CH.11. VARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC) - - PowerPoint PPT Presentation

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 : R X b  ( )   u x dx       3 : : m X u x R R   a b F u     , ( ), ( ) f x u x u x dx    The elements of are functions X u x a b of an arbitrary tensor order, defined in X   ( ) u x dx     R 3 a subset . u x a     a b  R : , u x    A functional is a mapping of the function space onto the X F u   :  set of the real numbers , : . R F u X R    It is a function that takes an element of the function space as X u x its input argument and returns a scalar. 9

Definition of Gâteaux Derivative  Consider :        3  : :  a function space m X u x R R   :   the functional F u X R  R  a perturbation parameter      a perturbation direction x X           The function is the perturbed function of in u x x X u x    the direction. x t=0 t   Ω u x P P’     Ω 0 x        u x x 10

Definition of Gâteaux Derivative     The Gâteaux derivative of the functional in the direction is: F u   d          ; : F F u u P’  d    0 F u t=0 t   Ω u x P P’   Ω 0   x        u x x REMARK not    u The perturbation direction is often denoted as .  u x ( ) Do not confuse with the differential . ( ) d u x  u x ( ) is not necessarily small !!! 11

Example Find the Gâteaux derivative of the functional              : F u u d u d   12

Example - Solution Find the Gâteaux derivative of the functional              : F u u d u d   Solution : d d d                           ; F u u F u u u u d u u d    d d d  0     0 0                       u u d u u u u d u u             d d         u d u d     0 0   u   u   ( ) ( ) u u            F u u d u d   u u   13

Gâteaux Derivative with boundary conditions  Consider a function space : V                 u     m * : : ; V u x u x R u x u x  x u  By definition, when performing the Gâteaux derivative on , V     . u u V  Then,          0 *     * u u u u x u u u     x x x x u u u u  u *    The direction perturbation must satisfy: u 0  x u 14

Gâteaux Derivative in terms of Functionals  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 family of functionals can be written as,   u                ; ( , ( ), ( )) ( , ( ), ( )) F u u E x u x u x u d T x u x u x u d   u 0     x u REMARK              : The example showed that for , the F u u u d d     ( ) ( ) u u     Gâteaux derivative is .        F u u d u d   u u   15

Extrema of a Function  A function has a local minimum (maximum) at x 0  Necessary condition: ( ) df x not      0 f x 0 dx  x x 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 has a minimum at F u V R u x V    Necessary condition for the functional to have extrema at : u x          ; 0 | F u u u u 0  x u  This can be re-written in integral form:                 ; ( ) ( ) 0 u F u u E u u d T u u d      u 0 Variational Principle  x u 17

11.3.Variational Principle Ch.11. Variational Principles 18

Variational Principle  Variational Principle :                 u ; 0 REMARK F E T u u u d u d       Note that u 0 u  x u is arbitrary.  Fundamental Theorem of Variational Calculus : The expression     u          ( , ( ), ( )) ( , ( ), ( )) 0 E x u x u x u d T x u x u x u d      u 0  x is satisfied if and only if u     ( , ( ), ( )) 0 Euler-Lagrange equations E x u x u x x     ( , ( ), ( )) 0 Natural boundary conditions T x u x u x x  19

Example Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional b                        , , with : , ; F u x u x u x dx u x a b R u x u a p    x a a 20