Chapter 2: Mathematical Modeling

### Introduction

As was discussed in the previous chapter, in order to apply optimization methods we must have a model to optimize. As we also mentioned, obtaining a good model of the design problem is the most important step in optimization. In this chapter we discuss some modeling concepts that can help you develop models which can successfully be optimized. We also discuss the formulation of objectives and constraints for some special situations. We look at how graphics can help us understand the nature of the design space and the model. We end with an example optimization of a heat pump.

### Physical Models vs. Experimental Models

Two types of models are often used with optimization methods: physical models and experimental models. Physical models are based on the underlying physical principles that govern the problem. Experimental models are based on models of experimental data. Some models contain both physical and experimental elements. We will discuss both types of models briefly.

#### Physical Models

Physical models can be either analytical or numerical in nature. For example, the Two-bar truss is an analytical, physical model. The equations are based on modeling the physical phenomena of stress, buckling stress and deflection. The equations are all closed form, analytical expressions. If we used numerical methods, such as the finite element method, to solve for the solution to the model, we would have a numerical, physical model.

#### Experimental Models

Experimental models are based on experimental data. A functional relationship for the data is proposed and fit to the data. If the fit is good, the model is retained; if not, a new relationship is used. For example if we wish to find the friction factor for a pipe, we could refer to the Moody chart, or use expressions based on a curve fit of the data.