Introduction to the Finite Element Method
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1 AL- F A L A H S C H O O L O F E N G I N E E R I N G A N D T E C H N O L O G Y D E P A R T M E N T O F M E C H A N I C A L E N G I N E E R I N G M A S T E R O F T E C H N O L O G Y ( M A C H I N E D E S I G N ) 1 Introduction to the Finite Element Method
2 M. Tech (FEM - Syllabus) 2 Paper Code: M-847-A Theory: 100 Marks Sessional: 50 Marks
3 FEM - BOOKS 3 1. Introduction to Finite Elements in Engineering Analysis by Tirupathi R. Chandruipatala and Ashok R. Belagundu. Prentice Hall. 2. The Finite Element Method in Engineering by S.S.Rao, Peragamon Press, Oxford. 3. Finite Element Procedures, by Klaus Jurgen Bathi, Prentice Hall. 4. The Finite Element Method by Zienkiewicz published by Mc Graw Hill. 5. An Introduction to Finite Element Method by J.N. Reddy published by Mc Graw Hill. 6. Fundamentals of the Finite Element Method for Heat and Fluid Flow by Roland W. Lewis, Perumal Nithiarasu, Kankanhalli N. Seetharamu by John Wiley.
4 Basic Concepts The finite element method (FEM), or finite element analysis (FEA), is based on the idea of building a complicated object with simple blocks, or, dividing a complicated object into small and manageable pieces. Application of this simple idea can be found everywhere in everyday life, as well as in engineering. 4 Examples
5 Best Example Of Fem 5 Blades in turbine engines are typical examples
6 FEM Examples 6
7 Numerical Methods: 1. Introduction to FEM 7
8 1.1 Applications 8 A diversity of specializations under the union of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.
9 Applications 9 Cont FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.
10 Applications 10 Cont This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications. The introduction of FEM has substantially decreased the time to take products from concept to the production line. It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated. In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.
11 1.2 Visualization (Applications) 11 Visualization of how a car deforms in an asymmetrical crash using finite element analysis
12 1.3 Structural Analysis 12
13 1.4 Thermal System Analysis 13
14 1.5 Flow Analysis 14
15 A. Rolling 1.6 Thermomechanical Process Analysis 15 B. Forging C. Injection Molding
16 Academics Research Work On 16 Industry B. Tech M. Tech Phd R&D Department Research Journals Conferences National Journals International Journals National Conferences International Conferences Concepts, Designs, Developments, Manufacturing, Renovation, Innovations
17 2 General Procedure of FEM 17
18 General Procedure of FEM 18
19 General Procedure of FEM 19
20 2.1 Finite Element Formulations Direct approach for discrete systems 20 Direct approach has the following features: It applies physical concept (e.g. force equilibrium, energy conservation, mass conservation, etc.) directly to discretized elements. It is easy in its physical interpretation. It does not need elaborate sophisticated mathematical manipulation or concept. Its applicability is limited to certain problems for which equilibrium or conservation law can be easily stated in terms of physical quantities one wants to obtain. In most cases, discretized elements are self-obvious in the physical sense.
21 2.1 Finite Element Formulations Coordinate Transformation approach 21 In many cases, one can introduce a local coordinate system associated with each element in addition to a global coordinate system. A local coordinate system can be defined in many cases in a self-obvious way inherent to the element itself. It is much easier to determine the stiffness matrix with respect to the local coordinate system of an element than with respect to the global coordinate system. The stiffness matrix with respect to the local coordinate system is to be transformed to that with respect to the global coordinate system before the assembly procedure. i) Vector Transformation in 2-D. ii) Transformation of stiffness matrix.
22 2.1 Finite Element Formulations Direct approach for Elasticity Problem (plane stress, plane strain) In this section, we are concerned about an elastic deformation problem in twodimensional continuous media (therefore, not a discrete system) Assembly Procedure The assembly procedure is based on compatibility and conservation law (e.g. force balance, mass conservation and energy conservation) Variational Approach in Finite Element Formulation Differential formulation, physical phenomena can be described in terms of minimization of total energy (or functional) associated with the problem, which is called variational formulation.
23 2.1 Finite Element Formulations Principle of Minimum Total Potential Energy 23 There is a very important physical principle to describe a deformation process of an elastic body, namely Principle of Minimum Total Potential Energy, which can be summarized as below:
24 Finite Element Method versus Rayleigh-Ritz Method 24 One of the historically famous approximate methods for this kind of problem is Rayleigh-Ritz Method, and the other modern method is the Finite Element method. Here we will discuss both methods with the comparison in mind. i) Rayleigh-Ritz Method: This method is very simple and easy to understand. However, it is not easy to find a family of trial functions for the entire domain satisfying the essential boundary conditions when geometry is complicated. ii) Finite Element Method: In this case, the shape functions can be found more easily than the trial functions without having to worry about satisfying the essential boundary conditions, which makes FEM much more useful than Rayleigh-Ritz Method. In this regard, the Finite Element Method is a modernized approximation method suitable for computer environment.
25 3 Shape Functions And Discretization 25 We will discuss the element types, shape functions and discretization in this section. It is important to be able to select an element type which is most suitable for the problem of interest, and to determine the shape functions for the chosen element type. Finally, automatic mesh generation techniques are, in practical sense, also important to finite element analysis applications. 3.1 Element Types i) One-dimensional Elements
26 3 Shape Functions And Discretization ii) Two-dimensional Elements 26
27 3 Shape Functions And Discretization iii) Three-dimensional Elements 27
28 4 Natural Coordinates 28 Note that N (x, y) in our current form is represented in terms of the nodal coordinates (xi, yi ) and a global coordinate (x, y). One can have a better form in terms of so-called Natural Coordinate, in particular for triangular type of elements (or Normalized Coordinate for a quadrilateral type of elements ). 1) One-dimensional case
29 4 Natural Coordinates 29 2) Two-dimensional case 3) Three-dimensional case
30 5 Normalized coordinate 30
31 6 Shape functions (Examples) Shape functions for several quadrilateral elements are summarized below: 31
32 6 Shape functions (Examples) 32
33 6 Shape functions (Examples) 33
34 7 Exercise 34
35 QUESTIONS 35
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