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# New Lec1 - MCE 561 Computational Methods in Solid Mechanics...

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MCE 561 Computational Methods in Solid Mechanics Introduction

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Need for Computational Methods Stress Analysis Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry Most Real World Problems 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, Boundary Element, Discrete Element Methods
Finite Difference Method Finite Difference Method (FDM) The finite difference method replaces the derivatives in governing field equations by difference quotients which involve values of the solution at discrete mesh points in the domain under study. For example, the first derivative with respect to x may be represented by Repeated applications of this representation set up algebraic systems of equations in terms of the unknown nodal values u i . The major difficulty with this method lies in the inaccuracies in dealing with regions of complex shape, although this problem can be eliminated through the use of coordinate transformation techniques. Finite Element Method (FEM) (i-1) x (i) (i+1) x x d u d x u u x b a c k w a r d d i f f e r e n c e u u x f o r w a r d d i f f e r e n c e u u x c e n t r a l d i f f e r e n c e i i i i i i - - - - + + - 1 1 1 1 2 . . . . . . . . .

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Finite Element Method Finite Element Method (FEM) The fundamental concept of the finite element method lies in dividing the body under study into a finite number of pieces (subdomains) called elements . Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or shape functions . This approximated variation is quantified in terms of solution values at special locations within the element called the 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 and shape are variable, the finite element method can handle problem domains of quite complicated shape, and thus it has become a very useful and practical tool.
Boundary Element Method Boundary Element Node Bounday Element Method (BEM) The boundary element method is based upon an integral statement of the governing equations of the problem under study. The integral statement may be cast into a form which contains unknowns only over the boundary of the body domain. This boundary integral equation may then be solved using concepts from the finite element method; i.e. the boundary may be discretized into a number of elements and the interpolation approximation concept may then be applied. This method again produces an algebraic system of equations to solve for the unknown

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## This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

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New Lec1 - MCE 561 Computational Methods in Solid Mechanics...

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