**FINITE ELEMENT METHOD (FEM)**

**Introduction:**

Finite Element Method (FEM) is nothing but a numerical technique to get the approximate solution to the boundary value problems consisting of a partial differential equation and the boundary conditions. The Finite element Method converts these typical equations into a set of algebraic equations which are easy to solve.

**Basic Aspects of FEM:**

- It can model a structure of arbitrary geometry by making use of various pipes, shapes and sizes.
- Arbitrary boundary conditions can be handled.
- Structures can be modeled by using different types of elements. For example: Plate elements or shell elements together can be analyzed.
- High Speed Computers are required.
- Only one solution is obtained at one time.
- Division of structures requires some experience.

**Basic Aspects needed for FEM:**

- Interpolation and Representation of Graphs (Lagrange Interpolation Method)
- Principle of Virtual Work and Minimum Potential Energy (P.E.)
- Classical Rayleigh Ritz Method
- Elements of theory of elasticity
- There are 2 broad finite element procedures :
- Variational Procedure
- Method of Weighted Residuals.

Principal of Virtual work and minimum PE is a particular form of Variational Procedure only.

**1. ****Lagrange Interpolation Method:**

In FEM, Lagrange interpolation method is used for the polynomial interpolation.

The formula was named after Joseph Louis Lagrange who published it in 1795, though it was first published by Edward Waring in 1779 and rediscovered by Leonhard Euler.

Polynomial *U*(x) of degree ≤ n – 1 passes through n points (x_{1,}u_{1 }= *f(*x_{1})), (x_{2,}u_{2 }= *f(*x_{2})),….., (x_{n,}u_{n }= *f(*x_{n})) is given by :

*U*(x) = ∑ *U _{j }(x) *where j = 1 to n.

Or *U*(x) can be written explicitly as,

*U*(x) =

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