Linkoping Studies in Science and Technology

Thesis No. 1389

Robust design

- Accounting for uncertainties in engineering

David Lonn

LIU–TEK–LIC–2008:47

Department of Management and Engineering, Division of Solid MechanicsLinkoping University, SE–581 83, Linkoping, Sweden

http://www.solid.iei.liu.se/

Linkoping, November 2008

Cover:The picture illustrates the tradeoff situation a design engineer is faced withwhen trying to choose the design variable optimally. Should the mean per-formance or the variance of the objective be minimised?

Printed by:LiU-Tryck, Linkoping, Sweden, 2008ISBN 978–91–7393–743–6ISSN 0280–7971

Distributed by:Linkoping UniversityDepartment of Management and EngineeringSE–581 83, Linkoping, Sweden

c© 2008 David Lonn

This document was prepared with LATEX, November 3, 2008

No part of this publication may be reproduced, stored in a retrieval system,or be transmitted, in any form or by any means, electronic, mechanic, pho-tocopying, recordning, or otherwise, without prior permission of the author.

Preface

The work presented in this thesis has been carried out at the Division ofSolid Mechanics at Linkoping University. It is a part of the ROBDES re-search program, which is funded by the Research Council of Norway, HydroAluminium Structures, Volvo Car Corporation, Scania, SSAB Tunnplat andGestamp Hardetech.

I am grateful for the support given to me by my supervisor Prof. Lars-gunnar Nilsson, providing me with guidance, appropriate tools to work withand fruitful discussions. I would also like to thank my assistant advisor Prof.Kjell Simonsson and my Ph D student collegues for letting me bother themwith questions at any hour of the day.

Finally, I thank my family and friends, for showing great interest in me ona non-solid mechanics level. Special thanks goes to Ylva who fills my heartwith a little bit of sunshine every day.

Linkoping, September 2008

David Lonn

All models are wrong, some are useful.

George Box, famous statistician

iii

Abstract

This thesis concerns optimisation of structures considering various uncer-tainties. The overall objective is to evaluate and develop methods which findsolutions that are optimal both in the sense of handling typical load casesand minimising the variability of the response, i.e. robust optimal designs.

Conventionally optimised structures may show a tendency of being sensi-tive to small perturbations in the design or loading conditions. These kindsof variations are of course inevitable. To create robust designs, it is necessaryto account for all conceivable variations (or at least the influencing ones) inthe design process.

This thesis is divided in two parts. The first part serves as a theoreti-cal background to the second part, which consists of two appended articles.The first part includes introductions to the concept of robust design, basicstatistics, optimisation theory and meta modelling.

The first appended paper is an application of an existing sensitivity anal-ysis method on a large industrial problem. A sensitivity analysis is performedon a Scania truck cab subjected to impact loading in order to identify themost influencing variables on the crash responses.

The second paper presents a new methodology that may be used in robustoptimisations, i.e. optimisations that account for variations and uncertain-ties. The methodology is demonstrated both on an analytical problem and aFinite Element example of an aluminium extrusion subjected to axial crush-ing.

v

List of Papers

In this thesis, the following papers have been appended:

I. D. Lonn, M. Oman, L. Nilsson, K. Simonsson, Finite Element basedrobustness study of a truck cab subjected to impact loading, acceptedfor publication in International Journal of Crashworthiness, 2008.

II. D. Lonn, Ø. Fyllingen, L. Nilsson, Robust optimisation methodologyusing random samples and meta modelling, submitted, 2008.

The papers have been reformatted to fit the layout of the thesis.

Own contribution

I have had the main responsibility regarding the writing of the two appendedpapers. For paper one, I have been involved in all parts of the work, whereasin paper two, the Finite Element model along with the random perturbationshave been created by Ørjan Fyllingen.

vii

Contents

Preface iii

Abstract v

List of Papers vii

Contents ix

Part I Theory and background 1

1 Introduction 3

2 Robust design 5

3 Statistical concepts 7

3.1 Basic statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Monte Carlo analysis . . . . . . . . . . . . . . . . . . . . . . . 10

4 Optimisation 11

4.1 Deterministic optimisation . . . . . . . . . . . . . . . . . . . . 114.2 Robust optimisation . . . . . . . . . . . . . . . . . . . . . . . 12

5 Meta model approximations 17

5.1 Response Surface Methodology . . . . . . . . . . . . . . . . . 185.2 Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . 205.4 MLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4.1 Local approximations . . . . . . . . . . . . . . . . . . . 255.4.2 Global approximations . . . . . . . . . . . . . . . . . . 26

6 Outlook 29

ix

CONTENTS

7 Review of included papers 31

Bibliography 33

Part II Included papers 35

Paper I 39

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.1 The Swedish cab strength test series . . . . . . . . . . 441.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.1 Sub-model . . . . . . . . . . . . . . . . . . . . . . . . . 452.2 Responses . . . . . . . . . . . . . . . . . . . . . . . . . 462.3 Meta model approximations . . . . . . . . . . . . . . . 472.4 Sensitivity analyses . . . . . . . . . . . . . . . . . . . . 492.5 Screening . . . . . . . . . . . . . . . . . . . . . . . . . 512.6 Effects due to the forming process . . . . . . . . . . . . 532.7 Final sensitivity analysis . . . . . . . . . . . . . . . . . 54

3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 553.1 Validation of the complete truck model . . . . . . . . . 563.2 Validation of the sub-model . . . . . . . . . . . . . . . 563.3 First screening . . . . . . . . . . . . . . . . . . . . . . 573.4 Second screening . . . . . . . . . . . . . . . . . . . . . 593.5 Third screening . . . . . . . . . . . . . . . . . . . . . . 603.6 Effects due to the forming process . . . . . . . . . . . . 603.7 Final sensitivity analysis . . . . . . . . . . . . . . . . . 61

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

References 65

Paper II 71

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.1 Random Fields . . . . . . . . . . . . . . . . . . . . . . 743.2 Meta modelling . . . . . . . . . . . . . . . . . . . . . . 763.3 Artificial Neural Network . . . . . . . . . . . . . . . . . 763.4 Robust optimisation . . . . . . . . . . . . . . . . . . . 78

4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Analytical example . . . . . . . . . . . . . . . . . . . . . . . . 80

x

CONTENTS

5.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 836 Square aluminium tube example . . . . . . . . . . . . . . . . . 86

6.1 Model description . . . . . . . . . . . . . . . . . . . . . 866.2 Response . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Problem formulation . . . . . . . . . . . . . . . . . . . 916.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 95

References 97

xi

Part I

Theory and background

1

1Introduction

A robust design is a design which is sufficiently insensitive to variations.An optimal design is a design which under given constraints performs opti-mally with respect to the objective. An optimal robust design can not bedefined equally simple. However, an optimal robust design performs nearlyoptimally for specific loading cases and is ”sufficiently” independent of ran-dom variations. For instance, the outcome of a car crash event is preferablyindependent of the angle of impact, variations in material properties, manu-facturing process variations, etc. The car should absorb the crash energy ina controlled manner, even though the weight of the car has been minimisedin order to decrease fuel consumption.

The ROBDES research project, Robust Design of Automotive Structures,aims at reducing the influence of variations in impact loading situations. Theobjective is

”To develop tools and guidelines for modelling of automotivestructures subjected to impact loading conditions, where focus isplaced on an optimal and robust design.”

Dealing with variations in computational engineering is often a complextask since the computational efforts are considerably increased when random-ness is introduced. A crash simulation with the use of the Finite ElementMethod (FEM) is no exception and, consequently, many simplifications aremade in order to retrieve approximate solutions faster. The introductorychapters contain basic statistical theory and theory for optimisation. It isfollowed by a chapter on approximation techniques, i.e. meta modelling, amethodology which is quite central for this research topic. Only the responsesurface and Artificial Neural Network meta models are used in the subsequent

3

CHAPTER 1. INTRODUCTION

appended articles, but as the other meta models presented also are interest-ing in the context of sensitivity analyses and robust optimisation problems,they have also been included in the discussion in the theory chapters. Fur-thermore, one of the meta models, based on the Moving Least Squares (MLS)technique, is a novel approach and its applicability to serve as a meta modelfor robustness applications has been investigated in this project. The imple-mentation of this meta model is still under development and the model hasnot been used in the work presented here.

4

2Robust design

Robust design, a term originally introduced by the Japanese engineer GenichiTaguchi [1], is a way of improving the quality of a product by minimisingthe effect of variations, without eliminating the causes themselves. A robustdesign is by this definition a design which is sufficiently insensitive to varia-tions. Since conventional Finite Element (FE) methodologies do not accountfor stochastic variations and since good solution techniques for deterministicproblems already exist, the concept of robust design is the next logical stepin the development of more advanced simulation based design processes.

A robust design has the property of being insensitive to variations, but itis usually not the most optimal one in the sense of handling the typical loadcases, for which the design is built. This is intuitively understood since therobust design also must account for variations, and thus must handle a widerange of loading cases apart from the typical ones. Finding an optimal robustdesign is therefore almost always a tradeoff between optimising the meanperformance of the typical loading cases and minimising the performancevariance due to uncertainties. The tradeoff situation is illustrated in Figure1 where different choices of the variable x imply different levels of meanperformance and robustness.

One should also have in mind that there is a distinct difference betweenrobust design optimisation and reliability based design optimisation (RBDO),see for instance Zang et al. [2]. The robust design optimisation rather aimsat reducing the variability of structural performance caused by fluctuationsin parameters, than to avoid a catastrophe in an extreme event. In the caseof RBDO, we can make a design that displays large variations as long asthere are safety margins against failure in the design.

5

Figure 1: Illustration of the tradeoff between minimising the mean perfor-mance and the robustness of a response.

3Statistical concepts

Stochastic variations are always present, i.e. the reality is never as deter-ministic as our model of it. There are variations in e.g. material properties,forces and geometries, which in turn produce variations in our objectives.The following statistical terminology can be found in e.g. Casella et al. [3].

3.1 Basic statistics

Variations are defined as deviations from the (arithmetic) mean value µ,defined as the sum of all values xa divided by the number of values n

µ =1

n

n∑

a=1

xa (1)

To measure how much a single stochastic variable deviates from its meanvalue, the variance σ2 is introduced as

σ2 =1

n − 1

n∑

a=1

(xa − µ)2 (2)

However, dispersions are usually expressed in terms of the standard devi-ation σ, which is defined as the square root of the variance

σ =

√√√√

1

n − 1

n∑

a=1

(xa − µ)2 (3)

It is always possible to calculate the mean value and the standard devia-tion of a stochastic variable.

7

CHAPTER 3. STATISTICAL CONCEPTS

For the sake of simplicity, it is very common to assume that a stochasticvariable is normally distributed according to Equation (4). This assumptionis a good approximation in many cases, even though it may sometimes be-come infeasible for some variables. The thickness, for instance, can never benegative and in some cases, one should consider to use a different distributionthat does not allow negative values, such as the β-distribution, the Rayleighdistribution or a logarithmic normal distribution.

The normal distribution is a symmetric distribution around the meanvalue. Also, it is more probable that the value of the variable is closer to themean value µ than far away from it. Many variables are likely to have thesedistribution properties, which makes the normal distribution approximationa good one in many situations.

The standard deviation σ tells us how wide the distribution is. A smallervalue of σ depicts a more narrow peak of the distribution and thus randomlypicked values are often closer to the mean value. This can be visualised bydrawing the probability density function (PDF) as illustrated in Figure 2.

−5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4µ=0,σ=1µ=0,σ=2dX(x)

x

Figure 2: Probability density functions for different normal distributions.

dX(x) =1

σ√

2πe−

(x−x)2

σ2 (4)

By integrating the PDF between two different values of x, one producesthe probability of a randomly picked value from the stochastic variable lying

8

3.1. BASIC STATISTICS

in the chosen interval. From this observation, the cumulative distributionfunction (CDF) is created, see Figure 3. The figure shows the probabilityof a randomly picked value from the variable lying in the interval ] −∞, x]as described by Equation (5). PDF:s and CDF:s are commonly used for allother distributions as well.

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1µ=0,σ=1µ=0,σ=2DX(x)

x

Figure 3: Cumulative distribution functions for different normal distribution.

DX(x) =1

σ√

2π

x∫

−∞

e−(t−x)2

σ2 dt (5)

Another measure of dispersion that is widely used is the coefficient ofvariation (COV), which is defined as follows:

COV =σ

µ(6)

The coefficient of variation is a dimensionless number that allows compar-ison of the variation of different variables or different responses with differentmean values. When the mean value is near zero, the coefficient of variationis sensitive to changes in the standard deviation, which limits its usefulness.

Finally, the stochastic contribution σf,a, see Equation (7), indicates howmuch each stochastic variable xa contributes to the standard deviation σf

in the response f . If all variables are uncorrelated (do not depend on eachother) and if the relations between the variables and the response are linear,

9

CHAPTER 3. STATISTICAL CONCEPTS

the sum of the stochastic contributions will equal the standard deviation ofthe response.

σf,a =

∣∣∣∣

∂f

∂xa

∣∣∣∣σxa

(7)

3.2 Monte Carlo analysis

Consider f being a function that depends on stochastic variables collectedin the vector x. As the variables vary stochastically, f is also bound tovary and thus have a distribution, mean and standard deviation. The mostdirect method of evaluating the mean and standard deviation of f is theMonte Carlo method. Here, each variable in x is randomly picked fromtheir distribution, respectively, and used in an evaluation of the function f .Thousands of these evaluations are performed, all yielding a response value.The mean value and standard deviation of f may then be approximated fromthese response values by Equations (1) and (3).

The approximate statistical measures will converge to the true valuesas the number of samples increases. This is called the strong law of largenumbers. For the mean value

fna.s.−→ µ for n → ∞ (8)

where fn is the approximated mean value of the response based on n samplesand µ is the true mean value. The error of the mean value estimation is arandom variable with standard deviation

σθ =σ√n

(9)

where σ is the true standard deviation of the response.Further details on Monte Carlo methods can be found in e.g. Robert et

al. [4].

10

4Optimisation

The following sections briefly describe different formulations of optimisationproblems. Solution strategies are not described here and the reader is referredto e.g. Nocedal et al. [5] for further information.

4.1 Deterministic optimisation

A deterministic optimisation model is a good starting point before extend-ing the problem formulation to include stochastic variations. A traditionaloptimisation problem is often stated as

find x

minimising f(x)subject to gi(x) ≤ 0 (i = 1, 2, . . . , k)

x−a ≤ xa ≤ x+

a (a = 1, 2, . . . , n)

(10)

The vector x denotes the vector of all design variables that we want tochoose in an optimal way. Different choices of the design variables producedifferent values in our objective function f .

The variables are furthermore subjected to constraints. Two types of con-straints are described above, the functions gi, that represent constraints onsome given responses, and the second group of constraints which representslimits for the variables themselves.

11

CHAPTER 4. OPTIMISATION

4.2 Robust optimisation

Which changes are needed in the formulation of the optimisation problemabove in order to achieve an optimal robust design? A major part is to in-troduce the variations that are to be considered. Once the variations havebeen introduced, the variations that they subsequently create can be stud-ied. There are several possible approaches to introduce the variations intothe optimisation formulation and how to evaluate the statistical measures.The task of robust design optimisation is to minimise the variability of theperformance while meeting the requirements of optimum performance andconstraint conditions. As discussed previously, these goals, i.e. optimumperformance for the typical load cases and robustness (minimum variabil-ity), very often conflict with each other and a method that in some senseminimises them both is sought.

Variations can be associated to some or all design variables, but otherparameters could also be subjected to variations. Parameters which are notchosen in an explicit way as the design variables, but still show a stochasticbehaviour, are commonly referred to as noise variables or random variables.Young’s modulus could for instance be assumed to have a stochastic behavior,which could affect the different responses. To account for these variations aswell, the random variables in the formulation presented here are collected inthe vector y.

An appealing formulation of robust design optimisation is presented byDoltsinis et al. [6] and Lee et al. [7], where the variations of the designvariables and the structural performance are introduced into the objectivefunction as well as the constraint conditions. The formulation resembles (10)with some small changes

find x

minimising E(f(x,y)), σ(f(x,y))subject to E(gi(x,y)) + βiσ(gi(x,y)) ≤ 0 (i = 1, 2, . . . , k)

σ(hj(x,y)) ≤ σ+j (j = 1, 2, . . . , l)

x−a ≤ xa ≤ x+

a (a = 1, 2, . . . , n)

(11)

This formulation indicates that both the mean value (expected value)of the performance function, E(f), and its standard deviation, σ(f) areminimised. The notation hj(x,y) represents the structural performancesto which constraints on standard deviations are applied. In other words,the j:th structural performance function has an upper limit on the standarddeviation that is given by σ+

j .

12

4.2. ROBUST OPTIMISATION

The quantity βi is a prescribed feasibility index for the i:th original con-straint. Thus, the constraint will not always be fulfilled. Depending on thedifferent choices of βi, the probability that the constraint is fulfilled will vary.Assume for instance that the function gi(x) is normally distributed and setβi to be 3.0, then the probability that the original constraint condition willbe satisfied is 0.9987 (99.87%).

Doltsinis et al. [6] take one further step in formulating a robust designoptimisation problem, by introducing a weighting factor α for the tradeoffbetween minimising the mean performance and its standard deviation.

find x

minimising f = (1 − α)E(f(x,y))/µ∗ + ασ(f(x,y))/σ∗

subject to E(gi(x,y)) + βiσ(gi(x,y)) ≤ 0 (i = 1, 2, . . . , k)σ(hj(x,y)) ≤ σ+

j (j = 1, 2, . . . , l)x−

a ≤ xa ≤ x+a (a = 1, 2, . . . , n)

0 ≤ α ≤ 1(12)

This is the simplest form of introducing weights to the conflicting objec-tives, namely by making them linearly weighted. α = 0 corresponds to apure mean value minimisation problem and α = 1 a pure standard deviationminimisation problem. This particular formulation can be useful when inves-tigating the tradeoff situation, simply by using different values of α from zeroto one. All the different choices of the parameter α constitutes the Paretooptimal set, a concept introduced by the Italian economist Vilfredo Paretoin the late 19th century. The basic idea here is that the problem will havea different optimal solution depending on what variances of the objectiveperformance we tolerate. This, of course, is the designers choice. Levi et al.[8] give one example of how to choose α, where the choice depends on thedesired objective for the objective function f .

There is a possibility that the absolute values of the mean and the stan-dard deviation of the response f differ quite a lot. If, for instance, theabsolute value of the mean value is much greater than that of the standarddeviation, it becomes more important to minimise the mean value, almostindependent of the choice of α. In order to make the tradeoff entirely de-pendent on the choices of α, it may be useful to introduce the normalisationfactors µ∗ and σ∗ so that the absolute values of the two entities are simi-lar. The normalisation factors could for instance denote the mean value andstandard deviation from the first iteration with the original values of x andy, respectively.

So far, the formulation of the robust optimisation problem is rather

13

CHAPTER 4. OPTIMISATION

straightforward. However, the mean value and the standard deviation ofthe responses are required and these entities are generally very computation-ally expensive to evaluate. When the mean values and standard deviationsof the components in x and y are known, it is possible to make estimates ofthe mean value and standard deviation of the responses. The Monte Carlomethod may be used for this, but it is often computationally too expensive,and an approximation is therefore needed.

One possible approximation is presented by Chen et al. [9], Lee et al.[7] and Levi et al. [8]. By neglecting the higher-order terms, the mean andthe standard deviation of the response f are linearly approximated by thefirst-order Taylor series as

E(f(x)) = f(µx)

σ2(f(x)) =

n∑

a=1

(∂f

∂xa

)2 ∣∣∣∣µ

x

σ2a

(13)

This approximation holds if the stochastic variables are fairly uncorre-lated, the objective function is not too nonlinear and the variances of thestochastic variables are small. One may also note that this estimate requiresknowledge of the gradient of f . The similarity to Equation (7) is also obvious.

Another way of estimating the stochastic response characteristics is tofirst create an approximation of the deterministic response, a so called metamodel. The idea is to generate a simpler model that describes the responsesufficiently well. Given the meta model, it is much cheaper to retrieve anapproximate value of a response from a set of variable values. Thus, a MonteCarlo simulation with evaluations on the meta model instead of using the fullmodel, becomes an efficient tool for estimating the mean value and standarddeviation of a response. This procedure is referred to as a meta model-basedMonte Carlo analysis. Of course, the estimations of the mean values andstandard deviations of the responses g and h are obtained in an identicalmanner.

The problem formulation above is easily extended to multi-objective opti-misation. Optimising both the mean and the standard deviation of a responseis of course also a multi-objective optimisation problem, but traditionally,the term multi-objective indicates several different structural responses inthe objective. When several objectives are present, e.g. in a frontal car crashsituation where the objectives are to minimise both passenger accelerationand passenger compartment intrusion, a set of Pareto optimal solutions willbe present1. Each structural response may be decomposed into a mean per-

1This will of course be a different set than the one discussed previously as this newPareto set describes the tradeoff between the different structural responses in the objective.

14

4.2. ROBUST OPTIMISATION

formance part and a standard deviation part and tradeoff parameters for therobustness and the different structural responses can be set.

An optimisation of the mean performance and robustness is accomplishedeither by introducing the variations in the objective function or setting a con-straint on the maximum allowed variability. In the latter case, the variabilityis minimised in order to satisfy the constraint. Approaches that only accountfor variations by using constraints similar to the first of the constraints inEquation (11), are commonly referred to as reliability based design optimi-sations (RBDO). As these approaches do not explicitly try to minimise thestandard deviation of the responses, these methods are not by definition ro-bust optimisation methods. The general interpretation of these approachesis that a safety margin has been added to the constraint, rather than a min-imisation of the variations. However, according to Beyer et al. [10], thereis no consensus in literature whether RBDO should be regarded as a robustoptimisation method or not.

For a good overview of the field of robust optimisation, it is recommendedto read the review by Beyer et al. [10].

15

5Meta model approximations

When an evaluation of the response with given variable values is compu-tationally expensive, it may be efficient to create approximations of the re-sponse, often named meta models or surrogate models. These approximationsare in turn based on a finite number of designs, that is, explicit choices ofthe variable values.

There are several techniques to create a meta model. The first step is tochoose evaluation points (designs) on which to perform actual evaluations ofthe response. This choice of evaluation points is called Design of Experiments(DOE). The second step is to utilise these evaluated responses, i.e. how toconstruct the approximation for the non-evaluated designs. This is the choiceof the type of meta model approach to be used.

The field of applications for meta modelling is very large, but the maininterest here is in optimisation and sensitivity analysis. A sensitivity analysisis an analysis of how sensitive a structure is to small changes and, in thiscontext, also uncertainties in the variables. Sensitivity analyses may forinstance be performed by meta model-based Monte Carlo analyses, where ameta model is created for the response containing the dispersion interval ofthe variables, followed by Monte Carlo analyses where the response is givenby the meta model instead of evaluations of each design. In optimisation,meta models can for instance be used in order to find gradient directions forthe objective function, i.e. indicate how to change the design variables inorder to improve the objective.

In the following, the theory of four different meta modelling techniquesare briefly presented, namely Response Surface Methodology (RSM), Kriging,Neural Networks (NN) and Moving Least Squares (MLS). Although Krigingand MLS are not used in the appended papers, their properties may verywell be suited for robust optimisation and are thus considered relevant for

17

CHAPTER 5. META MODEL APPROXIMATIONS

the research topic. Future studies of Kriging and MLS in conjunction withrobustness studies are considered, but for the understanding of the papersappended in this thesis, the theory for these methods is not needed. It shouldalso be noted that the use of a meta model based on MLS is a novel approachin the context of robustness studies. In conventional optimisation, an MLSmeta model has been used by e.g. Breitkopf et al. [11].

5.1 Response Surface Methodology

A surrogate response surface is a polynomial approximation of a responsebased on the DOE. An example of this is the quadratic response surface

yi = β0 +∑

j

βjxij +∑

j

∑

k

βjkxijx

ik + εi

i = 1, 2, . . . , Nj = 1, 2, . . . , Mk = 1, 2, . . . , M

(14)

where xi are the design points, εi is the sum of both modelling and randomerrors, N is the number of evaluations, M is the number of variables and yi

is the evaluated (true) response values. The approximation can be writtenin matrix form as

y = X(x)β + ε (15)

where the coefficients in β are found by minimising the error ε in a leastsquares sense. These optimal coefficient values β∗ are found to be

β∗ = (XTX)−1XTy (16)

In order to determine all parameters in β, i.e. to be able to construct theapproximation, at least an equal amount of evaluations as parameters arerequired. However, an over sampling of 50% is recommended, see Redhe etal. [12]. A more extensive description of RSM can be found in e.g. Myers etal. [13].

5.2 Kriging

The Kriging approximation is constructed as

y(x) = fT (x)β + Z(x) (17)

where y is the true response value to be approximated, f is a known poly-nomial approach with coefficients in β and Z is a random process. The

18

5.2. KRIGING

approximation can also be written in matrix form when assuming that Nresponse values have been evaluated.

y(x) = X(x)β + Z(x) (18)

The similarity to Equation (15) may be noted.Z(x) has zero mean value and covariance

cov(Z) = σ2R(xi,xj) = σ2

R(x1,x1) . . . R(x1,xN )...

. . ....

R(xN ,x1) . . . R(xN ,xN)

(19)

where σ2 is the process variance and R(xi,xj) is the correlation functionbetween the evaluated points xi and xj, which makes R symmetric, positivedefinite and with a unit diagonal.

Two commonly used correlation functions are

R =n∏

k=1

e−θk|dk |

R =

n∏

k=1

e−θkd2k

n: number of variables

dk = xik − xj

k

(20)

denoted exponential and Gaussian, respectively. Optimal values of the re-gression coefficients, β∗, are found by multiplying Equation (18) by a weightmatrix W = (cov(Z))−1 = σ2R−1, which leads to

β∗ =(XTR−1X

)−1XTR−1y (21)

However, R is not determined yet. The n values of θk, collected in θ,are determined by solving the optimisation problem of maximising the log-likelihood function

max L(θ) =−N ln(s2) + ln |R|

2subject to θk > 0 k = 1, . . . , n

(22)

where

s2 =(y − Xβ∗)T

R−1 (y − Xβ∗)

N(23)

is the estimate of the process variance, i.e. a measure of the deviance fromthe polynomial regression.

19

CHAPTER 5. META MODEL APPROXIMATIONS

The approximate response at a point x0 can finally be written as

y(x0) = fT (x0)β∗ + rT (x0)R

−1ZD (24)

where ZD is a vector of residuals for all evaluation points

ZD = y −Xβ∗ (25)

and

rT (x0) =

R(x0,x1)...

R(x0,xN)

(26)

From Equation (24) it is fairly easy to see that Kriging is a global poly-nomial approximation with a local correction term. For a more detaileddescription of Kriging, see e.g. Stein [14].

5.3 Artificial Neural Networks

A Neural Network, or more precisely, an Artificial Neural Network (ANN),may be used to approximate complex relations between input and outputdata, and thus serve as a meta model. An ANN consists of neurons, i.e.small computing devices, that are connected. The output yk from neuron kis calculated as

yk(x) = f

(d∑

i=0

wkixi

)∣∣∣∣∣x0=1

= f(a) (27)

where f is the activation function and wki is the weight of the correspondinginput signal xi. The latter is either a variable value or a previous outputvalue from a neuron in the network. The term wk0 corresponds to the biasparameter and it may be included in the summation by adding an inputsignal x0 = 1. An illustration of a neuron can be seen in Figure 4.

The nature of the connection topology between the neurons, the weightsand the type of activation functions f in the neurons determine the type ofANN used. The two most common approaches for function approximationare the multilayer feedforward (FF) network and the radial basis function(RBF) network. In a multilayer feedforward network, no information travelsbackward in the network, i.e. the output of each layer serves as an input tothe next, see Figure 5.

Furthermore, the activation functions in the hidden layers are usuallysigmoidal functions

20

5.3. ARTIFICIAL NEURAL NETWORKS

Σ

x0 = 1

x1

xd

...f

a yk

wk0

wk1

wkd

Figure 4: Illustration of neuron k.

x1

x2

x3

y

︸ ︷︷ ︸

Input layer

︸ ︷︷ ︸

Hidden layers

︸ ︷︷ ︸

Output layer

Figure 5: A multilayer feedforward network with two hidden layers (bias notshown). Each circle represents a neuron and the type of activation functionis indicated as a symbol in the neuron.

f(a) =1

1 + e−a(28)

and the input and output layers are usually linear, i.e. f(a) = a, see Figures5 and 6. The network is called a RBF network using Gaussian basis functionswhen the following mapping is used

yk(x) =

d∑

i=1

wkiφi(x) + wk0

φi(x) = exp

(

−||x − ci||222θ2

i

) (29)

21

CHAPTER 5. META MODEL APPROXIMATIONS

where || . . . ||22 denotes the square of the Euclidian distance, and where ci andθi are the center and width of the i:th Gaussian basis function, respectively.

−5 0 50

0.2

0.4

0.6

0.8

1

a

f(a)

(a) Sigmoidal function.

−5 0 5−5

0

5

a

f(a)

(b) Linear function.

Figure 6: Activation functions.

Regardless of which of the above types of networks that are used, thereare some free parameters for the network that must be set. This procedureof setting the weights and biases for a multilayer feedforward network, oralternatively also setting the center and width of the basis functions for aRBF network, is called training the network. Training the network is anoptimisation problem, typically choosing the free parameters in an optimalway in order to minimise some error measure such as the mean squared error(MSE).

More information regarding ANN in the context of function approxima-tion are found in e.g. Bishop [15].

5.4 MLS

The Moving Least Square (MLS) approximation y(x) of a function u(x), isconstructed as

y(x) =

n∑

i=1

pi(x)βi(x) ≡ pT (x)β(x) (30)

where p is the vector of basis functions for the approximation and β is thevector of coefficients for the approximation.

For a specific value of x, a polynomial is fitted according to the leastsquares method, where the influence of the surrounding points are weighteddepending on their distance to x. One polynomial fit is thus not valid over

22

5.4. MLS

the entire domain as in the case of RSM, but locally around the point x

where the fit is made.

Depending on the order of approximation, p will be different. Two com-mon examples of p, for the case of two variables x1 and x2, are

pT (x) =

[1 x1 x2] (n = 3 , linear)[1 x1 x2 x2

1 x1x2 x22] (n = 6 , quadratic)

(31)

In contrast to RSM, the MLS technique raises the importance of an eval-uated point that is closer to the point where the approximation is made.When comparing with Kriging, in MLS this distance correlation comes intothe polynomial fit, whereas in Kriging it is merely a local correction to apolynomial fit.

Figure 7: Domain of influence for different sampling points in Ω. FromHallquist [16].

The coefficients β(x) for the polynomial approach at a given point x

depend on the sampling points xI that have a domain of influence includingthe point x. The domain of influence for sampling point xI is represented by aregion ΩI , see Figure 7. The size, i.e. the radius of the region, is determinedby the parameter a, cf. Equation (32). The influences from the samplingpoints xI are collected by a weight function wa(x − xI). A commonly usedweight function is the cubic B-spline kernel function, cf. Equation (32) for a

23

CHAPTER 5. META MODEL APPROXIMATIONS

mathematical description and Figure 8 for a two-dimensional example.

wa(x − xI) =

2

3− 4

( ||x − xI ||a

)2

+ 4

( ||x − xI ||a

)3

for 0 ≤ ||x − xI ||a

≤ 1

24

3− 4

( ||x − xI ||a

)

+ 4

( ||x − xI ||a

)2

− 4

3

( ||x − xI ||a

)3

for1

2≤ ||x − xI ||

a≤ 1

0 otherwise

(32)

01

23

45

01

23

450

0.2

0.4

0.6

x1

x2

wa

Figure 8: Plot of a cubic B-spline kernel function.

The MLS technique is to minimise the weighted L2-norm, J

J =NP∑

I=1

wa(x − xI)

(n∑

i=1

pi(xI)βi(x) − u(xI)

)2

= (XT (x)β(x) − u)TWa(x)(XT (x)β(x) − u)

(33)

where NP is the number of points whose domain of influence includes thepoint x, uT = [u1 u2 . . . uNP ] is a vector of the true function values atsampling points xI , and X and Wa are matrices defined as

X =

(p(x1))T

...(p(xNP ))T

Wa = diag[wa(x − x1) . . . wa(x − xNP )]

(34)

24

5.4. MLS

If all weights are set to unity (and all points are considered to influencethe approximation), the method is reduced to the well known least squaresmethod used in RSM, where no consideration is made concerning the distanceto evaluated points in the design space. In RSM, one least square fit issufficient as the fitted polynomial is assumed to be valid over the entiredesign space.

To find the optimal coefficients β∗ in order to calculate the approximationy(x), the extremum of J is obtained by setting the partial derivative withrespect to β to zero.

∂J

∂β= 2XTWaXβ − 2XTWau = 0 (35)

The solution isβ∗(x) =

(XTWaX

)−1XTWau (36)

which yields the estimate

y(x) = pTβ∗ = pT(XTWaX

)−1XTWau (37)

The dependencies on x has been dropped in the above for notational con-venience. The similarity between Equations (16) and (36) is clear. However,as indicated on the left hand side of Equation (36), it is important to notethat β∗ will depend on the point where the approximation is made, x. Byfixating the value of β∗ on the right hand side of Equation (37), a localpolynomial fit around x is retrieved.

As several MLS fits are required for a global approximation, it is necessarythat the inversion of the n × n matrix XTWaX is reasonably cost effective.This is the case when a low polynomial order of the approximation is usedand the number of variables are correspondingly few.

Also, if for instance a linear approximation is used, there has to be at leastn influencing evaluated design points in order to construct the approximation,and in general, some over sampling is preferable. The number of influencingevaluated design points can be adjusted by changing the parameter a in theweight function.

5.4.1 Local approximations

This section shows why MLS provides a better local approximation than theordinary least square approximation used in RSM. Here, a simple quadraticfunction is to be locally linearly approximated using MLS. The quadraticfunction is given by

u(x) = (x1 − 2.5)2 + (x2 − 2.5)2 + 1 (38)

25

CHAPTER 5. META MODEL APPROXIMATIONS

The only information known when performing the local approximationwith the two methods is the regular 10 × 10 grid of points in Figure 9,representing the evaluated design points. The approximation is made atx = [1 4] and it is based on the four nearest evaluated points, the parametera chosen accordingly. The straightforward usage of the previously presentedformulas yield the vector β∗ = [−3.375 − 3 3]T , and the expression for thelinear approximation is thereby

y(x) = pT (x)β∗ = −3.375 − 3x1 + 3x2 (39)

01

23

45

01

23

450

5

10

15

x1

x2

u

Figure 9: Fitted plane to a quadratic surface.

Note that if a ordinary least square approximation had been used, suchas RSM, it would yield a plane with some constant response value over theentire domain, which would be a rough global approximation, but locallyit would be inaccurate. It should also be noted that a larger value of theparameter a yields a better global approximation, but consequently a worselocal approximation.

5.4.2 Global approximations

As the local approximation by MLS is a better approximation than the ordi-nary least square approximation, a set of local approximations by MLS also

26

5.4. MLS

constitutes a better global approximation than the ordinary least squaresapproach. Performing a set of local approximations and keeping only theapproximated value y(x) and not the entire surface, yields a facet surfacewith arbitrary shape over the design domain.

This global approximation may therefore capture the characteristics of anarbitrary varying response, given a sufficient number of function evaluations,which in turn depends on the rate of nonlinearity of the true function. Asthe number of evaluated points increases, so does the accuracy of the approx-imation. Previous evaluations can be kept and reused, and it is possible toachieve a greater accuracy in interesting regions of the domain by performingmore evaluations locally.

As different fits are retrieved at different designs, the coefficients of thepolynomial approximation will vary. Thus, automatically, approximations ofthe gradient, and Hessian for 2:nd order or higher polynomial approaches,will be retrieved at the approximation points. This information is generallyexpensive to evaluate. However, one must be cautious as the quality of theseentities depend on the ability of the MLS meta model to make a correctapproximation at every approximation point.

27

6Outlook

In the Linkoping University part of the ROBDES project, there are twoongoing sub-projects. One of the sub-projects is a sensitivity analysis ofa car bumper system. Gestamp HardTech, the industrial partner in thissub-project, is interested in ranking the importance of the parameters thatinfluence the crash responses of the bumper system. A test series consistingof 20 bumper systems is planned, and the statistical results from the testsare to be compared with a sensitivity analysis conducted with the existingFE model.

The second sub-project is completely simulation based. A robust optimi-sation of a front rail system in a complete model of a Volvo S80 passengercar is to be performed, using a full frontal impact as the loading case. VolvoCar Corporation is interested in developing a methodology to perform robustoptimisations on its large scale industrial applications. The methodology tobe used is presently not decided, however, it is probable that some sub-modelor surrogate model needs to be created in order to reduce computation time.This sub-project will serve as a large scale application of the presented robustoptimisation methodology.

Several methodological issues remain to be explored. The most costlyentity to evaluate is probably the standard deviation σ. It is needed inthe robust optimisation formulation used, and must be evaluated cheaplyand accurately. Some new ideas of how to evaluate this entity should bedeveloped.

The properties for the novel approach of using MLS as a meta modelare also promising, and further development should take place. Possible ap-plication areas for MLS meta model are sensitivity analyses using the moreaccurate MLS local approximation, sensitivity analyses using new MLS fitsfor every design generated with the Monte Carlo method (applicable when

29

CHAPTER 6. OUTLOOK

the dispersions of the variables are large), deterministic optimisation usingthe global MLS approximation, and finally, robust optimisations where ap-proximate gradient information may help to locate flat regions of the trueresponse, which are required for a robust optimum.

30

7Review of included papers

Paper I

Finite Element based robustness study of a truck cab subjectedto impact loading

The first paper uses the meta model based Monte Carlo analysis methodin conjunction with polynomial approximations, i.e. RSM is used as metamodel approach. The robustness of a truck cab subjected to impact loadingis studied and the sensitivity analysis is performed in order to identify the im-portant design parameters and study effects from test setup variations. Afterseveral steps in a screening process, where variables that show no influence onresponse variations are removed, a final and more detailed sensitivity anal-ysis is conducted. This analysis shows that the cab design is already quiterobust. It is concluded that test setup variations may contribute largely toexperimental variations and that it also is important to consider prior loadinghistory, such as the forming processes, on parts in the structure.

Paper II

Robust optimisation methodology using random samples and

meta modelling

The second paper proposes a new robust optimisation method, accountingfor variations in the optimisation. Random variations are introduced onan analytical and on an FE structural member problem. By performingseveral response evaluations at the same design point, but with different

31

CHAPTER 7. REVIEW OF INCLUDED PAPERS

random perturbations introduced, it is possible to make an estimation of thefirst and second statistical moments, i.e. the mean value and the standarddeviation. These entities are used in a robust optimisation formulation and itis concluded that the suggested method is well suited for problems with manyrandom variables. The new method requires no additional methodologiesin the field of optimisation and this is demonstrated by using the existingoptimisation strategies in LS-OPT, see [17]. Also, it is possible to introducerandom variations that cannot be described by a variable in a traditionalway, here exemplified by variations described by random fields.

32

Bibliography

[1] Taguchi G. (1993). Taguchi on robust technology development bringingquality engineering upstream, ASME, New York.

[2] Zang C., Friswell, M. I., Mottershead J. E. 2005. A review of robust opti-mal design and its applications in dynamics, Computers and Structures,83, pp. 315-326.

[3] Casella G., Berger R. L. (2002). Statistical Inference, Duxbury.

[4] Robert C. P., Casella G. (1999). Monte Carlo Statistical Methods,Springer-Verlag, New York.

[5] Nocedal J., Wright S. J. (1999). Numerical Optimization, Springer Sci-ence+Business Media, New York.

[6] Doltsinis I., Kang Z. (2004). Robust design of structures using optimiza-tion methods, Computer Methods in Applied Mechanics and Engineering193, pp. 2221-2237.

[7] Lee K.-H., Park G.-J. (2001). Robust optimization considering tolerancesof design variables, Computers and Structures 79, pp. 77-86.

[8] Levi F., Gobbi M., Mastinu G. (2005). An application of multi-objectivestochastic optimisation to structural design, Structural and Multidisci-plinary Optimization 29, pp. 272-284.

[9] Chen W., Allen J. K., Tsui K.-L., Mistree F. (1995). A procedure forrobust design: Minimizing variations caused by noise factors and controlfactors, ASME Journal of Mechanical Design, Paper Number 951012.

[10] Beyer H.-G., Sendhoff B. (2007). Robust optimization – A comprehensivesurvey, Computer Methods in Applied Mechanics and Engineering 196,pp. 3190-3218.

33

CHAPTER 7. REVIEW OF INCLUDED PAPERS

[11] Breitkopf P., Naceur H., Rassineux A., Villon P. (2005). Moving leastsquares response surface approximation: Formulation and meatl formingapplications, Computers and Structures 83, pp. 1411-1428.

[12] Redhe M., Forsberg J., Jansson T., Marklund P.-O., Nilsson L. (2002).Using the Response Surface Methodology and the D-optimality criterionin crashworthiness related problems, Structural and MultidisciplinaryOptimization 24(3), pp. 185-194.

[13] Myers R. H., Montgomery D. C. (1995). Response Surface Methodology,Wiley, New York.

[14] Stein M. L. (1999). Interpolation of Spatial Data, Springer-Verlag, NewYork.

[15] Bishop C. M. (1995). Neural Network for Pattern Recognition, OxfordUniversity Press, Oxford.

[16] J. O. Hallquist (2006). LS-DYNA Theory Manual, Livermore SoftwareTechnology Corporation, Livermore.

[17] Stander N., Roux W., Eggleston T., Craig K. (2006). LS-OPT User’sManual, Version 3.1, Livermore Software Technology Corporation, Liv-ermore.

34

Division of Solid Mechanics, Depart-ment of Management and Engineering

2008–12–15

xx

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-

15479

Robust design – Accounting for uncertainties in engineering

David Lonn

This thesis concerns optimisation of structures considering various uncertainties. Theoverall objective is to find methods to create solutions that are optimal both in thesense of handling the typical load case and minimising the variability of the response,i.e. robust optimal designs.Traditionally optimised structures may show a tendency of being sensitive to smallperturbations in the design or loading conditions, which of course are inevitable. Tocreate robust designs, it is necessary to account for all conceivable variations (or atleast the influencing ones) in the design process.The thesis is divided in two parts. The first part serves as a theoretical backgroundto the second part, the two appended articles. This first part includes the concept ofrobust design, basic statistics, optimisation theory and meta modelling.The first appended paper is an application of existing methods on a large industrialexample problem. A sensitivity analysis is performed on a Scania truck cab subjectedto impact loading in order to identify the most influencing variables on the crashresponses.The second paper presents a new method that may be used in robust optimisations,that is, optimisations that account for variations and uncertainties. The method isdemonstrated on both an analytical example and a Finite Element example of analuminium extrusion subjected to axial crushing.

Robust optimisation, robust design, robustness, meta model, sensitivity analysis.

ISSN0280–7971

ISRN

LIU–TEK–LIC–2008:47

ISBN

978–91–7393–743–6

Nyckelord

Keyword

Sammanfattning

Abstract

Forfattare

Author

Titel

Title

URL for elektronisk version

Serietitel och serienummer

Title of series, numbering

Sprak

Language

Svenska/Swedish

Engelska/English

Rapporttyp

Report category

Licentiatavhandling

Examensarbete

C-uppsats

D-uppsats

Ovrig rapport

Avdelning, InstitutionDivision, Department

DatumDate