CH.11. VARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC) - - - PowerPoint PPT Presentation
CH.11. VARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Introduction Functionals Gteaux Derivative Extreme of a Functional Variational Principle Variational Form of a Continuum Mechanics
Continuum Mechanics Course (MMC) - ETSECCPB - UPC
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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For any physical system we want to describe, there will be a
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
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This is essentially the same procedure one does for finding the
A variational principle is a mathematical method for determining
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In computational mechanics physical mechanics problems are
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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Numerical Methods use algorithms which solve problems through
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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Consider a function space : The elements of are functions
a subset .
A functional
It is a function that takes an element of the function space as
its input argument and returns a scalar.
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: :
m
u x X R R
u x
3
u F X R
u F
u x
u x
( )
b a
u x dx
( )
b a
u x dx
, ( ), ( )
b a
f x u x u x dx
u F
: , u x a b R
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Consider : a function space the functional a perturbation parameter a perturbation direction The function is the perturbed function of in
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: :
m
u x X R R
:
u F X 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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The Gâteaux derivative of the functional in the direction is:
u F
; : d d
u u F F
Ω0 Ω
t=0
P P’
t
u x
x
u x x
P’
F u
REMARK The 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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Find the Gâteaux derivative of the functional
: d d
u u u F
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Find the Gâteaux derivative of the functional Solution :
: d d
u u u F
; d d d d d d d d d d d d d d
u u u u u u u u u u u u u u u u u u F F
( ) ( ) d d
u u u u u u u F
u u
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Consider a function space : By definition, when performing the Gâteaux derivative on ,
Then, The direction perturbation must satisfy:
m *
: : ;
u
x
u x u x u x u x V R
u u V
*
u
x
u u u x
*
u u
x x
u u u
u
x
u
u
x
u
*
u
u
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Consider the family of functionals The Gâteaux derivative of this family
( , ( ), ( )) ( , ( ), ( )) 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
REMARK The example showed that for , the Gâteaux derivative is .
( ) ( ) d d
u u u u u u u F
: d d
u u u F
u
u
x
u u
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A function
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.
Local minimum
( )
not x x
df x f x dx
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A functional
Necessary condition for the functional to have extrema at : This can be re-written in integral form:
u F V R
u x V
u x
u
x
; ( ) ( ) d d
u u u u u u F E T
Variational Principle
u
x
u u
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Variational Principle: Fundamental Theorem of Variational Calculus:
; d d
u u u u F E T
REMARK Note that is arbitrary. u
( , ( ), ( ))
x u x u x x T ( , ( ), ( )) x u x u x x E
Euler-Lagrange equations Natural boundary conditions
( , ( ), ( )) ( , ( ), ( )) d d
x u x u x u x u x u x u E T
u
x
u u
u
x
u u
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Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional
, , : , ;
b x a a
u x u x u x dx u x a b u x u a p
F R with
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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 a a
u x u x u x dx u x u a p
F with
u x
|
not a
u x u x x x u x a u x u x x
, ,
b a
u x u x u x dx
F
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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:
;
b a
d d
u u dx u u
F F
; d d
u u u u F E T ( ) ( )
b b b b b a a a a a b a
d d dx dx dx u u dx u u u dx u
( (
, , ; ; ) ; ) [ ( )]
b a b b a b
u
u x u x u x dx u a p d u u u udx u u dx u u
a
, ,
b a x a
u x u x u x dx u x u a p
F
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Therefore, the Variational Principle takes the form If this is compared to , one obtains:
( ;
) [ ( )]
b b a b
d u u udx u u dx u u
; d d
u u u u F E T
a
u u
, , , d x u u x a b u dx u E Euler-Lagrange Equations Natural (Newmann) boundary conditions Essential (Dirichlet) boundary conditions ( ) ( )
x a
u x u a p
, ,
x b
x u u u
T
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Consider a continuum mechanics problem with local or strong
Euler-Lagrange equations with boundary conditions:
Natural or Newmann Forced (essential) or Dirichlet
( , ( ), ( )) V x u x u x x E
*
( , ( ), ( )) ( ) ( )
x u x u x u n t x 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
u x X
3 3
: :
( ): ( )
m u m u
V V
u x u x u x u x u x V R R V R R ( , ( ), ( )) ( ) ( , ( ), ( )) ( )
( )
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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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.
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Continuum mechanics problem for a body: Cauchy equation Boundary conditions
2 2
( ( ( , )))
, , ,
t
t t t V t
u x
u x x b x in
( ( ),t)
,t ,t ,t
u
x n x t x
s
, ,
u
t t
u x u x
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The variational principle consists in finding a displacement field
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 2
; [ ( )] ( )
V
dV d t
u u u b u t n u u W V
T
3
: , : , ,
m u
t V t t
u x u x u x V R R
3 0 :
:
m u
V u x u x V R R u
E
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The first term in the variational principle
Then, the Virtual Work Principle reads:
s
u u u
s
( d
V V
dV dV
u n u u
* s
;
V V
dV d dV
u u b a u t u u u
W V
2 2
; [ ( )]
V
dV d t
u u u b u t n u u W V
E T a
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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 , being the necessary condition to find a minimum of this functional.
u W
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The VWP can be interpreted as:
* s V V
; * dV d dV
u u b a u t u u b
pseudo - virtual body forces strains
W
Work by the pseudo-body forces and the contact forces. External virtual work Work by the virtual strain. Internal virtual work
int
ext
ext int
; u u u W W W V
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Engineering notation uses vectors instead of tensors: The Virtual Work Principle becomes
x x x y y y not z z z 6 6 xy xy xy xz xz xz yz yz yz
; ; 2 2 2 R R
:
* V V
dV dV d
b a u t u u W V
Total virtual work. Internal virtual work, .
int
External virtual work, .
ex t
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An explicit expression of the functional in the VWP can only be
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:
a
) G G
u u t u t u
) u u b u b u ˆ 1 ( ˆ( : : : 2 u u
ˆ(
V V
u dV dV G dS
u u u U
Elastic energy Potential energy of the body forces Potential energy of the surface forces
( ))
s u
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The variational form consists in finding a displacement field
This is equivalent to the VWP previously defined.
( , ) t u x V
u
u u in
*
;
d
V V
dV dV
u u
: b a u t u u U V
; u u W U
;
ˆ
S V V
G u dV dV d
u u
u u : u u u u u U
b * t
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The VWP is obtained as the variational principle associated with
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
U
*
1 ( ) ( ) ( ) ( ) 2 u u u b a u u t u
V V
dV dV d
U C
deriving from a potential
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Function space Functional Gâteaux derivative
In terms of functionals:
Necessary condition for the functional to have an extremum at :
u F X R
u x
( )
b a
u x dx
( )
b a
u x dx
, ( ), ( )
b a
f x u x u x dx
u F
: , u x a b R
; : d d
u u F F
; , , , , d d
u u x u x u x u x u x u x u F E T
u
x
u u
Variational Principle
; d d
u u u u F E T
u x
; |
u
x
u u u u F
3
: :
m
u x X R R
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Fundamental Theorem of Variational Calculus
is satisfied if and only if:
Virtual Work Principle
, , , , d d
x u x u x u x u x u x u E T
u
x
u u
, , x u x u x x T
, , x u x u x x E
Euler-Lagrange equations Natural boundary conditions
* s
; d d d
u u b a u t u u W
3
: , : , ,
m u
t t t
u x u x u x V R R
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Interpretation of the Virtual Work Principle Total potential energy:
* s
; *
V V
dV d dV
u u b a u t u u b
pseudo - virtual body forces strains
W
External virtual work Internal virtual work
int
ext
ˆ
V V
u dV dV G d
u u u U
Elastic energy Potential energy of the body forces Potential energy of the contact forces Total virtual work.
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