Engineering Optimization
Concepts and Applications
Fred van Keulen
Matthijs Langelaar
CLA H21.1
A.vanKeulen@tudelft.nl
Delft in The Netherlands
Delft
Background
Overview of research projects
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Optimization with Uncertainties
Approximate optimization
Topology Optimization
Multilevel optimization
Fast reanalysis
Buckling of submarine
Impregnation
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SMA actuators
Microactuation for Butterfly
Microactuators (them./electr)
MEMS packaging
MEMS surface effects
MEMS measurement structures
Electronic interface modeling
Modeling of MEMS
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Shoulder endoprosthesis
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MEMS optimization
Man-made Insect
5
Topology
Optimization
Submarines
Micro actuator
● 13 μm, ie 2.5% longitudinal strain
● at 2 V, 27 mW, Tmax = 200C
60 um
530 um
Who are you?
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”
● Newton, Leibniz, Bernoulli, De l’Hospital (1697):
“Brachistochrone Problem”:
?
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: variables with which the design
problem is parameterized:
x   x1 , x 2 ,
, xn 
● Objective: quantity that is to be minimized (maximized)
Usually denoted by:
( “cost function”)
f (x)
● Constraint: condition that has to be satisfied
– Inequality constraint:
g (x)  0
– Equality constraint:
h(x)  0
Optimization problem (cont.)
● General form of optimization problem:
min f ( x )
x
subject
to :
g (x)  0
h (x)  0
x X  
x  x  x 
n
Solving optimization problems
● Optimization problems are typically solved using an
iterative algorithm:
Constants
Responses
Model
Design
variables
x
Optimizer
f , g,h
Derivatives of
responses
(design sensitivities)
f
,
g
,
h
xi xi xi
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 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
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
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:
L
E, n
– Sizing optimization
– Material optimization
– Shape optimization
– Topology optimization
t
R
h
r
Shape optimization
Yamaha R1
Topology optimization examples
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, 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
SMA active catheter optimization
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
Constants
Responses
Model
Design
variables
x
Optimizer
f , g,h
Derivatives of
responses
(design sensitivities)
f
,
g
,
h
xi xi xi
Systems approach
Input
System function
Output
Environment
● Systematic way of thinking:
– What is input / output?
– What belongs to system / environment?
– What level of detail?
– Distinguish sub-systems, hierarchies
Example: cantilever beam
E, r
h
F, U
U(t)
E, r, h, L
F(t)
F(t)
wi
U(t)
Etc.
Model example
L
E, r
F, U
h, b
Steel
h
U(x), M(x), V(x)
b
Mathematical model:
U 
FL
3
3 EI

FL
3
 bh 3
3 E 
 12



1
Finite element model:
U  K ( E , L, b, h) F
Model example (2)
L
E, r
F, U
h, b
Steel
h
U(x), M(x), V(x)
b
● System (state) variables:
U(x), M(x), V(x)
● System parameters:
h, b, L
● System constants:
E, r
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
Exact
Numerical
model
Design variable
Finite difference sensitivities
● Straightforward way to compute sensitivities:
finite differences
df

f ( x  x)  f ( x)
dx
f
x
f ( x  x)
 f ( x)
Small!
x
● More later!
x
Einstein’s advice
“Everything
should be
made as
simple as
possible, but
not simpler”
● Model simplification important for optimization!
More in next lectures.