# Optimization in Practice with MATLAB for Engineering Students and Professionals (2015)

### PART 2

### USING OPTIMIZATION—THE ROAD MAP

### 4

### Analysis, Design, Optimization and Modeling

**4.1 Overview**

In this chapter, we introduce the important activities of analysis, design, optimization, and modeling. While we are all generally familiar with these terms, it is important to understand them in the context of how they relate to our optimization activities. We also develop an important understanding of how modeling system behavior is a distinct activity from modeling an optimization problem. These involve two distinct lines of expertise. In most of your courses (*e.g.*, structures, dynamics and finance) you focused on the former. In this book, we focus on the latter.

**4.2 Analysis, Design and Optimization**

*Analysis* and *optimization* are two activities integral to the process of design (Refs. [*1*, *2*, *3*, *4*, *5*]). In this chapter, as well as in the remainder of the book, we will primarily focus on these two activities in the context of engineering or systems design. However, the mathematical concepts, approaches, algorithms, and software tools that you will learn in this book could be applicable to diverse fields beyond engineering (*e.g.*, optimization of market portfolios) [*6*]. Although, from a technical perspective, *analysis* and *optimization* can be considered to be steps within the process of design, they can also be performed as stand-alone activities toward other end goals within the scope of academic research and industrial R&D. In this section, you will learn the definition of these activities and how they are related to each other. At the same time, you will have the opportunity to understand and appreciate the roles and responsibilities of the users, researchers, or engineers who execute these activities, whether individually or as a part of a team.

In order to help you better understand the practical essence of these activities, the following simple design example will be used throughout this section. You are designing a table that can carry as much weight (of objects placed on it) as possible, while fitting within a particular corner of your room. You are allowed a limited budget to construct the table.

*4.2.1 What Is Analysis?*

*Analysis* by itself is a broad term and generally refers to the process of dissecting a complex system, topic, phenomena, incident, or substance into smaller (and likely more tractable) parts to acquire a better understanding of it.*Engineering analysis**can* *be more specifically defined as the application* *of scientific principles and processes to reveal* *the properties and the state of a system, and also* *to understand the underlying physics driving the* *system behavior*. Now, in the case of the “table design problem,” you are primarily required to analyze how the different design decisions (*e.g.*, the table geometry and material) impact the capacity of the table to carry weight without breaking down. This capacity will be referred to as the *weight-holding capacity* of the table in the remainder of this chapter. Considering the problem in more technical detail, we realize that there are multiple modes of breakdown (*e.g.*, buckling of the table legs, fracturing of the table top, or failure of one or more joints). Hence, analyses of the the different modes of breakdown or failure becomes necessary, which again illustrates that *analysis* generallyinvolves decomposing a system/mechanism/problem into smaller parts to be studied. Now, if you are from a mechanical engineering or a related discipline, you might have already realized that the type of analysis needed for this problem is *structural analysis*. This realization brings us to another important aspect of *analysis*. The role of disciplinary knowledge in analysis and its implication in a research, industrial, or practical setting.

Analysis generally demands disciplinary knowledge pertinent to the system or mechanism being analyzed. More often than not, practical systems involve multiple disciplines. For example, designing an aircraft will require structural, aerodynamic, control, and propulsion analyses [*7*, *8*]. In a practical setting, the design team generally involves experts from different disciplines. Since disciplinary understanding may have reached different levels of maturity (with scientific progress), mathematical tools might be readily available for conducting certain types of analyses, thus alleviating the necessity for dedicated disciplinary expertise. On the other hand, in the case of mechanisms or phenomena that are not yet well understood, in-depth and fundamental analyses might be required -thereby demanding the involvement of a disciplinary expert. Now that the importance of analysis and the role of disciplinary knowledge therein has been established, the next question is - what are the basic approaches to *engineering analysis*?

*Analysis* is sometimes misinterpreted as a purely mathematical or theoretical activity. Analysis could involve “experiments - testing - mathematical inferencing” as an iterative process, especially for the following commonly-occurring scenarios:

•the underlying physics is not well understood;

•the fundamental disciplinary principles or theory do not directly apply due to geometrical complexities and inherent uncertainties; or

•lack of knowledge of the material properties (*e.g.*, thermodynamic or structural properties).

For example, in the case of the “table design problem,” ideally, the mathematical theory of solid mechanics (or mechanics or materials) can be used to fully analyze the system. However, in practice, you might not know the structural properties of the type of wood used for construction or the strength of the fasteners to be used at the joints. Appropriate experiments could be conducted in that case to fill this knowledge gap. Such scenarios are expected to be more common in designing new and innovative systems and, as such, a comprehensive understanding of the scope of “analysis” is important (for researchers/engineers) to effectively contribute to technological innovation.

*4.2.2 What Is Design?*

*Design*, in general terms, can be defined as the creation of a plan and/or strategy for constructing a physical system or process. *Engineering design* itself could be readily classified into multiple (often overlapping) categories based on the “object of design,” such as *product design*, *systems design*, *industrial design*, and *process design*. An entire dedicated book (or even sets of books) would be necessary to fully explain and demonstrate one of the “design” categories (*e.g.*, Ref. [*9*]). This book will primarily focus on engineering design in the context of mathematical optimization. From this standpoint, “engineering design” itself can be perceived as a multi-stage process, an example ofwhich is presented in Fig. *4.1*, which includes stages up to product delivery. In practice, the design process might not include all the stages shown in Fig. *4.1* or might include some additional unique stages.

Figure 4.1. Multi-stage Design Process

Considering the example of the “table design problem,” *conceptual design* will involve (i) planning the overall shape or configuration of the table (*e.g.*, round top or rectangular top; rigid or collapsible design), and (ii) choosing the class of material to be used (*e.g.*, metal, glass top, or wood construction). *Preliminary design* in this case will primarily involve determining the optimum dimensions of the different parts of the table and the material to be used for these parts. The objective of maximum weight-holding capacity and the constraints imposed by the budget and the size of the room will guide this preliminary design process. Detail design will finally involve determinimg the necessary modularity of the table and the joint mechanisms from the perspective of manufacturing.

It is important to note that, in practice, these design stages may not be distinctly defined in a linear fashion; significant overlap is common. Additionally, iterations among these stages could also become necessary. For example, if a feasible design or satisfactory value of the objectives could not be obtained in the *preliminary design* stage, you might need to go back and re-think your conceptual design.

*4.2.3 What Is Optimization?*

In the previous chapter, you were introduced to a philosophical and practical understanding of the role of optimization in academic research and industrial systems development. You were also presented the opportunity to appreciate the importance of learning optimization as an undergraduate student, a graduate student, or as a professional. In this section, we will take a step further into the world of optimization by looking at it from a quantitative design perspective without getting into the mathematical intricacies (which will be described in later chapters).

From the general standpoint of searching for the best available design, *optimization* can be defined as follows. *Mathematical optimization is the process of* *maximizing and/or minimizing one or more objectives* *without violating specified design constraints, by* *regulating a set of variable parameters that influence* *both the objectives and the design constraints.* It is important to realize that in order to apply mathematical optimization, you need to express the objective(s) and the design constraint(s) as quantitative functions of the variable parameters. These variable parameters are also known as design variables or decision variables.

To better understand the definition of optimization, we consider the preliminary design stage for the “table design problem.” For this problem, optimization of the table design can be articulated as the following process:

•maximize the weight-holding capacity of the table (*i.e.*, the total weight of objects that can be placed on the table), while

•satisfying (i) the geometrical constraints imposed by the size and shape of the room (where the table will be located) and (ii) the cost constraints imposed by the allowed budget for building the table,

•by varying the geometrical configuration (*e.g.*, length, breadth, and thickness of the table top) and the material used for constructing the table.

*4.2.4 Interdependence of Analysis, Design and Optimization*

Whether in academic research or in an industrial R&D setting, design, analysis, and optimization are generally undertaken as strongly interrelated activities toward developing better products and technologies. However, is no unique structure as to how they are related -the relational structure generally depends on the available human, computational, and physical resources and on the choices of decision-makers (*e.g.*, the design team leader). We use, as an example, one of the common relational structures to particularly focus on understanding how the activities interact with each other on a one-on-one basis. This structure is shown in Fig. *4.2*.

Figure 4.2. Relationship Between Design, Analysis and Optimization (A Representative Example)

*Design*, in general, is the enveloping process that includes *analysis* and *optimization* as sub-processes. Figure *4.2* expands on the preliminary design stage. The primary steps within this stage are (i) defining the design objectives, constraints, and variables, (ii) performing or using analysis, and (iii) performing optimization. In this case, *optimization* is the main driver for improving the preliminary design. However, in order to yield meaningful results in a time-efficient manner, *optimization* generally depends on a well-thought definition of the design objectives, constraints, and variables, which is also known as *effective problem* *formulation*. For the “table design problem,” this step calls for a clear definition of the following:

1.Which geometrical parameters will be considered as variables and what material choices are available for constructing the table?

2.What is the total budget/cost constraint, and what are the geometrical constraints imposed by the room shape and size?

3.What is the minimum needed weight-holding capacity of the table, and what is the desired maximum weight-holding capacity (if any)?

With a clear problem formulation, you are ready to perform optimization where the objective is to maximize the weight-holding capacity of the table. Optimization is a methodical process of changing the variable parameters (design variables) to determine better values of the objectives within the feasible design space (defined by the constraints). Thus, in order to implement optimization, you need a quantitative understanding of how the weight-holding capacity of the table is related to the table geometry and the table construction material. This necessity brings us to the relationship between *optimization* and *analysis*.

Analysis provides you with the knowledge or, more specifically, a mathematical model that accepts the variable values (defining a candidate design) and outputs the corresponding values of the objectives of interest and the values of constraint violations (if any). This knowledge is used by the optimization process in searching through the design space for the optimum results. In the case of the “table design problem,” a solid mechanics-based analysis of the table structure is required to provide a quantitative understanding of the stress distribution of the table as a function of the table geometry, table material, and the force acting on the table (attributed to the weight of the objects placed on it). This knowledge will, in turn, provide a strategy for quantifying the maximum weight-holding capacity for a table of a given geometry and material. Additionally, a cost analysis is necessary to estimate the total cost of constructing a table of any given geometry and material, which can include material costs, tool costs, utility costs, and labor costs. *Analysis* also provides the opportunity to investigate the performance of the final optimum design. Hence, in determining whether the optimum design satisfies the desired goals, input from both the optimization process and analysis is needed, as shown in Fig. *4.2*.

On the other hand, optimization could provide food for further analysis, especially in the context of practical optimization where the design process rarely stops at a single optimization run (instead generally requiring several iterations). For example, optimization could provide insight into which region of the design space (*i.e.*, a more focused range of designs than that initially allowed) show better promise in terms of objective values, thereby inciting more in-depth analysis of the system over that region of the design space. As a result, *optimization* and *analysis* are considered to be mutually contributive elements of the design process, and are linked with a bidirectional arrow in Fig. *4.2*.

Although *analysis* and *optimization* are two central elements of quantitative design, engineering design in itself could also involve qualitative elements that are often beyond the scope of *analysis* and *optimization*. Two typical qualitative elements include:

1.*Creativity- and aesthetics-driven design decisions*: For example, the overall configuration (*e.g.*, round top) and the color of the table surface could be conceived simply based on aesthetics and/or prior experience with tables. Although qualitative in nature, these decisions have important quantitative implications for the later stages of design where *analysis* and *optimization* are involved (*e.g.*, regulating the material options).

2.*Market-driven design decisions*: Design decisions can also be driven by an understanding of the market, especially in the case industrial and product design. Although, quantitative market analysis might be available in certaincases, such availability is not necessarily generic (*e.g.*, imagine the first Iphone or major changes to popular automobile models). A qualitative understanding of customer preferences or simply a clear vision for the product (*i.e.*, generating new customer preferences) is necessary to make design decisions in such cases.

**4.3 Modeling System Behavior and Modeling the Optimization Problem**

In the previous section, you learned about how *analysis* and *optimization* provides the necessary tools for performing engineering design. Now, in order to pursue such a quantitative design process, you require a tractablemathematical representation of the system analysis and of the optimization problem. This requirement brings us to the role of *modeling* in performing design through analysis and optimization. An introduction to this role is provided in this section. Further mathematical description of this role will be provided in later chapters of this book.

*4.3.1 Modeling System Behavior*

There are different definitions of a model in the context of systems design:

•Traditional definition: A model is a scaled fabricated version of a physical system.

•Simulation-oriented definition: A model is a symbolic device built to simulate and predict characteristics of the behavior of a system.

*Modeling* can be defined as *a process by which an engineer* *or a scientist translates the actual physical system* *under study into a mathematical model of the system*.

From the standpoint of design, one is generally concerned with quantifying certain parameters of interest through analysis and modeling. These parameters can be collectively termed as criteria functions. Thus, modeling system behavior boils down to developing a set of functions that represent the parameters of interest as functions of the variable parameters that can be controlled through design. Mathematically, modeling can be represented as:

(4.1) |

where the *P* is a parameter of interest that can be represented as a function of the design variables defined by the vector *X*. Therefore, the process of modeling system behavior is basically the process of determining the function *f*. Inpractice, *f* may not be a simple analytical function. It could be a collection of functions or a computational simulation [*10*].

Depending on the approach used to develop system behavior models, they can be classified into the following major categories:

1.*Physics-based Analytical Models*: These models are developed based on the physics of the system. If the physics of the system is defined by a set of differential equations, the analytical models represent the functional solution to those differential equations.

2.*Simulation-based Models*: These models generally leverage a discretized representation of the system in translating the system behavior to a set of algebraic equations that are solved using numerical techniques (by harnessing the number-crunching power of computers). Depending on modeling assumptions and the resolution of the discretization, the fidelity of these models can vary significantly. High-fidelity simulations, especially for complex systems, generally tend to be computationally expensive and more often than not require dedicated software for generating 3D geometries and performing the simulations (with limited portability). Examples of simulation-basedmodels include finite element models, finite volume models, and spectral analysis models.

3.*Surrogate Models*: Surrogate models are purely mathematical and/or statistical models with certain generic functional forms and coefficients that can be tuned. These models are trained (*i.e.*, the coefficients are tuned) using a set of input-output data (*i.e.*, [*P,X*] data) generated from a high fidelity source. The high-fidelity source could be comprised of experimental or simulations-based analysis. As a result, surrogate models by themselves lack any direct physical information of the system; however, they provide the advantage of being tractable, fast, and highly portable (generally not requiring any specialized software) (see Ref. [*11*]).

With the exception of surrogate models, the development of other types of models (*i.e.*, physics-based models) generally necessitates disciplinary knowledge, and that’s where your disciplinary courses come in handy in the design process. For example, in the case of the “table design problem,” your “Solid Mechanics,” “Mechanics of Materials,” or “Structures” course will prove helpful in developing a model of the maximum weight-holding capacity of thetable as a function of the table geometry and material.

At this point, you must be wondering about the challenges involved in designing real-life engineering systems (*e.g.*, an aircraft) where knowledge from multiple disciplines is required at a level which is unlikely to come from a single expert. Practical engineering design generally requires a team effort. Working with others to develop or use physics-based models that are outside of your field of expertise is a pervasive practice in industrial settings, where the expertise of one person is generally insufficient to model the global system. As a team contributor, you need to feel comfortable with the idea of understanding only part of the analysis.

*4.3.2 Modeling the Optimization Problem*

Modeling the optimization problem is also called problem formulation, a process that you will learn in more detail in later chapters. Essentially, it involves developing a clear definition of the design variables, design objectives, and design constraints. In this context, design variables and design constraints could be of different types (*e.g.*, continuous and discrete variables, and equality and inequality constraints). Problem formulation also involves defining the upper and lower bounds of the design variables, which are sometimes perceived as linear constraints.

Modeling the optimization problem is also strongly correlated with the choice of optimization algorithms. In other words, the class of optimization algorithms available to solve a problem depends on how that problem is formulated. This relationship often drives researchers to make important approximations in their problem formulation (*e.g.*, converting equality constraints to inequality constraints using a tolerance value) in order to leverage powerful algorithms that perform well in the absence of equality constraints.

*4.3.3 Interdependence of System Behavior Modeling and Optimization Modeling*

It is important to ensure that optimization problem formulation is coherent with the system behavior model. From Fig. *4.2* you can recall that analysis and optimization are interrelated. Therefore, if the optimization process demands a set of output parameters (criteria functions) to be estimated by the analysis model for a given set of input parameters (design variables), the analysis model should be able to provide the right outputs. Any discrepancy in this information exchange will crash the optimization process. In other words, the choice of objective and constraint functions and the choice of design variables should be made in view of the capabilities of the analysis model when accounting for the associated relationships. Alternatively, if the analysis model cannot meet the needs of the optimization formulation, new analysis models will need to be developed to represent the necessary functional relationships. When you put these issues in the context of practical design, where the analysis models are often developed by disciplinary experts and the optimization problem is formulated by a design expert (who may not have in-depth knowledge of the multiple disciplines involved), you will realize that there is often significant room for discrepancies. Effective communication is a necessary component of engineering design - essentially a collaborative effort.

There are also other practical considerations in harmonizing the optimization modeling and systems behavior modeling. For example, if you choose an optimization algorithm that requires a relatively high number of system evaluations, you would most likely require a fast (computationally-efficient) model of the system behavior to complete the optimization in a reasonable amount of time. Similarly, if the system behavior model is inherently highly nonlinear, you will need to formulate the optimization problem such that a nonlinear optimization algorithm can be used to solve the problem. To summarize, the characteristics of the optimization problem formulation and thesystem behavior model should be aligned with each other and with the overall objectives of the design effort.

**4.4 Summary**

This chapter introduced the key components of design optimization, namely analysis, design, modeling, and optimization. A holistic view of design, including the major activities involved in a design process (*e.g.*, preliminary design and detailed design), is provided. The importance of analysis, modeling, and optimization is then described in the context of engineering design. In doing so, this chapter also provided important insights into the relationship between these different components of design. The chapter ended with a bi-level perspective to modeling in the context of design. That is, modeling the behavior of the system being optimized, and modeling the optimization problem itself. This bi-level perspective essentially shows how modeling decisions in these two steps are distinct but strongly correlated, such as is terms of pertinent computational consequences. These thoughts will become ever clearer as we move along.

**4.5 Problems**

**4.1**Consider the table design problem discussed in this chapter and do the following:

1.Provide a hand-drawn sketch of a representative table (four-legged with rectangular table top) and clearly label and list the geometrical design variables. Be as comprehensive as possible.

2. Identify three differen types of analysis models (analytical, simulation-based, and surrogate models) that can be used to represent the weight-holding capacity of the table as a function of the geometrical variables. If you are not from the mechanical engineering, aerospace engineering, structural engineering or other related disciplines, feel free to discuss the problem with your peers who are from those disciplines in order to identify the analysis models.

**4.2**Compare and contrast the process of (i) conceptual design, (ii) preliminary design, and (iii) detailed design in general and in the development of an automobile. Feel free to refer to the appropriate literature for this purpose.

**4.3**Give an example of an original engineering design problem and clearly outline the objectives, the constraints, and the design variables.

**4.4**Outline two advantages and two limitations (each) of experiment-based analysis and simulation-based analysis.

**4.5**From your current understanding of design, analysis, and optimization, elaborate on the relationship of *analysis* and *optimization* in the context of *computational expense* (in 200-300 words).

**4.6**In 300-400 words, discuss the role of simulation software (*e.g.*, ANSYS, Abaqus, and Simulink) in engineering analysis and optimization, and compare and contrast the impact of licensed software and open-source software (*e.g.*, free codes) in performing engineering design in the 21st century.

**BIBLIOGRAPHY OF CHAPTER 4**

[1]M. N. Horenstein. *Design Concepts for* *Engineers*. Prentice Hall, 2010.

[2]M. Beaney. Conceptions of analysis in early analytic philosophy. *Acta Analytica*, pages 97-115, 2000.

[3]F. P. Brooks. *The Design of Design:* *Essays from a Computer Scientist*. Pearson Education, 2010.

[4]N. Cross, K. Dorst, and N. Roozenburg. Research in design thinking. Technical report, Delft, Netherlands: Delft University Press, 1992.

[5]A. Ertas and J. C. Jones. *The Engineering* *Design Process*. John Wiley and Sons, 1996.

[6]A. K. Dixit. *Optimization in Economic* *Theory*. Oxford University Press, 1990.

[7]I. M. Kroo, S. Altus, R. Braun, P. Gage, and I. Sobieski. Multidisciplinary optimization methods for aircraft preliminary design. Technical report, NASA Langley Technical Report Server, 1994.

[8]J. Sobieszczanski-Sobieski and R. T. Haftka. Multidisciplinary aerospace design optimization: Survey of recent developments. *Structural optimization*, 14(1):1-23, 1997.

[9]C. L. Dym, P. Little, and E. Orwin. *Engineering Design: A Project-Based Introduction*. John Wiley and Sons, 4th edition, 2013.

[10]J. T. Oden, T. Belytschko, T. J. R. Hughes, C. Johnson, D. Keyes, A. Laub, L. Petzold, D. Srolovitz, and S. Yip. Revolutionizing engineering science through simulation: A report of the national science foundation blue ribbon panel on simulation-based engineering science. *Arlington,* *VA: National Science Foundation*, 2006.

[11]G. J. P. Gigch. *System Design Modeling* *and Metamodeling*. Springer, 2014.