engineering optimization applications

Engineering Optimization Applications

柯春旭義守大學電機系

2010/5/21

Optimization Problem

Design Variables

ModelDesign Specs.

Outputs Objective function

Design and Fabrication of an Efficient Magnetic Microactuator

I Introduction

II Efficient Magnetic Microactuator

III Optimal Design of Efficient Magnetic Microactuator

IV Simulation of Magnetic Microactuator Using the

Macromodel

V Fabrication of Efficient Magnetic Microactuator

Introduction

• Magnetic microactuators have the advantages of

- large force and large deflection

- low driving voltage

• Micromachined microactuators achieve the needs of

- miniaturization,

- mass production and low cost

• Research activities of magnetic microactuator

- design and fabrication

- simulation and optimization

• MEMS applications

micromotors, microrelays, optics, printer.

Study of Magnetic Microactuator

Magnetic microactuator

• a micromachined electromagnet

as flux generator

• a movable microstructure

with the magnetic material

magnetic field interacts with magnetic material to product a force, the microstructure displaced

Permalloy plate

Planar coils

Magnetic core

attractive force

attractive and repulsive force


Magnetic flux

0

10

20

30

0 4 8 12n

forc

e (

N)

force vs. coil number n

Planar coils

Basic design of electromagnet is only using planar

coils

• The advantage

- easily fabricated

• The drawbacks

- leakage flux results in low efficiency

- truns n increases, force approaches to a

constant


External Flux

Using a non-micromachined external

magnetic field generator [6,10,23,24]

• The advantage

- force effectively increases

• The drawbacks

- filed generator is larger over all device

dimensions

- requiring additional assembling steps

- EM interference in array

Design of magnetic circuit

Chang Liu, et al.

Ranan A. Miller, et al.

Magnetic Circuits

Three types:

1. Spiral Type

• The advantages

- coils are easily fabricated

- accurate line pitch and width

• The drawbacks

- the resistance nonlinearly increases

- internal leakage flux, reduced by using

LIGA or high rMicrorelay, E. Fullin

Micropump, Ahn7 layers

2. Solenoid Type

• The advantages

- the magnetic circuit is easily constructed

- easily achieving the desired shape of

microactuator

• The drawbacks

- not suitable for the fine coil

- non-flat coil contacts raise the resistance,

increasing local high temperature

Microactuator, H. Ren, E. Gerhard

Micropump, SADLER et al.

5 layers

Magnetic Circuits

3. Meander Type

• The advantages

- the magnetic circuit is easily constructed

- the coils are flat

• The drawbacks

- not suitable for the fine coil

- non-flat core contacts increases magnetic reluctance, reducing the efficiency

Microrelay, Marc et al.

5 layers

Magnetic Circuits

Simulation of Magnetic Microactuator

• Detailed knowledge of all of the magneto-structural effects is a prerequisite for effective and efficient design

• Trying the simulation experiments only in hours instead of months, thus shorten the development cycle

• Utilizing the optimization algorithm to achieve the optimal performances of the devices

Magneto-structural coupled problem

(1) To design an efficient magnetic microactuator with the

magnetic circuit.

(2) To optimize the magnetic microactuator for applications.

(3) To develop an efficient macromodeling techniques for dynamic

coupled simulation of magnetic microactuator.

(4) To realize the efficient magnetic microactuator with

micromachining processes.

Objectives

Improved Designpole area can optimally enlarge1. Increasing the efficiency in producing magnetic force

2. Increasing the frequency of the microactuator

3. Increasing the rotation range.

4. Increasing the utilization area.

Efficient Magnetic Microactuator with An Enclosed Core

Efficient Magnetic Microactuator with An Enclosed Core

Improved Design

• 7 layers

- 2 coil layers

- 2 permalloy layers

- 3 insulators

• Dimensions of coils

and plate are dependent

• EM isolation

Design Problem

• Magnetic microactuators with magnetic circuits

- magnetic force varies with given designed dimensions

• This proposed microactuator allows

- a large variation in lengths of poles

The optimal design is to find the optimal values of the geometrical parameters to generate a maximum force

Table 3.1 The ranges of the geometrical parameters (m)

L g p1 p2 c tp tc ti

800 20 0 - 400 0 – 400 0-400 5-50 5-30 40-60

Design Procedure

• First, the effects of geometrical parameters on magnetic

force generation are analyzed by conducting a series of

finite element simulations.

• Then, those geometrical parameters which have critical

effect in optimization are found as design variables.

• Finally, the GA is applied to find the optimal values of the

design variables and the maximum magnetic force.

Model for Magnetic Force Computation

Maxwell equations in the magnetostatic case

The flux density expressed in terms of vector potential

The magnetic co-energy can be calculated as

Using the virtual work principle, magnetic force is

With the equations above, the finite element method is used to solve the problem.

0

1JA

AB

V

dVBdHW

s

WFm

'

Model for Magnetic Force Computation

FEM Model

• 2D axial symmetry element

• 2D infinite boundary element

• unsaturated, r is a constant

• ANSYS software

Initial Design

magnetic force is 512.2 N with the current of 0.08 A

L g p1 p2 c tp tc ti

800 20 200 100 100 10 10 50

Geometrical Parameter Analysis

• six geometrical parameters need to be determined

- pole length p1

- pole length p2

- radius c

- plate thickness tp

- core thickness tc

- insulator thickness h

• with all of six parameters as design variables in optimization, difficult

• analyze the effects of the geometrical parameters - take out those not so critical

- use the remaining critical ones as the design variables


Fig. 3.3 Magnetic fluxes at different pole lengths p1:(a) p1=150m, (b) p1=200m, (c) p1=250m, and (d) p1=300m.

Pole Length p1, p2• peak force occurs when pole

length p1 is about 250 m

• Similar results for pole length p2

Fig. 3.2 Relation between pole length p1 and the generated magnetic force.


Fig. 3.4 Relation between core radius c and the generated magnetic force.

Magnetic core radius c • peak force obtained when radius

c is about 115 m, the reasons are

in twofold.

- c increases, core reluctance

decreases that helps to increase

the magnetic force.

- c increases, the positions of all

coils move outward which leads

to magnetic force reduction


Magnetic core radius c • force generation on different

positions of the single coil

Fig. 3.6 Magnetic fluxes for different coil position(a) inner of the microactuator, (b) under the plate, (c) center of the microactuator, (d) outer of the microactuator.

Fig. 3.5 The influence of the position of a single coil on magnetic force generation.


Thickness parameters• larger tp, smaller plate reluctance

• magnetic core thickness tc has the

same phenomenon

• larger the insulator thickness ti,

less the internal leakage flux

The parameters affect magnetic force monotonically

The maximum force is obtained with the maximum thickness

parameters

Pole lengths p1, p2, and radius c are taken as the major

design variables design, to be found by using GA

Developed by Holland, the concept of biological evolution

• multiple search points, not a single point, the probability of

reaching for the global optimum is raised

• do not use any derivative or mathematical information

• nonlinear or unknown systems with a large search space

• Three operators: reproduction, crossover, and mutation

• drawbacks including premature convergence, low search

efficiency, and difficulty for parameter setting

Genetic Algorithm

A. Fitness scaling

to maintain diversity in the population

B. Population-elitist with rank selection reproduction

use the relatively good individuals from the previous

generation

C. Adaptation of operator probabilities

to avoid premature convergence and excessive diversity

Modified Genetic Algorithm

Step 1: Initialize the GA parameters, and generate initial population.

Step 2: Decode each chromosome for design variables and compute each fitness value.Step 3: Execute the fitness scaling.

Step 4: Evaluate each chromosome by performing the population-elitist with rank selection reproduction scheme.

Step 5: Perform the adaptation of the crossover and mutation probabilities.

Step 6: Create the new chromosomes by applying the operations of crossover

and mutation.

Step 7: If not convergent, go to step 2 for the next generation; otherwise, stop and output the optimal values.


• GA based optimizer that contains a simulator driver to interface with the FEM is developed

• the modified GA includes the three proposed operators,

while the SGA (simple GA) does not

• the modified GA can converge much

more quickly than the SGA


Fig. 3.9 Comparison of the evolution processes between the SGA and the modified GA.

Effects of GA parameters on the evolution


number of individuals is better selected as 20

crossover rate is better selected as 60%

mutation rate is better selected as 10%

Results

Fig. 3.11 Magnetic flux distribution for the initial and optimized geometry: (a) initial geometry and (b) optimal geometry.

• The optimal variables are found to be

• Magnetic flux flows much more through

the permalloy plate after optimization

• force is 589.2 N for the optimized model,

larger than 512.2 N for the initial design

the improvement can be achieved by only designing the layout of mask

p1 p2 c

Initial design 200 100 100

Optimal design 290.8 61.1 152.4

Thickness Design

• The maximum force increases as these thickness parameters increase, coincides the previous assumption

• The maximum force approaches the largest value when the plate thickness increases

• core thickness has the most evident effect on maximum force generation

• the relation between the maximum force and insulator thickness is approximately linear

set the thicknesses to be their maximum value simultaneously at 50, 30, and 60 m, maximum magnetic force is 1160.9 N, the largest among all of the models

Results

Macromodel Approach

Generate a Macromodel Directly from 3-D Geometry and Physics

( )( ( )) ( )r

r r

dx tF x t b u t

dt

( ) ( )Tr ry t c x t

Complicated Geometry, Coupled Elastics,

Magnetics

Low order state-space model which captures

input (u)/output(y) behavior

Fig. 4.1 Block diagram of the macromodel approach.

Macromodel Approach

0

ii q

L

q

L

dt

d

),,(),,(),,( tqqUtqqTtqqL

i

mim u

tuuUtuF

),,(

),(*

,

Theoretical Approach

Lagrange’s equations,

L is defined by

T is the kinetic energy and U is the potential energy.

Selecting the meshed nodal displacements u as the generalized coordinate,

and assuming u be the small displacements

M is mass matrix, and Fm is the nodally defined electromagnetic force with

0),( tuFKuuM m

qtqun

iii

1

)(

),( tqFqKqM mTTT

][ , 2i

TT diaqKandIM

),(2 tqFqq mTiii

)()(),( 2 qftItqF mm


Selecting the n-dimensional generalized coordinates,

By introducing the above Eq. into dynamic Eqs. and premultiplying the

result by by T

The basis functions can be determined by using the natural modes,

The dynamic equations become

Fm is proportional to the square of the input current


The equations can be expressed as

is the generalized force, referred to as the force macromodel

On the other hand, the equations derived with the magnetic co-energy [42],

is the magnetic co-energy with unit input current, referred to as the

energy macromodel

The force and energy macromodels are with different computation procedures

)()(22 qptIqq iii

)()( qfqp mTii

i

miii q

qutIqq

)(

)(*

22

)(* qum

Macromodel Generation

Building the approximate closed-form macromodels by

identification technique.

- Sampling a set of the FEM solutions as the fitting data

(experimental design)

- Selecting a model (FLM)

- Fitting the selected model to data (cluster estimation,

backpropagation)

Design of experiments

• n input variables

• The levels are used to adequately span the predetermined input, m levels.

• nm runs or Taguchi’s method

Sampling data

input outputMagnetic Analysisorthogonal array force, energy

Training data L25(56) Testing data L16(4

5)

• In Sugeno-type FLM, the ith rule is described as

• the representation is an integration of the rules rather than a single crisp correlation

• the Gaussian-type membership function

Fuzzy Logic Model

c - c c + Fig. 4.2 Gaussian type membership function.

isoutput rule then the, is and and , is and , is If 2211 innii FxFxFx

niniiini xpxpxppxxxy 2211021 ),,,(

2

2

1exp)(

ij

ijjjijF

cxx

• The weight for each rule’s output becomes

• For the FLM with r rules, the output

can be expressed as

• The differentiation of FLM output

can be analytically derived for

energy macromodel

• The parameters to be determined are

Fuzzy Logic Model

n

j ij

ijjninFiFiFi

cxxxxw

1

2

2211 2

1exp)()()(

r

ii

ii

r

iii

w

wvyvy

1

1

where

ijijiji cpp ,,,0

• Minimize the square of instantaneous error with respect to the unknown parameters

• Gradient-descent method

by applying the chain rule,

• Backpropagation method

Gradient-decent and Backpropagation methods

ijijiji cpp ,,,0

22

2

1)ˆ(

2

1kkkk eyyE kii

i

kii ekvkp

b

Ekpkp )()()()1( 00

0000

jkiijij

kijij xekvkp

b

Ekpkp )()()()1( 11

)(

)()()()()()1(

22*

02 k

kcxykyekvkc

b

Ekckc

ij

ijjkikiij

i

kijij

)(

)()()()()()1(

3

2

30

3 k

kcxykyekvk

b

Ekk

ij

ijjkikiij

i

kijij

2min

2 RR

Magnetic microactuator with complex structure for

demonstrating the efficiency of proposed approach

Geometry:

• an asymmetric 625 x 625 x 5m plate

• four beams 50m wide by 5m thick,

- the shortest 150m, the longest 300-m,

- the others 200m, and 250m

• a 500 x 500 x 5m permalloy

• 32-turn coil, 8 –m thickness for each layer

• 16-m gap

FEM:

• Structure, 411 elements, 799 nodes

• Magnetics, 17408 elements, 19602 nodes

• the resulting deformation is depicted

Simulation Results

• Modal analysis

• QR factorization

• Selecting 6 modes (> 99.5%)

Simulation Results

Mode 1

Mode 8

Mode 6

Mode 10

Mode 2

Mode 3

Mode # Contribution (m)

1 -5.2941

3 0.1789

2 0.0569

6 0.0142

8 0.0044

10 0.0031

7 0.0021

12 0.0021

5 0.0014

9 -0.0013

Mode # Frequency (kHz)

Period (s)

1 16.3 61.46

3 53.5 18.70

2 34.4 29.09

6 192.5 5.20

8 266.3 3.76

10 363.1 2.75

Macromodeling Computation Times

(Pentium III 850MHz)

• The total time would have been longer without experimental design

• FLM identification took only a few minutes, demonstrating its efficiency

Simulation Results

Computation Times (sec) Force macromodel Energy macromodel

Quasi-static analysis 4320 4320

Modal analysis 10 10

Modes selection 30 30

Data sampling 10660 10824

FLM identification 148 16

Total time (hours) 4.21 4.22

Static Simulation

• Force macromodel yielded an error of less than 1.5%

• Energy macromodel shows a much greater error

due to the differentiation of the fitted energy macromodel

Simulation Results

Dynamic Simulation

• Each mode response containing the ripple has the same timing as the applied square waves

• Mode 1 dominated the main response while the other modes reflected the general shape of the applied sawtooth wave

• Each simulation took

about 2 minutes

Simulation Results

0

20

40

appl

ied

curre

nt(m

A)

-5

0

5

mod

e 1

-0.2

0

0.2m

ode

3

-0.1

0

0.1

mod

e 2

-0.01

0

0.01

mod

e 6

-5

0

5x 10

-3

mod

e 8

0 100 200 300 400 500 600 700 800 900 1000-5

0

5x 10

-3

mod

e 10

time (s)

0

20

40

appl

ied

curre

nt(m

A)

-5

0

5

mod

e 1

-0.1

0

0.1

mod

e 3

-0.05

0

0.05

mod

e 2

-0.01

0

0.01

mod

e 6

-2

0

2x 10

-3

mod

e 8

0 100 200 300 400 500 600 700 800 900 1000-2

0

2x 10

-3

mod

e 10

time (s)

Fabrication of Electromagnet

Fig 5.1 A schematic processing sequence for electroplating Cu or NiFe

4. Electroplate Cu or NiFe1. Deposite SiO2

2. deposite seed layer

3. resist layer is spun, exposed, and developed

5.release resist

6. release seed layer

Fabrication of Electromagnet

Fig 5.2 A schematic processing sequence for the fabrication of the coil with enclosed core

6. coate 3rd insulation layer

1. electroplate NiFe

3. electroplate 1st coil

2. coate 1st insulation layer

5. electroplate 2nd coil

4. coate 2nd insulation layer7. electroplate NiFe

Fabrication of Permalloy Plate

Fig 5.3 A schematic processing sequence for the fabrication of the permalloy plate with a 4-suspended-beam structure

1. deposite SiO2 on the double side

2. coate polyimide membrane

3. deposite Au as the mask for the polyimide beams

5. pattern the SiO2 etching window on the backside wafer

4. electroplate NiFe

6. etch in KOH with the Teflon chuck

8. release the polyimide beam with polyimide etch

7. etch SiO2

Results and Discussion

Fig 5.4 Photograph of the electromagnet with the enclosed core: (a) perspective view and (b) top view

• NiFe permalloy is 10m thichness

• Insulator is 10m thichness

6st layer

7st layer, enclosed

• NiFe plate is 800m long and 10m thick.

• The length and width of beam are 1000m and 200m


Fig 5.5 Photograph of the NiFe plate supported by 4-suspended-beam structure: (a) perspective view and (b) enlarged view

• Each coil layer includes

- 17 turns

- line width and spacing are 25m

- thickness about 12m.

• The total electrical resistance is approximately 14 Ohm


Fig 5.6 Photograph of the Cu coils: (a) the resister pattern and (b) the electroplated coils

• A nanoindentor is used to measure the stiffness of the four-beam structure,

the stiffness k is about 45N/m

• B-H curve as magnetic properties of electroplated NiFe permalloy


Fig 5.7 The load-deflection curve of the 4-beam structure Fig 5.8 B-H curve of the NiFe permalloy

• The current-deflection curve is obtained using a laser displacement system

• The test results are well with numerical solution

• 27.6m displacement at 292mA and 4.5V

• The estimated force is bout 1240(N)


Fig 5.9 The comparison between the experimental and theoretical results of the current-deflection curves

),(

(2.4) Eq. ),(

(5.1) Eq.

IxFKx

IxFF

KxF

m

mm

s

Conclusions

The proposed magnetic microactuator is ready for practical applications

Crane Vibration Suppression

• Tower cranes are widely used in construction and transportation industries. • The payload acts as a pendulum, operating the crane inappropriately may

result in dangerous situations. • One way to suppress the oscillation is to move the payload slowly, which

leads to inefficiency.

• A control scheme to reduce the oscillation and increase the efficiency of the operation is important.

Tower Crane Model

The position of the payload ,

Lagrange’s equations

The energy T and V are

The velocity

kttLjtLitttLtxtp

(t)cos)(cos)( (t)sin)( ))(sin)(cos)()(()( 21121

21 coscos mgLV )(2

1vvmT

tpkttpv

iLLLLx

121221121 sincoscossinsinsincos

jLxLL

21111 sincoscossin

kLLL 21212211 coscossincoscossin

Tower Crane Model

L

x

L

xL

x

L

x

L

g

L

L

L

L

L

g

L

LL

L

22

1

2

2

2

1

2

1

2

2

2

2

1cos ,sin 111 1cos and ,sin 222

L

x

L

xL

x

L

xx

L

L

L

g

L

L

L

L

L

L

L

gx

22

2 20

0

222

222

1000

0100

TTxxxxx 21214321

Rotary Tower Crane Model

l

sl

s

l

gl

g

22

1

2

2

2

1

2

1

0

02

u

xl

sx

l

gxxxxxux

xl

suxxux

l

gxxx

x

uxf

6

26

3263621

4

6431261

2

2

2

),(

TTxxxxxxx 2211654321

u

Let

),,( uxfx

The state equation becomes

Crane Vibration Suppression

Generalized Input Shaping

Simulation Results

Experiments

engineering optimization applications

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