Fundamental Concepts of FEA

1. Fundamental Concepts of FEA Unit 1

2. Finite Element Analysis(FEA)

4. Methods to solve engineering problem 1.Analytical Method : • Classical Approach • 100% accurate results • Closed form solution • Applicable only for Simple Problems like cantilever and Simply Supported Beam, etc. • Complete in itself 2.Numerical Method : • Mathematical representation • Approximate, assumptions made. • Applicable even if a physical prototype is not available • Real life complicated problems • FEM comes under Numerical Method. 3.Experimental Method • Actual Measurement • Time Consuming and needs expensive set up • Applicable only if Physical prototype is available • Results cannot be believed blindly and a minimum of 3 to 5 prototypes must be tested

5. Procedure for Solving Any Analytical or Numerical Problem • There are two steps in solving analytical or numerical problems: • Step 1) Writing of the governing equation – Problem definition, or in other words, formulating the problem in the form of a mathematical equation. • Step 2) Mathematical solution of the governing equation. The final result is the summation of step 1 and step 2. The result will be 100 % accurate when there is no approximation at either of the steps (Analytical method). Numerical methods make an approximation at step 1 and at step 2, therefore all the numerical methods are approximate.

6. Brief Introduction to Different Numerical Methods 1) Finite Element Method (FEM) : FEM is the most popular numerical method. Applications - Linear, Nonlinear, Buckling, Thermal, Dynamic and Fatigue analysis. FEM will be discussed later. Are FEA and FEM different? Finite Element Method (FEM) and Finite Element Analysis (FEA) both are one and the same. The term “FEA” is more popular in industries while “FEM” is more popular at universities. Many times there is confusion between FEA, FEM, and one more similar but different term FMEA (Failure Mode Effect Analysis). FEA/FEM is used by design or R and D departments only, while FMEA is applicable to all of the departments.

7. 2) Boundary Element Method (BEM) • This is a very powerful and efficient technique to solve acoustics or NVH problems. • Just like the finite element method it also requires nodes and elements, but as the name suggests it only considers the outer boundary of the domain. • So when the problem is of a volume, only the outer surfaces are considered. If the domain is of an area, then only the outer periphery is considered. • This way it reduces the dimensionality of the problem by a degree of one and thus solving the problem faster.

8. 3) Finite Volume Method (FVM) : • Most Computational Fluid Dynamics (CFD) software is based on FVM. The unit volume is considered in Finite Volume Method (similar to element in finite element analysis). • Variable properties at the nodes include pressure, velocity, area, mass, etc. • It is based on the Navier - Stokes equations (Mass, Momentum, and Energy conservation equilibrium equations).

9. 4) Finite Difference Method (FDM) : • Finite Element and Finite Difference Methods share many common things. • In general the Finite Difference Method is described as a way to solve differential equation. • It uses Taylor’s series to convert a differential equation to an algebraic equation. In the conversion process higher order terms are neglected. • It is used in combination with BEM or FVM to solve Thermal and CFD coupled problems.

10. Is it possible to use all of the methods listed above (FEA, BEA, FVM, FDM) to solve the same problem (for example, a cantilever problem)? The answer is YES! But the difference is in the accuracy achieved, programming ease, and the time required to obtain the solution. When internal details are required (such as stresses inside the 3-D object) BEM will lead to poor results (as it only considers the outer boundary), while FEM, FDM, or FVM are preferable. FVM has been used for solving stress problems but it is well suited for computational fluid dynamics problems where conservation and equilibrium is quite natural. FDM has limitations with complicated geometry, assembly of different material components, and the combination of various types of elements (1-D, 2-D and 3-D). For these types of problems FEM is far ahead of its competitors.

11. FEM vs Classical method • 1. In classical methods exact equations are formed and exact solutions are obtained where as in finite element analysis exact equations are formed but approximate solutions are obtained. • 2. Solutions have been obtained for few standard cases by classical methods, where as solutions can be obtained for all problems by finite element analysis. • 3. Whenever the following complexities are faced, classical method makes the drastic assumptions’ and looks for the solutions: (a) Shape (b) Boundary conditions (c) Loading

12. 4. When material property is not isotropic, solutions for the problems become very difficult in classical method. Only few simple cases have been tried successfully by researchers. FEM can handle structures with anisotropic properties also without any difficulty. 5. If structure consists of more than one material, it is difficult to use classical method, but finite element can be used without any difficulty. 6. Problems with material and geometric non-linearitiescan not be handled by classical methods. There is no difficulty in FEM.

13. FEM vs FDM 1. FDM makes pointwise approximation to the governing equations. FEM make piecewise approximation to the governing equations. 2. FDM do not give the values at any point except at node points. FEM can give the values at any point. 3. FDM makes stair type approximation to sloping and curved boundaries as shown in Fig. FEM can consider the sloping boundaries exactly. If curved elements are used, even the curved boundaries can be handled exactly.

14. 4. FDM needs larger number of nodes to get good results while FEM needs fewer nodes.5. With FDM fairly complicated problems can be handled where as FEM can handle all complicated problems. NOTE: FDM makes pointwise approximation to the governing equations i.e. it ensures continuity only at the node points. Continuity along the sides of grid lines are not ensured. FEM make piecewise approximation i.e. it ensures the continuity at node points as well as along the sides of the element. 2. FDM do not give the values at any point except at node points. It do not give any approximating function to evaluate the basic values (deflections, in case of solid mechanics) using the nodal values. FEM can give the values at any point. However the values obtained at points other than nodes are by using suitable interpolation formulae.

15. Home Work • History of FEA

16. Finite Element Method – What is it? • The Finite Element Method (FEM) is a numerical method of solving systems of partial differential equations (PDEs) • It reduces a PDE system to a system of algebraic equations that can be solved using traditional linear algebra techniques. •  In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other.

17. Finite Element Terminology

19. What is DOF (Degree Of Freedom) ? • When can we say that we know the solution to above problem? If and only if we are able to define the deformed position of each and every particle completely. • The minimum number of parameters (motion, coordinates, temp. etc.) required to define the position of any entity completely in the space is known as a degree of freedom (dof).

20. Domain • In mathematics, a domain is set of independent variables for which a function is defined. • In FEA, a domain is continuous system (region) over which law of physics govern. Example • In structural engineering, a domain could be a beam or complete building frame. • In mechanical engineering, a domain could be a piece of machine part or thermal shield.

21. Fundamental concept

27. Typical FEA procedure by commercial software

31. Applications of FEM • Static-linear Analysis • Static-Nonlinear Analysis • Dynamic linear Analysis • Dynamic Non-linear Analysis • Thermal Analysis • Fluid flow Analysis

32. Advantages of FEM 1. Model irregularly shaped bodies quite easily 2. Handle general load conditions without difficulty 3. Model bodies composed of several different materials because the element equations are evaluated individually 4. Handle unlimited numbers and kinds of boundary conditions 5. Vary the size of the elements to make it possible to use small elements where necessary 6. Alter the finite element model relatively easily and cheaply 7. Include dynamic effects 8. Handle nonlinear behavior existing with large deformations and nonlinear materials

33. Disadvantages of FEM • FEM solutions are often approximate. The more refined the grid (mesh) the more accurate the FEM solution. • FEM solution may contain inherent computational error as a result of error accumulation during numerical computation. • FEM solution may contain fatal error as a result of incorrect modelling of structure loads and boundary conditions. • Skilled user, reliable program and computer are essential.

34. Boundary conditions • The values of variables prescribed on the boundaries of the region are called as boundary condition. • Types of boundary condition : • Geometric (Essential) boundary conditions • Force (natural) boundary conditions

35. 1.Geometric (Essential) boundary conditions • In a structural mechanics problem , the geometric or essential boundary conditions includes : prescribed displacement and slopes • The geometric or essential boundary condition are also know as kinematic boundary condition. • Example : 1. At x = L; y(displacement) = 0 2. At x= L; dy/dx(slope) = 0

36. 2.Force (Natural) boundary conditions • In a structural mechanics problem , the force or natural boundary conditions includes : prescribed force and moment • The force or natural boundary condition are also know as static boundary condition • Example : 1. At x = 0; EI d3y/dx3 (shear force) = 0 2. At x= 0; EI d2y/dx2 (bending moment) = 0

37. Strain-displacement relationships • u = displacement of a point in X direction • v = displacement of a point in Y direction • w = displacement of a point in Z direction • = strain in x direction • = strain in y direction • = strain in z direction • = strain in XY plane • = strain in YZ plane • = strain in ZX plane

41. Stress – strain relationship • = stress in X direction • = stress in Y direction • = stress in Z direction • = shear stress in XY plane • = shear stress in YZ plane • = shear stress in ZX plane

42. Plane stress • The plane stress condition is characteristics by very small dimensions in one of the normal direction. A thin planer body subjected to in plane loading on its edge surface • Plane stress is defined to be a state of stress in which the normal stress and the shear stresses and directed perpendicular to the X-Y plane are assumed to be zero. • i.e. = = = 0

44. PLANE STRESS Nonzero stresses: Nonzero strains: Isotropic linear elastic stress-strain law Hence, the D matrix for the plane stress case is

45. Plane strain • The problem involving a long body whose geometry and loading do not vary significantly in longitudinal direction are referred as plain stress problem. • Plane strain is defined to be a state of strain in which the strain normal to the x-y plane and the shear strains and are assumed to be zero. • = = = 0

46. PLANE STRAIN Nonzero stress: Nonzero strain components: Isotropic linear elastic stress-strain law Hence, the D matrix for the plane strain case is