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FINITE ELEMENT ANALYSIS

INTRODUCTION

The finite-element method originated from the needs for solving complex

elasticity, structural analysis problems in all fields of engineering and. Its

development can be traced back to the work by Alexander Hrennikoff (1941)

and Richard Courant (1942). While the approaches used by these pioneers are

dramatically different, they share one essential characteristic: mesh

discretization of a continuous domain into a set of discrete sub-domains.

Development of the finite element method began in earnest in the middle to late

1950s for airframe and structural analysis and picked up a lot of steam at the

University of Stuttgart through the work of John Argyris and at Berkeley

through the work of Ray W. Clough in the 1960s for use in civil engineering.,

and has since been generalized into a branch of applied mathematics for

numerical modeling of physical systems in a wide variety of engineering

disciplines, e.g., electromagnetism and fluid dynamics.

AIM

Aim of this paper is to femilarise students with FINITE ELEMENT ANALYSIS

FINITE ELEMENT METHOD (FEM)

is used for finding approximate solution of partial differential equations (PDE)

as well as of integral equations such as the heat transport equation. The solution

approach is based either on eliminating the differential equation completely

(steady state problems), or rendering the PDE into an equivalent ordinary

differential equation, which is then solved using standard techniques such as

finite differences, etc.

In solving partial differential equations, the primary challenge is to create an

equation that approximates the equation to be studied, but is numerically stable,

meaning that errors in the input data and intermediate calculations do not

accumulate and cause the resulting output to be meaningless. There are many

ways of doing this, all with advantages and disadvantages. The Finite Element

Method is a good choice for solving partial differential equations over complex

domains (like cars and oil pipelines), when the domain changes (as during a

solid state reaction with a moving boundary), or when the desired precision

varies over the entire domain

The finite difference method (FDM) is an alternative way for solving PDEs.

The differences between FEM and FDM are:

• The finite difference method is an approximation to the differential

equation; the finite element method is an approximation to its solution.

• The most attractive feature of the FEM is its ability to handle complex

geometries (and boundaries) with relative ease. While FDM in its basic

form is restricted to handle rectangular shapes and simple alterations

thereof, the handling of geometries in FEM is theoretically

straightforward.

• The most attractive feature of finite differences is that it can be very easy

to implement.

• There are several ways one could consider the FDM a special case of the

FEM approach. One might choose basis functions as either piecewise

constant functions or Dirac delta functions. In both approaches, the

approximations are defined on the entire domain, but need not be

continuous. Alternatively, one might define the function on a discrete

domain, with the result that the continuous differential operator no longer

makes sense, however this approach is not FEM.

• There are reasons to consider the mathematical foundation of the finite

element approximation more sound, for instance, because the quality of

the approximation between grid points is poor in FDM.

• The quality of a FEM approximation is often higher than in the

corresponding FDM approach, but this is extremely problem dependent

and several examples to the contrary can be provided.

Generally, FEM is the method of choice in all types of analysis in structural

mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics

of structures) while computational fluid dynamics (CFD) tends to use FDM or

other methods (e.g., finite volume method

Galerkin methods

In mathematics, in the area of numerical analysis, Galerkin methods are a

class of methods for converting an operator problems (such as a differential

equation) to a discrete problem. In principle, it is the equivalent of applying the

method of variation to a function space, by converting the equation to a weak

formulation. The approach is credited to the Russian mathematician Boris

Galerkin.

Rayleigh-Ritz method

In applied mathematics and mechanical engineering, the Rayleigh-Ritz method

is a widely used, classical method for the calculation of the natural vibration

frequency of a structure in the second or higher order. It is a direct variational

method in which the minimum of a functional defined on an normed linear

space is approximated by a linear combination elements from that space. This

method will yield solutions when an analytical form for the true solution may

be intractable.It is used for finding the approximate real resonant frequencies of

multi degree of freedom systems, such as spring mass systems or flywheels on a

shaft with varying cross section. It is an extension of Rayleigh's method. It can

also be used for finding buckling loads for columns, as well as more esoteric

uses.

FINITE ELEMENT ANALYSIS (FEA)

Is a computer simulation technique used in engineering analysis. It uses a

numerical technique called the finite element method (FEM). FEA consists of a

computer model of a material or design that is stressed and analyzed for specific

results. It is used in new product design, and existing product refinement. A

manufacturer is able to verify a proposed design will be able to perform to the

client's specifications prior to manufacturing or construction. Modifying an

existing product or structure is utilized to qualify the product or structure for a

new service condition. In case of structural failure, FEA may be used to help

determine the design modifications to meet the new condition.

.The finite element analysis was first developed in 1943 by Richard Courant,

who used the Ritz method of numerical analysis and minimization of variational

calculus to obtain approximate solutions to vibration systems. Shortly

thereafter, a paper published in 1956 [1]

established a broader definition of

numerical analysis. Development of the finite element method in structural

mechanics is usually based on an energy principle such as the virtual work

principle or the minimum total potential energy principle. By the early 70's,

FEA was limited to expensive mainframe computers generally owned by the

aeronautics, automotive, defense, and nuclear industries. Since the rapid decline

in the cost of computers and the phenomenal increase in computing power, FEA

has been developed to an incredible precision. Present day supercomputers are

now able to produce accurate results for all kinds of parameters.

HOW DOES FINITE ELEMENT ANALYSIS WORK?

FEA uses a complex system of points called nodes which make a grid called a

mesh (Fig 1).This mesh is programmed to contain the material and structural

properties which define how the structure will react to certain loading

conditions. Nodes are assigned at a certain density throughout the material

depending on the anticipated stress levels of a particular area. Regions which

will receive large amounts of stress usually have a higher node density than

those which experience little or no stress. Points of interest may consist of:

fracture point of previously tested material, fillets, corners, complex detail, and

high stress areas. The mesh acts like a spider web in that from each node, there

extends a mesh element to each of the adjacent nodes. This web of vectors is

what carries the material properties to the

object, creating many elements.

Fig 1: Mesh created on a structure.

Finite element analysis

In general, there are three phases in any computer-aided engineering task:

• Pre-processing – defining the finite element model and environmental

factors to be applied to it

• Analysis solver – solution of finite element model

• Post-processing of results using visualization tools

There are multiple loading conditions which may be applied to a system.

• Point, pressure, thermal, gravity, and centrifugal static loads

• Thermal loads from solution of heat transfer analysis

• Enforced displacements

• Heat flux and convection

• Point, pressure and gravity dynamic loads

An element has to face all the above loading condition which can be

successfully solve by FEA

Each FEA program may come with an element library, which is constructed

over a period of time. Some sample elements are:

• Rod elements

• Beam elements

• Plate/Shell/Composite elements

• She