Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M0

Optimization of Mechanical Systems

Prof. Dr.-Ing. Prof. E.h. P. Eberhard (eberhard@itm.uni-stuttgart.de) Dipl.-Ing. Alexander Held (held@itm.uni-stuttgart.de)

Institute of Engineering and Computational Mechanics

Pfaffenwaldring 9, 4th floor 70563 Stuttgart

Lecture information

Language: The lecture is taught in English.

Audience: Students of the interdisciplinary graduate program of study COMMAS and students of the programs of study Mechanical Engineering, Mechatronics, Engineering Cybernetics, Technology Management, Automotive and Engine Technology, Mathematics, etc.

Credits: 3 ECTS (2 SWS)

WS 11/12: 17.10.11-11.02.12, Christmas break: 23.12.11-06.01.12

Dates of class:

Wednesday, 9.45 - 11.15, weekly, first class on 19.10.11

Room: V7.12 (Pfaffenwaldring 7, Room 1.164).

Internet: The course web page can be found at

www.itm.uni-stuttgart.de/courses/optimization

Course material: Handouts can be downloaded at the course web page.

Exercises: Exercises are an incorporated part of the lecture.

Office hours: Tuesday: 13.00 – 14.00

Exam: The exam is scheduled for end of February 2012 or beginning of March 2012. Exact date and room will be announced later. The exam is mandatory for COMMAS students. All other students please register with Dipl.-Ing. Alexander Held before 11.02.2012.

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M1.1

Optimization of Mechanical Systems

- Content of lecture -

1 Introduction

1.1 Motivational Examples

1.2 Formulation of the Optimization Problem

1.3 Classification of Optimization Problems

2 Scalar Optimization

2.1 Optimization Criteria

2.2 Standard Problem of Nonlinear Optimization

3 Sensitivity Analysis

3.1 Mathematical Tools

3.2 Numerical Differentiation

3.3 Semianalytical Methods

3.4 Automatic Differentiation

4 Unconstrained Parameter Optimization

4.1 Basics and Definitions

4.2 Necessary Conditions for Local Minima

4.3 Local Optimization Strategies (Deterministic Methods)

4.4 Global Optimization Strategies (Stochastic Methods)

5 Constrained Parameter Optimization

5.1 Basics and Definitions

5.2 Necessary Conditions for Local Minima

5.3 Optimization Strategies

6 Multicriteria Optimization

6.1 Theoretical Basics

6.2 Reduction Principles

6.3 Strategies for Pareto-Fronts

7 Application Examples and Numerical Tools

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M1.2

Bibliography

Optimization:

S.K. Agrawal and B.C. Fabien: Optimization of Dynamic Systems. Dordrecht: Kluwer Academic Publishers, 1999.

M.P. Bendsoe and O. Sigmund: Topology Optimization: Theory, Methods and Applica-tions. Berlin: Springer, 2004.

D. Bestle: Analyse und Optimierung von Mehrkörpersystemen - Grundlagen und rechnergestützte Methoden. Berlin: Springer, 1994.

D. Bestle and W. Schiehlen (Eds.): Optimization of Mechanical Systems. Proceedings of the IUTAM Symposium Stuttgart, Dordrecht: Kluwer Academic Publishers, 1996.

P. Eberhard: Zur Mehrkriterienoptimierung von Mehrkörpersystemen. Vol. 227, Reihe 11, Düsseldorf: VDI–Verlag, 1996.

R. Fletcher: Practical Methods of Optimization. Chichester: Wiley, 1987.

R. Haftka and Z. Gurdal: Elements of Structural Optimization. Dordrecht: Kluwer Aca-demic Publishers, 1992.

E.J. Haug and J.S. Arora: Applied Optimal Design: Mechanical and Structural Systems. New York: Wiley, 1979.

J. Nocedal and S.J. Wright: Numerical Optimization. New York: Springer, 2006.

A. Osyczka: Multicriterion Optimization in Engineering. Chichester: Ellis Horwood, 1984.

K. Schittkowski: Nonlinear Programming Codes: Information, Tests, Performance. Lect. Notes in Econ. and Math. Sys., Vol. 183. Berlin: Springer, 1980.

W. Stadler (Ed.): Multicriteria Optimization in Engineering and in the Sciences. New York: Plenum Press, 1988.

G.N. Vanderplaats: Numerical Optimization Techniques for Engineering Design. Colorado Springs: Vanderplaats Research & Development, 2005.

P. Venkataraman: Applied Optimization with MATLAB Programming. New York: Wiley, 2002.

T.L. Vincent and W.J. Grantham: Nonlinear and Optimal Control Systems. New York: Wiley, 1997.

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M2.1

Optimization of mechanical systems

classical/engineering approach

analytical/numerical approach

intuition, experience of the design engi-neer and experiments and fiddling with hardware prototypes at the end of the de-sign process

intuition, experience of the design engineer and virtual prototypes based on computer simulations throughout the whole design process

sequential process

concurrent engineering

only small changes of the design pos-sible to influence the system behavior

more fundamental changes might re-quire the reconsideration of all previous design steps

“optimal” solution without quantitative objectives

hardware experiments are costly and time intensive (only few are possible)

provides many design degrees of free-dom at beginning of the design process

parameter studies can be executed fast and easy

systematic way to find optimal solution with respect to defined criteria

cost efficient shortening of the development time initial design for hardware experiments is

already close to optimum

idea

concept

CAD design

simulation

prototype

experiments product

optimization

production preparation

idea concept

design

calculation

prototype

experiments product

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M2.2

Optimization is a part of any engineering design process. Furthermore optimization is also extensively used in many other disciplines, such as e.g. industrial engineering, logistics or economics. The focus of this class is on optimization of mechanical systems using the analyti-cal/numerical approach. The concepts and methods are presented in a general manner, such that they can be applied to general optimization problems. The systematic formulation of an optimization problem requires the answers of three basic questions: 1. What should be achieved by the optimization? 2. Which changeable variables can influence the optimization goals? 3. Which restrictions apply to the system?

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M3

Iterative solution of the standard problem of parameter optimization

)( ),( ),( )()()( iii hgf ppp

initial design )0( ,0 pi

evaluation of the performance

propose a better design 1 ,)1( iiip

performance satisfying?

yes design “optimal”

no

evaluation of functions performance functions constraints

simulation model

calculation of gradients performance functions constraints

)(ip

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M4

Integrated Modelling and Design Process

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M5

Classification of optimization problems

optimization

topology function parameters

scalar optimization

multi-criteria optimization

)(min pfp P

unconstrained

)(min pp

fh

constrained

)(min pp

fP

ouhP ppp0)p(h0pgp , ,)(

scalarization

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M6.1

Geometric visualization in 2D

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M6.2

inequality constraints

equality constraints

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M7.1

Matrix Algebra and Matrix Analysis

vector nRx : n1 xx x , Rix

matrix nmR A :

mn

n1

1m

11

A

A

A

A

A , RijA ,

Basic Operations

operation notation components mapping

addition BAC ijijij BAC nmnmnm RRR

multiplication with scalar

AC ijij AC nmnm RRR

transpose TAC ijij AC mnnm RR

differentiation

ACdt

d

y

xC

ijij Adt

dC

j

iij y

xC

n mnm RR n mnm RR,R

matrix multiplication xAy

BAC

k

kiki xAy

k

kjikij BAC

mnnm RRR p mpnnm RRR

scalar product (dot product, inner p.)

yx k

kk yx RRR nn

vector product yxA jiij yxA nmnm RRR

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M7.2

Basic Rules

addition: CBACBA )()(

ABBA

multiplication with scalar: )()()( BABABA

BABA )(

transpose: AA TT )(

TTT)( BABA

TT)( AA

TTT)( ABBA

differentiation: BABAdtd

dtd

)(dtd

BABABA

dt

d

dt

d)(

dt

d

dt

d)(

dt

d y

y

xyx

matrix multiplication: CABACBA )(

CBACBA )()(

ABBA in general

scalar produkt: xyyx

xxx 0 , 0xxx 0

yxyx ,0 orthogonal

Quadratic Matrices

identity matrix

10

01

E

diagonal matrix

n

1

i

D0

0D

D

diagD

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M7.3

inverse matrix EAAAA 11

111)( ABBA

symmetric matrix TAA

skew symmetric matrix TAA

decomposition

)(2

1)(

2

1 TT AAAAA

TBB

TCC

skew symmetric 33 matrix

0aa

a0a

aa0~

12

13

23

a

baba ˆ~

abba ~~

Ebaabba )(~~

baabba ~

symmetric, positive definite matrix: 0xxAx 0

eigenvalues 0 , n)1(1

symmetric, positive semidefinite matrix: xxAx 0

eigenvalues 0 , n)1(1

orthogonal matrix T1 AA

~

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M8

Deterministic optimization strategies

Optimization algorithms are iterative and efficient strategies work in two-steps:

initial design )0(,0i p

evaluation of the performance of the current

design )(ip

propose a better design

1. search direction )i(s

2. line search )i(

no

performance satisfying?

yes design “optimal”

)i()i()1i( spp

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M9

Deterministic Optimization Strategies

optimization strategy

search direction

model order

information order

search parallel to the axes hmod

)(

es

gradient based method

)()( f s

conjugate gradient method

)0()0( fs

)(2)(

2)1()1()1(

f

ff

ss

Newton method

)(1)(2)( ff s

Example: quadratic criteria function

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M10

Line Search

possible requirements for line search

exact minimization:

0f)(f!

)1v()v()v( s

- many function evaluations inefficient

sufficient improvement:

in order to avoid infinitesimally small improvements, some conditions have been proposed, e.g. Wolfe-Powell conditions

1.0.g.e),1,(),0(f)(f

01.0.g.e),1,0(),0(f)0(f)(f!

!

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M11.1

Simulated Annealing

basic algorithm acceptance function cooling velocity

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M11.2

generation probability

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M12

Optimization by a Stochastic Evolution Strategy

from: P. Eberhard, F. Dignath, L. Kübler: Parallel Evolutionary Optimization of Multibody Systems with Application to Railway Dynamics, Vol. 9, No 2, 2003, pp. 143–164.

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M13

Particle Swarm Optimization

simulation of social behavior of bird flock (introduced by Kennedy & Eberhart in 1995) recursive update equation algorithm

k

ip position of particle i at time k

k

iΔp velocity of particle i at time k

]1,0[Ur,r 21 evenly distributed numbers

21 c,c,w control parameters

k

ip

1k

i

p

k,best

ip

k,best

swarmp

1k

i

k

i

1k

i

ppp

k

i

k,best

swarm

ki,22

k

i

k,best

i

ki,11

k

i

1k

ircrcw pppppp

tradition/ inertia

learning social behaviour

terminate?

initialization

recursive update equation

find best particle and

best solution k,best

swarmp

k,best

swarmp

yes

no

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M14

Karush–Kuhn–Tucker Conditions

If *p is a regular point and a local minimizer of the optimization problem

)(fmin pp P

with ,)(,)(h 0ph0pgp P

mhh , :h:g l ,

then Lagrange multipliers *λ and *μ exist, for which *p , *λ , *μ fulfill the following

conditions

0ppp

m

1j

jj

1i

ii

hgf l

0pg )(

0ph )(

0μ

0)(h jj p , m)1(1j

If we introduce the Lagrange function

m

1jjj

1iii )(h)(g)(f),,(L ppp:μλp

l

we can write the Karush–Kuhn–Tucker conditions as follows

0p

L

, 0λ

L

, 0μ

L

,

0μ , 0)(h jj p , m)1(1j

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M15.1

Lagrange-Newton-Method

= Sequential Quadratic Programming (SQP) = Recursive Quadratic Programming (RQP)

= Variable Metric Method

Here simplifying assumption: only equality constraints

})(|{Pwith)(fmin h

P0pgpp

p

Karush-Kuhn-Tucker Condition (KKT) for minimizer p

0

0

pg

pp

λ

pλpa

)(

)(g)(f

L

L

),( ii

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M15.2

)(T

)()1(

)()1(

)(T

ii22

fgf

g

λp

g

λλ

pp

0p

gp

g

)(

)(

)1(

)1(

)(

T)()(

f

gλ

p

0p

g

p

gW

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M15.3

In the case that the performance function and constraint equations are general nonlinear functions the parameter variation p is not necessarily the best possible parameter varia-

tion for the original optimization problem. In order to achieve a higher flexibility the method can be combined with a line search, sp .

)(

)(

)(

T)()(

f

gλ

s

0p

g

p

gW

equivalent to

0gs

p

gsssWs

s|Swithf

2

1min h)()(

S

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M16

Comparison of Various Deterministic Optimization

Algorithms for Nonlinear Constrained Optimization Problems

name algo-rithm

author availability mean number of function calls percent-age of failure

f ig , jh f ji h,g [%]

SUMT SUMT McCormick et al.

2335 24046 99 1053 69.9

NLP SUMT Rufer 1043 8635 111 957 15.6

VF02AD SQP Powell Harwell Subroutine Library

16 179 16 179 6.2

NLPQL (NCONF, NCONG)

SQP Schittkowski IMSL– Library

18 181 16 64 3.3

The results are based on 240 test runs with 80 test problems, using 3 different initial parameters for each problem.

See

Schittkowski, K.: Nonlinear Programming Codes. Information, Tests, Performance. Berlin: Springer, 1980. Schittkowski, K.: NLPQL: A FORTRAN Subroutine Solving Constrained Nonlinear Pro-

gramming Problems. Annals of Operations Res. 5 (1985/86) 485-500.

Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 11/12 M17

Principles of Reduction in Multicriteria Optimization