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**Unformatted text preview: **MCE 565 Wave Motion & Vibration in Continuous Media Spring 2005 Professor M. H. Sadd Introduction to Finite Element Methods Need for Computational Methods • Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry • Many Applicaitons Involve Cases with Complex Shape, Boundary Conditions and Material Behavior • Therefore a Gap Exists Between What Is Needed in Applications and What Can Be Solved by Analytical Closed- form Methods • This Has Lead to the Development of Several Numerical/Computational Schemes Including: Finite Difference, Finite Element and Boundary Element Methods Introduction to Finite Element Analysis The finite element method is a computational scheme to solve field problems in engineering and science. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields , etc. The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions . This approximated variation is quantified in terms of solution values at special element locations called nodes . Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool. Advantages of Finite Element Analysis- Models Bodies of Complex Shape- Can Handle General Loading/Boundary Conditions- Models Bodies Composed of Composite and Multiphase Materials- Model is Easily Refined for Improved Accuracy by Varying Element Size and Type (Approximation Scheme)- Time Dependent and Dynamic Effects Can Be Included- Can Handle a Variety Nonlinear Effects Including Material Behavior, Large Deformations, Boundary Conditions, Etc. Basic Concept of the Finite Element Method Any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains. Exact Analytical Solution x T Approximate Piecewise Linear Solution x T One-Dimensional Temperature Distribution Two-Dimensional Discretization-1-0.5 0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4-3-2-1 1 2 x y u(x,y) Approximate Piecewise Linear Representation Discretization Concepts x T Exact Temperature Distribution, T(x) Finite Element Discretization Linear Interpolation Model (Four Elem ents) Quadratic Interpolation M odel (Two Elem ents)...

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