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Unformatted text preview: 43 Introduction to Variational and Numerical Methods 3.1 INTRODUCTION TO VARIATIONAL METHODS Let u ( x ) be a vector-valued function of position vector x , and consider a vector- valued function F ( u ( x ) , u ( x ) , x ) , in which u ( x ) = u / x . Furthermore, let v ( x ) be a function such that v ( x ) = when u ( x ) = and v ( x ) = when u ( x ) = , but which is otherwise arbitrary. The differential d F measures how much F changes if x changes. The variation F measures how much F changes if u and u change at fixed x . Following Ewing, we introduce the vector-valued function ( e : F ) as follows (Ewing, 1985): (3.1) The variation F is defined by (3.2) with x fixed. Elementary manipulation demonstrates that (3.3) in which . If If then F = u = e v . This suggests the form (3.4) The variational operator exhibits five important properties: 1. ( . ) commutes with linear differential operators and integrals. For exam- ple, if S denotes a prescribed contour of integration: (3.5) 3 ( : ) ( ( ) ( ), ( ) ( ), ) ( ( ), ( ), ) e e e F F u x v x u x v x x F u x u x x = + + F = = e e , e d d F F u v F u v = + e e , tr = F u F v v e e ij ij u F u F u v = = = , . then e F u = , F F u u F u u = + tr . ( ( . ) ) dS dS = 2003 by CRC CRC Press LLC 44 Finite Element Analysis: Thermomechanics of Solids 2. ( f ) vanishes when its argument f is prescribed. 3. ( . ) satisfies the same operational rules as d ( . ) . For example, if the scalars q and r are both subject to variation, then (3.6) 4. If f is a prescribed function of (scalar) x , and if u ( x ) is subject to variation, then (3.7) 5. Other than for number 2, the variation is arbitrary. For example, for two vectors v and w , v T d w = 0 implies that v and w are orthogonal to each other. However, v T w implies that v = , since only the zero vector can be orthogonal to an arbitrary vector. As a simple example, Figure 3.1 depicts a rod of length L, cross-sectional area A, and elastic modulus E. At x = 0, the rod is built in, while at x = L , the tensile force P is applied. Inertia is neglected. The governing equations are in terms of displacement u , stress S , and (linear) strain E : strain-displacement stress-strain equilibrium (3.8) Combining the equations furnishes (3.9) The following steps serve to derive a variational equation that is equivalent to the differential equation and endpoint conditions (boundary conditions and constraints)....
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