AS 5850 Finite Element Analysis

Transcription

1 AS 5850 Finite Element Analysis INTRODUCTION Prof. IIT Madras

2 Goal of engineering computations Perform analysis and design of physical systems and processes subjected to imposed conditions (or loads) and help in engineering decision making Analysis: Determination of the behavioral response exhibited by a particular structural configuration under specific loads. (input, system) -> determine output Design: Process of altering dimensions, shapes, and materials to find the best structural configuration to perform a specific function or give a desired response. (input, output) -> determine the right system

3 Engineering Decision Process

4 Mathematical Formulation Identify the dependent and independent variables of the problem Obtain governing equations involving the dependent variables in terms of the independent variables - algebraic, differential or integral equations Identify primary and secondary variables and associated essential and natural boundary conditions Boundary/Initial Value Problems Partial differential equations model the behaviour of a wide range of problems in engineering.

5 Solution Methods Analytical solutions for most of the realistic PDE models are either too complex or non-existent. Numerical methods can be used to give approximate solutions. A general mathematically validated, computational efficient (programmable) method can give reasonably accurate approximate solutions for complex problems.

6 Numerical Methods Finite Difference Method (FDM) Point-wise approximation of the differential equation using array of grid points Finite Element Method (FEM) Integral approximation to the differential equation using an assembly of finite elements and satisfying boundary conditions exactly Boundary Element Method (BEM) Integral formulation of the equations satisfied exactly on the boundary and satisfying boundary conditions approximately. Finite Volume Method (FVM), spectral element method, meshless methods etc.

7 Advantages of FEM

8 Basic Idea of FEM 1. Divide whole into simple parts (finite element mesh) 2. Set up the `problem over a typical part of the domain. i.e. derive a set of relationships between primary and secondary variables 3. Assemble the parts to obtain the solution to the whole Divide and Conquer

9 Illustrating the idea of FEM Centre of Mass of a 3-D machine Component

10 Approximating a domain Total Area Under a Curve b I F( x) dx a

11 Example 2 (Contd) Approximation of the Curve a1 bx 1, a x x2 F( x) a b x, x x x a3 b3x, x3 x b

12 Example -2 (contd.) Total Area Under the Curve a1 bx 1, a x x2 F( x) a b x, x x x a3 b3x, x3 x b x i 1 i 1 I F( x) dx ( a bx) dx i i i i x x i I I1 I2 I3 x i

13 Finite Element Terminology Element A geometric sub-domain of the region being simulated, with the property that it allows a unique derivation of the approximation (interpolation) functions. Node A geometric location in the element which plays a role in the derivation of the interpolation functions and it is the point at which solution is sought. Mesh A collection of elements (or nodes) that replaces the actual domain. Weak Form An integral statement equivalent to the governing equations and natural boundary conditions.

14 Finite Element Discretization

15 Steps in Finite Element Modeling Begin with the governing equations of the problem Develop its weak form (weighted-integral statement) over a typical element Approximate the solution over each finite element Obtain relations among the quantities of interest over each finite element Assemble these relations over the entire domain Solve the resultant equations

16 Applications of FEA

17 Development of FEM Historical Landmarks 1943 R. Courant, Variational methods for the solution of problems of equilibrium and vibration, Bull. Amer. Math. Soc. 49 (1943), 1-23, MR 4, 200. Thirty years ago, Courant gave a remarkable lecture to this Society. To begin with, his idea was forgotten. Perhaps you have forgotten it too ; it had to do with approximation by piecewise polynomials Certainly there was an idea whose time was coming. When it finally came, fifteen years after Courant's lecture, it developed into what is now the most powerful technique for solving a large class of partial differential equations the finite element, method. The only sad part is that virtually the whole development took place as if Courant had never existed. G. Strang, American Mathematical Society, Apr M.J. Turner, R.W. Clough, H.C. Martin and L.J. Topp, Stiffness and deflection analysis of complex structures. J Aero Sci 23 (1956), pp This paper presented the idea of dividing the real continuum directly into elements of arbitrary shape and directly establishing their stiffness. This became known as the method of Finite Elements only in 1960, following a paper presented by Ray Clough. Ray Clough

18 History of FEM - Timeline 1943 R. Courant proposed a torsion solution using triangular subregions (not implemented since no computers were available) 1950s Digital computers which solved large systems of equations became available. Aerospace companies began solving structures problems. Classic Paper by Clough and others in term Finite Element Method first coined by R.W.Clough Early and mid 1960 s new elements formulated, method became accepted, extended to heat transfer and fluid flow problems Late 1960 s, early 1970 s Commercial availability of large general purpose software (NASTRAN, ANSYS, SAP) 1980s Inexpensive PC versions, interactive graphics 90 s and beyond Widespread application in numerous industries, Integration with CAD packages. Most recent emphasis Simulation of multi-physics problems

19 References 1. J. N. Reddy An to the Finite Element Method 2. O.C. Zienkiewicz et. al., Finite Element Method (3 vols) 3. K. J. Bathe Finite Element Procedures 4. T. J. R. Hughes The Finite Element Method Linear, Static and Dynamic Finite Element Analysis 5. S. S. Rao The Finite Element Method in Engineering 6. R. D. Cook, D. S. Malkus, M. E. Pleisha Concepts and Applications of Finite Element Analysis