Multidisciplinary Design Optimization in Computational Mechanics E-Book

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

Seitenzahl: 740

Veröffentlichungsjahr: 2013

Beschreibung

Multidisciplinary Design Optimization in Computational Mechanics

This book provides a comprehensive introduction to the mathematical and algorithmic methods for the Multidisciplinary Design Optimization (MDO) of complex mechanical systems such as aircraft or car engines. We have focused on the presentation of strategies efficiently and economically managing the different levels of complexity in coupled disciplines (e.g. structure, fluid, thermal, acoustics, etc.), ranging from Reduced Order Models (ROM) to full-scale Finite Element (FE) or Finite Volume (FV) simulations. Particular focus is given to the uncertainty quantification and its impact on the robustness of the optimal designs. A large collection of examples from academia, software editing and industry should also help the reader to develop a practical insight on MDO methods.

Leseprobe

Table of Contents

Foreword

Notes for Instructors

Acknowledgments

Chapter 1: Multilevel Multidisciplinary Optimization in Airplane Design

1.1. Introduction

1.2. Overview of the traditional airplane design process and expected MDO contributions

1.3. First step toward MDO: local dimensioning by mathematical optimization

1.4. Second step toward MDO: multilevel multidisciplinary dimensioning

1.5. Elements of an MDO process

1.6. Choice of optimizers

1.7. Coupling between levels

1.8. Post-processing

1.9. Conclusion

Chapter 2: Response Surface Methodology and Reduced Order Models

2.1. Introduction

2.2. Introducing some more notations

2.3. Linear regression

2.4. Non-linear regression

2.5. Kriging interpolation

2.6. Non-parametric regression and kernel-based methods

2.7. Support vector regression

2.8. Model selection

2.9. Introduction to design of computer experiments (DoCE)

2.10. Bibliography

Chapter 3: PDE Metamodeling using Principal Component Analysis

3.1. Principal component analysis (PCA)

3.2. Truncation rank and projector error

3.3. Application: POD reduction of velocity Þelds in an engine combustion chamber

3.4. Reduced-basis methods, numerical analysis

3.5. Intrusive/non-intrusive aspects

3.6. Double reduction in both space and parameter dimensions

3.7. The weighted residual method

3.8. Non-linear problems

3.9. General discussion and comparison of surrogates

3.10. A numerical example

3.11. Time-dependent problems

3.12. Numerical analysis of a linear spatio-temporal PDE problem

3.13. Related works and complementary bibliography

3.14. Bibliography

Chapter 4: Reduced-order Models for Coupled Problems

4.1. Introduction

4.2. Model reduction methods for coupled problems

4.3. Application 1: MDO of an aeroelastic 2D wing demonstrator

4.4. Application 2: MDO of an aeroelastic 3D wing in transonic flow

4.5. Application 3: multiobjective shape optimization of an intake port

4.6. Conclusions

4.7. Bibliography

Chapter 5: Multilevel Modeling

5.1. Introduction

5.2. Notations and vocabulary

5.3. Parallel model optimization

5.4. Multilevel parameter optimization

5.5. Multilevel model optimization

5.6. General resolution strategy

5.7. Use of the multiscale approach in multilevel optimization

5.8. A multilevel method for aerodynamics using an inexact pre-evaluation approach

5.9. Numerical examples

5.10. Conclusion

5.11. Bibliography

Chapter 6: Multiparameter Shape Optimization

6.1. Introduction

6.2. Multilevel optimization

6.3. Validation

6.4. Applications

6.5. Conclusion

6.6. Bibliography

Chapter 7: Two-discipline Optimization

7.1. Pareto optimality, game strategies, and split of territory in multiobjective optimization

7.2. Aerostructural shape optimization of a business-jet wing

7.3. Conclusions

7.4. Bibliography

Chapter 8: Collaborative Optimization

8.1. Introduction

8.2. DeÞnition of parameters

8.3. Notations and terminology

8.4. Different frameworks for multidisciplinary design optimization

8.5. Reduced order models and approximations

8.6. Application of MDO to conceptual design of supersonic business jets (SSBJ)

8.7. Comments and conclusions

8.8. Bibliography

Chapter 9: An Empirical Study of the Use of Confidence Levels in RBDO with Monte-Carlo Simulations

9.1. Introduction

9.2. Accounting for uncertainties in optimization problem formulations

9.3. Example: the two-bars test case

9.4. Monte-Carlo estimation of the design criteria

9.5. A simple evolutionary optimizer for noisy functions: introducing the confidence level

9.6. Effects of the step size, the Monte-Carlo budget and the confidence level on ES convergence

9.7. Conclusions

9.8. Bibliography

Chapter 10: Uncertainty Quantification for Robust Design

10.1. Introduction

10.2. Problem statement

10.3. Estimation using the method of moments

10.4. Metamodel-based Monte-Carlo method

10.5. Application to aerodynamics

10.6. Conclusion

10.7. Bibliography

Chapter 11: Reliability-based Design Optimization (RBDO)

11.1. Introduction

11.2. Numerical methods in RBDO

11.3. Semi-analytic methods in RBDO

11.4. Academic applications

11.5. An industrial application: RBDO of an intake port

11.6. An industrial application: RBDO of a simplified model of a supersonic jet

11.7. Conclusions

11.8 Bibliography

Chapter 12: Multidisciplinary Optimization in the Design of Future Space Launchers

12.1. The space launcher problem

12.2. Launcher design

12.3. Multidisciplinary optimization in the launcher preliminary design phase

12.4. Evolutionary optimization for space launcher design: an example

12.5. Bibliography

Chapter 13: Industrial Applications of Design Optimization Tools in the Automotive Industry

13.1. Introduction

13.2. Specific problems linked to manufacturing applications

13.3. Existing tools: objectives, functions and limitations

13.4. Using existing tools – Renault’s application

13.5. Expected developments

13.6. Conclusion

13.7. Bibliography

Chapter 14: Object-oriented Programming of Optimizers – Examples in Scilab

14.1. Introduction

14.2. Decoupling the simulator from the optimizer

14.3. The “ask & tell” pattern

14.4. Example: a “multistart” strategy

14.5. Programming an ask & tell optimizer: a tutorial

14.6. The simplex method

14.7. Covariance matrix adaptation evolution strategy (CMA-ES)

14.8. Ask & tell formalism for uncertainty handling

14.9. Conclusions

14.10. Bibliography

List of Authors

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Optimisation multidisciplinaire en mécanique published 2009 in France by Hermes Science/Lavoisier in two volumes © LAVOISIER 2009

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:

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© ISTE Ltd 2010

The rights of Piotr Breitkopf and Rajan Filomeno Coelho 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 Cataloging-in-Publication Data

Optimisation multidisciplinaire en mécanique. English

Multidisciplinary design optimization in computational mechanics / edited by Piotr Breitkopf, Rajan Filomeno Coelho.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-138-4

1. Engineering design. 2. Engineering mathematics. I. Breitkopf, Piotr. II. Coelho, Rajan Filomeno. III. Title.

TA174.O6813 2010

621.8'15--dc22

2010007329

British Library Cataloguing-in-Publication Data

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

ISBN: 978-1-84821-138-4

Foreword

This extraordinary volume on the state of the art of multidisciplinary design optimization (MDO) grew out of a French project of the same name started in 2005. I was present at one of the first organizational meetings of the project, and when I was asked to write an introduction to this volume, I was glad of the opportunity to view some of the results. I was impressed and gladly accepted.

However, this endeavor begs the question of how different is multidisciplinary design optimization from single discipline optimization? I have debated this question over the years, with my view ranging from “not at all” to “very different”. The reason for the ambiguity is that every challenge posed by MDO can also be found in examples of single discipline optimization. However, the additional complexity of MDO means that these challenges are encountered together, and they require the simultaneous use of more sophisticated methods.

The challenges include the need to combine very different disciplinary analyses that have traditionally been performed with little interaction. They include the frequent appearance of multiple objectives, and a more complex design space that may need a specialized combination of optimization methods. Finally, they include a heightened importance of modeling uncertainties because intuitive methods of handling these uncertainties by safety factor approaches may break down in the system design.

These challenges have been addressed by a host of methods. However, almost every MDO paper is likely to include surrogate (also known as metamodel, or response surface) methodology and/or decomposition methodology. I therefore welcome this volume of chapters on MDO methodology, which gives special attention to these two classes of methods while not neglecting others. The introductory chapter, Multilevel Multidisciplinary Optimization in Airplane Design, sets the stage for the volume, with the interaction between the structural and fluid flow disciplines providing most of the examples.

The importance of surrogate modeling in MDO is underscored by three leading chapters devoted to the topic. Chapter 2, Response Surface Methodology and Reduced-order Models, provides a thorough overview of surrogate methods. Unlike single disciplinary optimization, where it is enough to approximate a few functions (objective and constraints), in MDO we are often called upon to provide surrogates for disciplinary response fields that need to be used in another discipline (such as pressure fields generated by aerodynamics which are needed for structural analysis). Chapter 3, PDE Metamodeling using Principal Component Analysis, and Chapter 4, Reduced Order Models for Coupled Problems, deal with this special aspect. Together, these chapters account for more than one third of the volume.

The second major theme of MDO, decomposition, is dealt with in the next four chapters. Chapter 5, Multilevel Modeling, provides terminology and general background. Chapter 6, Multiparameter Shape Optimization, and Chapter 7, Two-discipline Optimization, provide info–rmation on specialized techniques and dealing with Pareto fronts. Finally, Chapter 8, Collaborative Optimization, provides a review of a variety of decomposition techniques (such as global sensitivity equations, concurrent subspace optimization, and collaborative optimization) that have been developed specifically with MDO in mind.

The next three chapters deal with design under uncertainty. Chapter 9, An Empirical Study of the Use of Confidence Levels in RBDO with Monte-Carlo Simulations, deals with the important problem that information about uncertainty is itself uncertain. Chapter 10, Uncertainty Quantification for Robust Design, deals mostly with the use of sensitivity methods and Monte-Carlo simulations for estimating the distribution moments needed for robust design. Chapter 11, Reliability-based Design Optimization, reviews methods such as the first- and second-order reliability methods (FORM and SORM) which address the need for estimating low probabilities of failure.

While the individual chapters provide illustrative examples, mostly from aircraft and automotive design, these examples are of limited complexity. Chapter 12, Multidisciplinary Optimization in the Design of Future Space Launchers, and Chapter 13, Industrial Applications of Design Optimization Tools in the Automotive Industry, provide more complex examples that a reader would be able to appreciate by the end of this volume.

Finally, the last chapter addresses the fact that in MDO the designer often requires more flexibility in configuring the optimization process. Chapter 14, Object-oriented Programming of Optimizers – Examples in Scilab, describes the working and advantages of the ask & tell optimization environment, which provides such flexibility.

I am sure that this volume will prove to be useful both for teaching graduate students and as a reference book for practitioners.

Rafael T. HAFTKA

Gainesville, FL, USA, March 24, 2010

Notes for Instructors

This book is suitable for advanced undergraduates, graduate students, researchers, and practicing engineers interested in all aspects of multidisciplinary design optimization. Readers should have a preliminary knowledge of numerical methods such as the finite element or the finite volume method.

The material presented in this book can be used for teaching specific subjects, such as metamodels and reduced-order models (1/2 semester, Chapters 2–4), optimization under uncertainty, including robustness and reliability (1/2 semester, Chapters 9−11), or multidisciplinary design optimization strategies (1 semester, Chapters 1, and 4–8). For a broader view, the whole book can be used for a two-semester course covering all current features of multidisciplinary design optimization, including applications and implementation issues (Chapters 1–14).

Acknowledgments

This work has been supported by the National Agency of French Research/National Network of Software Technologies (ANR/RNTL) within the framework of the OMD project, and by ANR within the framework of the OMD2 project.

The authors of Chapter 4 also acknowledge the Projet Pluri-Formations PILCAM2 at the Université de Technologie de Compiègne for providing HPC resources that have contributed to the research results (URL: http://pilcam2.wikispaces.com).

Finally, the author of Chapter 7 wishes to express his warmest thanks to his collaborators in the INRIA Opale Project-Team, and particularly to those whose results are used in Chapter 7: B. Abou El Majd (aerostructural optimization), N. Blaszka-Marco (two-point aerodynamic optimization by the NSGA) and Z. Tang (two-point aerodynamic optimization by a Nash game). Special thanks are also due to A. Habbal of the University of Nice–Sophia Antipolis and Opale, who originally proposed the formulation of the aerostructural concurrent problem and actively participated in fruitful discussions on Nash games.

Chapter 1

Multilevel Multidisciplinary Optimization in Airplane Design1

1.1. Introduction

MDO is the acronym for “multidisciplinary design optimization”. MDO can be defined as how to efficiently organize design, i.e. the art of optimizing the definition of our airplanes in order to find the best compromise between specifications and requirements. Mathematical optimization applied to multidisciplinary design provides an efficient context for researching the best solution, in all disciplines and at all levels of detail in the definition: from the system to the different subsystems and to the smallest components.

Beyond the search for a better response to the traditional needs of performance and competitiveness for our products, this methodology satisfies the need to adapt our design process as well as possible to the emergence of new design requirements for our airplanes (noise in the Falcons’ airport zone, conformance of military planes, etc.) and our ambition to design innovative products such as the supersonic Falcon and different types of UAVs (unmanned aerial vehicles). Finally, we propose a rational approach for design cooperation with our partners and subsystem manufacturers through a detailed process for systematizing and automating the exploration of the solution space, for synthesizing the results for decision support, and for evaluating the cost of a specification for the complete airplane cycle.

We will first present the traditional process of airplane design. The description of the traditional process will highlight the expected contributions of MDO methodology.

1.2. Overview of the traditional airplane design process and expected MDO contributions

For a very long time, aircraft manufacturers, as did all developers of complex products, implemented a multilevel and multidisciplinary organization for the definition of their products. This methodology is based on a tree structure by discipline (or trade) with increasing refinement of the global airplane definition to the definition of every detail in its components. A numerical modeling tree diagram for this approach in design of a civil airplane is shown in Figure 1.1.

Figure 1.1. Modeling tree for a civil airplane

In each discipline and at each level, we are today building two types of paired computer representations:

– a definition model, that is the CATIA definition of the geometry of elements involved;

– computation models, e.g. the finite model elements of resistance and vibrations, the finite model elements of aerodynamics, to show that the product satisfies specifications and good engineering practices of the different trades.

The traditional design process occurs by progressively building the model tree by going down the hierarchy of models, from global definition to detail definitions. The specifications of the different subsystems are generated in the same way. The design loop, in a traditional approach, can then be described in the following manner: after an initial choice of architecture determined by the engineer’s intuition and experience, the components are sized by iterations until the specifications are met.

In practice, for problems with numerous constraints or multidisciplinary coupling, the convergence of the traditional process can be difficult: mathematical optimization can provide optimization algorithms with efficient constraints. Level N + 1 specifications are the level N design parameters, and the verification of specifications can only be done when the level N + 1 system has been completely developed. This can be a drawback, especially for innovative projects for which subsystems cannot come from products already on the market. It is therefore necessary to implement a process of multilevel sensitivity analysis for managing uncertainties in the performance of subsystems and their impact on global product performance. Finally, intuition and experience are key elements in the traditional process. Breakthrough products cannot always rely on experience from previous projects and intuition can be faulty. A design process adapted to innovative products must enable rational comparisons to be made between optimized architectures, and provide decision support in design with the help of advanced visualization for exploration of the design space.

The complexity of the systems being designed makes it increasingly difficult for an engineer to have all the scientific and technical knowledge necessary to carry out the design process successfully. The global design problem is therefore broken down into subsets that can be controlled by experts from the disciplines involved. The two major challenges of MDO are complexity of physical modeling and complexity of the production organization necessary for its development. MDO offers a range of methodologies for coordinating the efforts and efficiently managing the interfaces between expert design groups. These methodologies can be described in the form of algorithms organizing the global design process broken down into subsystems. MDO must be viewed as an approach which integrates the engineer’s experience and intuition, and is based on the power of mathematical analysis and computer models to provide rational conclusions in a decision process.

1.3. First step toward MDO: local dimensioning by mathematical optimization

This was the first design process improvement. It was applied in the mid-1970s to structures, followed by other disciplines at different levels. The method is only applied to a single discipline at a time. Its principle is to reduce the dimensioning process to a mathematical optimization problem, i.e. to find a set of definition parameters that will maximize a performance while satisfying the specs.

We can cite two examples of optimizations commonly used today: structural dimensioning of an airplane, and aerodynamic dimensioning. Structural dimensioning optimization consists of determining all the zones in the structure – the thickness of panels, stringer sections, the number of folds and their composite material orientations – which minimize the structural mass, while respecting the criteria of mechanical resistance and aeroelastic failures in the flight envelope. An example of this type of optimization is presented for the case of a supersonic business plane in Figure 1.2.

Figure 1.2. Structural dimensioning of the airplane

In the case of the optimization of aerodynamic dimensioning, the idea is, for example, to determine the geometric form minimizing cruising drag while ensuring the airplane’s aerodynamic qualities in its flight envelope and respecting global or local geometric constraints for the cell setup. This process is illustrated in Figure 1.3.

1.4. Second step toward MDO: multilevel multidisciplinary dimensioning

This is the step in which we address optimal coupled dimensioning on all or a part of the modeling tree, while remaining within pre-established choices of system architectures and subsystems. Our goal is not only to be able to simultaneously optimize all the definitions of the system and subsystems for a given objective and criteria, but also to provide exchange rates among the different airplane performances (cost, flight radius, take-off field length, noise, pollution emissions, etc.) or between airplane performances and the performances of basic technologies (permissible constraint level for materials, motor turbine inlet temperature, etc.). The purpose is to know how to adjust the component specifications rationally to all description levels.

Figure 1.3. Aerodynamic plane dimensioning

An example of multidisciplinary optimization is to determine the optimal relative wing thickness of a supersonic business plane. The aerodynamicist will tend to choose the thinnest possible wing because, from a purely aerodynamic point of view, the wave drag increases with the relative thickness squared. From the point of view of structure, a too thin thickness can lead to aeroelastic flutter problems caused by the flexibility of the wing. Figure 1.4 illustrates the effects of this flexibility by showing the camber variations between the start of cruising where the airplane has maximum load (upper plate) and end of cruising where the airplane has used up most of its fuel (lower plate).

Figure 1.4. Camber variation

Multidisciplinary optimization makes it possible to find the relative thickness minimizing the supersonic drag, while respecting aeroelasticity constraints in the flight envelope.

The major challenge is in developing methods providing a good implementation complexity/efficiency compromise that can apply to numerous classifications of design problems. A more detailed analysis of the multilevel breakdown presented in Figure 1.5 reveals important characteristics. Level 1 considers the global system and its global performances in terms of the mission (take-off weight, flight radius, noise in the airport zone, etc.). At this level, the system is described with a limited number of design variables (100) and the models used to calculate global performances are rapid execution (a few seconds to a few minutes of CPU time). At level 2, we enter the field of high performance scientific computation. At this level, the calculation codes used are, for example, sophisticated CFD codes solving Euler or Navier– Stokes equations with a finite element general model (including aeroelasticity) to describe structural behavior. The number of design variables is much greater (a few thousand) and the computation times necessary for analyses are longer, taking hours of CPU time. Level 3 corresponds to the numerical prototype and details analysis. In this schema, the MDO process must be considered mainly as the art of efficiently managing design parameters between disciplines and levels.

Figure 1.5. Multilevel breakdown

1.5. Elements of an MDO process

Each design process is different. Its complexity and hard spots are unknown a priori, therefore the MDO process must be easily reconfigurable. The breakdown must take industrial divisions into account. MDO breakdown strategies can depend on the application, and methods of sophisticated breakdown are not yet totally validated in terms of their scaling for real applications. The use of high reliability analyses in an environment that is easily reconfigurable is still a challenge. That is why the identification of high/low couplings, multilevel breakdown, and surrogate models are central in an efficient MDO process. In addition, the complete MDO process must be powerful and reliable at the algorithmic level and in its technological implementation. Efficient optimizers, insensitive to numerical noise, are necessary to the system and each discipline must use the most adapted and detailed optimization strategies. Finally, the MDO process must be able to extend to optimization in uncertainties.

The level of detail and reliability for each discipline are unknown a priori. Dimensioning physical phenomena and critical constraints must be identified and integrated in the design process. The critical design points must be identified as it progresses and the identification of necessary reliability levels must be done through sensitivity analyses of the MDO process. The efficiency of the analysis and optimization for each discipline, and automatic generation of calculation models from the definition model, are crucial for the effectiveness of the global MDO process. The MDO process must directly benefit from convergence accelerations from each discipline and a deep parametric formulation is necessary to improve process automation. Coupling between the disciplines must be maintained strictly as the necessary level; experience plans and surrogate models are vital in the multilevel optimization strategy. Monodisciplinary optimizations using high reliability models must take into account global constraints and objectives. The MDO process must be used to specify the objectives and constraints of monodisciplinary optimizations and must be able to manage the propagation of uncertainties to all levels.

Different calculation environments are used simultaneously, for example CAD tools are in Windows and high performance calculation codes run on parallel UNIX machines. The MDO process must be able to be deployed throughout the different operating systems and calculation environments. A real design generates a large volume of data; parametric models paired with surrogate approximations are necessary to generate syntheses for decision support. The same data must enable different levels of synthesis, from the design engineer to the program director. The interactive use of sensitivities and surrogates is necessary for the interactive response to “what if?” scenarios. Finally, advanced visualization tools are necessary to make the analysis of results easier, and to help in decision support.

For industrial deployment, each discipline is responsible for the choice of its methods, its technological choices and the quality and integrity of the data in its own field. The MDO team, relying on the MDO process, is in charge of coordinating the multidisciplinary compromises, but the autonomy of disciplines must be preserved. The MDO process must facilitate the design process and make it more efficient, which is why the MDO process must be built on existing technologies and integrate monodisciplinary processes with as little change as possible.

Figure 1.6. Schema of Dassault-Aviation MDO process

The MDO process being deployed at Dassault-Aviation is presented in Figure 1.6. It is a two-level approach in which the MDO is performed at the global level and is based on detailed monodisciplinary optimizations. The monodisciplinary optimizations are managed by global optimization in terms of objectives, constraints, and research areas. Monodisciplinary optimization results are stored in a database, enabling the construction of surrogate models used for global optimization.

This two-level approach is the result of a pragmatic evaluation. Dassault-Aviation has been using monodisciplinary optimization for a long time and the MDO process immediately benefited from existing technologies. In addition, since the global synthesis is performed on a platform that is already integrated, this two-level breakdown seems natural in our existing industrial processes. Since the MDO is carried out at system level, it is possible to use the “one shot” approach presented in Figure 1.7.

Figure 1.7. Diagram of global optimization

In this approach, the MDO consists in putting the pre-project platform in a restricted optimization environment. Starting now, the system solver will be considered as a “black box”. As previously mentioned, the analysis of a set of design variables is performed with short CPU time, and we use this characteristic to the maximum. The three major elements of this approach are:

– choosing optimization strategies based on level and disciplines;

– coupling between levels;

– post-processing for the analysis of the design point.

1.6. Choice of optimizers

The choice of optimizers is determined by the nature of the problem to be solved, and different types of algorithms are necessary. We can classify them into two main categories: deterministic algorithms, and stochastic algorithms. We will briefly present the advantages and disadvantages of the different methods and propose a few strategies for using them in the context of multilevel multidisciplinary optimization. The list is not meant to be exhaustive and only covers the methods used by Dassault-Aviation.

1.6.1. Deterministic algorithms

Here again, we can distinguish two major classifications: algorithms requiring the gradients of the objective function and constraints, and algorithms using only the values of the objective function and constraints.

In gradient methods, we use mainly internal points algorithms or sequential quadratic programming algorithms. The internal points method is very effective for heavily constrained problems. This method requires an initialization from a feasible point (i.e. satisfying all constraints) but, in return, it generates iterates that are all feasible. This characteristic is very useful in design because even if we stop the optimization loop before reaching the optimum, the solution obtained verifies the constraints and provides an acceptable solution for the engineer. Quadratic sequential programming methods are able to reach a feasible field, but respect of constraints is only ensured at algorithm convergence.

Gradient methods have many advantages. They are particularly effective in the speed of convergence and in their capacity to process a significant number of design variables and constraints. The Karush–Kuhn–Tucker conditions provide a criterion of convergence that is mathematically solid, and the Lagrange multipliers make a direct sensitivity analysis of the solution possible. These methods unfortunately do not always apply; one of their major drawbacks is the necessity of providing gradients with the objective function and constraints. It is obviously possible to use automatic differentiation tools to calculate gradients but this requires access to the application’s source code which must be written in C or Fortran. Gradient calculation, when it is possible, is often resource intensive, even with automatic differentiation, and the investment must generally be justified.

The use of approximate gradients by finite differences is not optimal because, beyond the cost of evaluation for the terms necessary for the finite differences, gradients that are too approximate can compromise the convergence factor of the algorithm or even lead to premature stops. The gradient methods are local methods; several optimizations with different initializations are necessary to estimate a global optimum. In addition, these methods can turn out to be sensitive to numerical noise and are not well adapted for multi-objective optimization. In the context of multilevel optimization, these methods are more specifically adapted to monodisciplinary optimizations such as hydraulics or structure analysis because of their effectiveness and the sensitivity analysis that is possible. For system optimization, these methods can be very effective when the gradients can easily be calculated (in the case of models based on response surfaces, for example).

The method called “trust region” is our algorithm of choice for methods without gradient. This direct research method remains effective in terms of CPU and is not sensitive to numerical noise. This method is well adapted to optimizing “black boxes”, and, because of its capacity in reaching the feasible domain, the initialization point may not verify constraints. There is also a mathematical convergence criterion. The “trust region” method builds a polynomial approximation of the objective function and constraints, and, because of this, it is only adapted for a small number of optimization variables (less than 50 in practice). This method is a local method and therefore requires different starting points in the search for a global optimum. The algorithm does not automatically provide a sensitivity analysis at the end of optimization and may, in the case of highly constrained problems, not find a feasible domain. This method cannot be used directly for a multi-objective optimization. The “trust regions” method is nevertheless well adapted to optimization of new systems.

1.6.2. Stochastic algorithms

Genetic algorithms (GA) are the only stochastic algorithms considered here. GAs can be used for optimization problems that cannot be differentiated and enable the multi-objective optimization. Since GAs only require the values of objective functions and constraints, they are well adapted for “black box” optimizations. GAs are global optimizers naturally exploring the design space. Since GAs have a slow convergence, and a large number of evaluations of objective functions and constraints is necessary, their CPU cost can be prohibitive. In practice, and particularly for heavily constrained problems, modifications of the crossing and mutation operators may be necessary to improve convergence. Because of their design space exploration property and their potential for multi-objective optimization, GAs are particularly well adapted to optimizing new systems.

1.7. Coupling between levels

In our two-level MDO process, coupling between levels is ensured for scale models. The idea is to obtain parametric models simple enough to be easily used in the system’s global system, but nevertheless precise enough to guarantee a correct result. We use mainly two types of scale models: models based on a mathematical approximation, or simplified physical models.

1.7.1. Reduction of mathematical models

The construction of a mathematical scale model is generally done in three steps. A design of experiment (DOE) is built first. For each point in the DOE, the values of interest functions are evaluated using finite analyses or local optimizations. Finally, a mathematical approximation model is built to obtain a response surface which is a parametric model in relation to sampling variables. Different approximation models can be used interpolating exactly or otherwise the calculated values. The steps in the construction of a scale model are summarized in Figure 1.8.

Figure 1.8. Construction of a mathematical scale model

Among the different approximation techniques, we prefer radial functions for their capacity to simultaneously interpolate a function and its derivatives. This point is very interesting since many of our finite analyses provide the functions and their derivatives. Techniques such as kriging or kernel methods can be used for the construction of the approximate model. Kriging helps in estimating the model’s approximation error but is more restrictive in terms of interpolating functions. In fact, they must make it possible to build a covariance matrix, excluding splines for example, which is not the case with kernel methods.

Figure 1.9 shows in a one-dimensional example the advantage of using the function’s derivatives in approximately building a mathematical approximation model. Interpolation functions are cubic spline radial based functions.

The exact function is the black curve. The three sampling points are the same in all cases. The three grey curves show the approximate function rebuilt by one of the following approaches: (i) considering the function values at the sampling points; (ii) considering the function and gradient values at the sampling points; and (iii) considering the function, gradient and second-order derivative values at the sampling points. We can see that the quality of the approximation improves each time information coming from function derivatives is used to build the approximation. It is nevertheless important to note that mathematical approximation models are interpolation models, and the precision of these models is not ensured when we use them for extrapolations outside of the sampling field, and that is their main failure.

Figure 1.9. Scale models using derivatives

1.7.2. Simplified physical models

In this case, the idea is to simplify the physical model to obtain a model that is simple enough to be used in global multidisciplinary optimization, while maintaining the physical behavior. The simplest formulation is an analytical formula. For example, for aerodynamic coefficients, we can use a “polar” type formula connecting the drag coefficient to the lift coefficient: ; the polar coefficients can be interpolated using a mathematical scale model.

1.8. Post-processing

Post-processing is a vital element in the MDO process; it is the direct link between the mathematical algorithm and design. The use of appropriate post-processing can help the engineer understand the important design variables, what is the impact of a specification, what the significant exchange rates are, what compromises between disciplines must be monitored, etc. Post-optimal analysis can be considered the first step in the synthesis of information needed to feed the decision process. Post-processing, or, in a more general way, the analysis of exchange rates, can be carried out in different ways depending on the information provided by the optimization process. We will present three commonly used examples.

1.8.1. Lagrange multipliers

Optimizers using gradients calculate the value of Lagrange multipliers for saturated constraints. Lagrange multipliers directly provide the exchange rate between the objective function and a saturated constraint, whether it is a boundary constraint on a design variable or more generally a non-linear constraint. If x* and are respectively the solution and Lagrange multipliers of the following optimization problem:

and if is the solution of the disrupted problem:

a Taylor development makes it possible to directly obtain the value of :

Lagrange multipliers prioritize saturated constraints and quantify their impact in the objective function. Since the sensitivity is evaluated with the help of a Taylor development, Lagrange multipliers provide a precise local analysis for small variations in saturated constraint values.

1.8.2. Pareto fronts

Pareto fronts come from multi-objective optimization. They are very useful in visualizing the compromises between two objectives or between an objective function and a constraint. The Pareto fronts are particularly useful in the case of optimization with genetic algorithms. They are, however, limited to cases where the number of objectives or constraints is limited because their construction and their interpretation become increasingly difficult when the space dimension of observations increases. Pareto fronts enable a global visualization of exchange rates but the interpretation is more qualitative than quantitative.

1.8.3. Self-organizing maps

Exploration properties of the genetic algorithm design space enable post-processing of self-organizing maps (SOMs). SOMs are a category of neural network based on unsupervised learning methods. They are used to map a real space, i.e. to study the distributions of data in a large space. Self-organizing maps are made up of a grid. In each node of the grid, there is a neuron that is linked to a reference vector, responsible for a zone in the data space. Reference vectors provide a discrete representation of the input space. They are positioned in such a way that they retain the topological form of the input space. In addition, they retain neighborhood relations in the grid, thus they enable easy indexing by using the coordinates in the grid. All these characteristics turn out to be very useful for the visualization of multidimensional data. Figure 1.10 shows an example of maps applied to the optimization problem of a supersonic business plane.

Figure 1.10. Example of self-organizing maps

The idea is to minimize the maximum take-off weight (MTOW), while respecting a minimum flight radius and minimum initial cruise altitude. The variables represented are respectively the flight radius, the initial cruising altitude, thrust, fuselage length, wing area, and maximum take-off weight. The white point shows the result of the optimization. As discussed previously, each cell corresponds to the same reference vector and we can therefore directly calculate the maps. We thus notice a strong correlation between fuselage length and the take-off weight as well as a strong correlation between initial cruising altitude and net thrust.

Self-organizing maps enable us to make a qualitative analysis. Their main use is to help the engineer develop an intuition for the existing relations between the different dimensioning parameters.

1.9. Conclusion

For the new subsonic Falcon plane projects, the MDO approach will be an additional tool to confront the new challenges that powerful and less expensive airplane design, or “green” planes drastically decreasing their sound and polluting emissions, represent. In the field of military airplanes, for “post-Rafale” projects, as well as for UAV, projects, the MDO will be part of cost control for better compromise management between traditional airplane performances and operations performances, by enabling an optimization integrating aerodynamic and discretion constraints, “pulpit” spaces for sensors, etc. The MDO tool is also useful in directing our studies upstream of research and development by specifically identifying the most useful technologies for development by analysis of exchange rates between their basic performances and global airplane performances.

The ideal vision is that, with the help of reference pre-projects for the main potential products identifying innovative solutions validated by technological programs or demonstration flights if necessary, the launch of a “correct the first time” industrial program will only require an adjustment to the final specifications by a “MDO” process that has been broken in!

1 Chapter written by Michel RAVACHOL.