Engineering Optimization
What is Optimization?
● “Making things better”
● “Generating more profit”
● “Determining the best”
● “Do more with less”
● The determination of values for design variables
which minimize (maximize) the objective,
while satisfying all constraints”
Historical perspective
• Ancient Greek philosophers: geometrical
optimization problems
• Zenodorus, 200 B.C.:
“A sphere encloses the greatest
volume for a given surface area”
● Newton, Leibniz, Bernoulli, De l’Hospital (1697):
“Brachistochrone Problem”:
?
Brachistochrone curve , or curve of fastest descent, is the curve between two points that is covered in the least time by a point-like body that starts at
the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and assuming no
friction. - cycloid
g
Historical perspective (cont.)
•
•
•
•
•
•
Lagrange (1750): constrained minimization
Cauchy (1847): steepest descent
Dantzig (1947): Simplex method (LP)
Kuhn, Tucker (1951): optimality conditions
Karmakar (1984): interior point method (LP)
Bendsoe, Kikuchi (1988): topology
optimization
What can be achieved?
• Optimization techniques can be used for:
– Getting a design/system to work
– Reaching the optimal performance
– Making a design/system reliable and robust
• Also provide insight in
– Design problem
– Underlying physics
– Model weaknesses
Optimization problem
• Design variables: x   x1 , x2 , , xn 
variables with which the design problem is
parameterized:
• Objective: f (x)
• quantity that is to be minimized (maximized)
Usually called by “cost function”
• Constraint: condition that has to be satisfied
– Inequality constraint: g (x)  0
– Equality constraint:
h( x)  0
Optimization problem (cont.)
Formulation of a design improvement problem as a formal
mathematical optimization problem
To find components of the vector x of design variables:
F ( x )  min
g j ( x )  0,
Ai  xi  Bi ,
j  1,..., M
i  1,..., N
where F(x) is the objective function, gj(x) are the constraint functions, the last
set of inequality conditions defines the side constraints.
Solving optimization problems
Constants
Model
Design
variables
x
Optimizer
Responses
f , g, h
Derivatives of
responses
(design sensitivities)
f g h
,
,
xi xi xi
• Optimization problems are typically solved using an
iterative algorithm:
Curse of dimensionality
Looks complicated … why not just sample
the design space, and take the best one?
• Consider problem with n design variables
• Sample each variable with m samples
• Number of computations required: mn
Take 1 s per computation,
10 variables, 10 samples:
total time 317 years!
(1010=10,000,000,000)
Optimization in the design process
Conventional designdesign
Optimization-based
process:
process:
Identify:
1. Design variables
2. Objective function
3. Constraints
Collect data to describe
the system
Estimate initial design
Analyze the system
Check
Checkthe
performance
constraints
criteria
Does the design satisfy
Is
design satisfactory?
convergence
criteria?
Change
Changedesign
the design
based
using
on experience
an optimization
/
heuristicsmethod
/ wild guesses
Done
PROBLEM SPACE
2-D
3-D
N-D
?
f(x) = ….
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f(x,y) = ….
f(x1,x2,…,xN-1) = ….
LOCAL OPTIMIZATION
Deterministic methods
Optimality Criteria
- The mathematical conditions for an optimal solution are
Search Methods
- An initial trial solution is selected
- A move is made to a new point and the objective function is evaluated again
- The process is repeated until the minimum is found
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GLOBAL OPTIMIZATION
1. Deterministic approach (결정론적 또는 확정론적 방법)
2. Stochastic and Heuristic approach (확률론적 또는 경험론적 방법)
- Genetic Algorithms (유전자알고리즘)
- Simulated Annealing (냉각모사기법)
- Artificial Intelligence (인공지능)
- Knowledge Based System (지식기반시스템)
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Optimization popularity
Increasingly popular:
• Increasing availability of numerical modeling
techniques
• Increasing availability of cheap computer power
• Increased competition, global markets
• Better and more powerful optimization techniques
• Increasingly expensive production processes
(trial-and-error approach too expensive)
• More engineers having optimization knowledge
Structural optimization
• Structural optimization =
optimization techniques
applied to structures
• Different categories:
–
–
–
–
Sizing optimization
Material optimization
Shape optimization
Topology optimization
L
E, n
t
R
h
r
Shape optimization
Yamaha R1
Topology optimization examples
Practical example: Airbus A380
• Wing stiffening ribs
of Airbus A380:
• Objective: reduce weight
• Constraints: stress, buckling
Leading
edge ribs
Airbus A380 example (cont.)
• Topology and shape optimization
Airbus A380 example (cont.)
• Topology optimization:
• Sizing / shape
optimization:
Airbus A380 example (cont.)
• Result: 500 kg weight savings!
Other examples
• Jaguar F1 FRC front wing:
reduce weight
constraints on
max. displacements
5% weight saved
Other examples (cont.)
•
•
•
•
Design optimization of packaging products
(Van Dijk & Van Keulen):
Objective: minimize
material used
Constraints:
stress, buckling
Result: 20% saved
Shape Memory Alloy active catheter optimization
Optimization of a spanner
Optimization of an Airfoil
Problem formulation:
Objective function (to be minimized): drag coefficient at Mach 0.73 and Mach 0.76:
F0 (x) = 2.0 Cd total (M=0.73) + 1.0 Cd total (M=0.76)
Constraints: on lift and other operational requirements (sufficient space for holding fuel, etc.)
Optimization of a Bridge
Design of a negative Poisson's ratio material
Design of a negative Poisson's ratio material (expands vertically when
stretched horizontally) using topology optimization. Left: base cell. Centre:
Periodic material composed of repeated base cells. Right: Test beam
manufactured by Microelektronik Centret (Denmark)
Design of negative thermal expansion
Design of a material with negative thermal expansion. It is composed of two
materials with different thermal expansion coefficients 1 = 1 (blue) and 2 =10
(red) and voids. The effective thermal expansion coefficient is 0= - 4.17. Left:
base cell. Centre: thermal displacement of microstructure subjected to heating.
Right: periodic material composed of repeated base cells.
HSCT Wing design optimization
Min TOGW
Num.
1
2
3-20
21
22 T
23
24
25
26-30
31
32
33
Num.
34
35
36-53
54
55-58
59
60
61
62
63
64
65
66
67-69
required
Geometric Constraints
Fuel volume ≤ 50% wing volume
Airfoil section spacing at Ctip ≥ 3.0ft
Wing chord ≥ 7.0ft
LE break ≤ semi-span
E break ≤ semi-span
Root chord t/c ratio ≥ 1.5%
LE break chord t/c ratio ≥ 1.5%
Tip chord t/c ratio ≥ 1.5%
Fuselage restraints
Nacelle 1 outboard of fuselage
Nacelle 1 inboard of nacelle 2
Nacelle 2 inboard of semi-span
Aero. & Performance Constraints
Range ≥ 5, 500 naut.mi.
CL at landing ≤ 1
Section Cl at landing ≤ 2
Landing angle of attack ≤ 12◦
Engine scrape at landing
Wing tip scrape at landing
LE break scrape at landing
Rudder deflection ≤ 22.5◦
Bank angle at landing ≤ 5◦
Tail deflection at approach ≤ 22.5◦
Takeoff rotation to occur ≤ Vmin
Engine-out limit with vertical tail
Balanced field length ≤ 11,000 ft
Mission segments: thrust available≥ thrust
Initial and Optimal wing designs
Wing Shape Optimization Problem
 Flight Condition: Mach 0.84, Angle of attack 3.060
 Objective Function: Min Structure Weight
 Design Variables:
6.0 ≤ x(1) Aspect Ratio ≤ 8.0
0.2 ≤ x(2) Taper Ratio ≤ 0.35
200 ≤ x(3) Sweepback Angle ≤ 350
10 mm ≤ x(4) Upper Skin Thickness ≤ 15 mm
10 mm ≤ x(4) Lower Skin Thickness ≤ 15 mm
 Constraints:
G1(X) :
G2(X) :
G3(X) :
G4(X) :
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-CL/CD + 15 ≤ 0
Wing Tip Displacement(δ wing-tip) -4000mm ≤ 0
Skin Max Stress(σ)- 388MPa ≤ 0
Flange Max Stress(σ)- 588MPa ≤ 0
Optimization Result
Wing shape of Upper
Bound Design Variable
[X(1), X(2), X(3), X(4), X(5)]
= [9, 0.4, 40, 3, 40]
Wing shape of Lower
Bound Design Variable
[X(1), X(2), X(3), X(4), X(5)]
= [6, 0.2, 20, 2, 20]
Optimum wing shape
[X(1), X(2), X(3), X(4), X(5)]
= [6.73, 0.32, 40, 2, 20]
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OPTIMIZATION – Implementation is not easy
Thank You for Your Attention