engineering optimization concepts and applications fred van keulen matthijs langelaar cla h21.1...
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Engineering Optimization
Concepts and Applications
Fred van Keulen
Matthijs Langelaar
CLA H21.1
Overview of research projects• Optimization with Uncertainties
• Approximate optimization
• Topology Optimization
• Multilevel optimization
• Fast reanalysis
• Buckling of submarine
• Impregnation
• Shoulder endoprosthesis
• SMA actuators
• Microactuation for Butterfly
• Microactuators (them./electr)
• MEMS packaging
• MEMS surface effects
• MEMS measurement structures
• Electronic interface modeling
• Modeling of MEMS
• MEMS optimization
Micro actuator● 13 μm, ie 2.5% longitudinal strain
● at 2 V, 27 mW, Tmax = 200C
60 um
530 um
Course Objectives
● Understanding of principles and possibilities of
optimization
● Knowledge of optimization algorithms, ability to choose
proper algorithm for given problem
● Practical experience with optimization algorithms
● Practical experience in application of optimization to
design problems
Course overview
● General 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
Course material
● Main text: “Principles of Optimal Design
– Modeling and Computation”, P.Y.
Papalambros & D.J. Wilde, Cambridge
University Press
● Selected topics: “Elements of Structural
Optimization”, R.T. Haftka & Z. Gurdal, Kluwer
Academic Publishers
● Exercises and references
Examination
a) Report on practical exercises using Matlab and Optimization Toolbox (individual or in groups of 2 students)
b) Report on optimization project (individual or in groups of 2 students):
Definition of problem, approach (ca. 1 page A4, Deadline March 28, via email)
Final report
c) Oral exam (individual)
Course Schedule
● No lectures on: 19-2, 11-3 and 1-4
● How to find alternative time slots?
● Training lectures?
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”
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”:
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: variables with 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
Optimization problem (cont.)
● General form of optimization problem:
xxx
x
xh
xg
xx
nX
f
0)(
0)(
)(
:to subject
min
Solving optimization problems
● Optimization problems are typically solved using an
iterative algorithm:
Model
Optimizer
Designvariables
Constants Responses
Derivatives ofresponses(design sensi-tivities)
hgf ,,
iii x
h
x
g
x
f
,,
x
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!
Parallel computing
● Still, for large problems,
optimization requires lots
of computing power
● Parallel computing
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?
Change design based on experience /
heuristics / wild guesses
Done
Optimization-based design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Check the constraints
Does the design satisfy convergence criteria?
Change the design using an optimization
method
Done
Identify:1. Design variables2. Objective function3. Constraints
Optimization popularity
● 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
Increasingly popular:
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
weaknesses in models
● Optimization can result in very sensitive designs
● Some problems are simply too hard / large / expensive
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
Classification● Problems:
– Constrained vs. unconstrained
– Single level vs. multilevel
– Single objective vs. multi-objective
– Deterministic vs. stochastic
● Responses:
– Linear vs. nonlinear
– Convex vs. nonconvex (later!)
– Smooth vs. nonsmooth
● Variables:
– Continuous vs. discrete (integer)
Practical example: Airbus A380
● Wing stiffening ribs
of Airbus A380:
● Objective: reduce weight
● Constraints: stress, bucklingLeading edge ribs
Airbus A380 example (cont.)
● Topology optimization:
● Sizing / shape
optimization:
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
But also …● Optimization is also applied in:
– Protein folding
– System identification
– Financial market forecasting (options pricing)
– Logistics (traveling salesman problem),route planning, operations research
– Controller design
– Spacecraft trajectory planning
● This course: focus on (structural) design optimization
What makes a design
optimization problem interesting?
● Good 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 could be trivial!
The optimization model
Model
Optimizer
Designvariables
Constants Responses
Derivatives ofresponses(design sensi-tivities)
hgf ,,
iii x
h
x
g
x
f
,,
x
Systems approach
● Systematic way of thinking:
– What is input / output?
– What belongs to system / environment?
– What level of detail?
– Distinguish sub-systems, hierarchies
System functionInput Output
Environment
Model example
Mathematical model:
123
3 3
33
bhE
FL
EI
FLU
Finite element model:FhbLEKU 1),,,(
F, Uh, b
h
b
E, L
Steel
U(x), M(x), V(x)
Model example (2)
● System (state) variables: U(x), M(x), V(x)
● System parameters: h, b, L
● System constants: E,
F, Uh, b
h
b
E, L
Steel
U(x), M(x), V(x)
Features of computer models
● Finite accuracy due to:
– Discretization in time and space
– Finite number of iterations
(eigenvalues, nonlinear models)
– Numerical round-off errors, ill-conditioning
● Responses can be “noisy”:
– Due to different discretization in space and/or time
(e.g. remeshing)
Noisy response
● Example: effect of remeshing
Normalized stress constraint
Hole radius
Features of computer models (cont.)
● Computational models are (very) time consuming
● Often design sensitivities can be calculated
– Cost of design sensitivity analysis?
– Accuracy / consistency of sensitivities
Response
Design variable
ExactNumericalmodel
Finite difference sensitivities
● Straightforward way to compute sensitivities:
finite differences
● More later!
( ) ( )df f x x f x
dx x
Small!
f
x
( )
( )
f x x
f x
x
Einstein’s advice
“Everything should be made as simple as
possible, but not simpler”
● Model simplification important for optimization!
More in next lectures.