CH.11. VARIATIONAL PRINCIPLES
CH.11. VARIATIONALPRINCIPLESContinuum Mechanics Course (MMC) - ETSECCPB - UPC
Overview Introduction Functionals Variational Principle Virtual Work PrincipleInterpretation of the VWPVWP in Engineering NotationMinimum Potential Energy Principle 2Variational Form of a Continuum Mechanics ProblemVirtual Work Principle Gâteaux DerivativeExtreme of a FunctionalHypothesisPotential Energy Variational Principle
11.1. IntroductionCh.11. Variational Principles3
The Variational Approach For any physical system we want to describe, there will be aquantity 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 gravitationalpotential energy. 4To find the optimal configuration, small changes are made andthe configuration which would get less optimal under any changeis taken.
Variational Principle5 This is essentially the same procedure one does for finding theextrema (minimum, maximum or saddle point) of a function byrequiring the first derivative to vanish. A variational principle is a mathematical method for determiningthe state or dynamics of a physical system, by identifying it as anextrema of a functional.
Computational Mechanics 6In computational mechanics physical mechanics problems aresolved by cooperation of mechanics, computers and numericalmethods. This provides an additional approach to problem-solving, besides thetheoretical and experimental sciences. Includes disciplines such as solid mechanics, fluid dynamics,thermodynamics, electromagnetics, and solid mechanics.
Variational Principles in NumericalMethods Numerical Methods use algorithms which solve problems throughnumerical approximation by discretizing continuums. They are used to find the solution of a set of partial differentialequations governing a physical problem.They include: 7Finite Difference MethodWeighted Residual MethodFinite Element MethodBoundary Element MethodMesh-free MethodsThe Variational Principles are the basis of these methods.
11.2. FunctionalsCh.11. Variational Principles8
Definition of Functional Consider a function space X :X : u x : R R3mR The elements of X are functions u x of an arbitrary tensor order, defined in3 Ra subset.F u bu x a f x, u ( x), u ( x) dxb u ( x)dxau x : a, b RA functional F u is a mapping of the function space X onto theset of the real numbers , R : F u : X R . It is a function that takes an element u x of the function space X asits input argument and returns a scalar.9 u ( x)dxaXb
Definition of Gâteaux Derivative Consider : a function space X : u x : R3 Rm the functional F u : X Ra perturbation parameter Ra perturbation direction x XThe function u x x X is the perturbed function of u x inthe x direction.t 0tΩ010Pu x Ωu x x P’ x
Definition of Gâteaux Derivative The Gâteaux derivative of the functional F u in the direction is: F u; : dF u d 0F u t 0Ω0Ptu x Ωu x x REMARKnotThe perturbation direction is often denoted as u .Do not confuse u(x) with the differential du(x) . u(x) is not necessarily small !!!11P’P’ x
ExampleFind the Gâteaux derivative of the functionalF u : u d u d 12
Example - SolutionFind the Gâteaux derivative of the functionalF u : u d u d Solution : F u; u ddd u ud u u d F u u d d d 0 0 0 u u d u u u u d u u d d u ud d 0 0 u (u) (u) u d u d u u F u 13 u
Gâteaux Derivative with boundaryconditions Consider a function space V : V : u x *u By definition, when performing the Gâteaux derivative on V , u u V .Then, u u x u u u x : R ; u x x u x m u* x u x 0u x u x u *u u*uuThe direction perturbation must satisfy: u x 0u14
Gâteaux Derivative in terms ofFunctionals Consider the family of functionals u F u (x, u(x), u(x))d (x, u(x), u(x))d The Gâteaux derivative of this familyof functionals can be written as, F u; u E(x, u(x), u(x)) u d T(x, u(x), u(x)) u d u u x 0REMARKThe example showed that for F u : u d u d , the ( u) ( u) u d u d . u u Gâteaux derivative is F u 15u
Extrema of a Function A function has a local minimum (maximum) at x0Necessary condition:notdf ( x) f x0 0dx x x0Local minimum 16The same condition is necessary for the function to have extrema(maximum, minimum or saddle point) at x0 .This concept can be can be extended to functionals.
Extreme of a Functional. Variationalprinciple A functional F u : V R has a minimum at u x V Necessary condition for the functional to have extrema at u x : F u; u 0 u u x 0uThis can be re-written in integral form: F u; u E(u) u d T(u) u d 0 Variational Principle17 u u x 0u
11.3.Variational PrincipleCh.11. Variational Principles18
Variational Principle Variational Principle: F u; u E u d T u d 0 uREMARK u x 0 Note that uuis arbitrary.Fundamental Theorem of Variational Calculus:The expression E(x, u(x), u(x)) u d T(x, u(x), u(x)) u d 0 is satisfied if and only if19 u u x 0uE(x, u(x), u(x)) 0 x Euler-Lagrange equationsT(x, u(x), u(x)) 0 x Natural boundary conditions
ExampleFind the Euler-Lagrange equations and the natural and forced boundaryconditions of the functionalbF u x, u x , u x dx with u x : a, b R ; u x x a u a pa20
Example - SolutionFind the Euler-Lagrange equations and the natural and forced boundaryconditions of the functionalbF u x, u x , u x dx withu x x a u a paSolution :First, the Gâteaux derivative must be obtained. The function u x is perturbed:notu x u x x x u x a a 0u x u x x This is replaced in the functional:bF u x, u x , u x dxa21
bF u x, u x , u x dxExample - Solution au x x a u a pThe Gâteaux derivative will be F u; dd F u 0 dx u u a bThen, the expression obtained must be manipulated so that it resembles theVariational Principle F u; u E u d T u d 0: Integrating by parts the second term in the expression obtained: ba Theb db d dx ( ) dx b a ( ) dxa dx u u u a a dx u u b u abGâteaux derivative is re-written as:b u x, u x , u x dx ; a 0u a pa d (u; ) (u; u ) [ ( )] udx uba udx u u b ub22MMC - ETSECCPB - UPC
Example - SolutionTherefore, the Variational Principle takes the formb d (u; u) [ ( )] udx uba udx u u b u ua 0If this is compared to F u; u E u d T u d 0 , one obtains: E x, u , u 23 d 0 x a, b u dx u Euler-Lagrange Equations T x, u , u 0 u x bNatural (Newmann)boundary conditionsu ( x) x a u (a) pEssential (Dirichlet)boundary conditions
Variational Form of a ContinuumMechanics Problem Consider a continuum mechanics problem with local or stronggoverning equations given by, Euler-Lagrange equationsE(x, u(x), u(x)) 0 x V with boundary conditions: Natural or NewmannT(x, u(x), u(x)) ( u) n t* (x) 0 x 24Forced (essential) or Dirichletu x u x x uREMARKThe Euler-Lagrange equationsare generally a set of PDEs.
Variational Form of a ContinuumMechanics Problem The variational form of the continuum mechanics problem consistsin finding a field u x X where V u(x) : V R RV : u x : V R3 Rm u x u x on u3m0 u(x) 0 on u fulfilling: E(x, u(x), u(x)) u(x)dV V25 T(x, u(x), u(x)) u(x) d 0 u(x) V0
Variational Form of a ContinuumMechanics ProblemREMARK 1The local or strong governing equations of the continuum mechanicsare the Euler-Lagrange equation and natural boundary conditions.REMARK 2The fundamental theorem of variational calculus guarantees that thesolution given by the variational principle and the one given by the localgoverning equations is the same solution.26
11.4. Virtual Work PrincipleCh.11. Variational Principles27
Governing Equations Continuum mechanics problem for a body: Cauchy equation x, t ( ( u(x,t ))) 2u x, t 0b x, t 0in V2 tBoundary conditionsu x, t u x, t on u x, t n x, t t x, t on s( ( u),t)28
Variational Principle The variational principle consists in finding a displacement field, where V : u x, t : V R3 Rm u x, t u x, t on u such that the variational principle holds, 2u W u; u [ (b 2 )] u dV (t n) u d 0 u V0 t V E where V0 : u x : V R3 Rm u x 0 on u Note: 29 Tis the space of admissible displacements.is the space of admissible virtual displacements (test functions).The (perturbations of the displacements ) u are termed virtualdisplacements.
Virtual Work Principle (VWP) The first term in the variational principle a 2u W u; u [ (b 2 )] u dV t n u d 0 u V0 tV T EConsidering that u u s uand (applying the divergence theorem): u dVV s (n ud u dV VThen, the Virtual Work Principle reads: W u; u b a u dV t* u d s u dV 0V30 V u V0
Virtual Work Principle (VWP)REMARK 1The Cauchy equation and the equilibrium of tractions at the boundaryare, respectively, the Euler-Lagrange equations and natural boundaryconditions associated to the Virtual Work Principle.REMARK 2The Virtual Work Principle can be viewed as the variational principleassociated to a functional W u , being the necessary condition to finda minimum of this functional.31
Interpretation of the VWP The VWP can be interpreted as: W u; u b a u dV t * u d s u dV 0V b* Vpseudo body forces virtualstrainsWork by the pseudo-bodyforces and the contact forces.External virtual work WextWork by thevirtual strain.Internal virtualwork W int W u; u W ext W int 0 u V032
VWP in Voigt’s Notation Engineering notation uses vectors instead of tensors: x y z 6 R ; xy xz yz x x y y not z z 6 R ; 2 xy xy 2 xz xz yz 2 yz : The Virtual Work Principle becomes W dV b a u dV t* u d 0 u V0Total virtualwork.33VInternal virtualintwork, W . VExternal virtualex twork, W .
11.5. Minimum Potential Energy PrincipleCh.11. Variational Principles34
Hypothesis An explicit expression of the functional W in the VWP can only beobtained under the following hypothesis:1.Linear elastic material. The elastic potential is:uˆ ( 2.3. 1 :2: uˆ ( : Conservative volume forces. The potential is:(quasi-static problem, a 0) u b u u ) b uConservative surface forces. The potential is: G u t u G u) t uThen a functional, total potential energy, can be defined asU u uˆ ( dV u dV G u dSVs (u))35ElasticenergyV Potential energy of Potential energy ofthe body forcesthe surface forcesMMC - ETSECCPB - UPC08/01/2016
Potential Energy Variational Principle The variational form consists in finding a displacement fieldu(x, t ) V , such that for any u u 0 in u the followingcondition holds, u G u uˆ S U u; u : u dV u dV u d 0 u uVV * U u; u : dV b a u dV V V*t u d u V0 This is equivalent to the VWP previously defined. W U u; u 36 t b
Minimization of the Potential Energy The VWP is obtained as the variational principle associated withthis functional U , the potential energy.deriving from apotentialThe potential energy is1U(u) (u) C (u) dV b a(u) u dV t* u d 2VV This function has an extremum (which can be proven to be a minimum) forthe solution of the linear elastic problem.The solution provided by the VWP can be viewed in this case asthe solution which minimizes the total potential energy functional. U(u; u) 0 u V037
SummaryCh.11. Variational Principles38
Summary Function spaceRX : u x : R 3 R m FunctionalF u : X RF u u ( x)dxba f x, u ( x), u ( x) dxaGâteaux derivatived F u; : F u d 0 bXb u ( x)dxu x au x : a, b RIn terms of functionals: F u; u E x, u x , u x u d T x, u x , u x u d Necessary condition for the functional to have an extremum at u x : F u; u 0 u u x 0 u u x 0uVariational F u; u E u d T u d 0 Principle39u
Summary (cont’d) Fundamental Theorem of Variational Calculus E x, u x , u x u d T x, u x , u x u d 0 u u x 0uis satisfied if and only if:E x, u x , u x 0 x Euler-Lagrange equationsT x, u x , u x 0 x Natural boundary conditions Virtual Work Principle W u; u b a u d t * u d s u d 0 V : u x, t : R3 Rm u x, t u x, t on u40
Summary (cont’d) Interpretation of the Virtual Work PrincipleTotal virtualwork. W u; u Vb a u dV t * u d s u dV 0 b*Vpseudo body forces virtualstrainsExternal virtual work W ext Total potential energy: W intU u uˆ dV u dV G u d VElasticenergy41Internal virtualworkV PotentialPotentialenergy of the energy of thebody forces contact forces
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
Agenda 1 Variational Principle in Statics 2 Variational Principle in Statics under Constraints 3 Variational Principle in Dynamics 4 Variational Principle in Dynamics under Constraints Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Variational Principles 2 / 69
II. VARIATIONAL PRINCIPLES IN CONTINUUM MECHANICS 4. Introduction 12 5. The Self-Adjointness Condition of Vainberg 18 6. A Variational Formulation of In viscid Fluid Mechanics . . 25 7. Variational Principles for Ross by Waves in a Shallow Basin and in the "13-P.lane" Model . 37 8. The Variational Formulation of a Plasma . 9.
Action principles in Lagrangian/Hamiltonian formulations of electrodynamics Schwinger variational principles for transmission lines, waveguides, scattering specialized variational principles for lasers and undulators (e.g. Xie) Variational Principles are Perhaps Better Known in
Stochastic Variational Inference. We develop a scal-able inference method for our model based on stochas-tic variational inference (SVI) (Hoffman et al., 2013), which combines variational inference with stochastic gra-dient estimation. Two key ingredients of our infer
2. Functional Variational Inference 2.1. Background Even though GPs offer a principled way of handling ence carries a cubic cost in the number of data points, thus preventing its applicability to large and high-dimensional datasets. Sparse variational methods [45, 14] overcome this issue by allowing one to compute variational posterior ap-
entropy is additive :- variational problem for A(q) . Matrix of Inference Methods EP, variational EM, VB, NBP, Gibbs EP, EM, VB, NBP, Gibbs EKF, UKF, moment matching (ADF) Particle filter Other Loopy BP Gibbs Jtree sparse linear algebra Gaussian BP Kalman filter Loopy BP, mean field, structured variational, EP, graph-cuts Gibbs
of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected recent developments in hybrid and nonlinear variational principles. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations and natural boundary conditions.
us88685733 agma 1012-f 1990 us88685736 agma 2003-b 1997 us88685805 agma 6110-f 1997 us88685810 agma 9004-a 1999 us88685815 agma 900-e 1995 de88686925 tgl 18790/01 1972-09 de88686928 tgl 18791/01 1982-06 de88686929 tgl 18791/02 1983-07 us88687101 a-a-20079 2002-08-20 us88687113 a-a-50800 1981-04-23 us88687199 a-a-59173 1998-03-04 us88687222 a-a-55106 1992-07-15 us88687243 a-a-20155 1992-11-16 .
