Lecture 11 Euler-Lagrange Equations - ERNETsuresh/me256/Lectures/Lecture11.pdf · Lecture 11...
Transcript of Lecture 11 Euler-Lagrange Equations - ERNETsuresh/me256/Lectures/Lecture11.pdf · Lecture 11...
Lecture 11 Euler-Lagrange Equations and their Extension to Multiple Functions and Multiple Derivatives in the integrand of the Functional
ME 256 , Indian Inst i tute of Sc ience Va r i a t i o n a l M e t h o d s a n d S t r u c t u r a l O p t i m i z a t i o n
G . K . An a n t h a s u r e s h P r o f e s s o r , M e c h a n i c a l E n g i n e e r i n g , I n d i a n I n s t i t u t e o f S c i e n c e , B a n a g a l o r e
s u r e s h @ m e c h e n g . i i s c . e r n e t . i n
1
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Outline of the lecture Euler-Lagrange equations Boundary conditions Multiple functions Multiple derivatives What we will learn: First variation + integration by parts + fundamental lemma = Euler-Lagrange equations
How to derive boundary conditions (essential and natural) How to deal with multiple functions and multiple derivatives Generality of Euler-Lagrange equations
2
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
The simplest functional, F(y,y’)
3
First variation of J w.r.t. y(x).
The condition given above should hold good for any variation of y(x), i.e., for any But there is , which we will get rid of it through integration by parts.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Integration by parts…
4
We can invoke fundamental lemma of calculus of variations now.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Fundamental lemma…
5
The two terms are equated to zero because the first term depends on the entire function whereas the second term only on the value of the function at the ends.
The integral should be zero for any value of . So, by fundamental lemma (Lecture 10), the integrand should be zero at every point in the domain.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Boundary conditions
6
The algebraic sum of the two terms may be zero without the two terms being equal to zero individually. We will see those cases later. For now, we will take the general case of both terms individually being equal to zero. Thus,
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Euler-Lagrange (EL) equation with boundary conditions
7
and
Problem statement
Differential equation
Boundary conditions
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Example 1: a bar under axial load
8
Axial displacement =
Principle of minimum potential energy (PE)
Strain energy Work potential
= area of cross-section
Among all possible axial displacement functions, the one that minimizes PE is the stable static equilibrium solution.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Bar problem: E-L equation
9
Integrand of the PE
Governing differential equation
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Bar problem: boundary conditions
10
δ
δ
′ = = =
′ = =
0 or 0 at 0and
or 0 at
EAu u x
EAu u x L
This means that y is specified; hence, its variation is zero. This is called the essential or Dirichlet boundary condition.
This means that the stress is zero when the displacement is not specified. It is called the natural or Neumann boundary condition.
δ = 0u
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Weak form of the governing equation
11
( )δ δ δ δ′ ′= − =∫0
( ) ( ) ( ) ( ) 0 for anyL
uPE E x A x u x u p x u dx u
( ) ( )δ δ′ ′ =∫ ∫0 0
( ) ( ) ( ) ( )L L
E x A x u x u dx p x u dx
Internal virtual work = external virtual work
First variation is zero.
δuVariation of u is like virtual displacement.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Three ways for static equilibrium
12
( )δ
δ
′′ + =
′ = = =
′ = =
00 or 0 at 0
andor 0 at
EAu pEAu u x
EAu u x L
Minimum potential energy principle
Principle of virtual work; The weak form
Force balance; And boundary conditions. The strong form.
Q: What is “weak” about the weak form? A: It needs derivative of one less order.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc 13
From Slide 7 in Lecture 3
So, straight line in indeed the geodesic in a plane.
Example 2: is a straight line really the least-distance curve in a plane?
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Example 3: Brachistochrone problem
14
From Slide 11 in Lecture 2
H
g
B
A Minimize
L And we have Dirichlet (essential) boundary conditions at both the ends.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
A functional with two derivatives: F(y,y’,y’’)
15
First variation of J w.r.t. y(x).
We now need to integrate by parts twice to get rid of the second derivative of y.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Integration by parts… twice!
16
δ δ δ δ δ δ δ
δ δ δ δ
∂ ∂ ∂ ∂ ∂ ∂′ ′′ ′ ′′= + + = + + = ′ ′′ ′ ′′∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ′⇒ + − + − ′ ′ ′′ ′′∂ ∂ ∂ ∂ ∂
∫ ∫ ∫ ∫
∫ ∫
2 2 2 2
1 1 1 1
2 22 2
1 11 1
0x x x x
yx x x x
x xx x
x xx x
F F F F F FJ y y y dx y dx y dx y dxy y y y y y
F F d F F d Fy dx y y dx yy y dx y y dx
δ
δ δ δ
′ =
∂ ∂ ∂ ∂ ∂ ∂ ′⇒ − + + − + = ′ ′′ ′ ′′ ′′∂ ∂ ∂ ∂ ∂ ∂
∫
∫
2
1
2 22
1 11
2
2
0
0
x
x
x xx
x xx
y dxy
F d F d F F d F Fydx y yy dx y y y dx y ydx
= 0 gives differential equation by using the fundamental lemma.
Two sets of boundary conditions
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
E-L equation and BCs for F(y,y’,y’’)
17
Things are getting lengthy; Let us use short-hand notation.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Example 4: beam deformation
18
From Slide 27 in Lecture 3
When E and I are uniform, we get the familiar:
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Boundary conditions for the beam
19
( ) δ′′ ′ =0
0L
EIw w
Physical interpretation
Either shear stress is zero or the transverse displacement is specified.
Either bending moment is zero or the slope is specified.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Do we see a trend for multiple derivatives in the functional?
20
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Three derivatives… F(y,y’,y”,y’’’)
21
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc 22
Many derivatives… F(y,y’,y”,…y(n))
Most general form with one function and many derivatives
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
What if we have two functions?
23
Now, we need to take the first variation with respect to both the functions, separately.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
What if we have two functions? (contd.)
24
And, we will have two differential equations and two sets of boundary conditions. Two unknown functions need two differential equations and two sets of BCs. That is all!
and
and
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
Most general form: m functions with n derivatives.
25
The most general form when we have one independent variable x.
Variational Methods and Structural Optimization ME256 / G. K. Ananthasuresh, IISc
The end note
26
Thanks Eule
r-La
gran
ge e
quat
ions
and
thei
r ext
ensi
on
to m
ultip
le fu
nctio
ns a
nd m
ultip
le d
eriv
ativ
es
Dealing with multiple derivatives along with boundary conditions (need to do integration by parts as many times as the order of the highest derivative)
General form of Euler-Lagrange equations in one independent variable
Euler-Lagrange equations = first variation + integration by parts + fundamental lemma
Boundary conditions Essential (Dirichlet) Natural (Neumann)
Dealing with multiple functions (rather easy)