Anne Marsan: Software for Engineers

Currently I'm working on three different projects, and the one common component of all three is that they involve design optimization. Design optimization can be used in any discipline to find the "best" design for a particular system (the definition of "best" will vary from discipline to discipline, from system to system, and from one situation to another). Methodical, mathematic approaches to design optimization have been developed over the past 50 years, and their practical implementation, especially for large problems, has been made possible through the use of computers. A basic understanding of optimization is a good thing for software developers to have, especially those who are writing scientific and technical tools. This post is part one of a short primer on design optimization. I perused the book Principles of Optimal Design: Modeling and Computation by Panos Papalambros and Douglass Wilde before writing this, so their influence cannot be escaped.

Any system that can be modeled mathematically can be optimized. This means that disciplines as varied as engineering, physics, chemistry, biology, statistics, economics, finance, and others can make use of design optimization. Within engineering, systems such as automotive engines, building structures, HVAC systems, integrated circuits, etc. can be optimized. Two very simple systems that could be optimized are shown below. The first is a collection of 2D data points that are to be described by quadratic polynomial (a parabola). The second is a cantilevered beam that has a load applied at its tip. Different aspects of the beam can be described with mathematical equations, such as the geometry of the beam, the loading along the length of the beam, and the stresses in the beam at an point within it.

The next requirement for a design optimization problem is to identify a set of variables that describe design alternatives. These are parameters that the designer has control over, things that can be changed during the design process. For our data fitting problem above, the design variables would be the coefficients in the polynomial; the engineer can tweak their values to modify how the polynomial fits the data point. For the cantilevered beam problem, the designer has control over the length, height, and width of the beam (additionally, the beam material could be a design variable, but for this example we'll say that the material choice has been predetermined and the designer cannot change it).

Now we need to choose an objective function, expressed in terms of the design variables, that should be minimized. The objective function is frequently called a cost function. Note that if you ask any manager of a design optimization project what it is they are trying to minimize, they will say cost! It's up to the designers (and managers) to figure out what contributes to system cost and express it mathematically. In our beam example, clearly the larger the beam is, the more it will cost. But the beam has to perform a function, such as support a floor in a building, and therefore we can't minimize the size of the beam. Instead, a more typical objective would be to minimize beam deflection at its tip (if this beam is indeed supporting a floor in a building, it will make the building occupants feel a lot better if the floor doesn't sag when they walk over it). For our data fitting problem, the objective is to find the polynomial that best fits the data. A typical way of measuring the fit is to sum the squares of the distance of the given data points to the polynomial.

Designers next need to identify a set of constraints, expressed in terms of the design variables, that must be satisfied in order for their system to be valid. These can be practical limits on design variables, or more complicated relationships that describe the function of the system. For instance, in our data fitting problem, perhaps we want to ensure that the peak of our parabola is at x greater than some value d. Typical constraints for a beam design problem might be that the material properties are fixed because a certain material has been selected, or the stresses need to be below the limits of the material.

Now we have a complete specification for our design optimization problem: variables, objective function, and constraints. The formal way of expressing an optimization problem is:

minimize f(x)
s.t. (subject to)
g(x) < 0
h(x) = 0

The final step is to solve the problem, which is of course easier said than done! I will discuss this in the next installment.

But before I close, I want to point out that modeling a system using mathematical equations can itself be a challenging problem. In engineering, we spend much of our bachelors degree education learning how to do this, and it's something that doesn't always come easily, even to seasoned practitioners.