Page 17 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

4 • Using ansys for finite element analysis
1.1.2 Basic concePt of nuMerical Methods

The basic concept of these methods is based on the idea of building a
complicated object with simple blocks, or, dividing a complicated object
into small and manageable pieces. Application of this simple idea can be
found everywhere in everyday life, as well as in engineering. Examples
include Lego (kids’ play), buildings, and approximation of the area
of a circle:

Element S i

Area of one triangle = S i = 1 R sinq i
2
2

N 1  2p 
S
2
Area of onecircle = ∑ i = R N sin   N  
= i 1 2
Where N = total number of triangles (elements). The first step of any
numerical procedure is discretization. This process divides the medium of
interest into a number of small subregions and nodes.
There are two common classes of numerical methods: finite differ-
ence methods and finite element methods (FEMs). With finite difference
methods, the differential equation is written for each node, and the deriv-
atives are replaced by difference equations. This approach results in a set
of simultaneous linear equations. Although finite difference methods are
easy to understand and employ in simple problems, they become difficult
to apply to problems with complex geometries or complex boundary
conditions. This situation is also true for problems with nonisotropic
properties. By contrast, FEM uses integral formulations, rather than dif-
ference equations to create a system of algebraic equations. Moreover,
an approximate continuous function is assumed to represent the solution
for each element. The complete solution is then generated by connect-
ing or assembling the individual solutions, allowing for continuity at the
interelemental boundaries.
Thus, FEM is a numerical analysis technique for obtaining approxi-
mate solutions to a wide variety of engineering problems.