Short Course #2 Understanding the Finite Element Method
Bruce A. Finlayson, Instructor
This Short Course describes the finite element method as applied to transport processes using FEMLAB (Comsol Multiphysics) as a platform. Applications will include calculating primary and secondary potential distributions and the effect of flow on mass transfer rates at surfaces. When solving these problems, though, it is important to realize what the finite element method is doing – and not doing.
The basics of the finite element method include: trial function approximation (shape, degree of polynomial, when it matters, mesh refinement, and adaptive meshes). Iterative methods are used to solve large (10,000+) sets of linear or non-linear equations. These are described and strategies are given to insure a better chance for obtaining the solution. Techniques are described for finding parametric solutions as a parameter changes. Methods are given to estimate the error of the numerical solution and see how it depends upon the finite element choices (degree of polynomial, mesh refinement, etc.). The accuracy of primary variables (potential) and fluxes (current) is different, and it depends upon the type of finite element chosen.
To fully specify the problem, the physics must be known. The differential equations are specified, and possible boundary conditions are chosen. Illustrations will be given, using FEMLAB, which show how to modify the physics to give you the desired equations. The types of possible boundary conditions are limited to physical ones; and they are applied in conjunction with the differential equations. Sometimes differential equations are specified on a surface; or there is a surface layer that is very thin. Methods to detect and solve such problems are described. Free surfaces and moving domains are difficult to model, but methods exist. It is important to distinguish between laminar and turbulent flow. Sometimes a non-dimensional form of the equations is useful.
Strategies for detecting errors are described, as are strategies for approaching problems (start simple and build in the complications). Common pitfalls are described. One important problem arises for high speed flows or large Peclet numbers; and special methods have been developed for those cases. Considerations for choosing a commercial code are described.
About the Instructor
Bruce A. Finlayson received his BA and MS from Rice University, in 1961 and 1963, respectively. He received his PhD from the University of Minnesota in 1965. He spent two years at the U.S. Office of Naval Research. For 38 years, he taught at the University of Washington, a few of them joint in Applied Math; and for nine of those years, he was chair of Chemical Engineering. He is a member of the National Academy of Engineering. He received the Walker Award from AIChE; the Martin Award and Dow Lectureship, ASEE Ch.E. Division; and was AIChE President (2000) and Director (1992-94). He served on the Board on Chemical Science and Technology (NRC), Chemical Science Roundtable; and for six years on the Board of Petroleum Research Fund.
Prof. Finlayson is the author of four books; the first two give a concise description of the finite element method. His latest book, Introduction to Chemical Engineering Computing, was published in December 2005, by John A. Wiley & Sons; and it includes transport processes in 1D, 2D, and 3D using FEMLAB.
Prof. Finlayson has used the finite element method for 30 years. He wrote sections of the Finite Element Handbook; and has applied the method to polymer flow, heat transfer (including laser melting of metal), person swallowing, drug transfer and distribution, and ferrofluid flow.