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9780203009512.ch3

# 9780203009512.ch3 - 0749_Frmae_C03 Page 43 Wednesday 5:01...

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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 ) = 0 when u ( x ) = 0 and v ′( x ) = 0 when u ′( x ) = 0 , but which is otherwise arbitrary. The differential d F measures how much F changes if x changes. The variation δ F measures how much F changes if u and u change at fixed x . Following Ewing, we introduce the vector-valued function φ ( e : F ) as follows (Ewing, 1985): (3.1) The variation δ F is defined by (3.2) with x fixed. Elementary manipulation demonstrates that (3.3) in which . If If then δ F = δ u = e v . This suggests the form (3.4) The variational operator exhibits five important properties: 1. δ ( . ) commutes with linear differential operators and integrals. For exam- ple, if S denotes a prescribed contour of integration: (3.5) 3 Φ ( : ) ( ( ) ( ), ( ) ( ), ) ( ( ), ( ), ) e e e F F u x v x u x v x x F u x u x x = + + δ F = = e e , e d d Φ 0 δ 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 δ δ e F u = , δ δ δ F F u u F u u = + ∂ ′ tr . δ δ ( ( . ) ) dS dS = © 2003 by CRC CRC Press LLC

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44 Finite Element Analysis: Thermomechanics of Solids 2. δ ( f ) vanishes when its argument f is prescribed. 3. δ ( . ) 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 δ w implies that v = 0 , 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). FIGURE 3.1 Rod under uniaxial tension. δ δ δ ( ) ( ) ( ) qr q r q r = + . δ δ ( ) fu f u = . E du dx = S E = E d dx σ = 0 E A d u dx 2 2 0 = . E,A L P © 2003 by CRC CRC Press LLC
Introduction to Variational and Numerical Methods 45 Step 1: Multiply by the variation of the variable to be determined ( u ) and integrate over the domain.

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