Page 13 - Finite Element Analysis with ANSYS Workbench

4 Chapter 1 Introduction

solutions only for simple problems as taught in undergraduate
courses where differential equations, boundary conditions and
geometries are not complicated. Most problems are limited to one
dimensional problems so that their governing equations can be
simplified from partial to ordinary differential equations.

(b) Numerical Method. If the differential equations,
boundary conditions and geometry of a given problem are
complicated, solving with analytical method is not feasible. We
need to find an approximate solution from a numerical method.
There are many numerical techniques for finding solutions to
complex problems. The popular techniques widely used are the
finite element and finite volume methods. This is mainly because
both techniques can handle problems with complex geometry
effectively.

Both the finite element and finite volume methods
transform the governing differential equations into algebraic
equations. In the process, many numerical techniques are needed.
The techniques include solving a large set of algebraic equations,
understanding concepts of the interpolation functions, determining
derivatives and integrations of functions numerically, etc. Details
of these techniques are taught in undergraduate numerical method
courses and can be found in many introductory numerical method
textbooks.

1.2 Finite Element Method

Because most of CAE commercial software packages
employ the finite element method to solve for solutions, we will
introduce the method in this section.

1.2.1 What is the Finite Element Method?
The finite element method is a numerical technique for
finding approximated solutions of problems in science and
engineering. These problems are governed by the three
components including differential equations, boundary conditions
and geometries.