Visualization and Data Mining of Pareto Solutions Using Self-Organizing Map

Abstract

Self-Organizing Maps (SOMs) have been used to visualize tradeoffs of Pareto solutions in the objective function space for engineering design obtained by Evolutionary Computation. Furthermore, based on the codebook vectors of cluster-averaged values of respective design variables obtained from the SOM, the design variable space is mapped onto another SOM. The resulting SOM generates clusters of design variables, which indicate roles of the design variables for design improvements and tradeoffs. These processes can be considered as data mining of the engineering design. Data mining examples are given for supersonic wing design and supersonic wing-fuselage design.

Full text

Visualization and Data Mining of Pareto Solutions
Using Self-Organizing Map
Shigeru Obayashi and Daisuke Sasaki
Institute of Fluid Science, Tohoku University,
Sendai, 980-8577 JAPAN
obayashi@ieee.org, sasaki@reynolds.ifs.tohoku.ac.jp
Abstract. Self-Organizing Maps (SOMs) have been used to visualize tradeoffs
of Pareto solutions in the objective function space for engineering design
obtained by Evolutionary Computation. Furthermore, based on the codebook
vectors of cluster-averaged values of respective design variables obtained from
the SOM, the design variable space is mapped onto another SOM. The resulting
SOM generates clusters of design variables, which indicate roles of the design
variables for design improvements and tradeoffs. These processes can be
considered as data mining of the engineering design. Data mining examples are
given for supersonic wing design and supersonic wing-fuselage design.
1 Introduction
Multiobjective Evolutionary Algorithms (MOEAs) are getting popular in many fields
because they will provide a unique opportunity to address global tradeoffs between
multiple objectives by sampling a number of non-dominated solutions. To understand
tradeoffs, visualization is essential. Although it is trivial to understand tradeoffs
between two objectives, tradeoff analysis in more than three dimensions is not trivial
as shown in Fig. 1. To visualize higher dimensions, Self-Organizing Map (SOM) by
Kohonen [1,2] is employed in this paper.
SOM is one of neural network models. SOM algorithm is based on unsupervised,
competitive learning. It provides a topology preserving mapping from the high
dimensional space to map units. Map units, or neurons, usually form a two-
dimensional lattice and thus SOM is a mapping from the high dimensions onto the
two dimensions. The topology preserving mapping means that nearby points in the
input space are mapped to nearby units in SOM. SOM can thus serve as a cluster
analyzing tool for high-dimensional data. The cluster analysis of the objective
function values will help to identify design tradeoffs.
Design is a process to find a point in the design variable space that matches with
the given point in the objective function space. This is, however, very difficult. For
example, the design variable spaces considered here have 72 and 131 dimensions,
respectively. One way of overcoming high dimensionality is to group some of design
variables together. To do so, the cluster analysis based on SOM can be applied again.
Based on the codebook vectors of cluster-averaged values of respective design
variables obtained from the SOM, the design variable space can be mapped onto

another SOM. The resulting SOM generates clusters of design variables. Design
variables in such a cluster behave similar to each other and thus a typical design
variable in the cluster indicates the behaviour/role of the cluster. A designer may
extract design information from this cluster analysis. These processes can be
considered as data mining for the engineering design.
At first, SOM is applied to map objective function values of non-dominated
solutions in four dimensions. This will reveal global tradeoffs between four design
objectives. The multipoint aerodynamic optimization of a wing shape for SST at both
supersonic and transonic cruise conditions has been performed by using MOEAs
previously [3]. Both aerodynamic drags were to be minimized under lift constraints,
and the bending and pitching moments of the wing were also minimized instead of
imposing constraints on structure and stability. A high fidelity Computational Fluid
Dynamics (CFD) code, a Navier-Stokes code, was used to evaluate the wing
performance at both conditions. In this design optimization, planform shapes, camber,
thickness distributions and twist distributions were parameterized in total of 72 design
variables. To alleviate the required computational time, parallel computing was
performed for function evaluations. The resulting 766 non-dominated solutions are
analyzed to reveal tradeoffs in this paper. The resulting SOM is also used to create a
new SOM of the cluster-averaged design variables.
Second, SOM is applied to map entire solutions evaluated during the evolution of
two-objective optimization. Based on the wing design system mentioned above, an
aerodynamic optimization system for SST wing-body configuration was developed in
[4]. To satisfy severe tradeoff between high aerodynamic performance and low sonic
boom, the present objectives were to reduce CD at a fixed CL as well as to satisfy the
equivalent area distribution for low boom design proposed by Darden [5]. Wing shape
and fuselage configuration were defined in total of 131 design variables. The SOM of
the objective function values indicates the non-dominated front as edges of the map
and the SOM of the cluster-averaged design variables reveals the role of the design
variables for design tradeoffs.
2 objectives 3 objectives ?
4 objectives
?
4 objectives
Projection
Minimization problems
Fig. 1. Visualization of Pareto front

2 Evolutionary Multiobjective Optimization
2.1 MOGAs
The genetic operators used here are based on MOGAs [6,7]. Selection is based on the
Pareto ranking method and fitness sharing. Each individual is assigned to its rank
according to the number of individuals that dominate it. A fitness sharing function is
used to maintain the diversity of the population. To find non-dominated solutions
more effectively, the so-called best-N selection is employed.
For real function optimizations like the present research, however, it is more
straightforward to use real numbers for encoding. Thus, the floating-point
representation is used here. Accordingly, blended crossover (BLX-α) [8] is adopted at
the crossover rate of 100%. This operator generates children on a segment defined by
two parents and a user specified parameter α. The disturbance is added to new design
variables within 10% of the given range of each design variable at a mutation rate of
20%. Crossover and mutation rates are kept high because the best-N selection gives a
very strong elitism. Details for the present MOGA were given in Refs. 3, 4 and 7.
2.2 CFD Evaluation
To evaluate the design, a high fidelity Euler/Navier-Stokes code was used. Taking
advantage of the characteristic of GAs, the present optimization is parallelized on SGI
ORIGIN2000 at the Institute of Fluid Science, Tohoku University. The system has
640 Processing Elements (PE’s) with peak performance of 384 GFLOPS and 640 GB
of memory.
A simple master-slave strategy was employed: The master PE manages the
optimization process, while the slave PE’s compute the Navier-Stokes code. The
parallelization became almost 100% because almost all the CPU time was dominated
by CFD computations. The population size used in this study was set to 64 so that the
process was parallelized with 32-128 PE’s depending on the availability of job
classes. The present optimization requires about six hours per generation for the
supersonic wing case when parallelized on 128 PE’s.
2.3 Neural Network and SOM
SOM [1,2] is a two-dimensional array of neurons:
{
}
qp×
=mmM L
1 (1)
One neuron is a vector called the codebook vector:
[
]
n
iii mm L
1
=m (2)

This has the same dimension as the input vectors (n-dimensional). The neurons are
connected to adjacent neurons by a neighbourhood relation. This dictates the
topology, or the structure, of the map. Usually, the neurons are connected to each
other via rectangular or hexagonal topology. One can also define a distance between
the map units according to their topology relations.
The training consists of drawing sample vectors from the input data set and
“teaching” them to SOM. The teaching consists of choosing a winner unit by means
of a similarity measure and updating the values of codebook vectors in the
neighbourhood of the winner unit. This process is repeated a number of times.
In one training step, one sample vector is drawn randomly from the input data set.
This vector is fed to all units in the network and a similarity measure is calculated
between the input data sample and all the codebook vectors. The best-matching unit is
chosen to be the codebook vector with greatest similarity with the input sample. The
similarity is usually defined by means of a distance measure. For example in the case
of Euclidean distance the best-matching unit is the closest neuron to the sample in the
input space.
The best-matching unit, usually noted as mc, is the codebook vector that matches a
given input vector x best. It is defined formally as the neuron for which
[
]
i
i
cmxmx −=− min (3)
After finding the best-matching unit, units in SOM are updated. During the update
procedure, the best-matching unit is updated to be a little closer to the sample vector
in the input space. The topological neighbours of the best-matching unit are also
similarly updated. This update procedure stretches the best-matching unit and its
topological neighbours towards the sample vector. The neighbourhood function
should be a decreasing function of time. In the following, SOMs were generated in
the hexagonal topology by using Viscovery® SOMine 4.0 Plus [9].
2.4 Cluster Analysis
Once SOM projects input space on a low-dimensional regular grid, the map can be
utilized to visualize and explore properties of the data. When the number of SOM
units is large, to facilitate quantitative analysis of the map and the data, similar units
need to be grouped, i.e., clustered. The two-stage procedure --- first using SOM to
produce the prototypes which are then clustered in the second stage --- was reported
to perform well when compared to direct clustering of the data [10].
Hierarchical agglomerative algorithm is used for clustering here. The algorithm
starts with a clustering where each node by itself forms a cluster. In each step of the
algorithm two clusters are merged: those with minimal distance according to a special
distance measure, the SOM-Ward distance [9]. This measure takes into account
whether two clusters are adjacent in the map. This means that the process of merging
clusters is restricted to topologically neighbored clusters. The number of clusters will
be different according to the hierarchical sequence of clustering. A relatively small
number will be chosen for visualization (§3.2), while a large number will be used for
generation of codebook vectors for respective design variables (§3.3).

3 Four-Objective Optimization for Supersonic Wing Design
3.1 Formulation of Optimization
Four objective functions used here are
1. Drag coefficient at transonic cruise, CD,t
2. Drag coefficient at supersonic cruise, CD,s
3. Bending moment at the wing root at supersonic cruise condition, MB
4. Pitching moment at supersonic cruise condition, MP
In the present optimization, these objective functions are to be minimized. The
transonic drag minimization corresponds to the cruise over land; the supersonic drag
minimization corresponds to the cruise over sea. Lower bending moments allow less
structural weight to support the wing. Lower pitching moments mean less trim drag.
The present optimization is performed at two design points for the transonic and
supersonic cruises. Corresponding flow conditions and the target lift coefficients are
described as
1. Transonic cruising Mach number, M∞,t = 0.9
2. Supersonic cruising Mach number, M∞,s = 2.0
3. Target lift coefficient at transonic cruising condition, CL,t = 0.15
4. Target lift coefficient at supersonic cruising condition, CL,s = 0.10
5. Reynolds number based on the root chord length at both conditions, Re=1.0 x 107
Flight altitude is assumed at 10 km for the transonic cruise and at 15 km for the
supersonic cruise. To maintain lift constraints, the angle of attack is computed for
each configuration by using CLα obtained from the finite difference. Thus, three
Navier-Stokes computations per evaluation are required. During the aerodynamic
optimization, wing area is frozen at a constant value.
Design variables are categorized to planform, airfoil shapes and the wing twist.
Planform shape is defined by six design variables, allowing one kink in the spanwise
direction. Airfoil shapes are composed of its thickness distribution and camber line.
The thickness distribution is represented by a Bézier curve defined by nine polygons.
The wing thickness is constrained for structural strength. The thickness distributions
are defined at the wing root, kink and tip, and then linearly interpolated in the
spanwise direction. The camber surfaces composed of the airfoil camber lines are
defined at the inboard and outboard of the wing separately. Each surface is
represented by the Bézier surface defined by four polygons in the chordwise direction
and three in the spanwise direction. Finally, the wing twist is represented by a B-
spline curve with six polygons. In total, 72 design variables are used to define a whole
wing shape. A three-dimensional wing with computational structured grid and the
corresponding CFD result are shown in Figs. 2 and 3. See Ref. 3 for more details for
geometry definition and CFD information.

Fig. 2. Wing grid in C-H topology Fig. 3. Pressure contours on the upper surface
of a wing computed by the CFD code
3.2 Visualization of Design Tradeoffs: SOM of Tradeoffs
The evolution was computed for 75 generations until all individuals become non-
dominated. An archive of non-dominated solutions was also created along the
evolution. After the computation, the 766 non-dominated solutions were obtained in
the archive as a three-dimensional surface in the four-dimensional objective function
space. By examining the extreme non-dominated solutions, the archive was found to
represent the Pareto front qualitatively.
The present non-dominated solutions of supersonic wing designs have four design
objectives. First, let’s project the resulting non-dominated front onto the two-
dimensional map. Figure 4 shows the resulting SOM with seven clusters. For better
understanding, the typical planform shapes of wings are also plotted in the figure.
Lower right corner of the map corresponds to highly swept, high aspect ratio wings
good for supersonic aerodynamics. Lower left corner corresponds to moderate sweep
angles good for reducing the pitching moment. Upper right corner corresponds to
small aspect ratios good for reducing the bending moment. Upper left corner thus
reduces both pitching and bending moments.
Figure 5 shows the same SOM contoured by four design objective values. All the
objective function values are scaled between 0 and 1. Low supersonic drag region
corresponds to high pitching moment region. This is primarily because of high sweep
angles. Low supersonic drag region also corresponds to high bending moment region
because of high aspect ratios. Combination of high sweep angle and high aspect ratio
confirm that supersonic wing design is highly constrained.

Fig. 4. SOM of the objective function values and typical wing planform shapes
CDt
0.02 0.15 0.29 0.42 0.55 0.68 0.81 0.95
CDs
0.04 0.17 0.31 0.44 0.57 0.70 0.83 0.96
Mb
0.03 0.17 0.30 0.43 0.57 0.70 0.83 0.97
Mp
0.01 0.15 0.28 0.42 0.56 0.69 0.83 0.97
Fig. 5. SOM contoured by each design objective
3.3 Data Mining of Design Space: SOM of Design Variables
The previous SOM provides clusters based on the similarity in the objective function
values. The next step is to find similarity in the design variables that corresponds to
the previous clusters. To visualize this, the previous SOM is first revised by using
larger number of clusters of 49 as shown in Fig. 6. Then, all the design variables are

averaged in each cluster, respectively. Now each design variable has a codebook
vector of 49 cluster-averaged values. This codebook vector may be regarded to
represent focal areas in the design variable space. Finally, a new SOM is generated
from these codebook vectors as shown in Fig. 7.
This process can be done for encoded design variables (genotype) and decoded
design variables (phenotype). In the earlier study, the genotype was used for SOM.
However, the genotype and phenotype generated completely different SOMs. A
possible reason is because the various scaling appears in phenotype, for example, one
design variable is between 0 and 1 and another is between 35 to 70. The difference of
order of magnitude in design variables may lead to different clusters. To avoid such
confusion, the genotype is used for SOM here.
In Fig. 7, the labels indicate 72 design variables. DVs 00 to 05 correspond to the
planform design variables. These variables have dominant influence on the wing
performance. DVs 00 and 01 determine the span lengths of the inboard and outboard
wing panels, respectively. DVs 02 and 03 correspond to leading-edge sweep angles.
DVs 04 and 05 are root-side chord lengths. DVs 06 to 25 define wing camber. DVs
26 to 32 determine wing twist. Figure 7 contains seven clusters and thus seven design
variables are chosen from each cluster as indicated. Figure 8 shows SOM’s of Fig. 4
contoured by these design variables.
The sweep angles, DVs 02 and 03, make a cluster in the lower left corner of the
map in Fig. 7 and the corresponding plots in Fig. 8 confirm that the wing sweep has a
large impact on the aerodynamic performance. DVs 11 and 51 in Fig. 8 do not appear
influential to any particular objective. By comparing Figs. 8 and 5, DV 01 has similar
distribution with the bending moment Mb, indicating that the wing outboard span has
an impact on the wing bending moment. On the other hand, DV 00, the wing inboard
span, has an impact on the pitching moment. DV 28 is related to transonic drag. DV
04 and 05 are in the same cluster. Both of them have an impact on the transonic drag
because their reduction means the increase of aspect ratio. Several features of the
wing planform design variables and the corresponding clusters are found out in the
SOMs and they are consistent with the existing aerodynamic knowledge.
Fig. 6. SOM of objective function values with 49 clusters

v01
d
v
15
d
v
23
d
v
41
d
v
00
d
v
56
d
v
20
d
v
60
d
v
16
d
v
1
dv50 dv68
dv42 dv08 dv49 dv
0
dv10 dv43 dv29
dv54
dv30 dv66 dv36
3
4 dv48 dv25 dv
0
dv46 dv19 dv24
dv62 dv65 dv67
dv13 dv
5
dv58 dv06
dv64 dv14
v
18 dv21 dv71
dv26 dv22 dv
3
dv35 dv53
dv12 dv37 dv40
11 dv69 dv32 dv59
dv07dv52 dv70 dv47 dv
0
0
2 dv63 dv45 dv39 dv51
dv03
dv55
dv61
dv27
dv28
dv38
dv44
dv3
v01
d
v
15
d
v
23
d
v
41
d
v
00
d
v
56
d
v
20
d
v
60
d
v
16
d
v
1
dv50 dv68
dv42 dv08 dv49 dv
0
dv10 dv43 dv29
dv54
dv30 dv66 dv36
3
4 dv48 dv25 dv
0
dv46 dv19 dv24
dv62 dv65 dv67
dv13 dv
5
dv58 dv06
dv64 dv14
v
18 dv21 dv71
dv26 dv22 dv
3
dv35 dv53
dv12 dv37 dv40
11 dv69 dv32 dv59
dv07dv52 dv70 dv47 dv
0
0
2 dv63 dv45 dv39 dv51
dv03
dv55
dv61
dv27
dv28
dv38
dv44
dv3
dv01
dv11
dv02
dv00
dv28
04
05
Fig. 7. SOM of cluster-averaged design variables
dv02
0.2 0.4 0.6 0.8 1.0
dv11
0.0 0.3 0.5 0.8 1.0
dv01
0.04 0.49 0.95
dv28
0.01 0.50 0.99
dv51
0.0 0.3 0.5 0.8 1.0
dv00
0.0 0.2 0.5 0.7 1.0
dv04
0.00 0.44 0.88
Fig. 8. SOM contoured by design variables selected from clusters in Fig. 7

4 Two-Objective Optimization for Supersonic Wing-Fuselage
Design
4.1 Formulation of Optimization
In this study, SST wing-body configurations are designed to improve the aerodynamic
performance and to lower the sonic boom strength. Therefore, design objectives are to
reduce CD at Mach number 2.0 at a fixed CL (=0.10) and to match Darden’s equivalent
area distribution that can achieve low sonic boom. Multiblock Euler calculation was
used to evaluate aerodynamic performance [11]. For the evaluation of sonic boom
strength, an equivalent area distribution is matched to Darden’s equivalent area
distribution for 300 ft fuselage SST at Mach number 1.6 at CL = 0.125.
To evaluate aerodynamic performances, aerodynamic evaluation has to be
automatically performed for a given SST wing-body configuration. The wing
definition was almost same as the previous wing optimization. Then, 55 additional
design variables were used to define nonsymmetric fuselage configuration. Four more
design variables represented the wing lofting. The total number of design variables is
131.
As body length and wing area is fixed to 300 ft and 9,000 ft2, respectively, body
volume, minimum diameter of body and wing volume must be greater than values
given in Table 1. The other constraints are implemented to design variables as
boundaries. As a result, the present SST wing-body design problem has two objective
functions of minimization, three constraints and 131 design variables, and is
optimized by real-coded MOGAs. Master-slave type parallelization was again
performed to reduce the large computational time of each CFD evaluation in the
optimization process. Figures 9 and 10 show typical computational grid and
corresponding CFD result, respectively. See Ref. 4 for more details for geometry
definition and CFD information.
Table 1. Constraints of SST wing-body configuration
Body volume ≥ 30,000 ft3
Minimum diameter ≥ 11.8 ft (0.23≤x/L≤0.70)
Wing volume ≥ 16,800 ft3

X Y
Z
17
18
19 23
17
18
19 23
Fig. 9. Surface grid for SST wing-
fuselage configuration (numbers indicate
corresponding multiblock grids)
Fig. 10. Computed pressure distribution
on the upper surface of SST wing-
fuselage configuration
4.2 Visualization of Design Tradeoffs: SOM of Function Landscape
First, all the solutions obtained during the present evolutionary computation were
mapped onto SOM according to the scaled objective function values. The resulting
SOM is shown in Fig. 11. Several non-dominated solutions are indicated by * in the
figure. The map consists of eight clusters. The lower left cluster contains the extreme
non-dominated solution of the minimum drag. The upper right cluster contains the
extreme non-dominated solution of the minimum boom. The corresponding objective
functions values are then plotted in Fig. 12. Because only two objectives are used
here, the map coordinates approximately matches to the objectives. The vertical
direction corresponds to the drag and the horizontal axis corresponds to the sonic
boom. The lower edge and the right edge of the map indicate the non-dominated
front. Although the mapping is not essential to visualize tradeoffs here, the cluster
analysis may be used to generate clusters of design variables.
Fig. 11. SOM of the objective function values

DRAG
BOOM
Fig. 12. SOM coloured by each design objective
4.3 Data mining of Design Space: SOM of Design Variables
To generate SOM of the design variables, Fig. 11 was divided into 50 clusters as Fig.
13. Then, Fig. 14 was generated from codebook vectors of cluster-averaged design
variables in Fig. 13. Figure 14 shows SOM of the design variables in five clusters. In
Fig. 14, the labels indicate 131 design variables. Figure 14 can be interpreted from the
behaviors of the design variables representing the corresponding clusters. Figure 15
shows the map of Fig. 11 contoured by the five design variables indicated in Fig. 14.
A trend of the design variables in the left cluster of Fig. 14 is represented by DV 123
in Fig. 15. Its distribution appears the inverse of the sonic boom in Fig. 12. DV 123
determines the twist angle at the wing tip. It has an impact on the list distribution,
leading to influences on the equivalent cross sectional distribution and thus on the
sonic boom strength. The center cluster in Fig. 14 is represented by DV 2 and its
distribution in Fig. 15 appears the inverse of the drag in Fig. 12. DV 2 is one of the
design variables that define the sharpness of the nose of the fuselage. Blunt nose is
known to increase drag for supersonic aircraft. The right cluster in Fig. 14 is
represented by DV 28 and the corresponding distribution in Fig. 15 has a local
minimum in the middle of the left, upper edge of the map. This is one of the design
variables that determine the body radius distribution at the side of the fuselage, but it
does not seem primarily related to either objective here. DV’s 89 and 91 have
opposite trends, but they are not influential to the non-dominated front, either.
*
Pret
o
Pareto
Pareto
*
*
*
Pareto *
Fig. 13. SOM of objective function values with 50 clusters

DV32
DV61
DV30
DV91
DV69
DV60
DV82
DV88
DV55
DV37
DV2
DV4
DV20
DV52
DV28
DV1
DV93
DV2
V
72 DV8 DV110 DV62
DV128 DV108 DV122 DV18 DV81 DV
3
DV58 DV76
D
V27 DV96 DV94
DV125 DV67 DV106 DV48 DV9 DV56
V
77 DV36 DV39 DV73 DV78 D
V
DV102 DV109 DV121 DV97 DV45 DV26
DV115 DV16 DV104
D
V14 D
V
D
V64 DV33 DV57 DV63 DV116 DV113
DV46 DV34 DV51 DV7 DV53 DV95 DV40
DV119 DV126 DV66 DV9
DV
9
1 DV2 DV2
8
1
23 DV86 DV41 DV13 DV107 DV22
DV120 DV35 DV71 DV10 DV87
V
83 DV130 DV114 DV47 D
V
DV79 DV127 DV25 DV112 DV70 DV118 DV101 DV65
DV5 DV6 DV19 DV75 DV80 D
V
D
V15 DV105 DV12 DV38 DV129 DV29
DV21 DV11 DV103 DV99
DV12
3
D
V
V117
DV31
DV85
DV100
DV50
DV68
DV54
DV23
DV84
DV74
DV42
DV111
DV124
DV59
DV131
DV49
DV17
DV
89
Fig. 14. SOM of cluster-averaged design variables
DV123
*
Preto
Paret
o
Paret
o
*
*
*
Paret o
*
0.0 0.2 0.3 0.5 0.7 0.8 1.0
DV91
*
Preto
Paret
o
Paret
o
*
Paret o
*
0.00 0.25 0.50 0.74 0.99
DV2
*
Preto
Paret
o
Paret
o
*
*
*
Paret o
*
0.01 0.26 0.50 0.74 0.99
DV89
*
Preto
Paret
o
Paret
o
*
*
*
Paret o
*
0.00 0.24 0.48 0.72 0.97
DV28
*
Preto
Paret
o
Paret
o
*
Paret o
*
0.0 0.2 0.3 0.5 0.7 0.8 1.0
Fig. 15. SOM contoured by design variables selected from clusters in Fig. 14
5 Concluding Remarks
Design tradeoffs have been investigated for two multiobjective aerodynamic design
problems of supersonic transport by using visualization and cluster analysis of the
non-dominated solutions based on SOMs. The first optimization is to design

supersonic wings defined by 72 design variables with four objectives to be
minimized. The second optimization is to design supersonic wing-body
configurations represented by in total 131 design variables with drag and boom
minimization. Design data were gathered by MOGAs.
SOM is first applied to visualize tradeoffs between design objectives. In the first
design case, four objective functions were employed and 766 non-dominated
solutions were obtained. Three-dimensional non-dominated front in the objective
function space has been mapped onto the two-dimensional SOM where global
tradeoffs are successfully visualized. In the second design case, entire solutions
during the evolution have been mapped onto SOM to visualize function landscape,
and the non-dominated front was found at the edges of the map. The resulting SOMs
are further contoured by each objective, which provides better insights into design
tradeoffs.
Furthermore, based on the codebook vectors of cluster-averaged values for
respective design variables obtained from the SOMs, the design variable space is
mapped onto another SOM. Design variables in the same cluster are considered to
have similar influences in design tradeoffs. Therefore, by selecting a member (design
variable) from a cluster, the original SOM in the objective function space is contoured
by the particular design variable. It reveals correlation of the cluster of design
variables with objective functions and their relative importance. Because each cluster
of design variables can be identified influential or not to a particular design objective,
the optimization problem may be divided into subproblems where the optimization
will be easier to lead to better solutions.
These processes may be considered as data mining of the engineering design. The
present work demonstrates that MOGAs and SOMs are versatile design tools for
engineering design.
References
1 Kohonen T.: Self-Organizing Maps. Springer, Berlin, Heidelberg (1995)
2 Hollmen J.: Self-Organizing Map, http://www.cis.hut.fi/~jhollmen/dippa/node7.
html, last access on October 3, 2002
3 Sasaki D., Obayashi S. and Nakahashi K.: Navier-Stokes Optimization of
Supersonic Wings with Four Objectives Using Evolutionary Algorithm. Journal
of Aircraft Vol.39, No.4 (2002) 621-629
4 Sasaki D., Yang G. and Obayashi S.: Automated Aerodynamic Optimization
System for SST Wing-Body Configuration. AIAA Paper 2002-5549 (2002)
5 Darden, C. M.: Sonic Boom Theory: Its Status in Prediction and Minimization.
Journal of Aircraft, Vol.14, No.6 (1977) 569-576
6 Fonseca C. M. and Fleming P. J.: Genetic Algorithms for Multiobjective
Optimization: Formulation, Discussion and Generalization. Proc. of the 5th
ICGA (1993) 416-423
7 Obayashi S., Takahashi S. and Takeguchi Y.: Niching and Elitist Models for
MOGAs. Parallel Problem Solving from Nature – PPSN V, Lecture Notes in

Computer Science, Springer, Vol.1498, Berlin Heidelberg New York (1998)
260-269
8 Eshelman L. J. and Schaffer J. D.: Real-Coded Genetic Algorithms and Interval
Schemata. Foundations of Genetic Algorithms 2, Morgan Kaufmann Publishers,
Inc., San Mateo (1993) 187-202
9 Eudaptics software gmbh. http://www.eudaptics.com/technology/somine4.html,
last access on October 3, 2002
10 Vesanto, J. and Alhoniemi, E.: Clustering of the Self-Organizing Map, IEEE
Transactions on Neural Networks, Vol.11, No.3 (2000) 586 -600
11 Yang, G., Kondo, M. and Obayashi, S.: Multiblock Navier-Stokes Solver for
Wing/Fuselage Transport Aircraft. JSME International Journal, Series B, Vol.45,
No.1 (2002) 85-90