5b-WeakForm-2010s

5b-WeakForm-2010s - MAE M168/CEE M135C Introduction to Finite Element Methods MAE 168 Lecture 5b Weak Form of 1-D Elasticity Summary and Review 1

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Unformatted text preview: MAE M168/CEE M135C Introduction to Finite Element Methods MAE 168 Lecture 5b Weak Form of 1-D Elasticity Summary and Review 1. Strong form of 2 nd – order BVP in 1-D – 2 nd – order ODE – 1 BC at each end 2. BCs are essential or natural ; + = + = d du d dT EA q Ak Q dx dx dx dx Q – EBC specify fundamental fields. – NBC specify derivatives. (0) u = ( ) ( ) = = du L F L EA P dx P q (0) = = - = x dT q k q dx ( ) = L T L T q Variational BVP Formulation: The Weak Form • An alternative approach to formulation of BVPs. • More convenient framework than the strong form for developing approximate solutions. Method of Weighted Residuals • Derive weak form by the method of weighted residuals . Recall BVP: BC + = + d du EA q dx dx • Define the residual function of the bar BVP: • If u(x) is a solution to the BVP, then R(x) = for all ( ) d du R x EA q dx dx = + [ ] 0, . x L ∈ Residual measures error • Residual depends on displacement field ( ) d du R x EA q dx dx = + ( ) ) [ ( )] [ ] d d x L u x q L EA + = & • If u ( x ) not a solution, then • If u ( x ) is a solution, then...
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This note was uploaded on 08/09/2011 for the course MAE 168 taught by Professor Klug during the Spring '11 term at UCLA.

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5b-WeakForm-2010s - MAE M168/CEE M135C Introduction to Finite Element Methods MAE 168 Lecture 5b Weak Form of 1-D Elasticity Summary and Review 1

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