Page 359 - Practical Design Ships and Floating Structures

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We must therefore evaluate the fitness criterion of the approximate polynomial function before we
utilize it. And if the solution space is too large, it seems practical to subdivide the space into some
smaller regions and assign an accurate function to each of them. In this paper, we assume that the
region of interest, which includes the optimum inside point, is comparatively narrow, and that the
sub-division of the space is unnecessary.

3 EXAMPLES OF OPTIMIZATION BY RSM

3.1 Minimization of Bulkhead Structural Weight
3.1.2 Examination model

As a first example, we will show the results of the optimization of a transverse bulkhead of an oil
tanker center tank as shown in Figure 1. The goal in this example is the minimhition of structural
weight. The number of vertical stiffeners nvs and also the number of horizontal girders n,, are
considered design variables. For simplicity, those stiffening structures are assumed to be installed at
equal intervals. Scantlings of bulkhead plates and stiffeners are determined by the section modulus
requirements of NK (Nippon Kaiji Kyokai) rules, Le., plate thickness and section modulus for
stiffeners, taking into account of the space between the vertical stiffeners and that of the horizontal
girders. Furthermore, the cross-sectional shape of the vertical stiffeners and horizontal girders are
determined using the method described by Mano & Yoshida (1982) as the optimum shape (in view of
the weight minimum) that satisfies the required section modulus. For the edges of the vertical
stiffeners and horizontal girders, bracket plates of adequate size are installed. In this study, therefore,
the independent design variables are nvs and n,,, only, and the gross weight (W) of the transverse
bulkhead structures are taken as objective functions.

3. I 2 Results of weight minimum optimuation
There is no restriction in the order and the number of terms in the approximate polynomial used in this
study. However, if the numbers of orders and terms of the approximate polynomial are unnecessarily
increased, not only an increase in the calculation load but also instability of the approximation are
anticipated. Therefore, in this paper, we will use the second- and thkd-order polynomial and examine
the effect of those approximations.
The second-order model becomes:
w = Po + PA +An, + s3t 4- $An, 4. ps.:, (7)
And the third-order model becomes:
= flo ' PlnW + PZn@ ' fl,d + P,nVSn& flSnig
+ p6n:;nhg + + fl8%Sn& pgn:r (8)
For the number of combinations (Le., experimental points) of design variables necessary for
determining the unknown parameters, we used twice the unknown parameter number. In this paper, as
the first example, we compare the following three cases and will discuss the results.
Case 1 : Second-order model with 12 experimental points
Case2: Third-order model with 20 experimental points
Case3: Exact solution
The region of interest is 15 5 nvr 5 60 and 1 I n,, S 15 . Therefore, the number of candidate
experimental points in the region is 690. In Case 3, all the responses at these 690 experimental points
are evaluated. For Case 1 and Case 2, by using the @Optimal design, the effective experimental
points are selected, and the approximate response surfaces are computed using the least squares
method. The results are shown in Figures 2 to 4. The results of the second-order model and the