Multidisciplinary Design Optimization Supported by Knowledge Based Engineering

By Jaroslaw Sobieszczanski-Sobieski, Alan Morris and Michel van Tooren

Multidisciplinary Design Optimization supported by Knowledge Based Engineering supports engineers confronting this daunting and new design paradigm. It describes methodology for conducting a system design in a systematic and rigorous manner that supports human creativity to optimize the design objective(s) subject to constraints and uncertainties.  The material presented builds on decades of experience in Multidisciplinary Design Optimization (MDO) methods, progress in concurrent computing, and Knowledge Based Engineering (KBE) tools.

 Key features:

• Comprehensively covers MDO and is the only book to directly link this with KBE methods
• Provides a pathway through basic optimization methods to MDO methods
• Directly links design optimization methods to the massively concurrent computing technology
• Emphasizes real world engineering design practice in the application of optimization methods

Multidisciplinary Design Optimization supported by Knowledge Based Engineering is a one-stop-shop guide to the state-of-the-art tools in the MDO and KBE disciplines for systems design engineers and managers. Graduate or post-graduate students can use it to support their design courses, and researchers or developers of computer-aided design methods will find it useful as a wide-ranging reference.

• Language: English
• Publisher: Wiley
• Release date: Nov 6, 2015
• ISBN: 9781118897089

Book preview

1

Introduction

1.1 Background

Optimization formalizes the century’s old trial-and-error method which engineers have traditionally used to reason through the complexities of a design process where the merits and demerits of a large number of alternatives are evaluated and the best combination selected. Originally, this was done using hand-based calculation procedures but has evolved, in the modern design environment, into the application of sophisticated computer-based numerical algorithms. Whether done by hand calculation or by employing an advanced computer program, the underlying procedure is the same; the optimization process starts the search for a best solution from an initial guess and then iteratively seeks to find better alternatives. These alternative designs are generated by varying parameters that characterize the design problem. If the design is characterized by cost, these would be cost factors; if the design is to have minimum weight, structural parameters related to the volume of structural material would be used. These parameters are the design variables which are used as the defining terms in a design objective; for example, the cost of manufacture is defined in terms of economic cost factors; the total structural weight can be defined in terms of structural sizes. By the intelligent application of the trial-and-error process, a computer-based algorithm, or the engineer, evaluates the quality of the trial to decide on the next move. Employing a computer, the engineer can engage a numerical algorithmic process that brings the power of computational numerical methods into play which iteratively changes the values of the design variables to modify the numerical value(s) of the design objective(s) while adhering to the limitations on the design normally termed constraints. By proceeding in this manner, the algorithm is driving toward a design judged best for a given set of circumstances. While engineers naturally turn to computer methods to assist them in the design process, we should, nevertheless, not forget that the most innovative computer is the human brain and the best designs are always a result of the engineer thinking first and employing computers second.

In real world engineering where a large and complex system, for example, an aircraft, a ship, or a car, is being designed, a process involving trade-offs takes place both within disciplinary subsystem domains and across their boundaries. Optimization in this environment becomes multidisciplinary design optimization (MDO). The complexity of modern systems shows itself under a number of different headings: compositional, behavioral, modeling, and evaluative complexity. The compositional complexity relates to the high number of system elements in the design process and their connectivity; if we take into account manufacturing cost, structural mass, dynamic response, and so on, each of these interacts with each other and calls into play a wide range of associated software tools. The behavioral complexity comes from the many aspects that influence the behavior that the designer is looking for, or trying to avoid, and is well described by the adage that in a system everything affects everything. Modeling complexity is associated with the complex (physical) phenomena that need to be taken into account to analyze the system’s behavior such as major structural analysis programs, computational fluid dynamics software tools, and so on which also interact. Finally, evaluative complexity appears when conflicting design characteristics are aimed for and trade-offs are needed between disparate properties.

Many of the methods applied to design optimization originate from the world of operations research (OR) which aims at optimizing operations of existing systems while MDO extends the approach to the engineering system design process, explaining the D in MDO. However, as explained in Chapter 2, there is a long history to the development of optimization principles and methods that have migrated to the design environment from variety of mathematical sources. The totality of these inputs is made clear through the various chapters in this book.

MDO can be defined as an assemblage of methods, procedures, and algorithms for finding best designs measured against a set of specified criteria for complex engineering systems with interacting parts, whose behavior is governed by a number of coupled physical phenomena aligned with engineering disciplines. Such designs are brought to fruition by teams of engineers, often dispersed on a country or global scale, employing organization methods and processes that accommodate commercial realities which might involve human factors components, costs and profit considerations, market competitiveness, and so on. Within the design environment, uncertainties are always present, and handling them when employing optimizing methods is not always straightforward and currently a major research area. Coupled with the presence of uncertainties is the need to undertake reliability-based and robust (uncertainty tolerant) designs. It is in the resolution of this type of design problem, with its range of interactions and uncertainties, that MDO finds its application.

Knowledge-based engineering (KBE) aims at drawing together the knowledge required to construct an MDO system into a computer-based knowledge base which can be logically interrogated by an engineer. It supports those wishing to employ MDO methods by making knowledge directly available at each stage of the development and application of an MDO system—it cannot be expected that a designer is an expert in all aspects relating to this task. Currently, KBE tools are in a rapid state of development and as time passes will become directly linked with MDO in its successful support for generating optimized designs for complex products.

1.2 Aim of the Book

The aim of the book is to offer a basis for constructing a logical approach to the application and understanding of modern MDO methods and tools and provide a background to supporting MDO with KBE technology. This is an ambitious target, and it is not claimed the book gives a complete and totally comprehensive coverage of these major fields. Rather, it provides a door through which the reader is invited to step and after crossing the threshold absorb or possibly develop the ideas in these rapidly expanding areas. In essence, it provides a knowledge base that allows the reader to take advantage of this technology in engineering design. In the case of an inexperienced or new user of MDO/KBE technology, it represents a robust starting point. For an engineer experienced in the application of optimization tools for designing a product, we hope the book will give insight into a new set of optimization and optimization support tools for solving complex design problems.

In order to meet the book’s aim, we recognize the need to progress through the necessary background knowledge before launching into the complexities of the full MDO application. Before reaching the chapters devoted to multidisciplinary design, the book introduces and explains the basics of optimization and the method employed for single-discipline optimum design problems. Prior exposure to these basic optimization methods will assist the reader but is not a requirement as we start along the pathway to complex methods with a review of the necessary fundamentals. Readers familiar with the basics of optimization and optimization method may wish to pass by the earlier chapters. However, we all, from time to time, forget what we have previously learned, and in this situation, the early chapters can be viewed as a convenient aide-memoire that can be consulted when required. As regards prerequisite knowledge, we assume the reader is familiar with the vector and matrix calculus and the analysis methods commonly taught in undergraduate engineering courses.

Recent years have seen rapid development in computer technology leading to major increases in computer power and speed that have proved beneficial in general applications and for engineering design in particular. One development of particular importance in the field of MDO is massively concurrent data processing (MCDP) also popularly known as parallel computing. Therefore, throughout the book, we repeatedly point to the use of MCDP as an enabler for solving problems that, previously, were regarded as intractable.

Our objective is simple: to provide sufficient information for the reader to understand the basics of the MDO process rooted in the realities of engineering design practices and benefiting from the rapid advance in computer technology, to see how uncertainty can be incorporated, and to illustrate how KBE tools can give support to the implementation of an MDO design solution system.

1.3 The Engineer in the Loop

It is probably worthwhile to discuss the fact that no optimization process exists to take us out of the box set by the definition of the design space implicit in the initialization and the underlying design concept. In this context, we may note that optimization is always reductionist. For example, an aircraft optimization starting with a biplane could evolve into a monoplane with a low, mid, or high wing; but a biplane will not arise from a configuration initialized as monoplane. This underscores the importance of the initial design concept and why the engineer will remain the designer for the foreseeable future and MDO will remain his subordinate.

Once an initial design configuration has been selected, the engineer has the major task of setting up the optimization system to be used in the search for an improved design. It is tempting to think that the application of MDO methods is simply a question of selecting a method from the set of MDO recipes found in Chapter 8 supported by appropriate KBE tools as introduced in Chapter 9. This temptation should be resisted as complex design problems do not readily submit to being fitted into a set of preconceived methods. The engineer has, therefore, to come forward with a solution method that reflects the idiosyncrasies of the design problem. In configuring an MDO system, the reader should take account of the fact that the methods presented in this book are the best available and most often used in current mainstream applications of MDO. Our readers should recall that MDO is a major research field and, as a result, there is no doubt new methods will be developed in the future.

1.4 Chapter Contents

Most technical books are not read as a novel where it is essential to start at the beginning and proceed sequentially to the final chapter; rather, the reader selects those parts relevant to the technical issues being addressed. To assist the reader in making a judgment as to where relevant information can be found, we now describe what can be found in the other chapters of this book.

1.4.1 Chapter 2: Modern Design and Optimization

1.4.1.1 Aim

To introduce the role played by MDO/KBE methods in a modern design environment where products are complex and created by distributed design teams.

1.4.1.2 Outline

This chapter looks forward to what is to come in the ensuing chapters; it introduces the modern design environment and discusses the position and role to be played by MDO in this environment. It discusses the underlying realities of the design process then moves on to examine the role that optimization can play in achieving improved design solutions. Initially, this is done by focusing on optimal designs involving a single discipline and then moves on to discuss the multidisciplinary case. It also indicates that a role can be played by KBE tools in implementing MDO systems.

1.4.2 Chapter 3: Searching the Constrained Design Space

1.4.2.1 Aim

To introduce the fundamental mathematical principles on which the methods for solving optimization problems are based.

1.4.2.2 Outline

At the heart of any MDO system are the algorithms that guide the design process to an optimizing point, but before looking at them, we need to discuss and elaborate some of the characteristics and properties of the space within which an optimizing point lies. This chapter covers this broad topic area and can be considered as a precursor to later chapters which discuss optimizing strategies. The material presented is not comprehensive but covers the basic principles and methods that are used to solve constrained optimization problems. This includes the mathematical principles that underpin the application of optimization methods to solve design problems.

The Kuhn–Tucker constrained optimization conditions are developed together with the concepts of duality and dual bounding. Lagrange multipliers are introduced and linked to the concept of active and passive constraints. Readers familiar with the foundations of nonlinear optimization and optimization theory may wish to move directly to latter chapters. However, the chapter does include information which an engineer experienced in optimization methods might consider using as an aide-memoire.

1.4.3 Chapter 4: Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function

1.4.3.1 Aim

To describe the methods used in the solution of problems with a single-objective function.

1.4.3.2 Outline

The chapter supplies the reader with sufficient information to understand what the various mono-optimization methods and algorithms require in terms of gradients, update formulae, and so on. With this information, an engineer can make rational choices on the selection of appropriate optimization tools for use in a practical MDO design system. It does not attempt to provide a comprehensive set of methods covering the entire range of such software. The reader wanting to obtain a comprehensive description of the methods touched on should refer to the references found within the chapter. The chapter considers first unconstrained optimization problems with linear or nonlinear design objectives. It then moves on to review a range of constrained optimization solution methods. The concept of shadow prices is used to show how changes in the constraint limits can change the value of the optimized objective function. As with Chapter 3, an engineer experienced in employing optimization algorithms to solve engineering design problems may pass this chapter by but may want to employ it as a resource base.

1.4.4 Chapter 5: Guided Random Search and Network Techniques

1.4.4.1 Aim

To discuss the use of genetic algorithms and artificial neural nets in the solution of optimization problems.

1.4.4.2 Outline

There are a number of methods that attempt to find optimum designs using techniques that avoid following a specified search direction exploiting gradient or quasigradient information discussed in Chapter 4. The repetitious use of directional searches is replaced by a process which either exploits randomized variations in the design variables or avoids the direct variation of design variables altogether by using learning networks. The first group is called guided random search techniques, and this chapter uses genetic algorithms to represent this class of optimum seeking methods. The second group, introduced in the chapter, is learning-based methods which are trained to pick out optimal solutions to very complex problems. In covering this group, the chapter discusses network-based methods and specifically covers artificial neural networks. As with Chapter 4, it does not present the reader with a comprehensive treatment of these algorithms as there is extensive and comprehensive literature available. It gives the reader a sufficient understanding of how these work so that intelligent decisions can be made if these techniques are encountered in the development or use of an MDO system.

1.4.5 Chapter 6: Optimizing Multiple-Objective Function Problems

1.4.5.1 Aim

To describe the methods used in the solution of optimization problems with multiple-objective functions.

1.4.5.2 Outline

This chapter introduces the reader to methods that can be employed when a design has more than one objective function which is normal when real world design optimization problems are encountered. It indicates that the design engineer is confronted with a number of, possibly, conflicting design requirements that normally result in a need to undertake trade-off studies. The chapter covers Pareto-optimal solutions, the concept of the Pareto frontier, goal programming, weighted sum methods, and the application of the methods introduced in Chapter 5 to solve this type of complex design optimization problem.

1.4.6 Chapter 7: Sensitivity Analysis

1.4.6.1 Aim

To introduce the mathematical methods and procedures for generating sensitivities at both single and multiple levels for both complex and simple design problems.

1.4.6.2 Outline

At this stage, in the pathway to applying MDO methods in the solution of complex design problems involving several objective functions, sensitivity analysis emerges as a tool necessary for design in general and for optimization in particular. This chapter reviews the fundamentals of the sensitivity analysis based on analytic derivative methods, including the recently gaining attention method of computing the derivatives of the real-valued functions via complex numbers. It introduces a method for obtaining derivatives using adjoint methods as a means for reducing computational effort. Shadow prices, originally discussed in Chapter 4, are reintroduced in this chapter. The use of higher-order derivatives is considered, and the sensitivity of an optimum to the problem parameters is presented. A solution to the problem of the sensitivity of complex internally coupled systems is introduced.

1.4.7 Chapter 8: Multidisciplinary Design and Optimization Methods

1.4.7.1 Aim

To introduce and describe the methods that can be applied to the solution of problems where both the objective functions and the design constraints involve interacting disciplines or subsystems in the context of a large engineering optimization process conducted by a team that may be dispersed organizationally and geographically.

1.4.7.2 Outline

This chapter provides a detailed explanation of multidisciplinary design and optimization (MDO) and the methods for solving this class of optimization problems that represent the core topic of this book. It exploits the terminology, notation, and methods, introduced in the chapters preceding this chapter, augmenting them to support the discussion of MDO. The chapter describes how the large and difficult task of designing a complex engineering system can be decomposed into a set of smaller and simpler tasks that may be carried out concurrently by a geographically dispersed team with the aid of parallel computing technology and utilization of a formal data management. A review of a sample of methods applicable to engineering system optimization is then presented, contrasting the traditional sequential approaches that may be shown as leading to acceptable but suboptimal designs, with the MDO-based approach that has the potential for both reaching an optimum and reducing the cost and time of the design process. It is asserted that MDO supports design agility meaning an ability to quickly revise design decisions made early in the design process as required by new information obtained downstream. The chapter concludes with a summary of the key elements and features of MDO and assesses the current state of the art focusing on strength and weaknesses together with a forecast of future potential.

1.4.8 Chapter 9: KBE

1.4.8.1 Aim

To describe the role played by KBE methods in supporting the application of MDO methods.

1.4.8.2 Outline

This chapter discusses the fundamentals of KBE and illustrates how this technology can support and enable multidisciplinary design optimization of complex products. A definition of KBE is provided, and its application in engineering design, supporting the use of MDO, is discussed. The working principles and main features of KBE systems are described with particular focus on their embedded programming language. This language is the core element of any KBE system and allows the capture and reuse of the design knowledge necessary to model complex engineering products. Of particular importance in this book is the facility it offers to automate the preparation phase for the multidisciplinary analysis process. The main categories of design rules that can be embedded in a KBE application are described, and several examples are given that pinpoint the main differences between KBE and classical rule-based design system and, in particular, conventional CAD tools. Finally, a section is provided describing the major steps in the evolution of KBE and its current trend in the broader CAD panorama.

1.4.9 Chapter 10: Uncertainty-Based Multidisciplinary Design and Optimization

1.4.9.1 Aim

To introduce methods that allow MDO solution methods to incorporate uncertainties.

1.4.9.2 Outline

The scope of this chapter is to systematically introduce the uncertainty-based multidisciplinary design and optimization (UMDO) theory and present a concise introduction to the typical UMDO methods. The chapter specifically focuses on the fundamental theory and general UMDO approaches but does not include a comprehensive review of the state-of-the-art algorithms. The chapter is structured to first present the preliminaries of UMDO, including the basic concepts and the general process for solving UMDO problems. Second, the key steps of UMDO, including uncertainty analysis and optimization under uncertainty, are expounded. Finally, an example is presented to illustrate the application of UMDO methods.

1.4.10 Chapter 11: Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time

1.4.10.1 Aim

To describe methods that can be deployed to reduce the cost and time of an engineering design process based on the MDO methods.

1.4.10.2 Outline

This chapter reviews a variety of ways and means available for controlling and reducing the computational effort of both single-discipline and multidisciplinary design optimization. It takes into account the notion that computational effort may not be measured by a single metric. The chapter addresses the issues of speedup and cost reduction as scored by a number of different metrics that include the central processing unit (CPU) time, data transfer time, data storage, number of processors, and so on. Various techniques are presented that relate to these metrics in different ways and in varying degrees all utilizing the technology of parallel computing.

1.4.11 Appendix A: Implementation of KBE in Your MDO Case

1.4.11.1 A.1 Aim

The aim of the appendix is to outline the implementation of a KBE system and describe its role in supporting the creation of a working MDO system.

1.4.11.2 A.2 Outline

The appendix provides information that can assist in incorporating a multimodel generator (MMG) in a commercial or self-made framework. It covers the steps necessary to go from the stage where the design problem is analyzed to go the point where a working framework has been developed. This includes the capture of basic design knowledge, the codification of product knowledge, and the application of this to support the creation of a Design and Engineering Engine (DEE). The different stages are illustrated with simple examples.

1.4.12 Appendix B: Guide to Implementing an MDO System

1.4.12.1 B.1 Aim

To provide an overview of the basic structure and process necessary for implementing a working MDO system (a DEE).

1.4.12.2 B.2 Outline

This appendix addresses the problem of constructing a framework which can house a DEE and make it into a working reality. It approaches this from the viewpoint that a process is required which can handle, manipulate, and operate with data across the entire design spectrum from requirements definition to a design tool. It, therefore, focuses on the software and other requirements need to convert the basic MDO systems, introduced in Chapter 8, into a working system.

2

Modern Design and Optimization

2.1 Background to Chapter

This chapter sets the scene for the introduction of multidisciplinary design and optimization (MDO) methods and techniques in the design of modern products. It starts by exploring the nature of modern design then moves on to link this to the need for the application of optimizing methods which then leads on to the role played by MDO. We should emphasize that while, in the course of this book, we introduce and develop methods and models for MDO systems supporting the design of major engineering products, we are not advocating a rigid adherence to any of the methods and processes discussed. The key consideration in the selection or development of an appropriate design methodology is the nature of the design being addressed; it is always a question of horses for courses. It may be that a given design optimization problem is ideally suited to being solved by one of the methods to be presented in later chapters; but it may not be. In this latter situation, the reader may adopt a pick and mix approach taking components from several methods to generate an appropriate MDO system for the problem being confronted. Going farther down this pathway, the reader may decide to undertake a pick and add approach where some of the ideas presented are taken and added to by the introduction of new ideas specifically developed for the problem being addressed. This is an exciting path to go down as new ideas are required; however, if speed is of the essence, the engineer(s) adopting this approach needs to have well-honed arguments to avoid being publicly hair dried by senior management.

This chapter essentially sets the scene for what is to come in the ensuing chapters. We look at the modern design environment and discuss the position and role to be played by MDO in this environment. We start by looking at the underlying realities of the design process and then examine the role that optimization can play in finding an improved design.

2.2 Nature and Realities of Modern Design

The design of a large and complex engineering system such as an aircraft, an automobile, a ship, or a power plant may be described as a progression through a set of what-if questions and answers that engage a range of human, computing, and manufacturing resources, backed up by research and laboratory results and information. Underpinning this machinery is the need to find a solution to a set of design requirements, some of which will involve an optimum target, for example, that the weight or cost be minimized. In principle, the designer(s) must compile a design and computational architecture that is expected to create products that meet the requirements. Solving the design problem involves a sequential process employing a range of different analyses that create a set of trial designs whose performance can be compared with the required performance. This performance gap is then reduced through the sequential application of the design architecture until the gap closes. In an actual design environment, account has to be taken of the time and cost associated with the search. A design which is feasible but not optimal could very well be acceptable.

The team or teams involved in this design task may work in a single organization, but in today’s engineering environment, they are likely to be distributed across a number of cooperating organizations, some of which may be small and highly specialized. These teams work within a process that unfolds in time through a number of stages with increasing levels of complexity, as described in Section 2.3.1. MDO aspires to be the developer of the underlying methods and mathematical models employed in this process and provide the computing and data processing infrastructure. In achieving this objective, it must conform to the realities of the world in which engineers operate.

We now need to see how these realities map onto the MDO process which we do, in outline, under a number of key headings that make up the components in a working MDO system. These headings cover the multidisciplinary nature of the modern design process, introduce distributed design teams, and cover information growth and expansion as a design progresses down the time line. Some of the issues discussed in this section are addressed in greater detail in subsequent sections and in much finer detail in subsequent chapters. It should always be borne in mind that the type of formal methods being introduced in this book are focused on providing the design engineer with a computer-based architecture or infrastructure that supports, but does not replace, human ingenuity and creativity.

Multidisciplinary: The design of a major product in today’s world requires that individual disciplines interact and will not get disconnected. These include what might be called traditional disciplines, structures, aerodynamics, etc. but now encompass manufacturing, operations, disposal, and the associated economical business considerations. In normal practice, not all disciplines are taken into account to the same level of detail, and a key decision in setting out on an MDO design operation is what level of detail to consider for specific disciplines and, indeed, what to leave out.

Multiobjective Function: In the past, designers often tried to treat each design attribute as a separate function; today, attributes such as performance, quality, manufacturing cost, weight, safety, etc. must be grouped together in order to achieve a realistic optimum design and inform the designers about the trade-offs. In many cases, the sources of this information differ with respect to levels of fidelity, confidence, and completeness.

Multidisciplinary Multiteam Participation: The complexity of modern computer-based design support programs (such as FE, CFD, material design programs, etc.) means that the use of many of these tools has to be handled by separate specialists or specialist teams. These specialists expect to have authority in choosing their own tools and making design decisions in their domains as opposed to serving merely as analysts.

Distributed Design Team: Modern designs are usually the result of many teams, globally distributed, working on a common MDO problem. This team changes as the design advances; the depth of detail increases and is reflected in the composition of the team which normally grows from a relatively small group dominated by generalists into specialist groups, as discussed previously. The tools used in the design must also change as the level of complexity increases. At each stage of the design, information generated must be expeditiously available for use by all members of the design team(s). The participating specialist groups have authority to exercise judgment, use information from any source, intervene into the computational process, and override the computed information. An ability to view all results at all the cooperating sites is an essential capability with a distributed team. In the case of a major design, such as an aircraft, ship, or bridge, it is often not possible to keep all teams connected on the design variable level, and other means have to be used to avoid divergence. Systems engineering and program management techniques are required to cope with this issue. In practice, there is a considerable phase shift between requirement and solution updates across the teams.

Lead Time Domination: Competitive pressures always impose on a design team a need for speed in achieving a satisfactory design. Nevertheless, this has to be balanced against the requirement for the team to have opportunities to uncover new information late in the design process that may call for major revisions and a return to an earlier design. Acting on such information ought to be encouraged and facilitated so that the final design is of the highest quality. It is also important to have previous design solutions stored in a logical manner so that earlier best designs can be rapidly retrieved.

With these realities in mind, we can now move on to a discussion, once more in outline, of the major aspects of optimized design and how it has extended its scope over the last few decades.

2.3 Modern Design and Optimization

This section links the description of the design process given in Section 2.2 to the use and application of optimization methods to support the search for the best or an improved or at least a feasible design. In order to put this into perspective, we begin by looking at the early use of optimization in design. Before getting into modern applications of what is, essentially, an application of the calculus of variations, it is worth noting that this subject has a long history and some of the greatest mathematicians have been involved in its development. Many of the methods discussed in this book are based on this work but employ matrix methods and computer-based algorithms that were either unknown or unavailable to the earliest workers in the field; the interested reader might like to consult the classic work by Isaac Todhunter (1861).¹ The development of modern optimization methods for the solution of design problems has followed the same pathway as these early researchers; it started with problems involving a single discipline and a single design objective. After initial success, the developments moved on to take account of the added complexity that must be incorporated into this process in the design of an engineering product requiring the employment of many parts and engaging many disciplines. Before progressing to a discussion of the computational design optimization process, it is necessary to have an overview of the design task.

2.3.1 Overview of the Design Process

A design problem can be expressed as a list of requirements or a requirements specification. It is often useful to categorize requirements into three groups: functional requirements, performance requirements, and constraints. Functional requirements define what the design should do, performance requirements define how well it should do this, and the constraints define the barriers in the design space that should not be crossed for various reasons. These requirements are often constructed to reflect both business and design objectives and can be set by a senior management team to which the design teams are responsible. Because the design of a complex product cannot be fully defined at the beginning of the design process, some of the design requirements may not be complete and firm and represent a desire (the desirements) on the part of the senior team that may or may not be feasible or firmed up as the design advances down the time line. The MDO design process has to cope with this situation where a requirement is added or not firm and may be abandoned as the design progresses or is transformed into a firm requirement. Genuine uncertainty is a real problem that confronts the designers of major products, and we deal with this situation in Chapter 10.

From these requirements, the design team develops a set of design objectives for which it must seek minimum ² values. In the past, there would usually have been a single design objective, for example, requiring an aircraft structure to have a minimum weight. In modern design environments, it is likely to be an interconnected combination of a number of terms. Consider the total financial outlay involved in buying and using an automobile. This depends on the initial purchase price, the fuel consumption, the service cost, etc. These are all interconnected as reducing the fuel consumption might be achieved by a weight reduction program, but this could have a major impact on the manufacturing costs, particularly if new materials are introduced or expensive manufacturing steps are required to minimize material volume.

Thus, an attempt to create a design that reduces the total cost would require employing a design objective containing, at least, manufacturing and weight components in addition to the drivetrain fuel efficiency. This type of design problem has to be defined in terms of a set of parameters that are varied to allow a design to be generated that meets the requirements. These parameter variations are limited by the design constraints, specified by the requirements, resulting in a design space within which the design parameters can move subject to strict limitations.

The performance requirements are usually one of the main sources of the constraints as they set limits on the behavioral responses of the design which in turn set limits to the permitted variations of the design parameters. These behavioral responses are obtained as the outputs from a set of analysis modules. Thus, the deformations of a structure are found from the output of a structural analysis module, aerodynamic forces from an aerodynamics module, and so forth. These modules may employ anything from simple expressions for calculating the responses to large-scale and complex computational methods and may be interacting as components of a computational system representing the actual system being designed.

The previous paragraphs essentially defined a process that maps out a design framework whose architecture is universal and applies to all designs although its modules will differ to suit a specific application, that is, the same framework architecture setup for a ship design will be populated by different modules from that employed in automobile design. Within this framework, a design is defined as a set of variable parameters that are manipulated to search through a design space. This space is limited by the constraint barriers, and the search is to optimize a single or multiple design objectives. The process may, indeed, be broader than that discussed previously as the design may involve a set of designs, a product family, with variations on a common concept defined by values of the parameters. Although in practice purely monodisciplinary design problems do not exist, it is worth noting that engineers employ monodisciplinary optimization if a single discipline clearly dominates the analysis.

The design of a major product is usually a sequential process as the level of design complexity is so large that a design must progress down a time line where the level of complexity grows incrementally. It starts with an initial stage where the analysis modules employ relatively simple calculation procedures and then moves to an intermediate design stage where more complex analysis modules play a role, finishing with a phase where the most sophisticated computational analysis methods are brought into operation. These are not precisely defined phases of the process, and their boundaries may be blurred; they may also differ among the industrial companies and product lines. However, following the approach common to many industries, we define these terms as:

Phase 1: translates the requirements into an overall concept and a range of alternatives; extensive design trade-offs are examined including their dependence on the requirements.

Phase 2: starts with a single concept selected; it undertakes a comprehensive analysis and optimization coupled with any testing to acquire additional data and ends with a freeze on the major design variables.

Phase 3: involves the design of components at a detailed level including a specification for any off-the-shelf components—the design of the fabrication and manufacturing processes and associated tooling, equipment, and plants together with the testing of major components and final performance estimates.

In certain industries, particularly the aircraft industry, these phases are called conceptual, preliminary, and detailed design, respectively (see Torenbeek, 2013). While the previous classification has a long tradition, for the methodology presented in this book, it serves merely as a starting point and a background reference. The advent of contemporary massively concurrent computing has a potential to impact significantly the previous definitions and the description of the work in each of the phases. It is expected to enable entering into more detailed analysis sooner, keeping the design open to changes longer, and facilitate acting on the information acquired later in the process even if it implies a return upstream to revise decisions already made (later in the book, we shall see that capability referred to as the computational and design agility).

Whatever power of computing one has at one’s disposal, progression from conceptual to detailed design requires using a hierarchy of analysis modules that grow in complexity and detail as the design progresses down the time line, and questions can arise about the quality of the results generated at the early stages. As progress is made down the design time line, the level of detail increases as does the requirements list so that solutions valid for an initial requirements list may be invalidated when a new requirement enters at a later design stage. It is always to be hoped that conclusions drawn from earlier models are not contradicted by results from later analyses, but there is no guarantee that this will always be the case. The desire to bring the more sophisticated analysis modules into play as early as possible to avoid such conflicts is one of the major driving forces pushing the development of MDO methods. However, life is far from simple when a major design is being undertaken, and the application of MDO methods with high-fidelity analysis models is not a panacea, and engineering judgment and experience always have key roles.

2.3.2 Abstracting Design into a Mathematical Model

In Section 2.3.1, we outlined the design process which we wish to address and encapsulate in a program or set of programs that are able to generate a quality design. By quality, we mean that this design has the ability to perform satisfactorily in service and is suitable for its intended purpose. In order to talk about solving this type of problem, we need to introduce certain terms and concepts which are frequently referred to in the remaining chapters of this book. In essence, we are mapping the terms used in the discussion of the design process, from Section 2.3.1 into those used in the field of classical optimization. The mathematical details are not described here but covered in later chapters.

The design parameters discussed in Section 2.3.1 are now termed design variables as these are the variables we work with to alter the design. They enter as arguments in the functions evaluated in any of the analysis tasks supporting the MDO process. If these are differentiable with respect to the design variables, they are classified as continuous; otherwise, they are classified as discrete. In the category of the discrete variables, it is useful to recognize a subcategory of quasidiscrete variables that may appear as a continuous variable while being analyzed but can only be selected from a discrete set of commercially available gauges, for example, sheet metal thicknesses where the engineer must select values from a manufacturer’s catalog. In this situation, an approximate solution can be found by selecting the one gauge, in the proximity of the continuous solution, corresponding to the lowest feasible objective value. This approximation is usually adequate if the available set is dense; otherwise, the difference between the approximate continuous and discrete solutions may be substantial as shown in Fox (1971) where alternative approaches for finding a discrete solution are discussed.

In contrast to the continuous and quasidiscrete variables, discrete variables are not differentiable because their change translates into a qualitative change of the design and the selection of variables is made from a discrete list of items that can be mapped onto a sequential list of positive integers. For example, consider the number of engines in an aircraft or the bridge construction type being a truss or a suspension support. We return to this topic in Chapter 8.

The factors being optimized, previously referred to as design objectives, are now contained in an objective function which, from our previous discussion, can be a single term or a collection of terms, some of which may be in conflict requiring a process for conflict resolution. The design variables clearly span the design space which we introduced in the previous section, that is, there is no part of the design space which is not definable in terms of the design variables. The design space is not open but is constrained because a design has to satisfy certain constraints, defined by the set of requirements, which limit the domain of options available to a designer in generating new design solutions. This domain of available design options is known as the feasible region or domain.

The process of designing a new product is then a question of searching a design space for the best or optimal solution to a specific engineering problem. At this point, it is worth pausing to note that although we shall normally refer to the results of an optimization search as yielding an optimum design, this is often not the case in practice where engineers are seeking merely an improved or even just a feasible design.

A second set of quantities called state variables, behavior variables, or dependent variables are typical response quantities that are functions of the design variables. In the case of a weight optimization problem, the state variables might be the stresses or strains in the individual structural components or the displacement at specified points. State variables are, in the optimization world, quantities selected to represent the performance requirements and are subject to the design constraints. These might be single-sided limits, but not all state variables may be directly subject to such limits but may be indirectly controlled through being associated with another state variable which is subject to limits. For example, in one design problem, the temperature of a structure may be a state variable playing a direct role in the optimization process and subject to specified limits; for another design, it may be subject to an indirect limit because the stress levels induced in the structure by temperature variation are directly constrained but the temperature is not. In the first case, the temperature is a constrained variable and in the second case, the constraints are placed on the thermally induced stresses, and the temperature is controlled indirectly through these stress constraints.

Having described the features of a design optimization problem, the next question is, how is this problem solved? To achieve this, we need to have at our disposal a search engine that can sift the design options available in the feasible design space in a logical manner and attempt to