The Finite Element Method for Flow and Heat Transfer Analysis* Evan Mitsoulis and John Vlachopoulos Department of Chemical Engineering McMaster University Hamilton, Ontario, Canada ABSTRACT The finite element method (FEM) is discussed and a specific formulation for flow problems is outlined that can encompass non-Newtonian in- elastic and viscoehtic fluids. A temperaturefor- mulation is also considered t h t can be applied for nonisothermal analyses of fluid flow. Some illus- trative examples of the application of the method in polymer processing are also presented. INTRODUCTION he finite element method (FEM) is a numeri- cal analysis technique for obtaining approxi- mate solutions to a wide variety of engineer- ing problems (Huebner & Thornton, 1982). Like the better known finite difference method (FDM), the FEM is used to solve the appropriate differen- tial equations that describe these problems. The term “finite element analysis” is sometimes used incorrectly to describe macroscopic mass and energy balances performed for a specified number T *Financial assistance from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. ADVANCES IN POLYMER TECHNOLOGY of control volumes (i.e., “elements” of the total volume). In fact there exist several commercially available computer packages for injection mold design with such misleading labels. This is an unfortunate situation and it is time that the differ- ences between these methods are clearly under- stood by everyone in the field of polymer technol- ogy. The term “finite element” should be used to describe numerical techniques based on variational or weighted residual principles (Schechter, 1967; Finlayson, 1972). The FEM was originally developed for stress analysis in complex airframe structures during the 1960s. Because of its diversity and flexibility as an analysis tool, it has since been extended and ap- plied to the broad field of continuum mechanics. 107