In this chapter, we discuss the general finite element analysis procedure for 2-D and 3-D linear scalar field problems. A scalar field problem is a problem whose primary unknown physical quantity is a scalar at any spatial location in the computational domain. We demonstrate the finite element analysis procedure by solving 2-D and 3-D steady state heat transfer problems. The steady state heat transfer problems are solved step by step in the same fashion as solving 1-D problems. Strong and weak forms of the governing equations are derived from the law of energy conservation and the method of weighted residuals. 2-D and 3-D finite element approximations and elements are described in detail. Numerical integration over multi-dimensional elements is also described in detail. Convergence considerations are discussed. MATLAB codes for solving these problems are presented.
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