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Gâteaux Derivative  Extreme of a Functional CH.11. VARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Introduction Functionals Gâteaux Derivative Extreme of a Functional Variational Principle Variational Form of a Continuum Mechanics Problem Virtual Work Principle Virtual Work Principle Interpretation of the VWP VWP in Engineering Notation Minimum Potential Energy Principle Hypothesis Potential Energy Variational Principle 2 11.1. Introduction Ch.11. Variational Principles 3 The Variational Approach For any physical system we want to describe, there will be a quantity whose value has to be optimized. Electric currents prefer the way of least resistance. A soap bubble minimizes surface area. The shape of a rope suspended at both ends (catenary) is that which minimizes the gravitational potential energy. To find the optimal configuration, small changes are made and the configuration which would get less optimal under any change is taken. 4 Variational Principle This is essentially the same procedure one does for finding the extrema (minimum, maximum or saddle point) of a function by requiring the first derivative to vanish. A variational principle is a mathematical method for determining the state or dynamics of a physical system, by identifying it as an extrema of a functional. 5 Computational Mechanics In computational mechanics physical mechanics problems are solved by cooperation of mechanics, computers and numerical methods. This provides an additional approach to problem-solving, besides the theoretical and experimental sciences. Includes disciplines such as solid mechanics, fluid dynamics, thermodynamics, electromagnetics, and solid mechanics. 6 Variational Principles in Numerical Methods Numerical Methods use algorithms which solve problems through numerical approximation by discretizing continuums. They are used to find the solution of a set of partial differential equations governing a physical problem. They include: Finite Difference Method Weighted Residual Method Finite Element Method Boundary Element Method Mesh-free Methods The Variational Principles are the basis of these methods. 7 11.2. Functionals Ch.11. Variational Principles 8 Definition of Functional Consider a function space : X R b uxdx() 3 m ::ux a XRR b u F f xux,(),() u x dx The elements of are functions X ux a b of an arbitrary tensor order, defined in uxdx() 3 X a subset R . ux a ux :, ab R A functional Fu is a mapping of the function space X onto the set of the real numbers , R : FXR u : . It is a function that takes an element ux of the function space X as its input argument and returns a scalar. 9 Definition of Gâteaux Derivative Consider : 3 m a function space XRR::ux the functional FXRu : a perturbation parameter R a perturbation direction xX The function ux x X is the perturbed function of ux in the x direction. t=0 t ux Ω P P’ Ω0 x ux x 10 Definition of Gâteaux Derivative The Gâteaux derivative of the functional F u in the direction is: d FFuu;: P’ d 0 F u t=0 t ux Ω P P’ Ω0 x ux x REMARK not The perturbation direction is often denoted as u . Do not confuse ux () with the differential d ux () . ux()is not necessarily small !!! 11 Example Find the Gâteaux derivative of the functional uu : dd u F 12 Example - Solution Find the Gâteaux derivative of the functional uu : dd u F Solution : dd d FFuu; uu uu dd uu dd d 0 0 0 uu dd uu uu uu dd uudd 0 0 u u ()uu () Fuu dd u uu 13 Gâteaux Derivative with boundary conditions Consider a function space V : u m* VR::;ux ux ux u x xu By definition, when performing the Gâteaux derivative on V , uu V . Then, * * u 0 uu ux uuu x xu xxuu u u* The direction perturbation must satisfy: u0 xu 14 Gâteaux Derivative in terms of Functionals Consider the family of functionals u uxuxux(, (), ())d F (,xux (), ux ())d The Gâteaux derivative of this family of functionals can be written as, u u; u ( xux , ( ), ux ( )) udd ( xux , ( ), ux ( )) u FE T u0 xu REMARK The example showed that for uu : dd u , the F ()uu () Gâteaux derivative is F uu dd u . uu 15 Extrema of a Function A function has a local minimum (maximum) at x 0 Necessary condition: df() x not fx 0 dx 0 xx 0 Local minimum The same condition is necessary for the function to have extrema (maximum, minimum or saddle point) at x 0 . This concept can be can be extended to functionals. 16 Extreme of a Functional. Variational principle A functional FVRu : has a minimum at ux V Necessary condition for the functional to have extrema at ux : uu;0 u | u 0 F xu This can be re-written in integral form: uu;()()0 uu dd uu u FE T u0 Variational Principle xu 17 11.3.Variational Principle Ch.11. Variational Principles 18 Variational Principle Variational Principle: uu;0 udd u u REMARK FE T u0 Note that u x u is arbitrary. Fundamental Theorem of Variational Calculus: The expression (,xuxuxu (), ())dd (, xuxuxu (), ()) 0 u ET u0 x is satisfied if and only if u E(,xux (), ux ())0 x Euler-Lagrange equations T(,xux (), ux ())0 x Natural boundary conditions 19 Example Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional b u xuxu,, x dxwith ux :, ab ; ux ua p FR xa a 20 Example - Solution Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional b u xuxu,, x dxwith ux ua p F xa a Solution : First, the Gâteaux derivative must be obtained. The function ux is perturbed: ux ux x not xuxa | a 0 ux ux x This is replaced in the functional: b Fuxuxuxdx ,, a 21 b Fuxuxuxdx ,, a Example - Solution ux ua p xa The Gâteaux derivative will be b uu; d dx FFd 0 a uu Then, the expression obtained must be manipulated so that it resembles the Variational Principle uu ;0 u dd u : FE T Integrating by parts the second term in the expression obtained: b bbdd b dx() dx ba () dx aa a uudxuuudxuaba a 0 The Gâteaux derivative is re-written as: b uxuxuxdxuap ,, ; a b d ((uuu;) ;)[ ()] udxub a udxu u u b 22 MMC - ETSECCPB - UPC Example - Solution Therefore, the Variational Principle takes the form b d u (u;) u [ ( )] udx ub a udxu u b ua 0 If this is compared to uu ;0 u dd u , one obtains: FE T d Exuu,, 0 x ab , Euler-Lagrange Equations udxu Natural (Newmann) Txuu,, 0 boundary conditions u xb Essential (Dirichlet) ux() ua () p xa boundary conditions 23 Variational Form of a Continuum Mechanics Problem Consider a continuum mechanics problem with local or strong governing equations given by, Euler-Lagrange equations E(,xux (), ux ())0 x V with boundary conditions: Natural or Newmann * T(,(),xux ux ()) ( u ) n t () x 0 x Forced (essential) or Dirichlet ux u x x u REMARK The Euler-Lagrange equations are generally a set of PDEs. 24 Variational Form of a Continuum Mechanics Problem The variational form of the continuum mechanics problem consists in finding a field ux X where 3 m VRR::uxV ux u x on u 3 m VRR0 ux():V ux () 0 on u fulfilling: (,xux (), ux ()) ux ()dV (, xux (), ux ()) ux () d 0 ux() V ET V0 25 Variational Form of a Continuum Mechanics Problem REMARK 1 The local or strong governing equations of the continuum mechanics are the Euler-Lagrange equation and natural boundary conditions. REMARK 2 The fundamental theorem of variational calculus guarantees that the solution given by the variational principle and the one given by the local governing equations is the same solution. 26 11.4. Virtual Work Principle Ch.11. Variational Principles 27 Governing Equations Continuum mechanics problem for a body: Cauchy equation 2 ux ,t xbx,,tt00 in V t2 ( (ux ( ,t ))) Boundary conditions ux ,,tt u x on u xnxtx,t ,t ,t on s(( u ),t) 28 Variational Principle The variational principle consists in finding a displacement field 3 m , where VRR:,:uxtV ux , t u x ,on t u such that the variational principle holds, 2 u WVuu;[ ()]( b udV t n ) u d 0 u 0 t2 V T E 3 m where VRR0 ::uxV ux 0 onu Note: is the space of admissible displacements. is the space of admissible virtual displacements (test functions). The (perturbations of the displacements ) u are termed virtual displacements. 29 Virtual Work Principle (VWP) The first term in the variational principle a 2 u WVuu;[ ()] b2 udV t n u d 0 u 0 t V T E Considering that uuu s and (applying the divergence theorem): s unudV (d u dV VV Then, the Virtual Work Principle reads: uu;0 ba udV t*s u d u dV u WV 0 VV 30 Virtual Work Principle (VWP) REMARK 1 The Cauchy equation and the equilibrium of tractions at the boundary are, respectively, the Euler-Lagrange equations and natural boundary conditions associated to the Virtual Work Principle. REMARK 2 The Virtual Work Principle can be viewed as the variational principle associated to a functional W u , being the necessary condition to find a minimum of this functional. 31 Interpretation of the VWP The VWP can be interpreted as: *s W uu;0 ba udV t u d u dV VV b* pseudo - virtual body forces strains Work by the Work by the pseudo-body virtual strain. forces and the contact forces. Internal virtual External virtual work work ext int W W ext int WWWVuu; 0 u 0 32 VWP in Voigt’s Notation Engineering notation uses vectors instead of tensors: xxx yyy not 66zzz ;; : RR 2 xy xy xy 2 xz xz xz yz yz 2 yz The Virtual Work Principle becomes dV ba u dV t* u d 0 u WV 0 VV Total virtual work.
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