The document discusses the comparison between analytical and numerical methods, highlighting the advantages of each. It focuses on the finite element method (FEM) as a numerical approach used for solving differential equations related to physical problems, detailing a case study on stress analysis of a defective rotor. The case study illustrates how FEM was utilized to modify rotor geometry and reduce stress effectively within a short timeframe.

1. JK Lakshmipat University
Institute of Engineering and technology
Finite Element Method
Submitted to:
Prof R.K. Agrawal
Submitted by:
Bharat Sharma
2012BTechME008

2. Contents
• Analytical methods v/s Numerical method
• FEM
• FVM
• A Case study

3. Analytical Methods V/S
Numerical Methods
Find the root of f(x)=x−5
Analytical solution: f(x)=x−5=0
add +5 to both sides to get the answer x=5
Numerical solution:
Let's guess x=1: f(1)=1−5=−4. A negative number.
Let's guess x=6: f(6)=6−5= 1. A positive number.
And you get a range in which solution exists.

4. Why use Numerical method
• Can Solve Complex Problems with ease
• Example:- Higher Order Differential equations

5. Why use Analytical Method
• To validate results from Numerical Methods
• Analytical solutions can be obtained exactly with pencil and paper

6. What is FEM ?
• Finite Element Method
• A type of Numerical Method
• Used for solving a differential or integral equation.
• It has been applied to a number of physical problems, where the
governing differential equations are available.

7. A Simple Approach
Area ?

8. Area of geometry = A1 + A2 + A3 – A4
1
2
3
4
Finite Element method
It is not of course used to calculate areas, but
things like displacements, temperatures,
velocities, stresses and so on
Meshing

9. Is FVM related to FEM
• Finite volumetric method
• 3D version of FEM
• Most of the commercial analysis software work on the principle of
FVM

10. A Case Study

11. Stress Analysis of LP Rotor
Problem:-
• To determine stresses in the existing rotor
• To modify the defective Rotor geometry and limit the stress to the existing
value
Client : N.T.P.C. New Delhi Done by : IIT Madras

12. You can see that there is a small hole there. The
problem is that this small hole or defect which
has been created due to improper
manufacturing, it so happens that it is at a
place where the stresses are also high

13. Can’t you just manufacture another ?
• Rotor is very expensive, it is about 10 crores
• Lead time to manufacture this rotor is nearly 6 months
• A particular thermal power project worth about 3000 crores is going
to stand still.

14. And Rotor is very big

15. NTPC wanted to
• Check whether this manufactured rotor is tolerable
• If not then can we modify this

16. What IIT Did
• Compared stresses in good LP rotor and modified rotor

17. After some modifications and 4 iterations they were able to make the stresses equal in a matter of about 5 days
and they used FEM.