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  XRR::ux   a u b 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 Fu 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 FXRu:   a perturbation parameter R  a perturbation direction xX

 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 FFuu;:   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 FFuu;   uu    uu  dd    uu    dd d  0 0 0 uu  dd uu    uu   uu    dd  uudd  0  0  u  u

()uu  () Fuu  dd  u uu

13 Gâteaux Derivative with boundary conditions

 Consider a function space V : u  m*  VR::;ux ux ux  u x xu

 By definition, when performing the Gâteaux derivative on V , uu  V .  Then, * *  u 0 uu  ux  uuu   x xu xxuu  u  u*

 The direction perturbation must satisfy: u0 xu

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  xu 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 FVRu:  has a minimum at ux    V

 Necessary condition for the functional to have extrema at ux   :

 uu;0  u | u  0 F xu

 This can be re-written in integral form:

uu;()()0   uudd uu u FE T  u0 Variational Principle xu

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 dx with 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 dx with ux ua  p F   xa a Solution : First, the Gâteaux derivative must be obtained.  The function ux   is perturbed: ux ux   x  not  xuxa|  0    a ux ux  x    This is replaced in the functional: b Fuxuxuxdx     ,,     a

21 b Fuxuxuxdx   ,,    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 bb dd     b  dx() dx ba   () dx aa  a uudxuuudxuaba      

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  Exuu,,   0 x ab , Euler-Lagrange Equations udxu

 Natural (Newmann) Txuu,,   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::uxV 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 2ux ,t  xbx,,tt00  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:,:uxtV ux , t  u x ,on t  u such that the variational principle holds,

2  u  WVuu;[    ()]( b  udV t  n ) u d 0 u 0 t2 V   T  E 3 m where VRR0 ::uxV 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  WVuu;[    ()] b  udV t  n u d 0 u 0 t2 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  u dV  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 WWWVuu;     0 u 0

32 VWP in Voigt’s Notation

 Engineering notation uses vectors instead of tensors:

xxx        yyy    not    66zzz    ;;             :    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. Internal virtual External virtual int ex t work,  W . work,  W .

33 11.5. Minimum Potential Energy Principle

Ch.11. Variational Principles

34 Hypothesis

 An explicit expression of the functional W in the VWP can only be obtained under the following hypothesis: 1. Linear elastic material. The elastic potential is: 1(uˆ  uˆ(::   :  2  2. Conservative volume forces. The potential is: u) (quasi-static problem, a0  ) ubu  b u Gu) 3. Conservative surface forces. The potential is: Gutu   t u  Then a functional, total potential energy, can be defined as uuuudVˆ(    dVG   dS U VV   Elastic Potential energy of s Potential energy of  ())u energy the body forces the surface forces 35 MMC - ETSECCPB - UPC 08/01/2016 Potential Energy Variational Principle

 The variational form consists in finding a displacement field

ux(,)t V , such that for any  uu  0 in  u the following condition holds,

uˆ S uu G  Uuu;  :u  dV u dV  u d 0  uu VV    t*   b

uu; :bautuu dV dV * d UV  0 VV   This is equivalent to the VWP previously defined.

 WU uu; 

36 Minimization of the Potential Energy

 The VWP is obtained as the variational principle associated with this functional , the potential energy. U deriving from a  The potential energy is potential

1 * UC()uuu () () dV  bauutu() dV d 2 VV

 This function has an extremum (which can be proven to be a minimum) for the solution of the linear elastic problem.  The solution provided by the VWP can be viewed in this case as the solution which minimizes the total potential energy functional.

 UV(;uu ) 0 u 0

37 Summary

Ch.11. Variational Principles

38 Summary

 Function space R ::ux 3 m b XRR uxdx()  Functional a u b FXRu:  F  f xux,(),() u x dx a b  Gâteaux derivative X uxdx() d  ux a FFuu;:     d 0 ux :, ab R  In terms of functionals: u  u;,,  u xu x u x udd xu ,, x u x u FE T  u0 x  Necessary condition for the functional to have an extremum at ux   : u  uu;0  u | u  0 F xu

Variational uu;0   udd u FE T Principle 

39 Summary (cont’d)

 Fundamental Theorem of Variational Calculus

xu,, x u x udd xu ,, x u x  u  0u ET  u0 xu is satisfied if and only if: Exu,, x u x 0 x Euler-Lagrange equations

Txu,, x u x 0 x  Natural boundary conditions

 Virtual Work Principle

*s Wuu;0 ba  u dd  t u u d 

3 m  VRR:,:uxttt  ux ,  u x ,on  u

40 Summary (cont’d)

 Interpretation of the Virtual Work Principle

Total virtual *s work. Wuu;0  ba    udV   t  u d   u dV  VV   b*   pseudo - virtual body forces strains

Internal virtual External virtual work work ext int  W  W 

 uuu udVˆ     dVGd  Total potential energy: U VV   Elastic Potential Potential energy energy of the energy of the body forces contact forces 41