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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
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Ch.11. Variational Principles
11.1. Introduction
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.
The Variational Approach
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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.
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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.
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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.
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Ch.11. Variational Principles
11.2. Functionals
Definition of Functional
Consider a function space :
The elements of are functions of an arbitrary tensor order, defined in a subset .
A functional is a mapping of the function space onto the set of the real numbers , : . It is a function that takes an element of the function space as
its input argument and returns a scalar.
X
3: : m u xX R R
X u x
3R
: uF X R uF
RX
u x X
X u x
R( )
b
a
u x dx
( )b
a
u x dx
, ( ), ( )b
a
f x u x u x dx uF
: ,u x a b R
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Definition of Gâteaux Derivative
Consider : a function space the functional a perturbation parameter a perturbation direction
The function is the perturbed function of in the direction.
3: : m u xX R R : uF X R
R x X
u x x X u x x
Ω0
Ω
t=0
P P’
t
u x
x u x x
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Definition of Gâteaux Derivative
The Gâteaux derivative of the functional in the direction is: uF
0
; : dd
u uF F
Ω0
Ω
t=0
P P’
t
u x
x u x x
P’ F u
REMARKThe perturbation direction is often denoted as .Do not confuse with the differential .
is not necessarily small !!!
not u
( )u x ( )du x( )u x
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Example
Find the Gâteaux derivative of the functional
: d d
u u uF
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Example - Solution
Find the Gâteaux derivative of the functional
Solution :
: d d
u u uF
0 0
0 0
0
; d d dd dd d d
d dd d
d d
u u u u u u u u
u u u u u u u uu u
F F
( ) ( ) d d
u uu u u
u uF
u u
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Gâteaux Derivative with boundary conditions
Consider a function space :
By definition, when performing the Gâteaux derivative on , .
Then,
The direction perturbation must satisfy:
V m *: : ;
u
xu x u x u x u xV R
V u u V
*
u
xu u u x *
u u
x xu u u 0
u
xu
u
xu 0
* u
u
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Consider the family of functionals
The Gâteaux derivative of this family of functionals can be written as,
Gâteaux Derivative in terms of Functionals
( , ( ), ( ))
( , ( ), ( ))
d
d
u x u x u x
x u x u x
F
; ( , ( ), ( )) ( , ( ), ( ))d d
u u x u x u x u x u x u x u F E T
REMARKThe example showed that for , the
Gâteaux derivative is . ( ) ( ) d d
u uu u uu u
F
: d d
u u uF
u
u
x
uu 0
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A function has a local minimum (maximum) at
Necessary condition:
The same condition is necessary for the function to have extrema(maximum, minimum or saddle point) at .
This concept can be can be extended to functionals.
Extrema of a Function
0x
Local minimum
0
0( ) 0
not
x x
df x f xdx
0x
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A functional has a minimum at
Necessary condition for the functional to have extrema at :
This can be re-written in integral form:
Extreme of a Functional. Variational principle
: uF V R u x V
u x
; 0 |u
x
u u u u 0F
; ( ) ( ) 0d d
u u u u u uF E T
Variational Principle u
x
uu 0
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Ch.11. Variational Principles
11.3.Variational Principle
Variational Principle:
Fundamental Theorem of Variational Calculus:The expression
is satisfied if and only if
Variational Principle
; 0d d
u u u uF E T
REMARKNote that is arbitrary.
u
( , ( ), ( )) 0 x u x u x xT
( , ( ), ( )) 0 x u x u x xE Euler-Lagrange equations
Natural boundary conditions
( , ( ), ( )) ( , ( ), ( )) 0d d
x u x u x u x u x u x u E T
u
x
uu 0
u
x
uu 0
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Example
Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional
, , : , ;b
x aa
u x u x u x dx u x a b u x u a p
F Rwith
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Example - Solution
Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional
Solution :
First, the Gâteaux derivative must be obtained. The function is perturbed:
This is replaced in the functional:
, ,b
x aa
u x u x u x dx u x u a p
F with
u x | 0
not
a
u x u x xx u x a
u x u x x
, ,b
a
u x u x u x dx F
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Example - Solution
The Gâteaux derivative will be
Then, the expression obtained must be manipulated so that it resembles the Variational Principle :
Integrating by parts the second term in the expression obtained:
The Gâteaux derivative is re-written as:
0
;b
a
ddu u dx
u u
F F
; 0d d
u u u uF E T
( ) ( )b
b b b
b aa a aa b a
d ddx dx dxu u dx u u u dx u
( (
, , ;
; ) ; ) [ ( )]
b
a
b
babu
u x u x u x dx u a p
du u u udx uu dx u u
0a
, ,b
a
x a
u x u x u x dx
u x u a p
F
MMC - ETSECCPB - UPC22
Example - Solution
Therefore, the Variational Principle takes the form
If this is compared to , one obtains:
( ; ) [ ( )]b
bab
du u udx uu dx u u
; 0d d
u u u uF E T
0a
uu
, , 0 ,dx u u x a bu dx u
E Euler-Lagrange Equations
Natural (Newmann) boundary conditions
Essential (Dirichlet) boundary conditions
( ) ( )x a
u x u a p
, , 0x b
x u uu
T
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Consider a continuum mechanics problem with local or strong governing equations given by, Euler-Lagrange equations
with boundary conditions: Natural or Newmann
Forced (essential) or Dirichlet
Variational Form of a Continuum Mechanics Problem
( , ( ), ( )) 0 V x u x u x xE
*( , ( ), ( )) ( ) ( ) x u x u x u n t x 0 x T
u u x u x x REMARK
The Euler-Lagrange equations are generally a set of PDEs.
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The variational form of the continuum mechanics problem consists in finding a field where
fulfilling:
Variational Form of a Continuum Mechanics Problem
u x X
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: : on
( ) : ( ) on
mu
mu
V
V
u x u x u x
u x u x 0
V R R
V R R
0( , ( ), ( )) ( ) ( , ( ), ( )) ( ) 0 ( )V
dV d
x u x u x u x x u x u x u x u x E T V
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Variational Form of a Continuum Mechanics Problem
REMARK 1The local or strong governing equations of the continuum mechanics are the Euler-Lagrange equation and natural boundary conditions.
REMARK 2The 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.
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Ch.11. Variational Principles
11.4. Virtual Work Principle
Continuum mechanics problem for a body: Cauchy equation
Boundary conditions
Governing Equations
2
0 0 2
( ( ( , )))
,, ,
t
tt t V
t
u x
u xx b x
in
( ( ),t)
, t , t , t
u
x n x t x
s
on
, , ut t u x u x on
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The variational principle consists in finding a displacement field , where
such that the variational principle holds,
where Note: is the space of admissible displacements. is the space of admissible virtual displacements (test functions). The (perturbations of the displacements ) are termed virtual
displacements.
2
02; [ ( )] ( ) 0V
dV dt
uu u b u t n u uW V
T
Variational Principle
3: , : , , onmut V t t u x u x u xV R R
30 : : onm
uV u x u x 0V R R
u
E
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The first term in the variational principle
Considering that
and (applying the divergence theorem):
Then, the Virtual Work Principle reads:
Virtual Work Principle (VWP)
s u u u
s( dV V
dV dV
u n u u
* s0; 0
V V
dV d dV
u u b a u t u u u
W V
2
02; [ ( )] 0V
dV dt
uu u b u t n u u W V
E T
a
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Virtual Work Principle (VWP)
REMARK 1The 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 2The Virtual Work Principle can be viewed as the variational principle associated to a functional , being the necessary condition to find a minimum of this functional.
uW
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The VWP can be interpreted as:
Interpretation of the VWP
* s
V V
; 0
*
dV d dV
u u b a u t u u
b
pseudo - virtualbody forces strains
W
Work by the pseudo-body forces and the contact forces.
External virtual work
Work by the virtual strain.
Internal virtual work
intWextW
0ext int; u u 0 uW W W V
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Engineering notation uses vectors instead of tensors:
The Virtual Work Principle becomes
VWP in Voigt’s Notation
x x x
y y y
notz z z6 6
xy xy xy
xz xz xz
yz yz yz
; ;2
22
R R :
*00
V V
dV dV d
b a u t u u W VTotal virtual
work.Internal virtual work, .intW
External virtual work, .
ex tW
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Ch.11. Variational Principles
11.5. Minimum Potential Energy Principle
An explicit expression of the functional in the VWP can only be obtained under the following hypothesis:1. Linear elastic material. The elastic potential is:
2. Conservative volume forces. The potential is: (quasi-static problem, )
3. Conservative surface forces. The potential is:
Then a functional, total potential energy, can be defined as
Hypothesis
W
a 0
)GG
uu t u t
u
)
uu b u bu
ˆ1 (ˆ( : : :2
uu
ˆ(V Vu dV dV G dS
u u uU
Elastic energy
Potential energy of the body forces
Potential energy of the surface forces( ))s u
08/01/2016MMC - ETSECCPB - UPC35
The variational form consists in finding a displacement field , such that for any the following
condition holds,
This is equivalent to the VWP previously defined.
Potential Energy Variational Principle
( , )t u x V u0u u in
*0; d
V V
dV dV
u u : b a u t u u U V
; u uW U
;
ˆ 0S
V V
Gu dV dV d
u u
u u: u u u
u uU
b
* t
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The VWP is obtained as the variational principle associated with this functional , the potential energy.
The potential energy is
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.
Minimization of the Potential Energy
U
0( ; ) 0 u u uU V
*1( ) ( ) ( ) ( ) 2
u u u b a u u t uV V
dV dV d
U C
deriving from a potential
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Ch.11. Variational Principles
Summary
Function space
Functional
Gâteaux derivative
In terms of functionals:
Necessary condition for the functional to have an extremum at :
Summary
: uF X R
X u x
R( )
b
a
u x dx
( )b
a
u x dx
, ( ), ( )b
a
f x u x u x dx uF
: ,u x a b R
0
; : dd
u uF F
; , , , ,d d
u u x u x u x u x u x u x uF E T
u
x
uu 0
Variational Principle ; 0d d
u u u uF E T
u x ; 0 |
u
xu u u u 0F
3: : m u xX R R
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Fundamental Theorem of Variational Calculus
is satisfied if and only if:
Virtual Work Principle
Summary (cont’d)
, , , , 0d d
x u x u x u x u x u x uE T
u
x
uu 0
, , 0 x u x u x xT
, , 0 x u x u x xE Euler-Lagrange equations
Natural boundary conditions
* s; 0d d d
u u b a u t u u W
3: , : , , onmut t t u x u x u xV R R
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Interpretation of the Virtual Work Principle
Total potential energy:
Summary (cont’d)
* s; 0
*V V
dV d dV
u u b a u t u u
b
pseudo - virtualbody forces strains
W
External virtual workInternal virtual
workintWextW
ˆV Vu dV dV G d
u u uU
Elastic energy
Potential energy of the body forces
Potential energy of the contact forces
Total virtual work.
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