Uncertainty and Optimization in Structural Mechanics E-Book

Abdelkhalak El Hami

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 185

Veröffentlichungsjahr: 2013

Beschreibung

Optimization is generally a reduction operation of a definite quantity. This process naturally takes place in our environment and through our activities. For example, many natural systems evolve, in order to minimize their potential energy. Modeling these phenomena then largely relies on our capacity to artificially reproduce these processes. In parallel, optimization problems have quickly emerged from human activities, notably from economic concerns. This book includes the most recent ideas coming from research and industry in the field of optimization, reliability and the recognition of accompanying uncertainties. It is made up of eight chapters which look at the reviewing of uncertainty tools, system reliability, optimal design of structures and their optimization (of sizing, form, topology and multi-objectives) - along with their robustness and issues on optimal safety factors. Optimization reliability coupling will also be tackled in order to take into account the uncertainties in the modeling and resolution of the problems encountered. The book is aimed at students, lecturers, engineers, PhD students and researchers. Contents 1. Uncertainty. 2. Reliability in Mechanical Systems. 3. Optimal Structural Design. 4. Multi-object Optimization with Uncertainty. 5. Robust Optimization. 6. Reliability Optimization. 7. Optimal Security Factors Approach. 8. Reliability-based Topology Optimization. About the Authors Abdelkhalak El Hami is Professor at the Institut National des Sciences Appliquées, Rouen, France. He is the author of many articles and books on optimization and uncertainty. Bouchaib Radi is Professor in the Faculty of Sciences and Technology at the University of Hassan Premier, Settat, Morocco. His research interests are in such areas as structural optimization, parallel computation, contact problem and metal forming. He is the author of many scientific articles and books.

Leseprobe

Contents

Preface

1 Uncertainty

1.1. Introduction

1.2. The optimization problem

1.3. Sources of uncertainty

1.4. Dealing with uncertainty

1.5. Analyzing sensitivity

2 Reliability in Mechanical Systems

2.1. Introduction

2.2. A structure reliability problem

2.3. Modeling a structure reliability problem

2.4. Calculating the probability of failure in a structure

2.5. Reliability indices

2.6. Overview of the resistance–sollicitation problem

2.7. System reliability in mechanics

2.8. The finite element method and structural reliability

3 Optimal Structural Design

3.1. Introduction

3.2. Historical development of structural optimization

3.3. Classifying structural optimization problems

4 Multi-Object Optimization With Uncertainty

4.1. Introduction

4.2. User classification

4.3. Design classification

4.4. Multi-objective genetic algorithms

4.5. Robust multi-objective optimization

4.6. Normal boundary intersection method

4.7. Multi-objective structural optimization problem

5 Robust Optimization

5.1. Introduction

5.2. Modeling uncertainty

5.3. Accounting for robustness in optimum research

5.4. Robustness criteria

5.5. Resolution method

5.6. Examples of mono-objective optimization

6 Reliability Optimization

6.1. Introduction

6.2. Overview of reliability optimization

6.3. Reliability optimization methods

6.4. The reliability indicator approach

6.5. The single-loop approach

6.6. The sequential optimization and reliability assessment approach

7 Optimal Security Factors Approach

7.1. Introduction

7.2. Standard method

7.3. The optimal security factors (OSFs) method

7.4. Extension of the OSF method to multiple failure scenarios

8 Reliability-Based Topology Optimization

8.1. Introduction

8.2. Definitions in topology optimization

8.3. Topology optimization methods

8.4. Reliability coupling and topology optimization

8.5. Illustration and validation of the RBTO model

8.6. Application of the RBTO model to mechanics

Bibliography

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2013

The rights of Abdelkhalak El Hami & Bouchaïb Radi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2013930460

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISSN: 2051-2481 (Print)

ISSN: 2051-249X (Online)

ISBN: 978-1-84821-517-7

Preface

In recent years, engineers, scientists and officials have shown a strong interest in optimizing multi-objective and robust mechanical systems while accounting for uncertainty, a field which has received growing attention due to its challenges and industrial applications. Current deterministic models do not account for parameter variation, which is often poorly identified and provides an inaccurate picture of the problem in question. As a result, it is important to examine the context in which objective functions are uncertain and where it is often necessary to find reliable solutions for subsequent modifications likely to withstand decision variables. The final aim is a high-performing system. The two criteria, robustness and reliability, in mechanical systems are generally detrimental to performance criteria. The design problem in relation to the presence of uncertainty is a fundamentally multi-criteria optimization problem.

This book integrates the most recent notions in research and industry in the field of mechanical structure optimization, their reliability and their ability to account for uncertainty. This is described in Chapters 7 and 8 which focus on the different tools used to treat uncertainty, reliability and system optimization. The optimization–reliability combination is examined in order to account for uncertainty in modeling and resolving encountered problems. Each chapter explains the techniques being used and developed, with fully examined illustrative examples. The methods examined in this book can be applied to a number of different multi-physical systems.

Chapter 1 introduces the problems related to uncertainty. The optimization problems are treated, taking the uncertainties into account. We will, therefore, examine the main ideas relating to each theory and the propagation of uncertainty in the models studied.

Chapter 2 examines reliability in mechanical systems, demonstrating the basic elements involved in the calculation of the probability of failure in mechanical systems specifically and analyzing their reliability. The aim of modeling systems’ failure mechanisms is to identify improvements that are necessary to guarantee the system performance and reliability. In this context, the main failure mechanisms in a system are fully identified, modeled, analyzed and classed in order to guarantee the strength of the products’ design and, in addition, to identify necessary improvements to meet the increasing demand for reliability and durability.

Chapter 3 focuses on optimal structure design, an area of engineering which has been the focus of significant interest over the past 20 years. While as yet underapplied to standard technical research and development techniques, it has been gradually and progressively integrated, which has increased its reliability. In this chapter, we will first examine simplifying hypotheses in research used to mathematically explain the objective function and the constraints of our optimization problem. The different types of structure optimization problems will then be classified from a general perspective. We will conclude by describing a systematic approach in structural optimization.

Chapter 4 examines multi-objective optimization in uncertainty. The latter aims to optimize several components in a cost function vector. Unlike the single-objective optimization, the solution to a multi-objective problem (MOP) is not unique but is composed of a series of solutions, known as optimal Pareto solutions.

Chapter 5 considers robust optimization that has the same characteristics as deterministic optimization in terms of the treatment of data where there is uncertainty in terms of design variables and objective functions as well as treating constraints.

Chapter 6 discusses reliability optimization. The optimization of mechanical structures can be faulty, which means that the variability of parameters or random phenomena must be considered. An initial approach to be taken, taking uncertainties into account, consists of using security coefficients. However, this approach suffers from a lack of general applicability because security coefficients are closely linked to the existing situation and the engineer’s experience and, therefore, cannot be extended to new situations, particularly when there is not much experience and there are not enough previous faults. Analysis methods have been developed to respond to these difficulties. With this in mind, one of the first aspects examined is control in terms of reliability. We speak then of optimization, which accounts for reliability or reliability-based design optimization (RBDO).

Chapter 7 analyzes approaches of optimal security factors (OSFs). The security coefficient (a number associated with a given choice, a failure scenario and a sizing rule) generally results in a satisfactory design. It is validated by positive feedback. When an engineer designs a new form of structure, he is obliged to do several tests and make assumptions to identify the different extreme situations, which leads to an increase in the cost of the studied structures. The OSF method is a semi-numerical technique, which is based on the analysis of sensitivity of the limited state with respect to the variables of optimization and used in order to evaluate the influence of each parameter on the optimization process for the studied structure. The approach examined in this chapter to resolve the problem of reliable optimization is based on an analysis of sensitivity. This analysis plays a significant role in highlighting the influence of each parameter on the structure being studied.

Chapter 8 deals with reliability-based topological optimization (RBTO). Identifying structural components’ appropriate forms is of primary importance for the engineer. In all fields of structural mechanics, the impact of an object’s good design is highly important for its resilience, lifespan and application. In topology optimization, it is often required to solve large-scale problems. As such, there is a strong need to develop effective models and methods. The RBTO model examined in this chapter may have several solutions with different advantages, which allow us to select the best choice to fulfill a set of specifications.

Finally, this book constitutes a valued resource for both students and researchers. It is aimed at engineering students, practicing engineers and masters-level university students.

Acknowledgments

We would like to thank all those who have contributed to the completion of this book, in particular the engineering and PhD students at the INSA in Rouen (Normandy University – France) with whom we have worked over these past few years.

Abdelkhalak EL HAMIBouchaïb RADIFebruary 2013

1

Uncertainty

In a large number of optimization problems, simulation software, coupled with an appropriate mathematical optimization algorithm, is used for problems such as finance [LID 04], transport [CAP 03], manufacturing [CHE 08] and biochemical engineering and design engineering, [GAN 02]. This approach has proven to be much more effective than standard trial and error procedures. This is, for the most part, due to the development of faster digital computers, more sophisticated calculation techniques and the combination of simulation software based on the finite element method and mathematical optimization techniques [MAK 98]. In the case of deterministic optimization, the design variable can be accurately controlled and has a specific value. The input and output of the optimization procedure will be determinist in this case.

1.1. Introduction

Integrating uncertainty into the design process is a practice commonly used by engineers. This concerns the design of systems for critical values, the use of safety factors, and the more advanced techniques from the calculation of reliability. The aim is to design a system with statistically better performance that may often change according to uncertainty. For example, we may want to obtain a level of performance which is minimally sensitive to uncertainty. We may also not want to surpass a minimal performance threshold with a given probability. In addition, design problems are still constrained optimization problems. Where there is uncertainty, we want to identify it with a high degree of probability.

This uncertainty is a naturally inherent characteristic that cannot be avoided. We can, for example, cite exterior load point clouds and environmental conditions, such as variations in temperature and material properties. Alongside controllable design variables, processes are influenced by noise or stochastic variables. This type of variable cannot be precisely ordered and has either an unknown or known distribution. In the latter case, the variable may be commonly expressed by an average value and a corresponding standard deviation [JAN 08]. The input variation then translates the responding quantity that attaches a distribution rather than a deterministic value.

With continual demands on manufacturers to improve quality, quality control plays an increasingly important role in industrial procedures. One method that uses statistical techniques to monitor and control product quality is termed statistic process control. The different demands generally include three main tasks in the following order:

– a control procedure;

– process diagnostics;

– taking corrective measures.

What does it take to adjust the uncertain process so that it is evaluated on the basis of knowledge of the system and the experience [WER 07]?

With movement toward an integrated computer manufacturing environment, the need to develop applications that allow the implementation of various statistical process control tasks must be automatic. The ability to predict the response of a forming metal process to change in the number of input parameters is crucial. This is because, very often, an obtained deterministic optimum is found at the crossover between one and several constraints. The natural variation in materials, due to lubrication and process parameters, which could lead to an increased number of constraint violations, results in a higher amount of scrap iron [STR 10]. To avoid this unwanted situation, uncertainty must be specifically accounted for in the optimization strategy to avoid faults in the product such as wrinkling and material fractures and faults in form.

An initial approach of accounting for uncertainty in optimization problems has been carried out by considering security factors. The factor must compensate for the variation in yield caused by uncertainty in the system. The greatest security factors correlate with the highest levels of uncertainty. In the majority of cases, these factors are derived on the basis of past experience, but this does not absolutely guarantee security or a satisfying level of performance [BEN 02].

In recent years, several approaches have been developed to explicitly account for uncertainty [BEY 07, PAR 06]. This is examined in a special issue of the Review of Computing Methods in Applied Mechanics [HUG 05] and several research projects [PAD 03]. In addition, several modules (or courses) have already combined statistical control techniques for processes with simulations of finite elements to quantify robustness, such as, for example, Autoform-Sigma and LS-Opt [CLE 10]. However, these packets are mainly concentrated around quantifying reliability and robustness in a given solution, rather than optimization under uncertainty. The deterministic optimization strategy is extended to take design variables into consideration. This approach allows the quantification and optimization of a process or design performance.

1.2. The optimization problem

The basic idea of optimization consists of minimizing an objective function, f, by finding the optimum value of one or several design variables x. In addition, several types of restrictions or constraints can be present as equality constraints h, inequality constraints g or a box of constraints. This last type of constraint is often defined as boundaries that identify the area in which design variables are authorized to vary according to an upper and lower limit. These boundaries are respectively indicated by a and b. In general, an optimization problem must be mathematically described as follows:

[1.1]

The solution to an optimization problem entails finding one of the optimum design values or the variables that minimize the objective functions subject to different constraints. This process requires an optimization algorithm for specific problem. The inequality and equality constraints can be divided into linear or nonlinear and explicit or implicit constraints. Explicit constraints depend directly on the design variables while implicit constraints depend indirectly on the design variables. In the latter case, an evaluation of the constraint function is required to evaluate whether or not the constraint is satisfactory.

The aim of an optimization procedure is to find an optimal design with a high degree of precision. Some critical factors concerning optimization procedures are as follows:

– The optimization procedure is carried out using models and/or approximations of reality. In the majority of cases, we do not know what the error in the model is. If this is the case, we cannot be certain that the optimum model is the true optimum. This indicates that the optimum solution, even if calculated very accurately, may be difficult to replicate in a real world application. Verification of the optimum model against the physical process is therefore highly recommended.

– If the aim of optimization is to improve product quality, it is also helpful to verify the economic feasibility of improving products. There is, in general, a compromise between a potentially more complex and costly manufacturing procedure and an increase in the new design’s performance. It is therefore recommended to favor an optimum economic product or tend rather towards the qualitative optimum.

– The following optimum design for an optimum procedure is a static optimum. However, it is, in reality, dynamic. For example, the process or environmental parameters may change over time and therefore the static optimum is only valid for a limited duration.

– Both the objective function and implicit constraints require an evaluation of the response since they inherently depend on the design variables. Each evaluation of the response is carried out by calculating the nonlinear finite elements, which is therefore costly in terms of calculation time.

– The evaluation of the response relating to the obtained optimum, a series of finite element simulations, must be carried out. As a result, an effective algorithm must reduce the number of these simulations.

– In particular, optimization procedures that incorporate uncertainty are significantly less time effective in relation to their deterministic counterparts. This is because the evaluation of the objective functions and constraints are more costly in uncertain conditions. It is therefore necessary to use an appropriate and effective technique to manage uncertainty in the optimization procedure.

– To limit the computational load when there is a combination of optimization strategies with finite element simulations, only a limited number of parameters may be studied. Other procedures which make the design problem less costly in terms of calculation include the use of approximation and parallel calculation [SCH 08].

– A final, crucial remark regarding digital simulation in the optimization process is that the latter may introduce new sources of point clouds, also known as digital noise [DES 08].

1.3. Sources of uncertainty

Optimization under uncertainty requires information regarding the uncertainty influencing the system. There are different sources of variation. Each type of uncertainty requires a different approach for use in the optimization procedure.

There are different scenarios where the designer has to deal with uncertainty. A metal forming the procedure or the product has an output or a response, f, that depends on the input. The input can be divided into design variables x and design parameters p. The design parameters are governed by its environment such as temperature and humidity. The behavior being displayed by the system can be controlled by design variables such as, for example, process parameters and geometric tools. Uncertainty is the input that the designer cannot control in an industrial context, such as with a procedure forming metal while it creates the variation in the response. Different types of uncertainty can be present:

– Uncertainty in the design parameter: this type of uncertainty, denoted by zp, is caused by a change in the environmental and operational conditions. The environmental input can be classed as energy, information and material. Changes in temperature, etc., are examples of variations in the design parameters. Note that material parameters can also show variations. Having chosen a material, its parameters can no longer be controlled. Variation in the parameters must then be taken into account as a noisy variable.

– Uncertainty in the design variable: this type of uncertainty is caused by the limited degree of precision with which a design variable can be controlled. Examples of this include variations in the material’s thickness, geometric tolerances, inaccuracy in the actuator, variations in the strength of the implementation process, etc. Uncertainty in the design variable is often involved in the process where there is zx disturbance in the design variables x. This is conveyed by:

[1.2]

We can also use a different classification schema for uncertainty by differentiating between uncertainty in non-cognitive and cognitive sources. The previous source of uncertainty, also known as random uncertainty, is physical in nature. The random nature inherent in physical observations is a statistical uncertainty due to a lack of precise information regarding variation, etc. The last source of uncertainty, also known as epistemic uncertainty, reflects the designer’s lack of knowledge about the problem being examined.

Another classification system proposed in [KIM 10] describes different kinds of uncertainty according to the stage of process or lifecycle of the product displaying variation. For example, in the design phase, uncertainty may be caused by errors in the model as well as incomplete knowledge about the system. During the manufacturing stage, the fabrication tolerances of the material introduce uncertainty. Changes in temperature and fluctuations in load may be recognized as sources of variation in a product or procedure’s use. Finally, during aging, deterioration in the materials’ properties can lead to variability in performance.

1.4. Dealing with uncertainty

This section will describe the main approaches developed for accounting for uncertainty. The majority of these methods have been developed to be applied to technically complex problems. We will not attempt to provide an exhaustive overview of these different approaches and their applications. However, we will provide a general overview to give the reader an idea of the models available when dealing with uncertainty.

A random stochastic reliability-based description is used in the optimization stage and the robust optimization stage where uncertainty is managed probabilistically.

In these practical engineering problems, uncertain random parameters are often modeled as a set of discretized random variables. Let us suppose that X is a random variable and there are n observations of X and the specific occurrence of a random variable where the samples of X are given by x or x1, x2,…, xn. The statistical description of a random variable X can be fully described by a cumulative distribution function (CDF) or a probability density function (PDF), denoted by PX(x) and pX(x), respectively. To calculate the probability Pr[] of X having a value between x1 and x2, the area in the PDF between these two points must be calculated. This can be expressed as:

[1.3]

The PDF is the first derivative of the CDF, such that:

[1.4]

Now, taking a general expression to evaluate the mathematical mean value E(X), the variance var(X) and the asymmetry of the random variable are given by equations [1.5], [1.6] and [1.7], respectively. When these values are known, we can identify other parameters such as the standard deviation σX and μX:

[1.5]

[1.6]

[1.7]