Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

4M020 Design toolsOptimization problem formulation and visualization

L.F.P. Etman

Department of Mechanical EngineeringEindhoven University of Technology

Wednesday August 27, 2008

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Outline

1 Problem formulation

2 Two-bar truss example: one variable minimization

3 Problem properties

4 Two-bar truss example: two-variable minimization

5 Problem properties

6 Two-bar truss example: finite element analysis

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Design optimization

1 Select design variables

2 Select objective criterion in terms of design variables (tominimize 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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Two-bar truss example

F

SS

h

d

Minimizex

f (x)

subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn

Exercise 1

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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

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ii

ii

x

ff(x)

g2 = 0g1 = 0

• unconstrained optimum

• interior optimum

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Two-bar truss example

F

SS

h

d

Minimizex

f (x)

subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn

Exercise 2

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Visualization of a two-variable optimization problem

ii

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ii

ii

x1

x2

g1

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Visualization of a two-variable optimization problem

ii

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ii

ii

x1

x2

g1

g4

g5

g2 g4 g6

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Visualization of a two-variable optimization problem

ii

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ii

ii

x1

x2

g1

g4

g5

g2 g4 g6

feasible

domain

F

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Visualization of a two-variable optimization problem

ii

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ii

ii

x1

x2

g1

g4

g5

g2 g4 g6

optimum

F

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Problem formulation Truss example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

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 example 1 Problem properties 1 Truss example 2 Problem properties 2 Truss example 3

Two-bar truss example

F

SS

h

d

Minimizex

f (x)

subject to h(x) = 0g(x) ≤ 0x ∈ X ⊆ Rn

Exercise 3

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