Parameterized Mode Truncation Technique in MEMS · PDF fileParameterized Mode Truncation...

1

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Parameterized Mode Truncation Technique Parameterized Mode Truncation Technique

in MEMS Designin MEMS Design

V. Kolchuzhin, W. Dötzel and J. Mehner

Berlin, 4 Dec 2010

2

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Parameterized Mode Truncation Technique in MEMS Design

• Introduction and Motivation

• Parametric FE technique and pROM

• Numerical Examples

• Conclusion and Outlook

Outline

3

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What is the MEMS ?

KD M

V hu

Fel

F

Fu

uCVuKuDuM +

∂

∂=++

)(

2

2

&&&

VuCVuCtd

Qdi && )()( +==

Vibrationssensor TUC/ZfM

200um

2n

d o

rde

r O

DE

4

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MEMS Inertial sensors: application

Microsoft Freestyle Pro Gamepad(ADXL202)

HDD Active Protection System, 2003

InvenSense: Angle camera shake(ITG-3200)

Guidance and Navigation

Photo: Claudia Dewald/iStockphoto

Ford: Curve Control

Bosch: ESP

iPhone4STM: L3G4200D

Samsung YP-S1 TicToc

Wii Remote (ADXL330), 2005

SegwayScooter

Nike+iPod

5

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MEMS Design

TCAD SPICE, Simulink, VHDL-AMS,Verilog-A

Componentsimulations

System simulations

Process simulations

Lumped elements

Processsequence

RedesignRedesign

Behavioral modelBehavioral modelRequirementsRequirements

FDM, BEM, FEM, FVM

Mask design

L-EDIT, AutoCADMATLAB, MathCAD MATLAB, ANSOFTANSYS

D K

CM

Low-levelsimulations

Sentaurus Topography

Ma

cro

mo

delin

g

To Fab Virtual prototype

REM image

ANSYS, MATLAB

ROM, MOR

S

+

Nodal ports

Voltage ports

Modal ports

Auxillary ports(element load)

S

+

S

+

ROM144Electrostatic-Structural Reduced Order Model

q1

e1

u1+

V1

Pa

e3V2

q2q3

ROM140Fluid-Structural Interactions

Acel Pres

Package model

ROMq4

ASIC

@ master node

THM1

S

+

displ

contact

S

+ V2V1

ΘT1

u1-

r

u1-u1+

Curvature

R1

C1

Epol

The MEMS designing is a very challenging and interdisciplinary task!

Subsystem

6

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MEMS Simulations

),(),(2

2

txfx

R

t

Rtxp =

∂

∂+

∂

∂

PDE ODE

>105 DOF x n time steps DOF< 10

mq′′+dq′+kq=f(t)

BEM

FDM

FEMReduced order (2001-…)

SPICE, Simulink, VHDL-AMS,Verilog-A

Physical level System level

FDM, BEM, FEM, FVM

MATLAB, ANSOFTANSYS

ANSYS, MATLAB

ROM, MOR

S

+

Nodal ports

Voltage ports

Modal ports

Auxillary ports(element load)

S

+

S

+

ROM144Electrostatic-Structural Reduced Order Model

q1

e1

u1+

V1

Pa

e3V2

q2q3

ROM140Fluid-Structural Interactions

Acel Pres

Package model

ROMq4

ASIC

@ master node

THM1

S

+

displ

contact

S

+ V2V1

ΘT1

u1-

r

u1-u1+

Curvature

R1

C1

Epol

Subsystem level

Online-coupling

(1995-2000) Matlab/Simulink

VHDL-AMS

Verilog-AMS

gen

era

lized

Kir

ch

off

ian

law

s

PSPICE

-level-physics

-scale-material

multi

mq′′+dq′+kq=f(t)

dq′+kq=f(t)

Lumped elements

Electrical equivalent

Modal decomposition

Krylov based method

Control theory methods

1. Loading design parameters2. Material design parameters3. Geometrical design parameters 4. Boundary conditions

mechanicalelectrofluidicthermal

MFM

FVM

CVM

Subsystem

?

7

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Outline

8

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Taylor Series and HODM

1715

Brook Taylor

"Taylor's Theorem"

1993

The Intel Pentium supports 8 transcendentalfunctions through direct execution of microcodeusing polynomial approximation techniques.

…i

n

i

ipppf

ipf )()(

!

1)( 0

0

)(

0 −≈∑=

High order sensitivity analysis:

Transient analysis[M]a+[D]v+[K]u=F

Modal analysis[K-LM]X = 0

Static analysis[K]u=F

Harmonic analysis(-Ω2[M]+iΩ[D]+[K])u=F

Taylor expansion

Rational approximation

PDE → DAEMatrix EquationDifferentiation

Mode truncation

method

−= −

=

− ∑ )()(

1

)(1)( )()()()()( inin

i

i

n

nnppCppp uKFKu

−= −

=

− ∑ )()(

1

)(1)( )()()()()( inin

i

i

n

nnppCppp uAFAu

=+

=+−+−++

+++

)()1(

)()1()1()1(

)1(

)1()1()1(n

L

n

i

T

i

n

xi

n

i

n

ii

n

i

n

LnLnn

SMxx

SMxMx Kx

FEM

stiffnesspermittivity

displacementpotential

forcecharge

u = F = K =

9

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II. Parametric Method

Parametric FE mesh

2. 3.

4.n

n

p

u

∂

∂

Extraction

Assembling

design parameter

obje

ctive f

unction

1st

2nd

5th TS4th TS

in

i

i

ppi

pfpf )(

!

)()( 0

0

)(

0 −=∑=

n+1

][K

5

5][

p

K

∂

∂

3

3][

p

K

∂

∂ 4

4][

p

K

∂

∂

2

2][

p

K

∂

∂

p

K

∂

∂ ][

3rd TS

2. 3. 4.

FE Technique

2. 3. 4.

I. Ordinary Data Sampling Method

][ 1 FKu −=

FE mesh

Global matrix Solution

for i=1:n

Assembling

1.

CAD model

)( ipf

design parameter

computation computation

time?time?

accuracy?accuracy?

pi

∑=el

kK ][][

end

pi

…

obje

ctive f

unction

...)( 2

210 +++= papaapx

...)( 2

210 +++= pbpbbpy

...)( 2

210 +++= pcpccpz

10

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Data Structure

Arrays of polynomials terms (Taylor series coefficients) of 3 variables and 4th order

Pyramid array (Pascal tetrahedron)

∑∑∑∑====

++++====

n

j

j

j

jKR

1

)(µ

size = 35

2D array

+=

p

p

n

nordersize np = 3

order = 4

…

ji

000 R(µµµµ)

100

010

103013

004

0

1

2

32

33

34

µµµµ R(µµµµ)j

…

i

k

y

y2

y3

y4

x

x2

x3

x4

z2yzxz

xy

z3

y2z

yz2

x2z

yz2

xy2

xyz

x2y

x2y2

xy3x3y

y2z2x2z2

y3z

yz3xz3

x3zxyz2

xy2zx2yz

1

z4

z

µ 1

µ 3

µ 2

3D array

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Geometrically Parameterized Finite Element

∑=ngp

i

iii

T

i wJBDBk][

ngp – number of Gauss points

n

i

n

n

i

n

n

i

n

p

z

p

y

p

x

∂

∂

∂

∂

∂

∂,,

Nodal coordinates and its derivatives at p0

AD:

[f]=[u][v]

AD:

[f]=[u].v

AD:

[f]=det[u]

[f]=inv[u]

n

n

p

k

∂

∂ ][

Solid

element

),,(][ ZYXfJ =

])[(][ JinvfB ====

)(

)(

)(

pfZ

pfY

pfX

=

=

=|J|, (tuple)

[J-1], (3,3,tuple)

[B] = [J-1] [dN]

(3,8,tuple) = (3,3, tuple) . (3,8,tuple)

[BD] = [B]T[D]

(8,3,tuple) = (8,3, tuple) . (3,3)

ngp = 8 wi, ξi, ηi, ζi

Ni = (1 ± ξi)(1 ± ηi)(1 ± ζi) (8 x ngp)

dNi=[Ni /dξ; Ni /dη; Ni /dζ;] (24 x ngp)

[J] = [dN] . [XYZ]

(3,3,tuple) = (3,8, ngp) . (8,3,tuple)

[BDB] = [BD].[B]

(8,8,tuple) = (8,3, tuple) . (3,8,tuple)

[BDBJ] = [BDB]. |J|

(8,8,tuple) = (8,8, tuple) . (tuple)

[ke] = [ke] + [BDBJ].wi

(8,8,tuple) = (8,8, tuple) . (ngp)

Nodal Coordinates:

n

∑=ngp

i

iii

T

i wJBDBk][∫∫∫=V

TdxdydzBDBk][

=

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

][

PPP

OOO

NNN

MMM

KKK

JJJ

III

pzpypx

pzpypx

pzpypx

pzpypx

pzpypx

pzpypx

pzpypx

pzpypx

XYZLLL

1 2

34

5 6

78

I

K

J

ON

P

M

L

ζ

0

η

ξ

M(p0)

N(p0)

O(p0)P(p0)

I(p0)

J(p0)

K(p0)

K(p)

O(p) p

y

z

x

i:=i+1

Isoparametrictransformation

Cell

12

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Generation of MEMS Macromodels Using HODM

Test loads:couple domain run

),,( zyxu∑≈ ),,(),,( zyxwzyxu iiφ

0.0343.382

………

0.09420.067

18.2758.553

81.5834.031

Contribution

factor [%]

Frequency,

kHz

Mode #

Computation of basis functions:

iw

),,( zyxiφ

i

Modal analysis

Mode 1 Mode 3 Mode 7

),,( zyxiφ

E1 conductor 2

conductor 1

conductor 3

∑=

=3

1

),,()(i

iiii zyxqwqu φ

Static structural domain Electrostatic domain

Mode i

),( qkii Ω),( qcii Ω

Para

mete

rs e

xtra

ctio

n

Fluid domain

12C 13C 23C1 3

2

Strain energy Capacitance Damping coefficient

n

i

n

q

qqqW

∂

∂ ),,( 731

ROM database

iω

ji

ijqq

Wk

∂∂

∂=

2

n

i

n

q

P

∂

∂n

i

n

q∂

∂ ϕn

i

n

q

u

∂

∂

1ω3ω 7ω

0=iqiωω =0

Static parametric analysis at Static parametric analysis at Harmonic parametric analysis at0=iq

1. 2.

4.1. 4.2. 4.3.

( )∑ −∂

∂+=

∂

∂++

r

sk

i

ksq

i

SENEiiiiii tVtV

q

pqCptf

q

pqWtqpmppqtqpm )()(

),(

2

1),(

),()()()(),(2)()( &&& ωξ

Para

metr

ic r

uns

3.

n

ij

nc

ω∂

∂

n

i

r

n

q

qqqC

∂

∂ ),,( 731

13

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Outline

14

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Application: Parametric Linear Static Analysis of the FFB

L

B

H

y

z

x

FE stiffness matrix

FE-mesh

Load: Pressure=12.5 kPa

120x30x2mkm

8415×8415

Elements = 2000Nodes = 2805

Size(Kfull) = 540.25Mb

Size(Knz) = 26.37Mb

nz = 539081

E = 169 GPa, Poisson’s ratio ν = 0.066, and mass density ρ = 2.329 g/cm3

Design parameters: L, B, H

15

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Analytical Models of FFB

L-2Resonant frequency

L5Bending energy

L2Maximum stress

(under uniform pressure P)

L4Center deflection

(under uniform pressure P)

L-33rd order stiffness coefficient

L-3Stiffness

(point excitation at the center)

LMass

L-dependenceAnalytical modelProperty

ρLBHm =

ρπ

E

L

Hnf i

i 2

2

3=

3

3

425.10L

EBHK =

KB

2

767.0=α

PEH

L3

4

32=δ

PH

L2

2

2=σ

3

24

96L

EIW

δπ=

12

3BH

I =

16

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Application: Results of Parametric Linear Static Analysis

4th order TS

100 200 300 400

0.0

0.5

1.0

50 100 150 200

Evaluationpoint

0.0

0.5

1.0

100

FE solutions

50 100 150 200

0

50

2 4 6 8

0

50

100

0

Padé app.

4th order TS

1st order TS

50 100 150 200

0

20

40

60

Width, µm

4th order TS

0 2 4 6 8

0

20

40

60

Thickness, µm

3rd order

Padé app.

0 2 4 6 8

0.0

0.5

1.0

2nd order TS

Padé app.

100 200 300 400

0

50

100

Evaluationpoint

2nd order TS

100 200 300 400

0

20

40

604th order TS

Length, µm

Evaluationpoint

Str

ain

energ

y, p

J

Str

ain

energ

y, p

J

Str

ain

energ

y, p

J

Max. eqv. str

ess, M

Pa

Max. d

ispla

cem

ent, µ

m

Max. d

ispla

cem

ent, µ

m

Max. d

ispla

cem

ent, µ

mM

ax. eqv. str

ess, M

Pa

Max. eqv. str

ess, M

Pa

∝∝∝∝ H-3

∝∝∝∝ H-2

∝∝∝∝ H-3

∝∝∝∝ B0

∝∝∝∝ B0

∝∝∝∝ B1

∝∝∝∝ L4

∝∝∝∝ L2

∝∝∝∝ L5

17

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Application: Parametric Linear Static Analysis

eff

icie

ncy

of

HO

DM

1 2 3 4 5 6 7 8 9 100

100

200

300

400

Number of sampling points / Number of derivatives

Com

puta

tion

tim

e,

sec

Parametric FE technique (HODM)

FE runs for data sampling

∝ DOFsone FE run

one FE run ×5

Benefits of novel approach compared to ordinary data sampling procedures become obvious:

• for multi-parameter problems (np > 3)• for large problems (dof > 104).

The grid-like data sampling process is expected to grow as nk, where n is the number of parameters and k is the number of states for each parameter.

18

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Application: Parametric Electrostatic Analysis

2.580

2.581

2.582

2.583

2.584

2nd order TS

2.579

p4, degree

-5 0 5

1st order TS

Padé app. [2/2]

2.5

3.0

3.5

4.0

4.5

-2 0 2p2, µm

Evaluationpoint

Evaluationpoint

p5, degree

4th order TS

0 2-2-4 42.40

2.45

2.50

2.55

2.60

2.65

2.70

Padé app. [2/2]

p6, degree

0 2-2

2.5

3.0

3.5

4.0

4.5

4th order TS

2nd order TS

2.40

2.45

2.50

2.55

2.60

2.65

2.70

Capacitance, fF

p1, µm-10 -5 0 5

4th order TS

2nd order TS

2 40-2 6p3, µm

3rd order TS

2.40

2.45

2.50

2.55

2.60

2.65

2.70

Levitation

p1 – motion in operating directionp2 – horizontal shift in y-directionp3 – vertical shift in z-directionp4 – x-rotationp5 – y-rotationp6 – z-rotation

Conductor 3SubstrateConductor 1

Moving finger

Conductor 2 Fixed fingers

Vc

Capacitance stroke function of the movable finger with regard to six degrees of freedom C=f(p1,…, p6)

Linear electrostatic analysis

FE model

Solid model

C(p0)=2.57982e-15 F

=

0

c

e

c

ee

QV

KK

KK

ec

eccc

ϕ

p2

p3

p1

19

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Application: Parametric Electrostatic Analysis

Capacitance, fF

p1 – motion in operating direction

Conductor 3SubstrateConductor 1

Moving finger

Conductor 2 Fixed fingers

Vc

Comparison between mesh-morphing and remesh approaches

y

x

y

x

y

x

a) Initial FE-mesh p1= 0 µm

b) Mesh-morphing p1= −18 µm

c) Remesh p1= −18 µm

Solid model

p1

-20 0

2.0

2.5

3.0

p1, µm

4th order TS

FE-remesh

FE-morphing

20

Generally, HOD FE technique is limited to

moderate parameters variations ±±±±15% due to the mesh-morphing procedures!

• Multi-Point Taylor Expansions• BEM

20

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Application: Discrete Analysis of Perforated Beam

p2: element component 2





FE-model

0 111

12

13

14

15

16

Discrete parameter p

Str

ain

energ

y, p

J

3rd order TS

1st order TS

Evaluation point

0 150

55

60

65

Discrete parameter p

4th order TS

p

3rd order TS

0 1Discrete parameter p

-0.50

-0.45

-0.40

Evaluationpoint

Estimatedpoint

5th order TS

2nd order TS

4th order TS

y

z

x

Ma

x. e

qv.

str

ess,

MP

a

Ma

x. d

ispla

cem

ent,

µm

pEpE 0)( =

Linear Static Analysis

Perforated beam with circular holes having a radius of 25 µm

Young modulus:

Discrete parameter p corresponds to sets of elements in the model that can be turned 1 or 0.

Design parameter p ≡≡≡≡ number of holes

0,0,0 1,0,0 0,1,0 1,1,0 0,0,1 1,0,1 1,0,0 1,1,1

21

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∂

∂=

∆

∂

∂−

∆∇

g

z

tPa

P

tPa

PPag 22

12η

pPa

P→

∆zv

g

z

t→

∂

∂

zv

FC

Re

=

21 ippp +=

gapvz

Linearized Reynolds squeeze film equation

zv

FK

ωIm

=

is the pressure

Harmonic Problem

is the normal velocity componentzv

Damping Coefficient, Ns/mSqueeze Stiffness Coefficient, N/m

Real component of the pressureImaginary component of the pressure

Application: Parametric Squeeze Film Analysis

0.0

0.5

1.0

1.5

),,( 02 fyxP

×101

2nd order TS

1st order TS

air gap:3.8µm4.0µm4.2µm

-0.5 4.5

5.0

5.5

6.0

×10-6

),,( 01 fyxP

Operating frequency, MHz

0.0 0.5 1.0 1.5f0

gap = 4.2µm

gap = 3.8µm

gap = 4.0µm

6.5

Operating frequency, MHz

=

−

02

1Q

P

P

KC

CK

ω

ω

∫=A amb

TdA

P

gapNNc

η1210

01 3gap

=D

∫=A

TdABDBk

2nd order TS

0.0 0.5 1.0 1.5f0

∫= dAPF 2

Im

∫= dAPF 1

Re

22

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pROM Example: Fixed-fixed beam

FE Model of the coupled domains

Mode1 Mode2 Mode3 Mode4

MODAL CONTRIBUTION FACTOR84.45 % 2.74%

3. right conductor

1. leftconductor

bulk

anchor

2. beam

insulator

Vs

squeezeelementsmaster

node

L

y

zx

airelements

structuralelements

Solid Model

200×20×2 µm

Air-gap 4 µm

1033 elements 1767 nodes

23

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pROM Example: Parametric Modal Analysis and Mesh-Morphing

Mode 1 Mode 3

The relative error is less than 0.5 %.

160 180 200 220 2400.2

0.3

0.4

0.5

0.6

0.7

Evalution point

4th order TS

160 180 200 220 2401.0

1.5

2.0

2.5

3.0

3.5

4.0

Fre

quency,

MH

z

Evalution point

4th order TS

Length of the beam in µmLength of the beam in µm

Fre

quency,

MH

z

Perturbations of the nodes with respect to modal amplitude qj and design parameters L are given by:

∑ ⋅+=j

jzjiji wLqLzLqzi

)()(),( φz

x

0

1

Contour plot of design velocity v(x,y,z) for mesh perturbationsq1 q3

Structural mesh perturbation with q3 = 1.0µm

Air mesh perturbation with q3 = 1.0µmAir mesh perturbation with q1 = 2.0µm

Structural mesh perturbation with q1 = 2.0µm

)()( 00 LLaxLx iii −+=

24

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pROM Example: SIMULINK Parametric Model of the FFB

cap12

dcap 12

cap23

dcap 23

cap13

dcap 13

current 13

6

current 23

5

current 12

4

displacement at master node

3

modal displacements

2

modal velocities

1

v to u

1

s

modal restoring forces

MATLAB

Function

modal damping forces

MATLAB

Function

modal accelerations

MATLAB

Function

electrostatic

force 23

electrostatic

force 13

electrostatic

force 12

charge Q 23

charge Q 13

charge Q 12

cap 23 +dcap 23

MATLAB

Function

cap 13 +dcap 13

MATLAB

Function

cap 12 +dcap 12

MATLAB

Function

a to v

1

s

Q23 to I23

du /dt

Q13 to I13

du /dt

Q12 to I12

du /dt

Eigenvector

K*u

voltage 13

5

voltage 23

4

voltage 12

3

parameter L

2

modal external forces

1

5

10

15

20

25

increasing L

Cap

acit

ance

C12, fF

-3 -2 -1 0 1 2 3Modal amplitude q1, µm

-5

-4

-3

-2

-1


∂C12/∂

q1, fF

/µm

increasing L

-0.5

0.5


∂W

/∂q

1, nJ/

µm

0.0

increasing L

160 180 200 220 2405

6

7

8

9

Length L, µm

Modal m

ass

m1, kg

×10-12

∑−

∂

∂=

∂

∂++

r

sk

i

ks

i

iiiiii

tVtV

q

LqC

q

LqWtqLmLLtqLm

2

)()(),(),()()()()(2)()( &&& ωξ

25

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pROM Example: Results of Simulation

The transient responses for double step voltage excitation

0

50

100

150

200

Ele

ctr

icalexi

tation

, V

Time, µs

5 10 15 20 25 30 35 40 45 50

L = 240µm

-0.6

-0.4

-0.2

0.0

0.2

Dis

pla

cem

ent

, µ

m

Time, µs

5 10 15 20 25 30 35 40 45 50

L = 160µm

T1T2

Non-linear dynamics:

T1 > T2

),( Lqf ii =ξ

(FSEI)

26

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Conclusions and Outlook

A short conclusions of the work are given in the following:

• A parametric FE-solver based on the derivation of discretized FE equations and the computation of a Taylor polynomial of the solution from the high order derivatives has been prototyped

• Automatic differentiations technique for extraction high order derivatives of the FE matrixes, its inverse and determinant has been utilized

• A library of parameterized finite elements has been developed

• A simple mesh-morphing algorithm has been successfully implemented

• A large number of numerical tests have been performed in order to evaluate the accuracy and efficiency.

Further work will be focused on:

• Adaptation of the HODM to really industry problems (over 105 DOF)

• Fully automation of the algorithm

27

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The End

Спасибо за внимание!

Vielen Dank für ihre Aufmerksamkeit!

Thank You for Your Attention!

28

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Error Estimation

1.FEM Solution at the point p0:• Discretization errors • Computation errors• Formulation errors

2.Compute higher order Derivatives at the point p0:• The smoothness of the mesh perturbation• Method of differentiation (AD)

3.Build an approximation:Convergence Domain of the Taylor expansion

δ

δ = F(mesh perturbation , order derivatives, range)

-Exact FEM Solution

-Finite element method error

-Parametric FEM Solution

- Derivatives computation and approximation errors

possible to control

p

G(p)

p0

Validity range: p0 ± ∆ p

δ

Parametric solution

Evaluation point

δ0

Reference point

δ1 δ2

δ0 ≠ δ1 ≠ δ2

1

0

)1(

)()!1(

)()( +

+

−+

= mm

m ppm

fpR

ξ , where p0 < ξ < p.

The Lagrange form of the remainder is given by

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Parameterization of Linear FE Problem

1.Solution at the point p0:

u(p0) = [K(p

0)]-1.F(p

0)

[K] is stiffness matrixu is solution vector (unknown)F is load vectorp

0is initial parameter

Linear FEM Problem:

[K(p0)].u(p0) = F(p0)

2.Compute Higher Order Derivatives at the point p0:

u(p0)′ = [K(p

0)]-1′.F(p

0) + [K(p

0)]-1.F(p

0)′

u(p0)′′ = ([K(p

0)]-1′.F(p

0) + [K(p

0)]-1.F(p

0)′) ′

u(p0)′′′ = …

u(p0)n =

− −

=

− ∑ )(

0

)(

0

1

)(

0

1

0 )()()()( innn

i

i

n

npupKCpFpK

3.Build an approximation:• Taylor expansion: u(p - p0) = Σ u(p0)

(n). (p - p0)n / n!

• Pade approximation: u(p-p0) =Σ r m(p0)(∆p)m /Σ s k(p0)(∆p)k

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Parameterization of NL FE Problem

1.Solution at the point p0:

u(p0)i = u(p0)i-1 − [J(p

0)]-1.R(p

0)

2.Compute Higher Order Derivatives at the point p0:

u(p0)′ = −[J(p

0)]-1.R(p

0)′ (implicit function)

u(p0)′′ = [J(p0)]

-1.(R(p0)′′ − [J(p

0) ]′.R(p

0)′)

u(p0)′′′ = …

u(p0)n =

3.Build an approximation:• Taylor expansion: u(p - p0) = Σ u(p0)

(n). (p - p0)n / n!

• Pade approximation: u(p-p0) =Σ r m(p0)(∆p)m /Σ s k(p0)(∆p)k

[K] is stiffness matrixu is solution vector (unknown)F is load vectorp

0is initial parameter

NL FEM Problem:

R(u(p0)) = 0j

iij

u

RJ

∂

∂=

p

pu

RJ

∂

∂=][

− −

−

=−

−− ∑ )(

0

)(

0

1

1

1

)1(

0

1

0 )()()()( inin

i

i

n

npupJCpRpJ

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Differentiation Techniques

[ ] ∑∂

∂=

∂

∂ ngp

i

iii

T

i wJBDBp

kp

]][[][

3.1. AD relies on the fact that every function is a sequence of arithmetic operations: additions, multiplications and elementary functions.3.2. Evaluate numerically values at the point as (2) up to machine precision!3.3. There are two ways to implement:

source transformation;operator overloading.

Analytical Derivatives are usually more complicated than the function itself ! ! !

1. Analytical / Symbol

3. Automatic / algorithmic

Chain rules> 100 pages

308.017

37174)0(

2−=

⋅−⋅−=′f

4)0( −=′u 3)0( =′v

17)0( =v7)0( =u

)(

)()(

pv

pupf =

22

22

)5317(

)103)(1.047()5317)(2.04()(

pp

pppppppf

++

++−−+++−=′

21.047)( pppu +−=25317)( pppv ++=

p

ppfppf

dp

pdf

∆

∆∆

2

)()()( −−−−−−−−++++≈≈≈≈

2.1. Truncation and cancellation errors2.2. Partial and high order derivatives

n-point Finite Difference Formulas

2. Numerical

2)(

)()()()()(

pv

pvpupvpupf

′−′=′

Matrix rules:

MathCAD

4. Example: differentiation of the f(p):

4.1.

4.2.

4.3.f = uT.v

f = [u].v[ f ] = [u].v

[ f ] = [u].[v][ f ] = det[u][ f ] = inv[u]

at p = 0

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Overview of AD software

ADOGEN

ADiMat

FFADLib

ADIC

ADMIT-1

ADOL-C

ADIFOR

Tools

C/C++

MATLAB

C/C++

C/C++

MATLAB

C/C++

Fortran77

Language

source transformation


operator overloading






Technique

up to n

up to 2nd

up to n

up to 2nd

up to 2nd

up to n

1st

Order

patented, proprietary tools

for derivate matrices

2002 Jacobian and

Hessian

2000

1997

1996 / 98

1992

1991

Comment

33

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Automatic Mesh-Morphing Algorithm

Ω

Γ

x

Morphing process

x

y

L0

L = L0+∆L

Design velocity Perturbated mesh

Laplacian smoother: 2 it. Final mesh

Original mesh

Vs = 1V

∑

∑

=

==n

i

i

n

i

c

ii

k

A

xA

x

1

1

Γ′

Ω′

pxvxpOpp

xxpx )()(

)0,()0,(),( 2 +≈+

∂

∂+=

TTT

v(x)px′

parameter ≡≡≡≡ L

02

2

=∂

∂

ix

v

1. Laplacian smoother 2. Electrical analogy

How obtain the perturbation of the interior nodes?

T(x,p):Translation

Rotation

Scaling

v(x) ≡≡≡≡ Design velocity

p ≡≡≡≡ Time

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Polynomial FE Mesh

L=4*nKB=2*nKD=0.5*nK

nL=8 nB=4 nD=2

/prep7

ET,1,SOLID45

K,1,0,0,0 $K,2,L,0,0 $K,3,L,B,0 $K,4,0,B,0K,5,0,0,D $K,6,L,0,D $K,7,L,B,D $K,8,0,B,D

V,1,2,3,4,5,6,7,8

L,1,2,nLL,1,4,nBL,1,4,nB

VMESH,ALL

Parametric Design Language (APDL)

…………

0.00.00.12

0.00.00.01

zyx#

……………

8…211

n8…n2n1#

Elements Table

Nodal Table

EWRITE, Fname, ext ……………………

…

2

2

2

1

1

1

#

…

z

y

x

z

y

x

dof

…

…

…

…

…

…

…

…

…

3.7

1.0

0.5

1.3

0.2

0.0

an-1

0.6…1.00.1

5.4…0.10.0

2.0…0.50.0

6.1…1.30.0

…………

0.1…0.30.0

0.0…0.00.0

ana2a1a0

Polynomial Nodes Table

NWRITE, Fname, ext

n = tuple_size

Parametric FE model vs. Parametric polynomial FE model

L

D

Re-m

esh

n

n papapaapx ...)( 2

210 +++=n

n pbpbpbbpy ...)( 2

210 +++=n

n pcpcpccpz ...)( 2

210 +++=

Z

YX

0

?

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