Lecture 11
Euler‐Lagrange
Euler
Lagrange Equations
Equations
and their Extension to
Multiple Functions and Multiple Multiple
Functions and Multiple
Derivatives in the integrand of the Functional
in the integrand of the Functional
ME 256, Indian Institute of Science
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. Ananthasuresh
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
suresh@mecheng.iisc.ernet.in
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Outline of the lecture
E l
Euler‐Lagrange equations
Boundary conditions
Multiple functions
Multiple derivatives
What we will learn:
First variation + integration by parts + fundamental lemma = Euler‐
Lagrange equations
Lagrange equations
How to derive boundary conditions (essential and natural)
H
How to deal with multiple functions and multiple derivatives
t d l ith
lti l f
ti
d
lti l d i ti
Generality of Euler‐Lagrange equations
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The simplest functional, F(y,y’)
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
variation of y(x), i.e., for any But there is , which we will get rid of it through integration by parts.
g
yp
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Integration by parts…
We can invoke fundamental lemma of calculus of variations now.
now
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Fundamental lemma…
The two terms are h
equated to zero because the first term depends on the entire
depends on the entire function whereas the second term only on the value of the
the value of the function at the ends.
The integral should be zero for any value of
value of . So, by fundamental . So, by fundamental
lemma (Lecture 10), the integrand should be zero at every point in the domain
domain.
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Boundary conditions
The algebraic sum of the two terms may 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.
individually being equal to zero.
Thus,
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Euler‐Lagrange (EL) equation with boundary conditions
boundary conditions
P oble
Problem statement
tate e t
and
Differential equation
Differential equation
Boundary conditions
Boundary conditions
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Example 1: a bar under axial load
Axial displacement = Strain energy
Work potential
Principle of minimum potential energy (PE)
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= area of cross‐section
Among all possible axial
possible axial displacement functions, the one that
one that minimizes PE is the stable static q
equilibrium solution.
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Bar problem: E‐L equation
Integrand of the PE
Governing differential equation
Governing differential equation
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Bar problem: boundary conditions
u  0
EAu  0 or  u  0 at x  0
and
EAu or  u  0 at x  L
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This means that y is specified; hence, its variation is zero. This is called the essential or
is called the essential or Dirichlet boundary condition.
This means that the stress is zero Thi
th t th t
i
when the displacement is not specified. It is called the natural
or Neumann boundary or Neumann
boundary
condition.
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Weak form of the governing equation
First variation is zero.
u
Variation of u
is like virtual di l
displacement.
Internal virtual work = external virtual work
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Three ways for static equilibrium
Minimum potential energy principle
i i l
Principle of virtual work;
The weak form
 EAu  p  0
Force balance;
EAu  0 or  u  0 at x  0 And boundary conditions.
and
The strong form.
EAu or  u  0 at x  L
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Q: What is “weak”
about the weak form? b
h
kf
A: It needs derivative of one less order.
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Example 2: is a straight line really the least‐
distance curve in a plane?
distance curve in a plane?
From Slide 7 in Lecture 3
Min L 
y( x )
y(x)
(x1, y1 )
ds
((x2 , y2 )
x2

1 y dx
2
x1
Data : x1 , x2 , y(
y(x1 )  y1, y(
y(x2 )  y2
x
So, straight So
straight
line in indeed the geodesic in
geodesic in a plane.
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Example 3: Brachistochrone problem
From Slide 11 in Lecture 2
A
L
Minimize T  
y(x)
( )
H
0
1  y 
2
2g(H  y)
dx
y(x))
y(
g
L
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B
And we have Dirichlet (essential) boundary conditions at both the ends.
y
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A functional with two derivatives: F(y,y’,y’’)
First variation of J w.r.t.
w.r.t. y(x).
y(x).
We now need to integrate by parts twice
W
d i
b
i
to get rid of the second derivative of y.
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Integration by parts… twice!
= 0 gives differential equation by using the fundamental lemma.
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Two sets of boundary conditions
y
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E‐L equation and BCs for F(y,y’,y’’)
Things are getting lengthy;
Let us use short‐
hand notation.
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Example 4: beam deformation
From Slide 27 in Lecture 3

 1  d 2 w 
Min PE    EI  2   qqw  dx
w(x)
( )
2
d
dx


0


Data : q(x), E, I
L
2
2
1  d w
F  EI  2   qw
2  dx 
2
w(x )
q  0   EIw   0
  EIw   q
Wh
When E
and I
d are uniform, we get the familiar: if
h f ili
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EI  q
EIw
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iv
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Boundary conditions for the beam
2
1  d w
F  EI  2   qw
2  dx 
2
w(x )
Physical interpretation
Either shear stress Ei
h
h
is zero or the transverse displacement is
displacement is specified. Either bending moment is zero or the slope is h l
i
specified.
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Do we see a trend for multiple derivatives in the functional?
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Three derivatives… F(y,y’,y”,y’’’)
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(n)
Many derivatives… F(y,y’,y”,…y )
Most general Most
general
form with one function and many d i ti
derivatives
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What if we have two functions?
Now, we need to take the first variation with respect to both the
respect to both the functions, separately.
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What if we have two functions? (contd.)
And, we will have two differential differential
equations and two sets of b
boundary d
conditions.
Two unknown ou
o
functions need two differential
differential equations and two sets of BCs. That is all!
and
and
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Most general form: m functions with n
functions with n derivatives.
derivatives
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The most general form when we have one independent variable x.
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Eu
uler‐Lagra
ange equ
uations an
nd their eextension to multiple function
ns and mu
ultiple deerivatives
The end note
Euler‐Lagrange equations = first variation + integration by parts + fundamental lemma
Boundary conditions
Essential (Dirichlet)
Natural (Neumann)
Natural (Neumann)
Dealing with multiple derivatives along with boundary conditions
(need to do integration by parts as many times as the order of the highest
(need to do integration by parts as many times as the order of the highest derivative)
Dealing with multiple functions (rather easy)
g
p
(
y)
General form of Euler‐Lagrange equations in g
g q
one independent variable
ME256 / G. K. Ananthasuresh, IISc
Variational Methods and Structural Optimization
Th k
Thanks
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