module 1: lecture 1 classification of optimization problems ......solving two problems concerning...
TRANSCRIPT
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Lecture 16a
Size optimization of beams for stiffness and flexibility
ME 260 at the Indian Inst i tute of Sc ience , Bengaluru
Str uc tur a l O pt imiz a t io n : S iz e , Sha pe , a nd To po lo g y
G . K . Ana ntha s ur 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 @ i i s c . a c . i n
1
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Outline of the lectureSolving two problems concerning size optimization of beams for stiffness and flexibility with volume constraint.
What we will learn:
How to apply the eight steps we had used for bars to the case of beams.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Problem 1
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( ) ( )
( )0
*
0
2 *
( ) : 0 0
: 0
: ,q( ), , ,12
L
A x
L
Min MC q wdx
Subject to
x EIw q E Aw q
Adx V
tData L x E V
=
− = − =
−
=
We assume here that only b(x) is variable.
( )I x A=2
; 112
t
= =
Minimize the mean compliance of a beam for given volume of material.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Steps in the solution procedure
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Step 1: Write the Lagrangian
Step 2: Take variation of the Lagrangian w.r.t. the design variable and equate to zero to get the design equation.
Step 3: Take variation of the Lagrangian w.r.t. state variable(s) and equate to zero to get the adjoint equation(s).
Step 4: Collect all the equations, including the governing equation(s), complementarity condition(s), resource constraints, etc.
Step 5: Obtain the optimality criterion by substituting adjoint and equilibrium equations into the design equation, when it is possible.
Step 6: Identify all boundary conditions.
Step 7: Solve the equations analytically as much as possible.
Step 8: Use the optimality criteria method to solve the equations numerically.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Solution
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( ) ( )
( )0
*
0
2 *
( ) : 0 0
: 0
: ,q( ), , ,12
L
A x
L
Min MC q wdx
Subject to
x EIw q E Aw q
Adx V
tData L x E V
=
− = − =
−
=
Minimize the mean compliance of a beam for given volume of material.
( )( ) *0
L
L q w E Aw q A dx V = + − + −Step 1
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Expand the Lagrangian
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( )( ) *0
L
L q w E Aw q A dx V = + − + −Step 1
( )( )
( )
*
0
*
0
*
0
2
L
L
L
L q w E Aw q A dx V
qw E A w Aw q A dx V
qw E A w E A w E Aw q A dx V
= + − + −
= + + − + −
= + + + − + −
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Design equation
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*
0
2
L
L qw E A w E A w E Aw q A dx V = + + + − + −Step 2
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
0 0
2 0
2 0
0
0
A
F F FL
A A A
E w E w E w
E w E w E w
E
E
E
w
w
E w E
E
w w E w
Ew w E w
E
= − + =
+ − + =
+ − + + =
+ − + =
+ − − + +
=
+
( ) 0w =
2F qw E A w E A w E Aw q A = + + + − +
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Adjoint equation
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( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
0 0
0 2 0
2 0
0
w
F F F F FL
w w w w w
q E A E A E A
q E A E A E A E
E EA
A
q E A E A E A
q
= − + − + =
− + − + =
+ − + + =
+ − + =
+ − ( ) ( ) ( ) ( )
( )
0
0
A E EE A A
A
A
q E
− + + =
+ =
2F qw E A w E A w E Aw q A = + + + − + Step 3
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Collect all equations
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Step 4
( ) 0E w + =
Unknowns
Adjoint equation ( ) 0q E A + =
Governing equation ( ) 0E Aw q − =
Feasibility condition*
0
0
L
Adx V−
Complementarity condition
*
0
0, 0
L
Adx V
− =
w = −
2E w =
Strain energy density is uniform throughout the beam.
And, cannot be zero.
So, the volume constraint is active.
Step 5Design equation
( ), ( ), ( ),A x w x x
Three functions and one scalar variable.
We have three differential equations and one scalar equation.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Identify all boundary conditions
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Step 6
( )
− =
=
0
0
0
0
and
L
A A
L
A
F F A
F A
Boundary conditions for ( )A x
2F qw E A w E A w E Aw q A = + + + − +
( )
0
0
0
2 0
0
0
L
L
L
E w E w A
E w E w A
E ww E w w A
− =
− =
− + = since = −w
0
0 0
0
0 0
L
L L
E w A
E ww A E ww A
=
− = =
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Identify all boundary conditions
11
Step 6 Boundary conditions for ( )x
2F qw E A w E A w E Aw q A = + + + − +
( ) ( ) ( )
( ) ( )
( )
− + − =
− + =
− =
=
0
0
0
0
0
0
0
0
L
L
w w w w
w w w
L
w w
L
w
F F F F w
F F F w
F F w
F w
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Identify all boundary conditions
12
Step 6 Boundary conditions for ( )x
2F qw E A w E A w E Aw q A = + + + − +
( ) ( ) ( )
( ) ( )
( )
− + − =
− + =
− =
=
0
0
0
0
0
0
0
0
L
L
w w w w
w w w
L
w w
L
w
F F F F w
F F F w
F F w
F w
( )0
0L
E A w =
( ) 0
2 0
L
E A E A w − =
( ) ( ) 0
2 0
L
E A E A E A w − + =
( ) ( ) ( ) 0
2 0
L
E A E A E A w − + − =
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Simplify 1st adjoint boundary condition
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( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0
0
0
0
0 0
2 0
2 0
0
0
0 0
L
L
L
L
L L
E A E A E A w
E A E A E A E A w
E A E A E A w
E A E A E A E A w
E A E A w E A E A E A w
E
− + − =
− + − − =
− + − =
− + + − =
− = − − =
−( ) ( ) 00
00L
L
E AA w E A w + = =
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Simplify 2nd-4th adjoint boundary conditions
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( )0
0L
E A w =
( ) ( )0
002 0
LL
E A E A wE A E A w = −− =
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
0
0
0
00
2 0
2 0
0
0 0
L
L
L
L
L
E A E A E A w
E A E A E A E A w
E A E A E A w
E w E AE A wA
− + =
− + + =
− + =
− + = =
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
All four adjoint boundary conditions
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( )0
0L
E A E A w + =
( )0
0L
E A w =
( )0
0L
E A w =
( )0
0L
E A E A w − =
PinnedFixed0w w = = 0w w = =
0A =
0A A − =
0A =
No BC for ( )x No BC for ( )x
No BC for ( )x
No BC for ( )x
0A =
Notice that BCs of the state variable transfer to adjoint variable (most often). But be sure to keep the BCs on the design variable in mind.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
All four adjoint boundary conditions
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( )0
0L
E A E A w + =
( )0
0L
E A w =
( )0
0L
E A w =
( )0
0L
E A E A w − =
Transversely guided Free
0w w = =
0w w = =
0A A − =
0A =No BC for ( )x
No BC for ( )xNo BC for ( )x
No BC for ( )x
0A A + = 0A A + =
Notice that BCs of the state variable transfer to adjoint variable (most often). But be sure to keep the BCs on the design variable in mind.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Solving for a particular beam BCs
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Step 7Transversely guided
0w w = =
Fixed
0w w= =
( ) 0E Aw q − =*
0
0
L
Adx V−
2E w
wE
=
=
0A A − =0A A + =0A = 0A A − =& &
0
0L
E ww E w w A − + = 0
0L
E ww A = &
0( )q x q=
( )2
00 1 0
2
q xA E q A C x C
E
= = + +
Take 0 = = Take 0A = = =
Satisfied at x = 0 and x = L
Satisfied at x = 0 and x = L
Solve for
01
q LC
E= −
using
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Reconciliation of BCs for A(x)
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20 0
02
q x q LA x C
E E = − +
From the previous slide, we haveStep 7
We can use the active volume constraint to solve for
( )
3 3*0 0
0
30
*0
6 2
3
q L q LC L V
E E
q LE
V C L
− + =
= −−
for + sign( )
30
*0
2
3
q LE
V C L = −
−for - sign&
( ) ( )
( ) ( )
2 * *0 0
03 2
2 * *0 0
03 2
3 3
2( )
3 3
4 2
x V C L V C Lx C
L LA x
x V C L V C Lx C
L L
− −− − +
= − −
− +
How do we choose the two possibilities (+ or -) and determine C0?
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
Validating with the numerical solution…
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( ) ( )
( ) ( )
2 * *0 0
03 2
2 * *0 0
03 2
3 3
2( )
3 3
4 2
x V C L V C LA x C
L LA x
x V C L V C LA x C
L L
+
−
− − = − − +
= − −
= − +
We need to see which part of the domain should have the “+ curve” and which part should have “-curve”.
+ curve
- curve
- curve+ curve
x̂x
L
( )A x
0C
We need to find using
0ˆ andx C
ˆ ˆ( ) ( ) 0A x A x+ −= =
*
min max 020; 1; 50; 210; 1 3; 10; 1L d V E A E A q= = = = = − = =
( )A x
x
-
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Problem 12
( )
( )
( )0
*
0
2 *
0
2 *
( ) : 0
( ) : 0
: 0
1: 0
2
: ,12
L
A x
d d
L
L
*d
Min V Adx
Subject to
x E Aw q
x E Av q
E Aw v dx
E Aw dx SE
tData L,q(x),q (x), = ,E,Δ SE
=
− =
− =
− =
− =
Minimize the volume of material of a beam (statically determinate or indeterminate) for a deflection constraint in its span with an upper bound on the strain energy.
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Structural Optimization: Size, Shape, and TopologyG. K. Ananthasuresh, IISc
The end note
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Thanks
Siz
e o
pti
miz
atio
n o
f b
eam
s
Iterative numerical solution, when it is needed, remains the same.
We follow essentially the same eight steps
Identifying the optimality criterion is the highlight.
See the correlation between BCs and optimal profiles
Boundary conditions for the adjoint variable need to be carefully done.
Analytical solution may be segmented with multiple possibilities because of + and – of constant strain.