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 tractable mathematical 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. In practice, 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-based models 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 the table 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.

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 the system 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: