The design problem is to minimize the tip deflection of a stepped cantilever beam subject to constraints, as shown in Figure 4-5. The problem is a modified version of that from [101,102].
Figure 4-5: Stepped cantilever beam with d steps
To test the proposed algorithm, a cantilever beam with = 10 step is chosen with g = 50000 H force on the tip. ñ = 200 þgÇ and •GG; = 35 × 10è gÇ are selected as the
properties for the used material. For each step beam, there exist three variables for the optimization, the width (Û ), height (ℎ ) and length (h ) of each step of the beam. Therefore, 30 input variables exist in the minimization problem that the following order is chosen for defining the problem [103]:
µ = [Û , ℎ , h , Û , ℎ , h , … , Û , ℎ , h ] ( 4-12)
The objective is to minimize the tip deflection (O), expressed as the summation of all the deflections
O = 5G\g ñ + 5 g ( ñ€ + h ) € € G\ô& + ⋯ + 5G&g ( + h + hñ—+ ⋯ + h ) = gh— 3ñ ÚÛ ℎ12 Ý— + g 3ñ ÚÛ € ℎ—€ 12 Ý l(h € + h )—− h —n + ⋯ + g 3ñ ÞÛ12 ß— [(h + h + ⋯ + h ) —− (h + h—+ ⋯ + h )—] =3ñg Û ℎ12— h! ! — − h! ! ¾ — ( 4-13)
where g is the concentrated load, ñ is the material elasticity modulus, and is the moment of inertia about the neutral axis. There are 2 × + 1 constraints in the problem, where is the number of steps. First, the bending stress in all steps should be less than the allowable bending stress ( •GG; ). The constraints for each step beam can be described as:
6g ∑ h! !
Û ℎ ≤ •GG; / = 1,2, … , ( 4-14)
In addition, the aspect ratios of all cross sections form another set of constraints that can be shown as:
ℎ
where Ÿ‡ is the aspect ratio. The last constraint is that the total length of the cantilever beam should be more than a specified value:
h ≥ ¸‹ s ( 4-16)
¸‹ s is the minimum required length that is expected for the beam. In brief, the
minimization problem can be rewritten as:
O = (µ) =3ñg 12 — € — € ! —! — − —! ! ¾ — ( 4-17) Subject to: 6g — € — € ! —!≤ •GG; — € — € ≤ Ÿ‡ / = 1,2, … ,10 —! ! ≥ ¸‹ s / = 1,2, … ,10 ( 4-18)
Ÿ‡ = 25 and ¸‹ s = 6 are selected as the aspect ratio and minimum length for the
0.01 < Û < 0.05
0.30 < ℎ < 0.65
0.50 < h < 1.00
/ = 1,2, … ,10
( 4-19)
Using the standard format of optimization problems:
0.01 − — € < 0 — € − 0.05 < 0 0.30 − — € < 0 — € − 0.65 < 0 0.50 − — < 0 — − 1.00 < 0 / = 1,2, … ,10 ( 4-20)
In total 1471 sample points are used for RBF-HDMR metamodeling. The SM and CCM matrices are omitted due to the size of the matrices. Figure 4-6 shows the number of groups versus r‹ s values.
Figure 4-6: Number of groups after decomposition versus value for cantilever beam problem
Similar to Problem 4, for r‹ s = 0.0020 it reaches to 29 groups and for r‹ s = 0.0029, 30 decomposed groups are obtained. It means that the correlations are very weak and can be neglected. Therefore, two cases, undecomposed and completely decomposed, are tested for this problem using TRMPS optimization method, and the results are compared. Note that the number of function evaluations for the decomposed case includes the sample points that are used for RBF-HDMR.
Table 4-5: Optimization results of the stepped cantilever beam problem (Average for 10 runs)
f∗ (meters) NFE
Decomposed 0.0163 1768.7
Undecomposed 0.0188 2946.5
As can be seen from Table 4-5, the obtained optimum function value in decomposed case is less than that in the undecomposed case. Moreover, the decomposed case uses less NFE’s in comparison with the undecomposed one. The metamodel-based decomposition strategy [11], has five main disadvantages:
1. RBF-HDMR metamodeling needs structured samples, which are not always available for engineering problems and not amenable for optimization.
2. To find the structure of the objective function and the sensitivity of the correlations, a certain number of sample points is needed, which is in addition to the optimization costs.
3. When the problem is complicated, to find all the correlations, RBF-HDMR needs an excessive number of sample points that may go even beyond the allowed sampling budget.
4. In cases when all correlations are strong and the problem is deemed non- decomposable, the metamodeling basically wastes the valuable sample points without helping the optimization.
5. In cases when the decomposition is not accurate, optimization may miss the global optima and there is no chance to improve the decomposition scheme.
The mentioned disadvantages are addressed in next section by developing a metamodel-supported iterative decomposition.
4.3. Summary
In this chapter, a new metamodel-based decomposition approach is introduced. RBF- HDMR and RS-HDMR approaches are integrated to quantify the variable correlations to aid the problem decomposition for HEB problems. RBF-HDMR is used for building a surrogate of the black-box function and RS-HDMR is used for sensitivity analysis and predicting the intensities of the variable correlations. Based on the quantified correlations, a wider range of problems could be possibly decomposed and solved than otherwise. An accompany optimization scheme is also developed to facilitate the optimization on decomposed sub-problems as a system. For problems with all weak or a mix of weak and strong correlations, the proposed method is very effective in reducing the total number of function evaluations while reaching a similar accuracy as optimization on the original problem. For problems with all strong correlations, the proposed method meets its limit. There are few limitations in using the proposed metamodel-based decomposition method for real engineering problems. The structured
sampling in RBF-HDMR and the two stage decomposition/optimization strategy without iterative feedback are the most important limitations of the proposed method. In the next chapter, a new optimization strategy is proposed that combines decomposition and optimization phases. The proposed method iteratively improves the decomposition and does not need structured sampling.