engineering optimization
DESCRIPTION
An introduction to Engineering Optimization.TRANSCRIPT
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Engineering Optimization
Concepts and Applications
Matthijs Langelaar
Fred van Keulen
3mE-PME 34-G-1-300
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Outline
● Course information
● Introduction to optimization: what, why, how?
● Basics
– Problem formulation
– Solution approach
– Optimization & (structural) design
● Practical examples
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Course Objectives
● Understanding of principles and possibilities of
optimization
● Knowledge of optimization algorithms, ability to
choose proper algorithm for given problem
● Practical experience with
1. Optimization algorithms
2. Application of optimization to design problems
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Course overviewGeneral introduction, problem formulation, design space / optimization terminology
Modeling, model simplification
Optimization of unconstrained / constrained problems
Single-variable, zeroth-order and gradient-based optimization algorithms
Design sensitivity analysis (FEM)
Topology optimization
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Assessment
a) Optimization project (case study)individual or in pairs:
Definition of problem, approach (ca. 1 page A4, deadline March 3, via email)
Final project report: due Apr 25
b) 3 online self-assessment tests (individual)(round-off bonus)
c) 10 exercises (indiv./pairs)(round-off bonus)
Bb: ‘Assignments\Online tests’
Final grade
Bb: ‘Assignments’
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Exercises
● 9+1 practical exercisesusing Matlab and Optimization Toolbox (individual or in pairs)
● Practice techniques, learn how to use Matlab Optimization Toolbox
– Recommended practice for final case study!
● Due dates: see planning
● Hand in via Bb
● Discussed during following lectures, based on your contributions
● First one: today (due before next lecture)
Bb: ‘Assignments’
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Course material
● Main text: “Principles of Optimal Design
– Modeling and Computation”, P.Y.
Papalambros & D.J. Wilde, Cambridge
University Press
[also available as e-book]
● Selected topics: “Elements of Structural
Optimization”, R.T. Haftka & Z. Gurdal, Kluwer
Academic Publishers
● Slide handouts, exercises and material on Bb
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Outline
● Course information
● Introduction to optimization: what, why, how?
● Basics
– Problem formulation
– Solution approach
– Optimization & (structural) design
● Practical examples
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What is optimization?
● “Making things better”
● “Generating more profit”
● “Determining the best”
● “Do more with less”
● Papalambros: “The determination of values for design
variables which minimize (maximize) the objective,
while satisfying all constraints”
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Historical perspective
● Ancient Greek philosophers: geometrical optimization
problems
Zenodorus, 200 B.C.:
“A sphere encloses the greatest
volume for a given surface area”
? g
● Newton, Leibniz, Bernoulli, De l’Hospital (1697):
“Brachistochrone Problem”:
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Historical perspective (cont.)
● Lagrange (1750): Constrained minimization
● Cauchy (1847): Steepest descent
● Dantzig (1947): Simplex method (LP)
● Kuhn, Tucker (1951): Optimality conditions
● Karmarkar (1984): Interior point method (LP)
● Bendsoe, Kikuchi (1988): Topology optimization
● …
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What can be achieved?● Optimization techniques can be used for:
– Reaching the optimal performance
– Getting a design/system to work (goal attainment)
– Making a design/system reliable and robust
● Also provide insight in
– Design problem characteristics
– Underlying physics
– Model weaknesses
● Provides a systematic problem solving approach
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Optimization problem
● Design variables: variables by which the design
problem is parameterized:
● Objective: quantity that is to be minimized (maximized)
Usually denoted by:
( “cost function”)
● Constraint: condition that has to be satisfied
– Inequality constraint:
– Equality constraint:
( ) 0g x
( ) 0h x
( )f x
1 2, , , nx x xx
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Optimization problem (cont.)
● General form of optimization problem:
min ( )
subject to: ( ) 0
( ) 0n
f
X
xx
g x
h x
x
x x x
s.t.
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Example: cheap chair
Minimize Material usage
Constraints:
(strength) Stress in legs ≤ 0.5*(Yield stress)
(comfort) Deflection of seat @ 70 kg = 3 mm
Ø x1
↕ x2
x
10 mm ≤ x1 ≤ 50 mm, 1 mm ≤ x2 ≤ 10 mm
Stress in legs - 0.5*(Yield stress) ≤ 0
Deflection of seat @ 70 kg - 3 mm = 0
= f (x)
= g (x)
= h (x)
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Solving optimization problems
● Optimization problems are typically solved using an
iterative algorithm:
Model
Optimizer
Designvariables
Constants Responses
Derivatives ofresponses(design sensitivities)
hgf ,,
iii x
h
x
g
x
f
,,
x
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An easier way?
Looks complicated … why not just sample many points
in the design variable space, and take the best one?
Take 0,001 s per computation,
10 variables, 10 samples:
total time 116 days! Please wait …
● Consider problem with n design variables
● Sample each variable with m samples
● Number of computations required: mn
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The Challenge
The challenge that optimization algorithms face:
Find the ‘best’ solution,
using the smallest number of function evaluations!
x1 x2 f(x1,x2) g(x1,x2)≤ 0 h(x1,x2)= 0
1 1 10.5 -2.1 0.7
1 2 4.3 1.3 -2.8
3 1.5 5.9 4.5 -2.2
? ? ‘best’ ≤ 0 0
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Parallel computing
● Still, for large problems,
optimization requires lots
of computing power
● Parallel computing
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Optimization in the design process
Conventional design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Check performance criteria
Is design satisfactory? Done
Change design based on experience / intuition
/ guesses
“Design me a rear-view mirror”
J.S. Arora
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Optimization in the design process
Optimization-based design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Identify:1. Design variables2. Objective function3. Constraints
Check the constraints
Does the design satisfy convergence criteria?
Change the design using an optimization method
Done“Design me a rear-
view mirror”
Model
Optimizer
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Optimization popularity
● Increasing availability of numerical modeling techniques
● Better and more powerful optimization techniques
● Increasing availability of cheap computer power
More and more adopted in industry:
● More engineers having optimization knowledge
● Increased competition, global markets
● Increasingly expensive production processes (trial-and-error approach too expensive)
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Optimization pitfalls!
● Proper problem formulation critical!
● Choosing the right algorithm
for a given problem
● Many algorithms contain lots
of control parameters
● Optimization tends to
exploit model weaknesses
● Optimization can result in very sensitive designs
● Some problems are simply too hard / large / expensive …
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Structural optimization
● Structural optimization =
optimization techniques applied to structures
● Different categories:
– Sizing optimization
– Material optimization
– Shape optimization
– Topology optimization
t
E, R
r
L
h
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Truss sizing
http://www.bloodhoundssc.com
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Shape optimization
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Topology optimization examples
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Classification
● Problems:
– Constrained vs. unconstrained
– Single level vs. multilevel (nested)
– Single objective vs. multi-objective
– Deterministic vs. stochastic
min ( )
subject to: ( ) 0
( ) 0n
f
X
xx
g x
h x
x
x x x
min ( )
subject to: ( ) 0
( ) 0n
f
X
xx
g x
h x
x
x x x● Responses:
– Linear vs. nonlinear
– Convex vs. nonconvex (later!)
– Smooth vs. nonsmooth
● Variables:
– Continuous vs. discrete (e.g. integer)
– Deterministic vs. stochastic
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Outline
● Course information
● Introduction to optimization: what, why, how?
● Basics
– Problem formulation
– Solution approach
– Optimization & (structural) design
● Practical examples
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Practical example: Airbus A380
● Wing stiffening ribs
of Airbus A380:
● Objective: reduce weight
● Constraints: stress, bucklingLeading edge ribs
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Airbus A380 example (cont.)
● Topology and shape optimization
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Airbus A380 example (cont.)
● Topology optimization:
● Sizing / shape
optimization:
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Airbus A380 example (cont.)
● Result: 500 kg weight savings!
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Other examples
● Jaguar F1 FRC front wing:reduce weight
constraints on
max. displacements
5% weight saved
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Other examples (cont.)
● Design optimization of packaging products
(Van Dijk & Van Keulen):
● Objective: minimize
material used
● Constraints:
stress, buckling
● Result: 20% saved
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But also …● Optimization is also applied in:
– Financial market forecasting (options pricing)
– Protein folding
– System identification
– Logistics (traveling salesman problem),route planning, operations research
– Controller design (SC4091 course!)
– Spacecraft trajectory planning
● This course: focus on (structural) design optimizationbut covered techniques apply to other problems too
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What makes a design
optimization problem interesting?
● Non-trivial design optimization problems often show a
conflict of interest / contradicting requirements:
– Aircraft wing: stiffness vs. weight
– F1 car: idem
– Oil bottle: stiffness / buckling load vs. material usage
● Otherwise the problem may be trivial or ill-posed!
Example:
Cost minimization without demands on performance.
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Exercise 0
Exploring applications of optimization
● Find an example of optimization of … <your choice>
● Write a brief summary (~ 1 A4 max, minimal text)
– Title, website where you found it
– What is optimized, what are the variables
Bb: ‘Assignments\Exercise 0’
(Challenge: optimize your contribution: the most interesting, most unexpected, most relevant, …)
Minimization of ..Maximization of .. Optimization of ..Optimized …
● And include some pictures, if possible
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Summary
● WB1440 Course outline, examination, assignments
● Optimization: what, why, how
● Basics: problem formulation, solution approach
● Optimization & (structural) design
● Practical examples
● Next lecture: model, problem characterization
Friday 13:45 – 15:30h
Room D (James Watt room, 3mE)Discussion of
Exercise 0