Finite Element analysis
The following three methods are coming under numerical
solutions or approximate methods in FEA.
1.
Functional Approximation
2.
Finite difference method (FDM)
3.
Finite Element Method (FEM)
1.
Functional Approximation:
·
The Classical method such as Rayleigh –Ritz
methods (variational approach) and Galerkin methods (weighted residual methods)
are based on functional approximation but vary in their procedure for
evaluating the unknown parameters.
·
Rayleigh-Ritz method is useful for solving
complex structural problems, encountered in finite element analysis.
·
Weighted residual method is useful for solving
Non-structural problems.
2.
Finite Difference Method(FDM):
·
Finite difference method is useful for solving
heat transfer, fluid mechanics and structural mechanics problems. It is general
method. It is applicable to any phenomenon for which differential equation
along with the boundaries parallel to the coordinate axes.
·
The starting point in the finite difference
method is that the difference method is that the differential equation must be
known before solving. After that, the region is subdivided into a convenient
number of divisions. The differential equation is applied successively at the
various points of the subdivided region, a set of simultaneous equations are
generated which upon solving lead to approximate solution to the problem. This
is the essence of finite difference method.
·
This method is difficult to use when regions
have curved or irregular boundaries and it is difficult to write general
computer programs.
3.
Finite Element Method (FEM)or Finite Element
Analysis(FEA):
·
Finite element method is a numerical method for
solving problems of engineering and mathematical physics.
·
In this method, a body or a structure in which
the analysis to be carried out is subdivided into smaller elements of finite
dimensions called finite elements. Then the body is considered as an assemblage
of these elements connected at a finite number of joints called nodes or nodal
points. The properties of each type of finite element is obtained and assembled
together and solved as whole to get solution.
·
In other words, in the finite element method,
instead of solving the problem for the entire body in one operation, we
formulate the equation for each finite element and combine them to obtain the
solution of the whole body.
·
Finite element
method is used to solve physical problems involving complicated
geometrics, loading and material properties which cannot be solved by
analytical method. This method is extensively used in the field of structural
mechanics, fluid mechanics, heat transfer, mass transfer electric and magnetic
fields problems.
Based on application, the finite element
problems are classified as follows:
1.
Structural problems.
2.
Non-structural problems.
1. Structural problems:- In structural problems, displacement
at each nodal point is obtained. By using these displacement solutions, stress
and strain in each element can be calculated.
2. Non-structural problems:- In non-structural problems,
temperature or fluid pressure at each nodal point is obtained. By using these
values, properties such as heat flow, fluid flow etc., for each element can be
calculated.
