The Finite Element Method in Three Dimensions

This chapter explains the discretization process, element integration, assembly, and solution by showing how to apply the finite element method (FEM) in three spatial dimensions, using the transient heat conduction equation as an example. The objective of the chapter is to use the FEM to solve the partial differential equation. The element matrices MM, KM, and the element load vector F contain the shape functions or their derivatives, both expressed in terms of physical coordinates, which must be integrated over the volume represented by finite hexahedra. The integrals are computed numerically using Gauss‐Legendre quadrature. The script listed in the chapter computes the numerical solution to the 3D heat conduction equation. Comparison of the results computed with the FEM with the exact solution shows excellent agreement. Although visualization is performed using Matlab, more advanced plotting can be performed using software such as ParaView.