The Smart
Global Optimization Technology
SmartDO eNews Nov 6, 2008
: The Global Optimization Technology in SmartDO
Introduction
One of the unique features of SmartDO, is it overcomes the traditional
local optimization technology, and provides the practical Smart Global
Optimization Technology. For an amateur user of design optimization software,
it is really difficult to tell the significant difference between local
optimization and global optimization. Even when the vendor already announces
that only local optimization is available in the package, the user can
still be unaware of the possibly misleading direction from the local optimum
solution.
In this issue of the SmartDO eNews, we will introduce some Global Optimization
Technology in SmartDO, and also few basic concept of global optimization.
The Gradient-Based
Response Smoothing Technology That Will Overcome the Numerical Noise
Almost all FVM or FEA-based approaches produce numerical noise. The
common sources of numerical include
(1) The mesh perturbation/variation when the shape of the model changes,
especially when the mesh is built in free form.
(2) The jumping of the location with the maximum response in the model.
For example, in stress analysis, the location of the maximum stress point
can jump from one position to another.
(3) Integration over the discretized domain or mesh. For example, calculating
the flow rate in CFD.
(4) During dynamics analysis, the differentiation and integration over
the time domain.
The phenomena of numerical noise can be explained by Figure 1. When
parametric study is performed to understand the variation of design response
with respect to the design variable, ideally the response curve should look
like the blue line in Figure 1. However due to the influence of numerical
noise as stated above, the actual response will appear like the red line
in the Figure 1.
Figure 1 The Ideal Response Curve and the One with Numerical Noise
The phenomena of numerical noise was observed in
as early as 1980, however it did not get a lot attention until recent years.
Figure 2 shows two figures from Reference 1, by Professor Hamasaki
of Hiroshima University, Japan, and
Professor Vassili V. Toropov of University of Leeds, United Kingdom. The
figure on the left shows the physical situation to be simulated by CAE,
and the figure on the right shows the parametric study with two design variables.
The numerical noise observed in the paper was far more serious than we can
imagine.
Figure 2 The Numerical Noise Document in Reference 1
When the numerical noise exists in a CAE model, several local minimum
will appear as shown in Figure 3. When the traditional gradient-based approach
is used, the solution will converge to the nearest local minimum. However
this solution is a misleading and artificial local trap, which may not even
present a useful local minimum.
In order to overcome this numerical phenomena, Dr. S-Y. Chen developed
the Response Smoothing Technology for SmartDO (Reference 2). This approach
is gradient-based, yet can still escape from the trap of local artificial
optimum, and search for the better design (shown as the green line in Figure
3). With this approach, SmartDO can overcome the numerical noise.
Figure 3 The Search Path (Green Line) of the Response Smoothing Technology
The
Robust Genetic Algorithms (RGA)
The Genetic Algorithms
(GA) has been widely accepted since 1995. One of the reasons is the simplicity
of the algorithms itself, which allows the code to be written without much
effort. However, they are many drawbacks of GAs that can be easily overlooked,
which includes
(1) The convergence of GA is not guaranteed. There is even no mathematics
to prove GA's convergence in the early days.
(2) GA is very computational expensive.
(3) GA has difficulties handling constraints.
In around 1997 to 2000, Dr. S-Y. Chen and Dr. Rajan proposed the Robust
Genetic Algorithms (RGA), which attempts to solve the problems stated above
with the following tools
(1) Adaptive Penalty Function
(2) Automatic Schema Representation
(3) Automatic Population and Generation
Number Calculation
(4) Adaptive and Automatic Cross-Over Probability
Calculation
(5) Absolute Descent
Because of its unique
feature, RGA can be used to solve the problem of simultaneous sizing, shaping
and topology optimization as indicated in Figure 4. This is still considered
very difficult, if not impossible, for other packages.
Figure 4 Simultaneous Sizing, Shaping and Topology Optimization with the
RGA in SmartDO
The Robust Genetic Algorithms (RGA) is one of the standard solver in
SmartDO. It was tested by both academic research and real life application.
For more information about the Robust
Genetic Algorithms (RGA), please see our Paper Archive.
The
Smart Particle Swarm (SPS)
The Smart particle Swarm (SPS) is the latest global optimization technology
in SmartDO. It combines the technology of RGA, NLP and the traditional
Particle Swarm, with the following advantages
(1) Using the design space exploration technology in RGA.
(2) Multiple design population. It is a global optimization algorithm
in nature.
(3) User can opt to define no initial design, one initial design or multiple
initial designs. When multiple initial designs are defined, SPS will also
consider the trend and difference between designs.
(4) The search direction is similar to gradient-based approach, which
is more accurate and efficient.
(5) More efficient than traditional GA and Particle Swarm.
The behavior of SPS can be explained with a simple example. Figure 5
shows a function with 2 design variables and multiple local minimums. When
the initial design is not defined, SPS will distribute the initial population
(automatically) as Figure 6. Figure 7 shows the distribution of designs
after the first iteration, and Figure 8 shows the distribution of designs
after the second iteration. It can be seen that the SPS quickly goes over
the local minimum and converges near the global minimum. SPS does not need
huge amount of gradient information and/or intensive population distribution
to find the search path.
SPS will be available in the latest version of SmartDO to be published
soon. Please see our future press release.
Figure 5 A Simple Example for The Smart Particle Swarm (SPS)
Figure 6 The Initial Design Distribution of The Smart Particle Swarm (SPS)
Figure 7 The Design Distribution After The First Iteration of The Smart
Particle Swarm (SPS)
Figure 8 
Conclusion
and Remarks
In this issue of the
SmartDO eNews, we introduced few global optimization technology of
SmartDO. We will continue to develop more Smart Global Optimization Technology,
and will keep you posted in our eNews.
For details about SmartDO, please visit our web site at http://www.fea-optimization.com/SmartDO/index_e.htm.
Reference
1. STOCHASTIC OPTIMIZATION OF TENSION LEVELING
PROCESS FOR PRODUCTION OF FLAT METALLIC STRIPS, Dr. Hiroshi Hamasaki, Dr.
Ryutaro Hino, Prof. Fusahito Yoshida (Hiroshima University, Japan), Prof.
Vassili V. Toropov (University of Leeds, United Kingdom), WCSMO7, May
2007, South Korea.
2. S-Y. Chen,
2007, Gradient-Based Structural and CFD Global Shape Optimization with
SmartDO and the Response Smoothing Technology, Proceedings of the 7th World
Congresses of Structural and Multidisciplinary Optimization (WCSMO7), COEX
Seoul, 21 May – 25 May 2007, Korea.
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All brand or product names are trademarks
or registered trademarks of their
respective holders. Copyright of all materials
in the links belongs to their respective authors.
I am not responsible for any contents inside any links.
(c)Copyright, 1998-, Shen-Yeh
Chen, Ph.D. All rights reserved.
www.FEA-Optimization.com
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