Saturday, October 2, 2010

GENERAL PROCEDURE OF FINITE ELEMENT METHOD

GENERAL PROCEDURE OF
FINITE ELEMENT METHOD

As stated in the previous chapter, following are the basics steps involved in the FEA
1.DISCRITISATION OF THE DOMAIN
The discretization of the domain or solution region into sub-regions(finite elements) is the first step in the finite element method .The process of discretization is essentially an exercise of an engineering judgement.The shapes,sizes,number and configuration of the body have to be chosen carefully so that ,the computational efforts needed for the solution are minimized.
In this step, following factors are considered chiefly:
BASIC ELEMENT SHAPE-
Mostly ,choice of the type of element is dictated by the geometry of the body and the number of independent spatial co-ordinates necessary to describe the system .The element may be one ,two and three dimensional
When the geometry, material properties and other parameters (stress, displacement, pressure and temperature) can be described in terms of only one spatial co-ordinate,we can use the one dimensional element . Although this element has a cross-sectional area ,it is generally shown schematically as a line segment .When configuration and other details of the problem can be described in terms of two independent spatial co-ordinates ,we can use the two dimensional elements.The basics element useful for the two dimensional analysis is the triangular element.Rectangular and parallelogram shaped elements or quadrilateral (combination of two or four triangular elements) element can also be used .
If the geometary,material properties and other parameters of the body can be described by three independent spatial co-ordinates, we can idealize the body by using three dimensional element.Tetrahedron element is the basics three dimensional element.Hexahedrogon can also be used advantageously.
The problems that possess axial symmetry like pistons, storage tanks, valves, rocket nozzles fall into this category. For the discritisation of the problem involving curved geometries finite elements with curved size are useful. The ability to model curved boundaries has been made possible by the addition of mid-side nodes.
Finite element with straight sides is known as linear elements, while those with curved sides are called higher order elements.

2.DISCRITISATION PROCESS-
(a) TYPE OF ELEMENTS- The type of element to be used will be evident from the physical problem itself ,
For example- If the problem involves analysis of a truss structure under a given set of load condition , the type of elements to be used for idealization is line or bar element
Similarly for stress analysis of short beam the elements are three-dimensional solid elements. Generally while selecting elements, following factors are considered : Degree of freedom needed ,ease with which the necessary equations can be modeled without approximation .
In certain problems where the body cannot be represented as assemblage of only one type of elements , we have to use two or more types of elements for idealization , for example : Aircraft wing analysis (fig) Here for the covers , rectangular shear panels are used and frame elements are used for flanges.
(b) SIZE OF ELEMENT:
Generally, small sized elements gives accurate final solution but here the computational time increases. Sometimes, we may have to use elements of different sizes in the same body. In the stress analysis of a plate with hole stress concentration is expected around hole. Therefore, finer mesh or smaller sized elements are used around the hole as compared to far away places.
Another characteristic, which is related to the size of element and affects the final solution, is ‘Aspect ratio’. Aspect ratio decreases shape of the element in the assemblage of elements. For two dimensional elements the aspect ratio is taken as the ratio of the largest dimension of element to the smallest dimension .Element with an aspect ratio nearly unity generally yield best results.
(c)LOCATION OF NODES:
If the body has no abrupt changes in the geometry ,material properties and external conditions (load,temperature etc.) the body can be divided into equal sub- divisions and hence spacing of the nodes can be uniform.On the other hand ,if there are any discontinuities in the problem ,nodes have to be introduced at these discontinuities .
In other words where there is a possibility of stress concentration ,node is introduced at that point on the body .
(d)NUMBER OF ELEMENTS:
The no elements to be chosen for idealization is related to accuracy desired,size of elements and the no. of degree of freedom involved .Although an increase in no. of elements generally means more accurate results,for any given problem,there will be a certain no. of elements beyond which the accuracy can not be improved any significant amount .If we increase no. of elements beyond this limits ,unnecessarily computation
time goes on increasing .Also we may not be able to the resulting matrices in the available computer memory .
(e)SIMPLIFICATION OFFERED BY PHYSICAL CONFIGURATION OF THE BODY:
If the configuration of the body as well as external conditions are symmetric
,we may only half of the body for the finite element idealization .The symmetry conditions , however have to be incorporated in the solution procedure .In certain cases,depending upon physical configuration of the body ,one quarter of the plate can be considered for analysis .
(f)FINITE REPRESENTATION OF INFINITE BODIES:
In some cases, where the boundaries of the body are not clearly defined,
For example dam, unit slice of the dam be considered for idealization and analyzed as a plane stain problem.

(g)NODE NUMBERING SCHEME:
Bandwidth (B)=(Maximum difference between the numbered degrees of freedom at the ends of any number +1)
This can be generalized as,:
Bandwidth (B)=(D+1)*f
Where,
D=maximum largest difference in the nodes numbers occurring for all elements of assemblage,
f=degrees of freedom at each node.
For good results ,bandwidth should be minimum and this can be achieved by minimizing D,which in turn can be achieved by numbering the nodes across the shortest dimension of the body .
The advances in the finite element analysis of large practical system have been made possible largely due to the banded nature of matrices .Further ,since most of the matrices .Further ,since most of the matrices involved are symmetric ,the demands on the computer storage can be substantially reduced by storing only the elements involved in the half bandwidth instead of storing whole matrix .
If we can minimize the bandwidth , the storage requirements as well as solution time can also be minimized .Since degrees of freedom is generally fixed for any given type of problem, bandwidth can be minimized by proper numbering scheme .
(2)INTERPOLATION POLYNOMIALS:
After dividing the body into elements ,the next step is to approximate the solution over each subregion by a simple function .The functions used to represent the behavior of the solution within an element are called interpolation functions/approximating functions or interpolating models .Mostly ,polynomial type of interpolation functions have been used because of following reasons:
a)It is to formulate and computerize the finite elements equations with polynomials type of interpolation functions.
b)It is possible to improve the accuracy of the results by increasing the order of the polynomials .Generally polynomials of infinite order corresponds to exact solution ,but in practice we take finite order polynomials as an approximation.