# 1.4: Modeling and Engineering Analysis

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Engineering analysis and modeling are intimately related. In fact, modeling is something we do daily as we solve the problems of every day life. In most engineering science courses, you will be developing your ability to construct mathematical models that serve as a basis for design decisions.

## What is a model?

Rather than answer this question directly, let's explore what a model is by doing some modeling. Imagine that you have been asked to describe a typical engineering course to a college friend who is not an engineering student. Surely, you have some model in your mind. Based upon your experience you might describe a typical engineering course as follows:

- Forty 50-minute classes (including three 50-minute exams).
- One 4-hour comprehensive final exam.
- Each class has 50 minutes of lecture interspersed with questions.
- Two (really hard) homework problems are assigned and collected daily.
- 10 to 20 pages of reading are assigned daily.
- Homework must be worked and presented following a prescribed format.
- Engineering classes are very hard and time consuming.

Now suppose your friend is really impressed and wants to know how hard the course is, specifically, how much time would you expect to spend weekly on one of these courses? Take a minute and predict how much time you would expect to spend.

- How much time did you predict? (16 hours, 24 hours, 32 hours, 40 hours?)
- How did you come up with your number?
- How did you model the courses to answer this question?

There are several ways to answer this question. One approach might be to estimate the time using the relationship \[ t = N \times (T+1) \nonumber \] where \(t\) is time spent on the course, \(N\) is the total number of credit hours, and \(T\) is the hours per week spent outside of class per credit hour. Assuming one 4-credit hour course and say 3 hours per week outside of class for every hour in class, you get a total of 16 hours per week. (If a typical Rose student takes 16 credit hours per quarter, how many hours a week should he or she devote to the course? C'mon try calculating how many hours this would be. Hmmm! This almost sounds like a full-time job.)

Before proceeding, we should recognize that both our models are useful representations of an engineering course. The first model, a *descriptive* model, is a list of attributes of an engineering course. The second model, a *predictive* model, is a mathematical formula to predict the time spent during the week on a course.

How can you have two different models? Why are they different? Don’t they both describe the same engineering course? Herein lies the basis for our definition of a model. True, both models describe the same engineering course, but they were developed for different purposes. They were constructed to answer different questions. In both cases, the modeler took an engineering course in the "real world" and constructed a model of an engineering course in the "model world."

As illustrated in Figure \(\PageIndex{1}\), the real world and the model world appear to be separated. In passing from the real world to the model world, the modeler must carefully apply Occam's razor to carve away all but the essential elements. Occam's razor is named after William of Occam, a fourteenth century English philosopher, who challenged philosophers to keep only the essential elements in any problem.\(^{1}\)

So what is a model? A **model** is a *purposeful* representation.\({2}\) The phrase "purposeful" is an essential part of the definition. As we have shown above, the very nature of a model depends upon its purpose — the reason it was constructed.

Reexamining our two models, we see that they each capture different features of an engineering course. In this way they are both *incomplete*.

Figure \(\PageIndex{1}\): Occam's Razor

All models are incomplete to some extent. The best models capture only features of the "real world" that are essential to accurately answer the questions being posed. The best engineers judiciously apply Occam's razor to develop models that provide answers within the constraints of the available resources.

## Types of Models

There are many different ways to classify models. Three useful classifications are shown in Figure \(\PageIndex{2}\). The first two classifications are fairly self explanatory; however, the third one may require some explanation.

Figure \(\PageIndex{2}\): Ways to Classify Models

Engineers often make use of mathematical and of physical models. With the advent of modern day computers, mathematical models have become increasingly powerful. **Mathematical models** are descriptions of real systems using mathematical expressions that can be used to predict their system behavior. **Physical models** are scale or full-size representations of real systems whose performance is usually measured in the laboratory. Physical models are often used to verify the predictions of mathematical models. In many important technological applications, such as the flow of liquids in a pipe or flow over an aircraft wing, physical models are the best way to predict the behavior of the real system.

Mathematical models can be further classified as either deterministic or stochastic. **Deterministic models** will give you the same answer each time if the inputs are unchanged. A stochastic model, on the other hand, has the element of chance built into it and is only repeatable on average. **Stochastic models** are commonly used in fault-analysis or reliability analysis where a sequence of events must occur for something to happen and each event has a certain probability of occurrence.

While the focus of this course is on developing deterministic, mathematical models of engineering systems, the important role of physical models should not be forgotten. Later courses will illustrate the use of physical models as an engineering analysis tool.

## Modeling Heuristics and Algorithms

What makes modeling and engineering problem solving such a challenge is that it is impossible to give you one set procedure that will always give a solution. Any set of steps that will always give you an answer is called an **algorithm**. Unfortunately, there are no algorithms for engineering problem solving.

Successful engineering problem solvers develop the necessary models by applying heuristics. A heuristic is "a plausible or reasonable approach that has often (but not necessarily always) proved to be useful; it is not guaranteed to be useful or to lead to a solution."\(^3\) Some people call this a rule of thumb.

**Consider:** When you ask your parents for money, do you rely on an algorithm or a heuristic?

During the course of your engineering education you should begin to collect heuristics. In this course we will introduce many heuristics without explicitly calling them by name; however, we will often reflect on why we did something in a particular problem. This is when you should be on the lookout for a useful heuristic.

One of the most useful heuristics we will be using in this course is the problem solving method described in Appendix A. Every problem you solve in this course should be solved using this approach. Occasionally you will find a problem that is so simple that the full approach is unnecessary. But in most cases, it should be followed.

### Sources

\(^1\) A. M. Starfield, K. A. Smith, and A. L. Bleloch. *How to Model It: Problems Solving For the Computer Age*. Burgess Publishing 1994, p. 19.

\(^2\) Ibid., p. 8.

\(^3\) Ibid., p. 21.