About Numerical Methods for Engineers

Numerical methods have been used to solve mathematical expressions of engineering and scientific problems for at least 4000 years (for some historical discussion you may wish to browse the Ethnomathematics Digital Library or the MacTutor History of Mathematics Archive from St. Andrews University).* Such methods apply numerical approximation in order to convert continuous mathematical problems (for example, determining the mechanical stress throughout a loaded truss) into systems of discrete equations that can be solved with sufficient accuracy by machine. Numerical methods provide a way for the engineer to translate the language of mathematics and physics into information that may be used to make engineering decisions. Often, this translation is implemented so that calculations may be done by machines (computers). The types of problems that you encounter as an engineer may involve a wide variety of mathematical phenomena, and hence it will benefit you to have an equally wide range of numerical methods with which to approach some of these problems. This course will provide you with an introduction to several of those numerical methods which you may then find opportunity to practice later in the curriculum. Let us consider a few examples in which numerical methods might offer a net benefit to the engineer. 1. A new problem results in a numerical expression involving logarithmic, polynomial, and trigonometric terms. Not only do you wish to find the zeros of this particular expression, but you need to find the zeros of 100,000 similar expressions per day. Hence, you need an automated procedure for doing such that can be implemented on a computer. 2. Consider the manufacture of an oddly shaped part out of very expensive materials. The manufacturing process involves cutting, heating, cooling, and bending of the part. In order to optimize the manufacturing process, you wish to understand the details of the heating, cooling, and shaping processes. Modeling of these processes requires consideration of transient heat flow in three dimensions and transient deformation in three dimensions; further, the part is an oddly-shaped domain. One approach to improve understanding of the process is to numerically simulate the heating, cooling, and deformation; this simulation involves the numerical solution of systems of partial differential equations. 3. You have an empirical model of a process that predicts temperature and oxygen content as a function of time. The empirical model involves four parameters that need to be estimated from measurements under a variety of conditions. There are several nonlinear regression or optimization algorithms which might be suitable for this task. As you progress through the curriculum, you will encounter many more problems which might benefit from analysis by numerical methods. Indeed, you may gain much insight by applying numerical methods to a variety of problems that you encounter later in the curriculum. This course will consist of ten units: an introduction to the numerical properties of machine computations; numerical differentiation; solution of non-linear equations; linear algebra (or the solution of systems of linear equations); interpolation; regression and optimization; numerical integration; numerical solution of ordinary differential equations; numerical solution of partial differential equations; and some miscellaneous numerical tools. Each unit is accompanied by lectures, readings, and exercises. *Terms of Use: Please respect the copyright and terms of use on the webpages above.