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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
4M020 Design toolsOptimization problem formulation and visualization
L.F.P. Etman
Department of Mechanical EngineeringEindhoven University of Technology
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Outline
1 Problem formulation
2 Two-bar truss exercise: one variable minimization
3 Visualization & Problem properties
4 Two-bar truss exercise: two variable minimization
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Design optimization
Design optimization is the selection of the best designwithin the available means
[Papalambros & Wilde 2000: Principles of optimal design]
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Design optimization
1 Select design variables
2 Select objective criterion in terms of design variables(to minimize or maximize)
3 Determine constraints in terms of design variables, whichmust be satisfied
4 Determine design variable values which minimize (maximize)the objective while satisfying all constraints
[Papalambros & Wilde 2000: Principles of optimal design]
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Mathematical problem formulation
Minimizex
f (x) x = (column) vector of design variables
subject to hj (x) = 0 j = 1; : : : ;mh
gk (x) ≤ 0 k = 1; : : : ;mg
x ∈ X ⊆ Rn
[Papalambros & Wilde 2000: Principles of optimal design]
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Negative null form
Minimizex
f (x) x = (column) vector of design variables
subject to h(x) = 0 h = [h1;h2; : : : ;hmh ]T
g(x) ≤ 0 g = [g1; g2; : : : ; gmg ]T
x ∈ X ⊆ Rn
Other formulations:
• positive null form (g(x) ≥ 0)
• negative unity form (g(x) ≤ 1)
• positive unity form (g(x) ≥ 1)
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Two-bar truss example
F
SS
h
d
Minimizex
f (x)
subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn
• Analysis equations
• Constant parameters
• Design variables
• Objective function
• Constraint functions
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Two-bar truss exercise
F
SS
h
d
Minimizex
f (x)
subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn
Exercise: single variable minimization
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a one-variable optimization problem
ii
“plot1Db˙temp” — 2006/6/26 — 16:21 — page 1 — #1 ii
ii
ii
x
f
f(x)
x
g
g3(x)
0
g6(x)
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a one-variable optimization problem
ii
“plot1De˙temp” — 2006/6/26 — 16:21 — page 1 — #1 ii
ii
ii
x
f
f(x)
x
g
g3(x)
0
g6(x)
g3 = 0
g6 = 0
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a one-variable optimization problemii
“plot1Df˙temp” — 2006/6/26 — 16:21 — page 1 — #1 ii
ii
ii
x
f
f(x)
g3 = 0
g6 = 0
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Constrained versus unconstrained optimumi
i“opt˙bnd˙temp” — 2006/6/15 — 21:36 — page 1 — #1 i
i
ii
ii
x
f f(x)
g = 0
• constrained optimum
• bounded optimum
ii
“opt˙unc˙temp” — 2006/6/15 — 21:35 — page 1 — #1 ii
ii
ii
x
ff(x)
g2 = 0g1 = 0
• unconstrained optimum
• interior optimum
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Multimodality (multiple local minima)ii
“multi˙temp” — 2006/6/15 — 17:12 — page 1 — #1 ii
ii
ii
x
f
f(x)
global
local
x
ff(x)
globallocal
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a two-variable optimization problemi
i“plot2Da˙temp” — 2006/6/15 — 21:15 — page 1 — #1 i
i
ii
ii
x1
x2
objective function
6 78
9 10
f
x1
x2
constraint g1
g1= 0
g1< 0
feasible
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a two-variable optimization problem
ii
“plot2Db˙temp” — 2006/6/15 — 21:29 — page 1 — #1 ii
ii
ii
x1
x2
g1
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a two-variable optimization problem
ii
“plot2Dc˙temp” — 2006/6/15 — 21:16 — page 1 — #1 ii
ii
ii
x1
x2
g1
g4
g5
g2 g4 g6
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a two-variable optimization problem
ii
“plot2Dd˙temp” — 2006/6/15 — 21:29 — page 1 — #1 ii
ii
ii
x1
x2
g1
g4
g5
g2 g4 g6
feasible
domain
F
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Visualization of a two-variable optimization problem
ii
“plot2De˙temp” — 2006/6/15 — 21:26 — page 1 — #1 ii
ii
ii
x1
x2
g1
g4
g5
g2 g4 g6
optimum
F
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Constraint activity
A constraint gj (x) ≤ 0 is active at the optimum implies that
1 if gj (x) ≤ 0 is left out, the location of the optimum changes
2 the constraint is satisfied with strict equality: gj (x) = 0
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Problem formulation Truss exercise 1 Visualization & Problem properties Truss exercise 2
Two-bar truss exercise
F
SS
h
d
Minimizex
f (x)
subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn
Exercise: optimization problem formulation and visualization withtwo design variables
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