrane, suspended over electrode

Squeeze film damping, modelled using Rayleigh model

CMOS compatible process, directly on top of the electronic circuits

Membrane has ca. 30 MPa tensile stress ➔ Tunes resonance frequency

Deflection measured using an interferometric profilometer

FEM model has ca. 45000 DoF Needed is small accurate circuit

model

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Model Reduction Procedure: From 45’000 DoF to 15

Simplified capacitor model yields 9 nonlinear gap

forces, distributed over the membrane

FEM model in ANSYS Includes stress stiffening

(geometrical stiffness matrix)

Block Arnoldi for MOR 15 DoF Comparison with

experiment used as stop criterion

Simulation of system response VHDL + Cadence

8

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Model Reduction Procedure: From 45’000 DoF to 15

Simplified capacitor model yields 9 nonlinear gap

forces, distributed over the membrane

FEM model in ANSYS Includes stress stiffening

(geometrical stiffness matrix)

Block Arnoldi for MOR 15 DoF Comparison with

experiment used as stop criterion

Simulation of system response VHDL + Cadence

8

Operation Tool Computationaltime

Initial stress state & element matrices computation

ANSYS 68.1s

Reduced order model extraction Mor4fem 500s

Electrostatic forces computation (with FEM) ANSYS 59.2s

Dynamic simulation Cadence 660ms

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Model Reduction Procedure: From 45’000 DoF to 15

Simplified capacitor model yields 9 nonlinear gap

forces, distributed over the membrane

FEM model in ANSYS Includes stress stiffening

(geometrical stiffness matrix)

Block Arnoldi for MOR 15 DoF Comparison with

experiment used as stop criterion

Simulation of system response VHDL + Cadence

8

Operation Tool Computationaltime

Initial stress state & element matrices computation

ANSYS 68.1s

Reduced order model extraction Mor4fem 500s

Electrostatic forces computation (with FEM) ANSYS 59.2s

Dynamic simulation Cadence 660ms

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Model Reduction Procedure: From 45’000 DoF to 15

Simplified capacitor model yields 9 nonlinear gap

forces, distributed over the membrane

FEM model in ANSYS Includes stress stiffening

(geometrical stiffness matrix)

Block Arnoldi for MOR 15 DoF Comparison with

experiment used as stop criterion

Simulation of system response VHDL + Cadence

8

Operation Tool Computationaltime

Initial stress state & element matrices computation

ANSYS 68.1s

Reduced order model extraction Mor4fem 500s

Electrostatic forces computation (with FEM) ANSYS 59.2s

Dynamic simulation Cadence 660ms

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

IBM Millipede Memory Device

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 10

MEMS Many-Parts Challenge

Many parts – each is complex (10many potentially nonlinear equations)

Teams develop parts

Challenge: how to test entire system?

(without building it!)

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

IBM scanning-probe data storage device

Writing

Reading

Write currentResistive heater

Scan direction

PolymerSubstrate

Read current

© IBM Research Center, Zürich11

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Modelling issues

Tip actuated electrostatically

→ electrostatic/structural coupling

Use of transducer elements Nonlinear 1/x2 force law

(parallel plate capacitor) Coupled equation after

discretization (ANSYS):

Transient solution very time consuming

→ requires MOR Goal: Use in system

simulator→ Verilog-A Model order reduction

allows to cosimulate with circuitry

Mx+Ex+Kx = f [x,u]

12

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Assume nonlinear stiffness matrix, convert to polynomial

Idea: Use projection on polynomial matrices:

Advantages: Form of the equations remains the same Simulation free

Open questions: How to choose V? One possibility: Use only the linear part of K

and hope that you catch all the important information How to estimate the accuracy of the reduced model? Where is

the trust region?

Nonlinear Model Order Reduction

Ci jx j +K1i jx j +K2

i jkx jxk +K3i jklx jxkxl = Bimum

Ci jVinVjoxr,o +K1i jVinVjoxr,o +K2

i jkVinVjoVkpxr,oxr,p + . . .

= BimVinum

13

Idea fromChen & White

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Results: From 9440 DoF to 19Model order reduced versusfull solution

Model order reduced versuspolynomial approximation

14

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Results: From 9440 DoF to 19Model order reduced versusfull solution

Model order reduced versuspolynomial approximation

14

CCIC’06

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Automatic parametric MOR for MEMS Design

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametrisation – Simple beam model

Stationary model 2 Materials, different

Young‘s modulus Defined displacement at

both sides Ratio of Young‘s moduli

gives shape of loaded beam -> shape(A:B)

dB

A

16

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametrisation – Simple beam model

Stationary model 2 Materials, different

Young‘s modulus Defined displacement at

both sides Ratio of Young‘s moduli

gives shape of loaded beam -> shape(A:B)

dB

A

16

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametrisation – physical

Equation is Parameter is Young‘s

modulus Dependencies in are

linear, do not influence mesh

Approach: Assemble elements A and

B in 2 matrices Matrix A is constant Matrix B can be scaled

BA

!g · "[Y,#] · !g = !f

Y (i!!)(1!2!)(1+!)

,Y

2(1+!)

KA KBK = KA +KB

17

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametrisation – geometric

Parameters influence discretisation Assembly often not controllable

in commercial codes (e.g. Ansys) Approach:

Influence of parameter is known Generate set of matrices with

varying parameters [i,j] entry of matrices change

with parameter p Define mapping function:

Least squares fit finds matrices a and b

Assemble to build K[p]

Z... d!

KA,KB, ...,KN

fi j[p] = ai j + p!bi j

Eg. 2 x 2 Matrix

18

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope

Gyroscope: Sensor for angular velocity Coriolis acceleration

Coupling between y- and z-Axes -> Signal(v)

„simple“ design Cheap fabrication Good performance Needs good electronics

-> system simulation Optimal parameters?

Foucault demonstrates Earth’s rotation, Paris, Pantheon, 1851

vaCor ×= ω2

19

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope

Gyroscope: Sensor for angular velocity Coriolis acceleration

Coupling between y- and z-Axes -> Signal(v)

„simple“ design Cheap fabrication Good performance Needs good electronics

-> system simulation Optimal parameters?

Foucault demonstrates Earth’s rotation, Paris, Pantheon, 1851

Working principle of the Imego Butterfly Gyroscope

vaCor ×= ω2

19

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope

Gyroscope: Sensor for angular velocity Coriolis acceleration

Coupling between y- and z-Axes -> Signal(v)

„simple“ design Cheap fabrication Good performance Needs good electronics

-> system simulation Optimal parameters?

Foucault demonstrates Earth’s rotation, Paris, Pantheon, 1851

Working principle of the Imego Butterfly Gyroscope

vaCor ×= ω2

Butterfly Gyro: www.monolitsystem.se

19

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope

Gyroscope: Sensor for angular velocity Coriolis acceleration

Coupling between y- and z-Axes -> Signal(v)

„simple“ design Cheap fabrication Good performance Needs good electronics

-> system simulation Optimal parameters?

Foucault demonstrates Earth’s rotation, Paris, Pantheon, 1851

Working principle of the Imego Butterfly Gyroscope

vaCor ×= ω2

Butterfly Gyro: www.monolitsystem.se

19

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope - parametrisation

Modelling (IMS): ~1’000 quadratic elements ~18’000 DoF

5 Parameters: Physical parametrisation:

• Angular velocity ω• Material density ρ • Raleigh damping

parameters α, β Geometric parametrisation:

• Width of bearing d• Param. error: 10-21

Gyroscope model generated with the Imtek Mathematica Supplement: http://www.imtek.de/

simulation/mathematica/IMSweb/

M[!,d]x+D[",d,#,$]x+S[d,1/d]x = Bu20

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope - parametrisation

Max deflection is 0.1 - 1 µm Output is difference signal Value range of output is

10-7 - 10-4 µm

21

d

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Butterfly Gyroscope - parametrisation

Max deflection is 0.1 - 1 µm Output is difference signal Value range of output is

10-7 - 10-4 µm

21

d

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametric MOR - overview

Requirements: Multi-variable Padé – type

moment matching approach Implicit moment matching

Approach: discard mixed moments Error will increase rapidly

with distance from expansion point

Use several expansion points to cover complete parameter space

How many expansion points? How many expansions at

each point?

22

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametric MOR - overview

Requirements: Multi-variable Padé – type

moment matching approach Implicit moment matching

Approach: discard mixed moments Error will increase rapidly

with distance from expansion point

Use several expansion points to cover complete parameter space

How many expansion points? How many expansions at

each point?

22

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametric MOR - overview

Requirements: Multi-variable Padé – type

moment matching approach Implicit moment matching

Approach: discard mixed moments Error will increase rapidly

with distance from expansion point

Use several expansion points to cover complete parameter space

How many expansion points? How many expansions at

each point?

22

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 23

Parametric MOR – Local stop criterion

stop criterion – convergence check: define output compute output near

expansion point when:

change < threshold value ➔ converged

Mproj global

expansion point

Expansion direction

add next moment

Mproj local

add next moment

Mproj global

check convergence

NoYes

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 24

Parametric MOR – Global stop criterion Find next expansion point Stop expansions

compute supporting points & solutions

set 1st exp. point

for all expansion directions: perform

expansion at selected point with convergence

check

compute solution & error of reduced model

parametric reduced System

new exp. point

> Limit

< Limit

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 24

Parametric MOR – Global stop criterion Find next expansion point Stop expansions

compute supporting points & solutions

set 1st exp. point

for all expansion directions: perform

expansion at selected point with convergence

check

compute solution & error of reduced model

parametric reduced System

new exp. point

> Limit

< Limit

Mproj global

expansion point

Expansion direction

add next moment

Mproj local

add next moment

Mproj global

check convergence

NoYes

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 24

Parametric MOR – Global stop criterion Find next expansion point Stop expansions

compute supporting points & solutions

set 1st exp. point

for all expansion directions: perform

expansion at selected point with convergence

check

compute solution & error of reduced model

parametric reduced System

new exp. point

> Limit

< Limit

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 25

Parametric MOR of Gyro

Raleigh damping built from stiffness and mass not needed in reduction

Parameterized ODE in Laplace space (no DRaleigh) 7 separate matrices but 6

parameter combinations ➔ expand along 6 directions

Alternative approach: only expand along s and ω use „snapshots“ for d to

build projection matrix project completely

parameterized system

System1

Parametric SystemMproj

Reduced Parametric System

System2 System3 System4

DRaleigh = ! M +" K

! s2 (Mb +d Mv) · x+! s (Db +d Dv) · x+

(Kb +1d

Kv,1 +d Kv,2) · x = B · u

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametric MOR of Gyro: From 18’000 DoF to 196

expansion along all directions reduced system size:

196 DoF 360 supporting points

solution

error(Arg) error(Abs)

26

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Parametric MOR of Gyro: From 18’000 DoF to 92

“snapshot” approach reduced system size:

92 DoF 4 snapshots

(d = 1, 1.33, 1.66, 2) 4x200 supporting

points

solution

error(Arg) error(Abs)

27

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Conclusion

MOR works well for IBM’s millipede tip & the IRST MEMS Switch Still too much hand tuning, e.g., nonlinear force, stop criterion

Algebraic parametrisation allows for geometric parametrisation enables parametrisation of commercial software models

Parametric MOR semi-automatic tested up to 6 terms

Snapshot technique reduces number of parameters for expansion

Open questions remain28

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink 29

Institute of Microsystem TechnologyLaboratory for Microsystem Simulation

C. Moosmann, J. Lienemann, J. G. Korvink

Small Demo: Realtime Solving