The number of possible solutions rapidly approaches infinity and the probability of randomly finding the right solution to try approaches zero. We are forced to abandon trial-and-error and seek a better problem-solving method. For people in STEM (science, technology, engineering and math) fields, the expression rapidly transforms into "If at first you don't succeed, simplify, simplify again."

If the system we are working with is too complex to analyze, we analyze a simpler system instead. If a problem is too difficult to solve, we solve a simpler problem instead. This does not result in the answer, but it does result in an answer. This answer may not be perfect, but often it's enough.

By far, the most famous and humorous example of simplification in engineering analysis is the spherical cow. I have never seen a spherical cow, but I can assume that my cow (or farm animal of choice) is spherical and do some calculations in a matter of minutes and get a reasonable answer to my problem. Or I can go out into a field, take measurements of a cow, use those measurements to create a solid model of my cow, import the cow geometry into a finite element analysis package, and then analyze the cow. If the cow-related system is relatively simple, this might take me a few days. The answers that I get probably won't be different enough to justify the field work.

We often assume that animals and people are made of water when doing heat transfer analysis. After all, muscle tissue is 75 percent water. Blood is 83 percent water. The difference in thermal conductivity between tissue and water is small, so it's a pretty good assumption. We like to assume that friction doesn't exist, that entropy doesn't increase, and that temperature is constant. None of these things are true, but their effects are often small enough to neglect at first. The trick is in knowing whether or not you really can neglect those things and in estimating how good your simplifications are. Computers can solve the system once you set it up, but we still need engineers with good judgment to make the decisions about how good the initial assumptions were and how good the final results are.

Simplifications in analysis often follow the 80/20 rule: You will spend 20 percent of your effort getting 80 percent of the answer. You will spend your last 80 percent of your effort getting the last 20 percent of the results. Sometimes this is necessary, but not usually. Ask yourself: How soon do you need an answer? The answer is usually "as soon as possible." In that case, the first 80 percent of the solution will help you direct your work, steering you away from random probability and towards a better world.

There is always the occasional individual who seems determined to bask in the complexity of the problem instead of solving it. A (potentially apocryphal) story is told about a researcher who was trying to model weather patterns. They claimed to have created a perfect model which contained no simplifications or assumptions. Unfortunately, the model was too large to run on any existing computer and it was uncertain if the model would ever yield results. I would prefer to know the short range weather forecast with a 15 percent error rate than not know anything at all. You can bask in the complexity of your problem for a few minutes, but don't wallow. Definitely do not use the complexity as an excuse to not do your work. Get out there and solve it! Ruthlessly rip away the complexity until you have something you can solve. Be undeterred by farm animals shaped like geometric primitives. Be creative. Be relentless. And try to have a little bit of fun with it if you can.

Once you have an answer, you can (and should!) go back and review the assumptions that you have made and start taking them back out. Remove the assumptions that are easy to put into the model. Remove the assumptions that give you the largest error. Work forward from an answer as you try to make your way towards the answer if you need it. Time, constantly improving technology, and discussions with colleagues and friends will help you.

There may come a time when we can get the answer to any problem immediately without any simplifications or assumptions, but that time is still in the future. Today we have to do the best we can.

Mary Kathryn Thompson, Ph.D., is an assistant professor in the Department of Civil and Environmental Engineering, the Korea Advanced Institute of Science and Technology. She can be reached at mkthompson@an.kaist.ac.kr. - Ed.

2008.08.11