Abstract
This chapter presents the details of formulating the solution to the steady-state heat conduction problem. The approach is general, however, and by redefining the physical quantities involved the formulation is equally applicable to other problems involving the Poisson equation. The physical behavior governing a variety of problems in engineering can be described as scalar field problems. That is, where a scalar quantity varies over a continuum. One usually needs to compute the value of the scalar quantity, its gradient, and sometimes its integral over the solution domain. Typical applications of scalar fields include: electrical conduction, heat transfer, irrotational fluid flow, magnetostatics, seepage in porous media, and torsion stress analysis. Often these problems are governed by the well known Laplace and Poisson differential equations. The analytic solution of these equations in 2D and 3D field problems can present a formidable task, especially in the case where there are complex boundary conditions and irregularly shaped regions. The finite element formulation of this class of problems by using Galerkin or variational methods has proven to be a very effective and versatile approach to the solution. Previous difficulties associated with irregular geometry and complex boundary conditions are virtually eliminated.
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