Optimization of mixed discrete/continuous design variable systems using neural networks
R. Sellar, Stephen M. Batill, John E. Renaud · 5th Symposium on Multidisciplinary Analysis and Optimization · 1994
Emerging techni ues must he ca ablc of dealing with a wide variety of Yisciplines ,ecif?c rllethodologies Of particular conceln is the development of methods which can effective1 integrate continuous and discrete variables. d i s paper documents a study in which material selection, structural arrangement and component sizing are considered for a sinfe, simple structural system. A hierarchical ro lem is formulated in which com onent sizing baseif;lpon finite element analysis is pergrmed a t the s i~bsyster i~ level. Artificial neural networks are used to provide response surface mapping of t,he subsystem for use a t t,he system level. The system level problem which includes the discrete material selection variables and continuous structural arrangement variables is formulated as a discrete and optimum designs were identified usin g simul%ed annealin algorithms for two clifferri~t merit functions. 'rhis stu$y identifies requirernmts for the effective use of neural network m a ping of the and how this representati,on of tE sl~lbil~z[:e irlflurnrcs the system level o p t i m ~ z a t ~ o n . (I ~timizatiori methods have a long history and these me tho i s hale pla ed inlportarit roles in engineering anali s I i r$hr foundation for many of the current o jtir~lization methods used in en ineering are analytic or numerical techniques \v%ich are well-suited for specific classes of problems. As the engineering communit,y nttcrnp!s to expand the influence of rnetl~otl:; into the realm of multidisciplinary optimization it is faced with the problem of adapting optirnization 111cthods to problerris which are more complex and reprosrnt a variety of problem classes. l ' h e current s tudy was concerned wit,ti just one case in which both c o r ~ t , m ~ ~ o l ~ s and discrete desiqn variables were required for a single, problem. rhere are a variet of techniques tha t can be used to find t i e solution t o a problem when the syst,em in question is cornposecl solely of either continuous (spar cap cross srctional area. wing sweep angle,. . . ) or discrete (rnater ~ n l choice frolri a f in~ te w t , number of spars, ) karlnbles 7'11esc tcchnlques, however, are in many cases *Graduate Research Assistant, Member AIAA t Prof-sor. Associate Fellow AIAA $Clark Equipment Assistant Professor, Member AIAA Copyright @I094 bv Stephen M. Batill. Published by the American Tnstit,iite OF Aeronautics and Astrona~~tics, Inc. with p c ! n m i ~ < i o n . inadequate or a t least untested when faced with the task of automating or optimizing the of mixed discrete/continuous systems. A second and sometimes equally important issue is the vast amount of information required for the desi n of complex, en ineering systems. T f e c o m p ~ t a t ~ i o n a l capabilities provi$ed by current corn o t ers can allow the designer the o portunity to prolucr vast amounts of da ta and cons ig r many potential variations. Providing a framework for these studies and a method of storage of the information generated so tha t it can be useful in practical desi n decision making is critical t o the success of automatefdes ign efforts. These issues result in questions as to how one can generate optimal or a t least improved designs given real constraints on t ime and computational resources. This is particular1 important in combinatorial desi n problems mvolving d;screte variables which often require an extreme1 large number of objective function evaluations. For txese roblems t,he reduction of the nnmber or cost of these otjective function evaluat,ions is a high priority: The purpose of this study was to address issues ertaining t o deslgn space representation and evaluation For structural deslgn problerris which contain both discrete and continuous variables. The method by which the space definition (response surface rna.pping) for the mixed discrete/continuous variable problem was addressed in this study was through the use of artificial neural networks. These networks have been shown to provide a useful tool for storage and manipulation of da ta obtained through conventional analysis techniques for systems of either continuous[l, 2, 31 or discrete[l, 4 , 5, 61 variables, but problems containing both continuous and discrete variables have only recently been considered in some detail [7, 81. This paper employs feedforward, back propagation-neural networks to provide an approximation t,o the mixed discretje/continuous space and to replace conventional numerical analysis methods a t the system level in the optimizat,ion process. The problem considered in this paper is an extension of a preliminary study initially presented in Reference 191. This earlier work indicated the ~ o t e n t i a l for usL , ing neural networks in the combined continuous/discrete problem but the space considered did not provide a particularly demanding application and the problem was rather limited. The methods considered in the present study were evalnated bv application to a relative1 straightforward structural problem. This probyem was selected since it contained the basic characteristics of both contlnuous and discrete variables. The space was rather complex in character but the itself was simple enough to allow for ra hicafpresentation of various results. The sections w%icf follow briefly describe the specific tlesi n problem, the way in which neural networks were use3 to map the space for this problem, and the methodology used to obtain optimal solutions through the use of neural network representations of the s ace T h e paper concludes with a detailed discussion o?results for the application of these methods to the sample structural deslgn problem. TIIF, DESIGN PROBLEM The problem presented in this p a er is the conceptual of a structure for which tRe system desi n vector is comprised of both discrete and continuous Resign variables. T h e design roblem is an example of a two-step hicmrchic design. $he first step, referred t o as t,he o timization, contained contiriuous variables and t P le second step, or system level optirnization, contained both continuous and discrete variables. T h e of a structure typically involves the selection of the basic geometric arrangement (configuration), the selection of the materials to be used in the structure ( the material system), and finally the sizing of the components which make up the structure. The sequence in which these decisions are made can significant1 influence the desi n There have been many methods dYeveloped which aRok for the sizing of st,ructural components for a given confi uration and material propR erties. These are often base u on finite element representations of the structure. i n t i i s paper the selection,of the component sizes for a glven structure and, material combination is referred to as a subspace and is the first step in this process. Including the selection of the inaterial system and confi uration in the may allow for evenehetter designs but these types of variables are more difficult t o include in many of the current structural schernes. This is often the result of the fact t ha t the finite element method used to model and analyze the ~ t ~ r u c t u r e is not as easily adaptable t.0 variations in configuration and material properties as it, is t o variations in component sizing. An addl t~onal complexity is introduced to the problem due to the discrete nature of some of the configuration and material property desi n variables and the difficulty associated wlth using eit a er optimality criteria or gradient-based optirnization algorithms in the process. Combinatorial al orithn?s, usually associated with discrete variab f es re ulre large numbers of analyses and are often quite cos?ly when using finite element st,ructural analysis methods. In the current study, finite element analysis techniques were used to model and analyze the,structure a t the level. For a given configuration and material cornposit~ion, a gradient-based a1 orithm was used to deternline the least weight sufject tor ield, local buckling and rninlnlum gage coiistraints. $hen the desi 11 space represented by the configuration and mater i5s variables was evaluated to determine the configuration and materials which yielded the best of the least weight designs. Two measures of merit were used a t the s sten1 level to evaluate the suitabilit of the designs. Geight , the same merit function used a t t he subsystein level was used as well as a hy othetical ,co~tn function which was based upon a n a n a g t i c comb~nat ion of weight a n d performance as measured by structural deformation. Details on this cost function are provided below. pounds