Explanation about Finite Element Method Basic Introduction

1. Finite Element Method
(FEM)
By,
Mr. M. Sasi Kumar
Assistant Professor
Department of Aeronautical Engineering
Kalaignarkarunanidhi Institute of Technology
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2. Syllabus
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3. Text Books & References
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4. Overview
• Objectives
• Methods of Engineering Analysis
• Methods under Numerical solutions
• What is FEM?
• Historical background
• General steps of the Finite Element Analysis
• Advantage, Disadvantage and Application of FEM/FEA
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5. Objectives
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6. Methods of Engineering analysis
Engineering analysis
Classical Methods Numerical methods
Experimental Analytical Energy
Boundary
Element
(BVM)
Finite
Difference
(FDM)
Finite
Element
(FEM)
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7. CONDITIONS EXPERIMENTAL ANALYTICAL NUMERICAL
Applicable
If physical prototype is
available
For simple problems like
cantilever, simply
supported beams, etc.
For Real life complex
problems
Result Accuracy
Cannot be believed blindly
and a minimum of 3 to 5
prototypes must be tested
100% Accurate results
Approximate but
acceptable solutions
Time consuming High High Low
Process cost
Expensive man power and
materials
Medium Low
Example
Strain gauge
measurements, Vibration
measurement, etc.
Theory of bending FEM, FDM, FVM, BEM
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8. Methods under Numerical solutions
1. Functional approximation :
• Rayleigh-Ritz methods (Variational approach) and Galerkin methods (Weighted
residual methods) are based on functional approximation but vary in their
procedure.
• Rayleigh-Ritz method is useful for solving complex structural problems.
• Weighted residual methods is useful for solving non-structural problems.
1. Functional approximation
2. Finite Difference Method (FDM)
3. Finite Element Method (FEM)
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9. 2. Finite Difference Method:
• It is numerical method for solving differential equations by approximating
derivatives with finite differences.
• It is useful for solving heat transfer fluid mechanics and structural mechanic
problems.
• It is applicable to any phenomenon for which differential equation along with
the boundary conditions are available.
3. Finite Element Method (FEM) or Finite Element Analysis
(FEA):
• In finite element method, instead of solving the problem for the entire body,
in one operation, we formulate the equations for each finite element and
combine them into the solution of the whole body.
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10. What is FEM?
• The finite element method is the most widely used method for solving problems of
engineering and mathematical models.
• It is a numerical method used to calculate approximate solutions to differential equation.
• In this method, a body or a structure is subdivided into smaller elements of finite
dimensional called finite elements. Then the body is considered as an assemblage of these
elements connected at a finite number of joints called ‘Nodes’. The properties of each type of
finite element is obtained and assembled together and solved as a whole to get solution.
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11. • FEM can be applied to both Structural and Non Structural problems
1. Structural problems:
In this type, displacements at each nodal point is obtained. By using this
Stress and strain in each element can be calculated
2. Non structural problems:
In this type, temperature or fluid pressure at each nodal point is obtained.
By using these values, properties such as heat flow, fluid flow etc., for each element
Can be obtained.
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12. Historical background
• 1940 - Basic idea of FEA were developed by aircraft engineers.
They used matrix methods
• 1945 - Hrennikoff - Field of structural engineering
• 1947 - Levy - Introduce force method
• 1953 - Levy – Stiffness method for analysis aircraft
structures
• 1954 - Argyris & Kelsey – Matrix structural analysis
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13. • 1960 - Clough - Introduced the term ‘finite element’ in plane stress
analysis.
• 1961 - Turner - Large deflection and thermal analysis
problem.
• 1962 - Gallagher - Material non-linearity problems.
• 1968 - Zinkiewicz - Visco elasticity problems.
• 1969 - Szabo, Lee - Weighted Residual Method for structural analysis
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14. • 1970 - Zinkiewicz, Parekh - Weighted Residual method for transient
field problems.
• 1970s - Applications extended to shell
bending, plate bending, heat
transfer analysis, fluid flow
analysis, etc.
• 1967 - Zinkiewicz - First FEM book -
“The Finite Element Method”.
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15. General Steps of the FEM/FEA
• STEP 1 : Discretization of Structure
• STEP 2 : Numbering of Nodes & Elements
• STEP 3 : Selection of Displacement Function
• STEP 4 : Define the Material Behavior
• STEP 5 : Derivation of Element stiffness matrix
• STEP 6 : Derivation of Global stiffness matrix
• STEP 7 : Applying Boundary Condition
• STEP 8 : Solution for the unknown displacements
• STEP 9 : Computation of the element strains and stresses
• STEP 10: Interpret the Results
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16. STEP 1 : Discretization of Structure
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17. STEP 1 : Discretization of Structure
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18. STEP 2 : Numbering of Nodes & Elements
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19. STEP 3 : Selection of Displacement Function
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20. STEP 4 : Define the Material Behavior
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21. STEP 5 : Derivation of Element stiffness matrix
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22. STEP 6 : Derivation of Global stiffness matrix
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23. STEP 7 : Applying Boundary Condition
From the global stiffness matrix [K] is a singular matrix because its determinant is
Equal to zero. In order to remove this singularity problem, certain boundary
Conditions are applied.
STEP 8 : Solution for the unknown displacements
These equations can be solved and unknown displacements {u} are calculated
by using Gaussian Elimination method or Gauss Seidal method
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24. • From the solution of displacement vector {u}, stress and strain value can be calculated.
STEP 9 : Computation of the element strains and stresses
STEP 10: Interpret the Results
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25. Advantages of FEM :
• FEM can handle irregular geometry in a convenient manner.
• It handles general load conditions without difficulty.
• Non-homogeneous materials can be handled easily.
• All the various types of boundary conditions are handled.
• Higher order elements may be implemented.
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26. Disadvantages of FEM :
• It requires a digital computer and fairly extensive software.
• It requires longer execution time compared with finite difference
method.
• Output result will vary considerably, when the body is modeled
with fine mesh when compared to body modeled with course
mesh.
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27. Applications of FEM :
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29. THANK YOU
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