{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

9780203009512.ch3

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

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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.
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern