B.5
Table B.1
Scaling of Design Variables and Constraints
Summary of Some Structural Optimization Packages
Software system
(program)
Source
(developer)
Capabilities and
characteristics
ASTROS (Automated
STRuctural
Optimization System)
Air Force Wright Laboratories
FIBRA
Wright-Patterson Air Force
Base, OH 45433-6553
ANSYS
Swanson Analysis Systems,
Inc.
P.O. Box 65
Johnson Road
Structural optimization with
static, eigenvalue, modal
analysis, and flutter
constraints;
approximation concepts;
compatibility with
NASTRAN; sensitivity
analysis
Optimum design based on
curve-fitting technique to
approximate the response
using several trial design
vectors
Houston, PA 15342-0065
B.5
787
MSC/NASTRAN
MacNeal Schwendler
Corporation/NAsa
STRuctural ANalysis)
MacNeal-Schwendler Corporation
15 Colorado Boulevard
Los Angeles, CA 90041
NISAOPT
Engineering Mechanics
Research Corporation
P.O. Box 696
Troy, MI 48099
GENESIS
VMA Exngineering Inc.
Manderin Avenue, Suite F
Goleta, CA 93117
Structural optimization
capability based on static,
natural frequency, and
buckling analysis;
approximation concepts
and sensitivity analysis
Minimum-weight design
subject to displacement,
stress, natural frequency
and buckling constraints;
shape optimization
Structural optimization;
approximation concepts
used to tightly couple the
analysis and redesign
tasks
SCALING OF DESIGN VARIABLES AND CONSTRAINTS
In some problems there may be an enormous difference in scale between variables
due to difference in dimensions. For example, if the speed of the engine (n) and the
cylinder wall thickness (t) are taken as design variables in the design of an IC engine,
n will be of the order of 103 (revolutions per minute) and t will be of the order of
1 (cm). These differences in scale of the variables may cause some difficulties while
selecting increments for step lengths or calculating numerical derivatives. Sometimes
the objective function contours will be distorted due to these scale disparities. Hence it
is a good practice to scale the variables so that all the variables will be dimensionless
and vary between 0 and 1 approximately. For scaling the variables, it is necessary to
establish an approximate range for each variable. For this we can take some estimates
(based on judgment and experience) for the lower and upper limits on x i(ximin and
788
Some Computational Aspects of Optimization
ximax ) , i = 1, 2, . . . , n. The values of these bounds are not critical and there will not
be any harm even if they span partially the infeasible domain. Another aspect of
scaling is encountered with constraint functions. This becomes necessary whenever the
values of the constraint functions differ by large magnitudes. This aspect of scaling
(normalization) of constraints was discussed in Section 7.13.
B.6
COMPUTER PROGRAMS FOR MODERN METHODS
OF OPTIMIZATION
Fuzzy Logic Toolbox.
Matlab has a fuzzy logic toolbox for designing systems based
on fuggy logic. Graphical user interfaces (GUI) are available to guide the user through
the steps of fuzzy interface system design. The toolbox can be used to model complex
system behaviors using simple logic rules and then implement the rules in a fuzzy
interface system. Fuzzy optimization can be implemented using fuzzy logic toolbox in
conjunction with an optimization program such as fmincon.
Genetic Algorithm and Direct Search Toolbox.
The genetic algorithm and direct
search toolbox, which can be used to solve problems that are difficult to solve with
traditional optimization techniques, is available with Matlab. The genetic algorithm of
the toolbox can be used when the function, such as the objective or constraint function,
is discontinuous, highly nonlinear, stochastic, or has unreliable or undefined derivatives.
In this toolbox also, graphical user interfaces (GUI) are available for quick setting up of
problems, selecting algorithmic options, and monitoring progress. Naturally, the options
of creating initial population, fitness scaling, parent selection, crossover and mutation
are available in the toolbox. The Matlab optimization programs (using direct search
methods) can be integrated with the genetic algorithm.
Neural Network Toolbox.
The neural network toolbox is available with Matlab for
designing, implementing, visualizing and simulating neural networks. The GUI available with the toolbox helps in creating, training and simulating neural networks. It
permits modular network representation to have any number of input-setting layers and
network interconnection and a graphical view of the network architecture. Optimization
programs can be used in conjunction with the functions of the neural network toolbox
to accomplish neural network-based optimization. The neural network toolbox can also
be used to apply neural networks for the identification and control of nonlinear systems.
Simulated Annealing Algorithm.
An m-file to implement the simulated annealing
algorithm to solve function minimization problems in the Matlab environment was
created by Joachim Vandekerckhove. The link is given below:
http://www.mathworks.com/matlabcentral/fileexchange/10548
Particle Swarm Optimization.
An m-file to implement the particle swarm optimization method in the Matlab environment was created by Wael Korani. The link is given
below:
http://www.mathworks.com/matlabcentral/fileexchange/20205
References and Bibliography
789
An m-file to implement the ant colony optimization
Ant Colony Optimization.
method in the Matlab environment for the solution of symmetrical and unsymmetrical
traveling salesman problem was created by H. Wang. The link is given below:
http://www.mathworks.com/matlabcentral/fileexchange/14543
Multiobjective Optimization.
An m-file to implement multiobjective optimization
using evolutionary algorithms (based on nondominated sorting genetic algorithm, abbreviated NSGA) in the Matlab environment was created by Arvind Seshadri. The link is
given below:
http://www.mathworks.com/matlabcentral/fileexchange/10429
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B.7
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B.10
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790
Some Computational Aspects of Optimization
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J. L. Kuester and J. H. Mize, Optimization Techniques with Fortran, McGraw-Hill, New
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J. J. More´ and S. J. Wright, Optimization Software Guide, Society of Industrial and
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