Finite element method introduction

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 08-11-2021 1 M.SASI KUMAR, AP/AERO

2. Syllabus 08-11-2021 2 M.SASI KUMAR, AP/AERO

3. Text Books & References 08-11-2021 3 M.SASI KUMAR, AP/AERO

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 08-11-2021 4 M.SASI KUMAR, AP/AERO

5. Objectives 08-11-2021 5 M.SASI KUMAR, AP/AERO

6. Methods of Engineering analysis Engineering analysis Classical Methods Numerical methods Experimental Analytical Energy Boundary Element (BVM) Finite Difference (FDM) Finite Element (FEM) 08-11-2021 6 M.SASI KUMAR, AP/AERO

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 08-11-2021 7 M.SASI KUMAR, AP/AERO

8. Methods under Numerical solutions 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. Functional approximation Finite Difference Method (FDM) Finite Element Method (FEM) 08-11-2021 8 M.SASI KUMAR, AP/AERO

9. 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. 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. 08-11-2021 9 M.SASI KUMAR, AP/AERO

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. 08-11-2021 10 M.SASI KUMAR, AP/AERO

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. 08-11-2021 11 M.SASI KUMAR, AP/AERO

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 08-11-2021 12 M.SASI KUMAR, AP/AERO

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 08-11-2021 13 M.SASI KUMAR, AP/AERO

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”. 08-11-2021 14 M.SASI KUMAR, AP/AERO

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 08-11-2021 15 M.SASI KUMAR, AP/AERO

16. STEP 1 : Discretization of Structure 08-11-2021 16 M.SASI KUMAR, AP/AERO

17. STEP 1 : Discretization of Structure 08-11-2021 17 M.SASI KUMAR, AP/AERO

18. STEP 2 : Numbering of Nodes & Elements 08-11-2021 18 M.SASI KUMAR, AP/AERO

19. STEP 3 : Selection of Displacement Function 08-11-2021 19 M.SASI KUMAR, AP/AERO

20. STEP 4 : Define the Material Behavior 08-11-2021 20 M.SASI KUMAR, AP/AERO

21. STEP 5 : Derivation of Element stiffness matrix 08-11-2021 21 M.SASI KUMAR, AP/AERO

22. STEP 6 : Derivation of Global stiffness matrix 08-11-2021 22 M.SASI KUMAR, AP/AERO

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 08-11-2021 23 M.SASI KUMAR, AP/AERO

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 08-11-2021 24 M.SASI KUMAR, AP/AERO

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. 08-11-2021 25 M.SASI KUMAR, AP/AERO

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. 08-11-2021 26 M.SASI KUMAR, AP/AERO

27. Applications of FEM : 08-11-2021 27 M.SASI KUMAR, AP/AERO

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29. THANK YOU 08-11-2021 M.SASI KUMAR, AP/AERO 29