Industry depends on physical phenomena. This much is easy to assert. Any manufacturing technique, any technology, any product is dependent on an interplay of various physical phenomena. Two questions arise — first, are mathematical models of such phenomena important, and, second, don't they already exist in textbooks anyway?
Importance of Models
A model is useful if it can predict the behavior of a certain object under certain conditions — whether it is an organ under a surgical knife, a bridge under heavy traffic, or a submarine under water. A model is useful, if it is easier, cheaper or more feasible to predict behavior using the model than it is to “just try it out”. For example, if we had an accurate model, would we predict the earthquake resistance of a new stadium we are about to build using the model, or would we just try it out?!
Though the case for using mathematical models may not always be as obvious as the one for earthquake resistance modeling, the industry has been using modeling more and more strongly and crucially in the past few decades. One reason is the coming into existance of powerful yet cheap computers that can “run models” (called simulation). Another is what use such models can be put to — cheaply trying out literally millions of design combinations to find one that optimally solves the problem at hand (called optimization).
Don't Models for Everything Already Exist?
Don't models for everything already exist? After all, physics and engineering textbooks are full of equations, and each equation is a particular model of some aspect of reality. Is there any need for building new models?
The answer is a resounding “Yes!”. Following are just some of the reasons that modeling is an important and relevant art to engineering and technology today:
A] Many phenomena in daily use in the industry have very old or understudied models. Sometimes they are just rules of thumb, sometimes received knowledge, and many times, just the attitude of “if it ain't broke, don't fix it!”. In these cases, no one can know exactly how to modify the product or process to suit new market needs, or even if the current methodology is optimal in any sense. Modeling can improve such situations tremendously.
B] New technologies are being continuously developed, based on phenomena which are not well understood. Better models of new phenomena, new materials and new technologies will enable faster research and development of products based on them.
C] Individual, well modeled phenomena, are sometimes put together in novel ways. Composite models which can predict the outcome of such complex interactions need to be built.
D] The traditional models, even if they exist, may be too hard to compute, even on a supercomputer. In this case, models which are less accurate, but very fast, can be very useful!
E] A traditional model may require a tremendous amount of data to predict accurately. For example, a detailed model of the Earth's weather would be one which computes the behavior of each molecule of air. But it is impossible to ever have the present location of each air molecule, so as to be able to predict future locations! Thus, models of “emergent” physics, which will model the phenomenon based on emergent properties (in the case of weather, properties such as pressure, wind velocity, etc. which are emergent — statistical — properties) need to be created.
In short, mathematical modeling is and will be increasingly relevant for various engineering and technology applications.