Coupled Boundary/Finite Element Solution of Thermal and Electrodynamic Problems in Materials Processing

"This paper describes an on-going research on developing coupled boundary element and finite element algorithms for the solution of thermal and electrodynamic problems relating to materials processing. The coupled methods are uniquely suitable for field problems involving composite regions, in which detailed information on field distributions within one particular region is of more interest and yet the solution can be obtained only if the effects of other regions are all considered. Problems of this type include heat transfer in multiple regions and also magnetic induction involving an infinite boundary. The hybrid boundary and finite element formulation, algorithm development and the applications are discussed. Two examples are given to demonstrate the capability and accuracy of the coupled algorithm for heat transfer problems. Various treatments of corner discontinuity are discussed. It is found that the double flux treatment is an ideal choice amongst those tested as it is easy to implement and also gives the most accurate solution. 1. INTRODUCTIONElectrodynamic and transport problems occur frequently in materials processing systems. Accurate solution of these problems is of great importance not only for our fundamental understanding of physical principles governing these phenomena in the processing systems but also for optimization of the existing processes as well as development of new processes. For this purpose, many numerical algorithms have been developed. These computational schemes in general can be classified into three categories, that is, the finite element and finite difference schemes and the boundary element scheme. The former two are domain based numerical algorithms while the latter the boundary-based algorithm.The finite element method, which is based on the variational calculus, is a very useful and yet popular engineering numerical method for boundary value problems defined over irregular boundaries. The use of the method requires the discretization of the entire domain. It can be applied to solve both linear and nonlinear problems, but is particularly powerful for nonlinear problems that involve the field variables defined within the computational domain. On the other hand, the boundary element method, with its root in the Green's function method, is very efficient for linear boundary value problems or nonlinear problems that involve field variables only along the boundaries. The primary advantage of the boundary element method lies in the fact that only the boundaries of the computational domain need to be discretized, and hence the dimension of the problem is reduced by one, compared with the finite element method."