First Step of Finite Element Method!!!

 The first step to Finite Element Method (FEM) is " Problem Formulation.

Three fundamental steps of  Finite Element Method (FEM)

1. Divide the whole domain into parts i.e. subdomains or elements.                                                                  
2. Over each element (part) , seek an approximation to the solution as a linear function or combination of nodal values and approximate function, and drive the algebraic relations among the nodal values of the solution over each element.                                                                                        
3. Assemble the elements and obtain the solution to the whole.

Basic Features of  Finite Element Method (FEM)

1. FEM can handle any engineering science problem whose physical phenomenon can be described by equation of calculus basically in the form of normal differential equations with any irregular geometry, non-uniform loading & non-linear properties.           
2. FEM based general purpose software can be developed easily because it has a very systematic procedure.                                                                                                                                                                                                                   
3. The subdivision of a whole domain into parts has advantages:
• It allows accurate representation of complex geometry and inclusion of dissimilar material properties.
• Enable easy representation of the total solution by functions defined within each element that capture local effects.

Mathematical Model

A mathematical model can be broadly defined as a set of equations that expresses the essential features of a physical system in terms of variables that describe the system.

The mathematical models of physical phenomenon are often based on fundamental scientific laws of Physics such as " Conservation of Energy, Mass or Linear Momentum etc."

Mathematical Modelling

Mathematical abstract of physical phenomenon of any problem called "Mathematical Modelling"

Mathematical model has 3 types of solutions:

1. Exact Solution:

Mathematical model solution is represented in simple equation. (Purely Analytical)

                                                                y(i) = f [x(i)]

We will get real solution.

Aim: To find the real system behavior.

2. Numerical Solution:

Here , using numerical techniques the complex mathematical real system problem is converted into simple algebraic equations. 

The common methods are:

    (i) Finite Difference Method (FDM)

    (ii) Finite Volume Method (FVM)

    (iii) Boundary Element Method (BEM)

    (iv) Finite Element Method (FEM)

3. Stochastic Solution:

Those things which are going in probabilistic way. In this condition mathematical model is more and more complicated.

example : Monto-Carlo method