An application of the finite element method to three dimensional heat conduction problems

Abstract

A finite element solution scheme for three-dimensional transient conduction heat transfer problems is devised and its validity is substantiated through application to representative problems. The governing matrix equation is first derived by applying two different methods, namely the variational approach and the Galerkin approach, and its interpretations are stated for the cases of steady state and transient conduction heat transfer. Using this model, the solution is reached when different types of finite elements are used where derivation of the element matrices and element load vectors are introduced. For the transient case, a specific cube problem is discussed for which the exact series solution and the finite element solutions are compared. On the basis of the comparisons and results obtained above, it is concluded that, given certain conditions, the finite element method is safely applicable to three-dimenisonal heat conduction problems.