UNIT - 2

FEM Procedure

2.1 FEM Procedure

• The finite element method (FEM) is the most widely used method for solving problems of engineering and mathematical models.
• Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
• The finite element method is a numerical procedure for obtaining solutions to boundary-value problems. The principle of the method is to replace an entire continuous domain by a number of subdomains in which the unknown function is represented by simple interpolation functions with unknown coefficients.
• Thus, the original boundary-value problem with an infinite number of degrees of freedom is converted into a problem with a finite number of degrees of freedom, or in other words, the solution of the whole system is approximated by a finite number of unknown coefficients. 
• Therefore, a finite element analysis of a boundary-value problem should include the following basic steps:-

1. Discretization or subdivision of the domain
2. Selection of the interpolation functions (to provide an approximation of the unknown solution within an element)
3. Formulation of the system of equations (also the major step in FEM. The typical Ritz variational and Galerkin methods can be used.)
4. Solution of the system of equations  (Once we have solved the system of equations, we can then compute the desired parameters and display the result in form of curves, plots, or color pictures, which are more meaningful and interpretable.)

• The first step and the last step are most relevant with the visualization process. These 2 steps are described in greater detail below.

Domain Discretization

• The discretization of the domain is the first and perhaps the most important step in any finite element analysis because the manner in which the domain is discretized will affect the computer storage requirements, the computation time, and the accuracy of the numerical results. The subdomains are usually referred to as the elements.
• For a 1D domain which is actually a straight of curved line, the elements are often short line segments interconnected to form the original line [Fig2 (a)]. For a 2D domain, the elements are usually small triangles and rectangles [Fig2 (b)].
• The rectangular elements are, of course, best suited for discretizing rectangular regions, while the triangular ones can be used for irregular regions. In a 3D solution, the domain may be subdivided into tetrahedral, triangular prisms, or rectangular bricks [Fig2(c)], among which the tetrahedral are the simplest and best suited for arbitrary-volume domains.

Fig.-1 Basic finite elements. (a) 1D (b) 2D (c) 3D

• Note that the linear line segments, triangles, and tetrahedral are the basic one-, two-, and three-dimensional elements. Figure3 shows the finite element discretization of a 2- and 3- dimensional domain.

Fig.2 - Examples of finite element discretization

(a) 2-D with triangular elements

(b) 3-D with tetrahedral elements

Solution of the system of equations-

• Once we have solved the system of equations, we can then compute the desired parameters and display the result in form of curves, plots, or color pictures, which are more meaningful and interpretable. This final stage, often referred to as post-processing, can also be separated completely from the other steps.

2.2 Derivation of FEM equations by variation principle polynomials

Variational method- principles

Note:

the presence of summetric coeffecient matrics for baove eqn. is one of the primary merits to this approch

natural and contrived – variational principles

Natural:

1. Variational forms which aries from physical aspects of problem itself.
2. Eg- mimnimum potential energy- equilibrium mechanical system .

Contrived-

1. Lagrance multipliers:-extending number of unknown by addition of variable.
2. Least square problems- procedures imposiong higher degree of continuity requirment.

2.3 Concept of shape functions-

• The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions. In this work linear shape functions are used.
• In FEM analysis, the displacement model be assume the variation of displacement within the element since the two variation of displacement are not known.
• But in higher engineering mathematics analytical solution of some problem is either not known or difficult to find out. In such cases we replaced that function by another function which is easy to solve mathematically that function is called as Shape function or Interpolation Function.

• Notes-

1. The magnitude of shape function at each node is unity.
2. Number of shape function is equal to number of nodes.
3. The sum of shape function is always unity.

• Methods of deriving shape functions-

1. Shape function using polynomials in Cartesian coordinate system.
2. Shape function using polynomials in natural coordinate system.
3. Shape function using Lagrange interpolation function in natural coordinate system.

2.4 Derivation for linear simplex element

1. shape function for two noded bar elemnent in (x,y) co-ordinates :- let consider two noded bar element of length L.   are the certesian co-ordinaters of nodes 1&2 respectively . are displacement of nodes 1and 2 u is the displacement of any point in x-directions

Fig.3

2.5 Need for integral forms

• The integral forms are most useful when dealing with macroscopic problems with high degrees of symmetry (e.g. spherical or axial symmetry; or, following on from comments below, a line/surface integrals where the field is either parallel or perpendicular to the line/surface element)
• Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves.
• To find the area between two curves defined by functions, integrate the difference of the functions.

2.6 Interpolation polynomials in global and local coordinates

Fig.4

The relationship between a global coordinates X & a loacal coordinates x.

Natural coordinates in 1-D

1. natural coordinates are basically local coordinates in a dimensionless farm.
2. Natural coordinates simplify numerical integration.
3. Rotation is lower case
4. The objective is for to be 1 at I & 1 at j .

Fig.5

2.7 Weighted residual Methods: Concept of weighted residual method

• The weighted residual method is employed to obtain approximate solutions to linear and nonlinear mostly non- structural problems whose characteristics are expressed in terms of differential equations.
• In weighted residual method, efforts have been taken to minimize the error by some kinds of procedure. Based on the methods of minimizing the error from approximate solution, the weighted residual technique can be adopted I four methods such as-

1. Point collocation method
2. Sub-domain method
3. Least square method
4. Galerkin’s method

• In Practice, the weighted residual methods are mostly preferred for non-structural problems whereas for structural problems Rayleigh-ritz method is adopted.

Point collocation method-

• The residual R(x ,ai) is set equal to zero at n specific points X1,X2,X3…….Xn.
• The weighted function can be expressed as in this method.
• 

Sub-domain method

• In this method, the domain is subdivided into n sub-domains and the integral of the residual over each sub –domain is then required to be zero.
• That is, the weighted residual function is selected as unity (Wi=1) over the domain (i.e., some specific region).

Least square method-

• In this method, the integral of the weighted square of the residual over the domain is required to be a minimum.

Galerkin's method-

• In this method, the trial function y(x) itself is considered as the weighting function. The domain integral of the product of trial function with the residual is then set equal to zero.

2.8 Derivation of FEM equations by Galerkin's method

2.9 Solving cantilever beam problem by Galerkin’s approach

Cantilever beam problem by galerkin’s method

Fig. 6

Using galerkin’s method find the expression of a cantilever beam subjected to point load ‘p’ at the end having moment of inertia I . youngs modulus “E” & length of beam “L” .

Fig.7

2.10 Derivation of shape functions for CST triangular elements

Fig.8

2.11 Shape functions for rectangular element

2.12 Shape functions for quadrilateral element

For 8 noded quadratic element .

                                                    Fig.10

References-

(1) Finite element method with application in engineering by Chandrupatla & Belegundu, Pearson Publication.

(2)Finite Element Method Basics concept & Applications by Alawala

(3). Fundamental of Finite element Analysis by Devid V. hutton

(4)Finite element Methods is fundamentals an application in engineering by Chen Z