A Comparison of Search Techniques on a Wing-Box Optimisation ...

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Abstract. This paper describes a thorough comparison of ten di er- ent search techniques applied to a wing-box design optimisation prob- lem. The techniques ...
A Comparison of Search Techniques on a Wing-Box Optimisation Problem Malcolm McIlhagga, Phil Husbands and Robert Ives School of Cognitive and Computing Sciences, University of Sussex, Brighton, BN1 9QH, UK

Abstract. This paper describes a thorough comparison of ten di er-

ent search techniques applied to a wing-box design optimisation problem. The techniques used vary from deterministic gradient descent to stochastic Simulated Annealing (SA) and Genetic Algorithms (GAs). The stochastic techniques produced as good solutions as the best found by the deterministic techniques. However, only the stochastic techniques consistently produced very good solutions every run. Signi cantly, only a distributed genetic algorithm (DGA) and hybrid methods (SA with gradient descent, DGA with gradient descent) had a reliable fast decent to good regions of solution space. Of these the hybrid DGA was signi cantly better than anything else. The issue of generating solutions stable to perturbations of the problem variables, without greatly increasing the runtime of the objective function, is also discussed. We describe a method for producing highly stable solutions with the DGA while increasing the run time of the objective function by a factor of only 4. No explicit term dealing with stability was added to the objective function.

1 Introduction Many engineering design optimisation problems involve search spaces with a highly complex structure. Stochastic techniques such as simulated annealing and genetic algorithms have been successfully applied to a number of these [4, 10], and folklore states that these are among the sorts of problems to which these techniques are best suited, having advantages over other methods [6, 4]. However, with a few notable exceptions [7], there has been a tendency to neglect large-scale comparative studies of many optimisation techniques applied to problems of this nature. We believe that studies of this kind are necessary in order to gain greater insight into the relative merits and weaknesses of di erent techniques, and to form an understanding of what kinds of problems they are most appropriate for. This is particularly important for genetic algorithms, where the body of empirical and theoretical work is still relatively small. As a contribution to knowledge in this area, we made a thorough (at least 5  107 objective function calls per technique) comparative study of ten optimisation techniques applied to an industrially-based structural design optimisation problem. The search techniques used were: random search; brute force search; local gradient descent; gradient descent; Powell's method; simulated annealing; a

distributed genetic algorithm (DGA); the Alopex method; hybridized simulated annealing and gradient descent; and hybridized DGA and local gradient descent. The next section describes the structural design problem. Section 3 explains the solution representation used in this study. Section 4 outlines the implementation of each of the techniques used. This is followed by a presentation of the results. The paper continues with a discussion of the issue of stability of solutions, and a simple DGA-based technique that produces highly stable solutions is described. Finally, conclusions are drawn.

2 Description of the Wing Box Problem Research in the design of aircraft structures, such as airframes and their components, often considers problems in which it is necessary to develop structures of minimum weight that can withstand a given load. For this study a simpli ed wing-box design optimisation task, based on industrially-realistic data, was formulated with the help of British Aerospace. The problem was simpli ed in such a way as to greatly reduce the computational cost of the evaluation function, but at the same time preserve many of the most salient features of the real problem. Hence a solution to the simpli ed problem gives a good `ball park' indication of areas of the design space of the full problem to investigate in detail. The wing box problem can be simply described as the task of nding the number of panels needed in an aircraft wing topskin and the thickness of each of those panels while minimising the mass of the wing and ensuring that none of the panels buckle under maximum operational stresses. FigSchematic View of a Wing ure 1 shows a schematic plan view of the wing discussed in this document. It was assumed that the topskin panels Fig. 1. Schematic view of wing can be modelled as at plates, and that a panel undergoes uniform compressive stresses. It was also assumed that the spacing between ribs (rib pitch) is uniform and that the thickness of all ribs is the same. Hence the aim is to nd the rib pitch, dictated by the number of panels, and the thickness of the top panels, such that the structure has minimum weight and will not buckle under the compressive stresses produced by the bending moments of a 2g manoeuvre. Rib Pitch

Wing Length (fixed)

2.1 The Objective Function The objective function (to be minimised) used throughout is given in Equation 1: C=

i=n X

=1

i

Mi (Pi + 1) + RM

(1)

Where n is the number of panels, Mi is the mass of the ith panel, Pi is the penalty term described in Equation 2 and RM estimates total rib mass.  i if i > ti ; (2) Pi = 0ti otherwise: Where i is the compressive stress in the ith panel and ti is a threshold stress that must not be exceeded in that panel; i is calculated using Equation 3. 

i =

BMi EHi



 Rs

(3) CSi Where BMi is the mean bending moment on the ith section, found by interpolating from tables of empirical data [1]; EHi is the height of the ith section, found by interpolating from tables; Rs is a safety factor and CSi is the cross sectional area of the ith section. Referring back to Equation 2, ti is calculated using Equation 4. p

ti = K  Et  LIi (4) Where K, the material structural eciency, is a constant; Et is the tangent modulus of elasticity which is calculated from Equation 5; and LIi is the loading index on the ith section calculated using Equation 6. 1 Et = (5) E 1 +  i m?1 n Where m is an index corresponding to the material shape factor, n is a reference stress and E is the Young's modulus of the material. Equation 5 is often referred to as the Ramberg-Osgood equation [9]. Equations 4 and 5 incorporate the assumption that the top skin can be treated as a at plate in which the thickness is small compared to the other dimensions. In these circumstances the tangent modulus of elasticity has been shown to give conservative values for buckling stress corresponding closely to test results [9]. 

LIi =

 BMi  Rs EHi wi  li

(6)

Where wi is the width of the middle of the ith section and li is the length of the section (distance between ribs). The mass of a panel is calculated using Equation 7. Mi = ti   Ai   (7) Where ti is the thickness of the ith panel, is a correcting factor for the sti ening action of stringers which e ectively increases the skin thickness. This is based on the procedure described in [5]. Ai is the planform area of the ith panel, taking into account wing tapering, and  is the density of the skin material.

Finally, RM, the total rib mass estimate, is calculated using Equation 8. RM = (a  Rp ? b)  Rp + c (8) Where Rp is the rib pitch and a, b and c are constants. This Equation was derived by tting a quadratic to empirical data [1]. The rib mass is scaled from a datum aircraft based on rib pitch and equivalent top skin thickness. As the rib pitch increases, the total rib weight decreases. The values used in this study for the various constants mentioned above were: Rs = 1:5; K = 0:7746; E = 71000N=mm2; n = 511N=mm2 ; m = 45:6; = 1:116;  = 2800Kg=m3 ; a = 0:021; b = 37:56; c = 23385:92.

3 Solution representation - the delta encoding A solution to the problem is represented by de ning for each section, apart from the one nearest to the fuselage, a term by which the thickness of that section should be increased/decreased from the previous one. Figure 2 illustrates the encoding. n?th panel is the amount by which the thickness of the n-th panel is bigger or smaller than that of the (n-1)th panel. Most of the search techniques used in this study could not handle variable dimensionality search spaces, so they were run on separate problems de ned for each possible Fig. 2. Delta Encoding used by all number of panels in the range 45{65. However, the GA easily handled variable numSearch Techniques bers of panels in a single run by having a xed length encoding allowing a maximum of 65 panels, but specifying how many panels to consider as one of the variables. The thickness of the initial panel was de ned to be in the range 5-15 mm (discretized into multiples of 0.125 mm), 0:25; ?0:125; 0:0;0:125;0:25g. This gives a search space P and10n 2 f? n?1, where n is the number of panels. of size nn=65  5 =45 0:125 Number of Ribs Thickness of 1st Panel

±∂ 2nd Panel

...

±∂ Nth Panel

4 Overview of Search Techniques Random Search simply consisted of generating a random solution and checking if it was better than the current best. This process was continued until the stop condition of 5  107 evaluations was met. Brute Force Search consisted of a systematic search through every possible combination of panel thicknesses until 5  107 evaluations were reached. A simple implementation of the standard Gradient Descent technique was used. A move from the current solution to the next solution is de ned by taking the set of solutions that are one increment and decrement of each dimension of the space away from the current solution, and moving to the solution with the best evaluation. The stop condition of one run of the search is to nd no better solution within that set of immediate neighbours to the current solution.

Local Gradient Descent is a simple variant on gradient descent. When evaluating all immediate perturbations of the current solution, instead of taking the best solution of that set, the algorithm moves to the rst solution found that is better than the current solution. Powell's Direction Set Algorithm is detailed in [11]. It can be brie y described as follows: given a set of conjugate directions (initially the basis vector directions), and a current position, record the tness gain of movement in each dimension. From the directional tness gain information gleaned, extrapolate a new vector direction that should lead to movements of greatest value. Minimise along this direction, using a single variable technique. If there is a signi cant gain, then add it to the direction set, replacing the direction of greatest descent from the last move. Simulated Annealing is a well know stochastic technique fully described in the literature [8]. In the implementation used in this study, a new solution was generated from the current solution as follows: with a probability of 0.6 a simple mutation was applied, with a probability of 0.4 a heuristic mutation was applied. The heuristic mutation favoured changes being made to the thickest part of the topskin. After a thorough empirical investigation of possible annealing schedules, an accept:reject ratio1 of 1.6N:16N was adopted for the main study, where N is the number of dimensions of the problem. The stopping condition for a run was 7,500 drops in temperature or 200 drops in temperature without any improvement, whichever came sooner. The initial temperature was set at 2:8  1012. Distributed Genetic Algorithms di er from standard generational GAs [6] in their use of an underlying conceptual model of a population spread out over a 2D grid with local asynchronous mating. There is mounting evidence that this sort of GA, whether implemented in parallel or sequentially (essentially simulating parallelism) gives very good reliable results over a range of problems [3]. The algorithm used was as follows: 1. Create random genotypes and place in cells on a 2D toroidal grid 2. Pick cell on grid, C, this holds genotype Cg 3. Create a set of cells, S, in neighbourhood of C 4. Select (proportional to tness) a genotype, m, from one of the cells in S 5. Create o spring, O, from m and Cg 6. Select a genotype, d, at a cell in S 7. Replace d with O 8. Goto 2. The set creation at step 3 simply used the nearest 12 neighbours to cell C. Selection at step 4 used xed linear selection probabilities determined by the rank of elements of set S. The least t element of S was replaced at step 7. This sort of algorithm allows very erce local selection, but because neighbourhoods overlap, there is rapid di usion around the grid of better than average genotypes. In this 1

For an accept:reject ratio of A:R, the temperature is lowered after A accepted moves or R rejected moves, whichever comes sooner.

study a population of 1500 on a 50x50 grid was employed. Two point cross-over and mutation were the genetic operators used. Each run was terminated after 1  106 evaluations. The Alopex Method is relatively unknown (see [12]), so it will be brie y described here. It is a stochastic algorithm that causes the simultaneous convergence of a number of parameters, using a biased random walk. On a given iteration, movement is made in every dimension of the search space by a small increment. Whether the move is positive or negative in the dimension considered will depend on: the previous move made in that dimension, correlated with the overall move. Formally, xni = xin?1 + in ,  tness gain of the previous n with probability pi ; n?1 1 n n where in = ? with probability 1 ? pni : , pi = 1+e T ni , and i = (xi ? xin?2)(F n?1 ? F n?2). Here xni is the value on the nth iteration of the ith member of the vector of values making up a solution to the problem; in is the amount to alter xin?1 by in producing the next solution; T is the `temperature'; and F n is the cost of the solution on the nth iteration. As the tness changes on each iteration decrease towards a converged solution, the temperature drops, thus causing automatic annealing: T = hjiji: In the implementation used here, the initial temerature was set at 1000 and the step size in all dimensions was xed at 0.0625. The maximum number of iterations performed on any run was 1  106. Hybrid Simulated Annealing with Gradient Descent involved gradient descent to a local optima after every SA move. In order to cut down the potential number of evaluation function calls, it was decided to Hybridise the DGA with Local Gradient Descent rather than ordinary gradient descent. Local gradient descent was run for Ne evaluations, or to the nearest minima { whichever came sooner, on each new o spring before placing it in the population. Their genetic material was altered during the local search, and their cost was taken to be the value returned by the gradient descent phase. Each run terminated after 2  106 evaluations, and Ne decreased linearly from 200 to 0 during the course of a run. A Re-evaluating DGA was developed to deal with stability issues and is dealt with in Section 6.

5 Results This section presents the data generated by the various search techniques. Each technique was run for 5  107 evaluations to allow water-tight statistics. Note that a test with this number of evaluations for a single technique comprised many runs of the method; each run was started from a new random position in the search space. Table 1 summarises the results of the comparative study. All of the deterministic methods (with the exception of Brute Force search) found comparatively good solutions within the liberal time-constraints allowed. However, the very large standard deviations for most of these techniques indicates that although they can nd good solutions, this is very unlikely and in

Search Technique

Random search Brute Force Search Local Gradient Descent Gradient Descent Powell SA DGA Alopex Hybrid SA and Gradient Descent Hybrid DGA and Local Grad. Desc.

Legal Best Solution Mean SD Solutions Solution ? (Kg) Solution (Kg) (Kg)

Table 1.

YES NO YES YES YES YES YES YES YES YES

13345.8 9:6  108 12761.9 12761.6 13015.8 12763.2 12859.5 13077.3 12759.5 12815

13663.8 2  1011 1:9  109 3  109 13456 12770 12921 5313578.3 12764.4 12856

336 6:6  1011 7:1  1010 9:8  1010 145.7 9.1 26.6 36588508 5.3 40.6

Comparison of all techniques

general these techniques have to be run many many times for very large numbers of evaluations before a high quality solution is found. It should be noted that the random search results are for one single run of 5  107 evaluations. The mean and standard deviation refer to the set of best-solution-so-far found during the course of the run, which necessarily monotonically decreases. Hence these gures should not be compared with those given for the other techniques. Results from the gradient descent methods give empirical credence to the belief that the search space is highly convoluted with many local minima. The GA and SA methods proved much better at avoiding the many local optima around the 13000?14000 Kg region of the search space. Both the DGA and SA consistently found very good solutions, i.e. had very low standard deviations of best solution. The DGA and SA methods were also able to handle variable numbers of panels rather than assuming a xed number as was the case with the other techniques. The Alopex method did not fare well. This appears to have been because there are complex interactions between the large number of variables in the problem. Always changing all variables at once is unlikely to retain the valuable parts of previous solutions. The DGA descended into the region of very good solutions much faster than SA; it typically reached solutions within 5% of the best found after only 100,000 evaluations, whereas SA needed several million calls to the objective function to proceed that far. Hybridized SA and Gradient Descent was as reliable as SA and found comparably good solutions. However, it produced them much faster. The performance of the hybridized DGA and Local Gradient Descent was very similar to that of the hybrid SA method in terms of reliability and quality of solution found. However, it was signi cantly faster again at descending into very low cost regions of the search space. Clearly the GA and SA based methods were head and shoulders above the other techniques tried, with the hybrids providing quality, reliability and speed.

6 Calculating the Stability of a Solution All the results reported so far have assumed a simpli ed and idealised evaluation function. Idealised in the sense that exact oating point values are given for panel thicknesses. This does not take into account real-world contingencies, such as manufacturing tolerances, which mean that exact values cannot be assumed. This section reports on an investigation into the stability of solutions found under random perturbations of the panel thicknesses. In many cases it is found that the best solutions obtained are highly unstable, i.e. very slight changes in just one panel thickness renders the solution untenable due to constraint violation. A promising DGA method for overcoming this problem is introduced. In order to simulate tolerances in the manufacturing of wing-boxes, the thickness of each panel was probabilistically perturbed. Each solution was perturbed 100 times and some summary statistics calculated. For each search algorithm employed, we took the best design (from the fty million evaluation runs) and subjected it to a number of `manufacture and test' phases. To gain some measure of stability, we then calculated the mean di erence between the mass of the original design speci cation and the mass of the `manufactured' wing. Additionally, the standard deviation of these di erences is assessed. The following formulas should sum this up. If manufacturing stability is S and the number of panels in a wing is L, then Pi=100L i=1 Mi

(9) 100L Where Mi is the mass returned from the objective function (Equation 1) when passed a solution formed by perturbing the original solution whose stability it is desired to measure. Each perturbated solution is formed by producing new panel thicknesses according to: S=

tn = to + 

(10)

Where  is a random variable with normal distribution about zero and standard deviation equal to half the minimum manufacturing tolerance (0.0625mm). Should a modi ed wing buckle the evaluation function adds a penalty `mass' to its cost. This component is somewhat misleading as far as a calculation of mass stability is concerned. Therefore, in cases where a design leads to manufactured wings which buckle during evaluation, their cost will not be included in the statistics, but rather the percentage of wings that buckle will be given. Table 2 gives stability results for the best solutions found with each of the ten search techniques used in this study. Surprisingly, we found that even very small tolerances lead to the manufacture of wings which regularly break buckling constraints. This can be explained, at least in terms of the problem at hand, as the result of fully optimising the wing-box up to the edge of the buckling constraint (i.e. the search algorithms are discovering points at the bottom of narrow tness precipices).

Search Technique Mean Deviation SD from Solution % buckling Random search 3.52 2.67 13.1 Brute Force Search NA NA 100 Local Gradient Descent 6.00 3.24 98.9 Gradient Descent 4.54 3.18 96.4 Powell 4.80 3.35 95.1 SA 5.67 3.41 99.0 DGA 4.29 3.07 93.0 Alopex 3.63 2.73 59.8 Hybrid SA and Gradient Descent 4.69 3.18 97.14 Table 2.

Stability of Best Solutions Results

As a general solution to this, our recent work centres on extending the distributed GA in such a way as to avoid potentially unstable solutions. The method employed here is novel, and exceptionally simple (compared with techniques such as cluster analysis etc), and as such is worth considering for a range of other manufacturing stability problems. On evaluating a given solution during the running of the GA, we employ the following technique: { Generate n `manufactured' versions of the design, complete with small tolerances as outlined above. { Cost each of these separately, and of these evaluations consider the worst to be the cost of the solution being tested. The DGA employing this strategy is referred to as a Reevaluating DGA. The re-evaluating idea is based on techniques used in evolutionary robotics [2]. We nd that small values of n are sucient to dramatically reduce the instability of solution space, typically in a run we take the worst of 4 trials. Because of the asynchronous geographic nature of the DGA, good solutions tend to remain on the grid for substantial time periods undergoing repeated evaluations. Hence an implicit pressure for stable solutions is maintained. The re-evaluating DGA operates with no explicit criterion of stability, but it will nevertheless eciently avoid unstable solution space. Fig. 3. Re-evaluating DGA, wing The pro les shown in gure 3 depict the pro les lowest mass solution found by the re-evaluating GA and the most stable solution. As would be expected, the lowest mass solution achieved is far more stable than those found by other methods. Importantly, the most stable solution has 0% buckling under the criteria explained at the beginning of this section while at the same time having only fractionally more mass than the lowest mass solution. On each run many very stable low cpst (and di erent) solutions were present in the nal population. Re-evaluating

DGA

Best Most Stable

14

(mm)

12

10

Thickness

8

6

4

2

0

1

7

13

19

25

Position along Wing

31

37

43

Best

49

7 Conclusions As far as we are aware, this is one of the most extensive comparative studies of di erent optimisation methods applied to a problem of this nature. However, because there are so many parameters associated with the various techniques that could be altered, it is inevitably far from exhaustive. Nevertheless, it does suggest strongly that hybridized stochastic search is the correct route for problems such as the simpli ed wingbox design task. One obvious direction for future work, then, is a more thorough investigation of hybrid algorithms. We have also described a DGA model where stability can be a factor driving selection, but which does not need an index of stability explicitly. This research was supported by EPSRC grant GR/J40812. Many thanks to Phil Green and colleagues from BAe Airbus, and to A. Wright of BAe Sowerby Research Centre for help with the work discussed in this paper.

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