CHAPTER 2 Revew of appromaton technques 2. Introducton Optmzaton problems n engneerng desgn are characterzed by the followng assocated features: the objectve functon and constrants are mplct functons evaluated by a numercal method whch usually nvolves a large amount of computer resources (e.g. the fnte element method), functon values present a certan level of nose and can only be estmated wth a fnte accuracy, especally when some teratve technques are appled because of the lmted accuracy of calculatons or for physcal reasons (Toropov et al. 996). An alternatve to such prohbtve computatonal costs s the use of appromaton concepts whch are much less epensve to compute (Schmt and arsh, 974). Ths chapter revews the basc appromaton technques (Barthelemy and Haftka, 993). Dependng on the range of ther applcablty, the appromatons can be classfed as local (vald n a vcnty of a desgn pont), global (vald n the
Revew of appromaton technques 8 whole regon defned by sde constrants) and md-range (also called multpont appromatons) whch s the combnaton of the two basc approaches. Attenton s focused on the possblty of the use of appromaton functons for flterng out the numercal nose. 2.2 Local appromatons Local appromatons are vald n the vcnty of the pont at whch they are generated. These functons are varatons on the Taylor seres epanson and, therefore, based on the dervatves of the objectve functon and constrants wth respect to the desgn varables. The smplest form s the lnear appromaton defned as follows: N ( ) ( ) + ( ) (2.) A dsadvantage of epresson (2.) s the possble naccuracy for ponts even close to. It was notced that n many structural optmzaton problems better accuracy was acheved wth an alternatve formulaton called the recprocal appromaton, defned as y ; N ( ) ( ) ( ) + (2.2) In the case of truss elements where the desgn varables are the cross sectonal areas, epresson (2.2) makes the constrants less nonlnear.
Revew of appromaton technques 9 The combnaton of epressons (2.) and (2.2) gves a new appromaton called the conservatve appromaton (Starnes and Haftka, 979). Barthelemy and Haftka (993) gve the followng defnton for the conservatve appromaton: ( ) ( ) ( ) ( ) + otherwse f n (2.3) If, epresson (2.3) s dentcal to the lnear appromaton (2.), and f / epresson (2.3) s dentcal to the recprocal appromaton (2.2). The most popular local appromaton technques also nclude CONLIN (leury, 989) and MMA (Svanberg, 987). Hgher order appromatons are rarely used because the hgher order dervatves are very dffcult to obtan. or eample, a quadratc appromaton based on the Taylor seres epanson s: ( ) ( ) ( ) ( ) ( ) + + + n n j j j j n 2 (2.4) It requres calculaton of the elements of the matr of second-order dervatves. Local appromatons help to reduce the complety of the problem, but due to ther local characterstcs, they lack a global perspectve of the problem wth the rsk
Revew of appromaton technques of convergng to a local optma. Also, these appromatons do not address the ssue of numercal nose n the functon value. 2.3 Global appromatons Global appromatons are vald n the whole desgn space. They allow the study of the regon for the locaton of optmum ponts. Most typcally, global appromaton technques nclude the response surface methodology (RSM), neural networks and the desgn and analyss of computer eperments (DACE). The performance of global appromatons was revewed by Rou et al. (996) and Sobeszczansk-Sobesk and Haftka (997) among others. 2.3. Response surface methodology Response surface methodology (RSM) s a method of constructng appromatons of the system behavour usng results of the response analyss carred out at a seres of ponts n the desgn varable space. The appromaton functons are obtaned by the least-squares method. The strength of the technque s n applcaton to problems where the desgn senstvty nformaton s dffcult or mpossble to obtan, as well as n cases where the response functon values contan some level of computatonal nose. A complete descrpton of RSM s gven n Chapter 3. 2.3.2 Desgn and analyss of computer eperments (DACE) The polynomal appromatons used n RSM were orgnally developed for responses obtaned from the desgn of physcal eperments, whch nvolves random
Revew of appromaton technques errors due to nose and human errors (Bo and Draper, 987). Later, these technques mgrated to the feld of computer eperments where there s no random error, although ths pont s subject to debate (van Campen et al., 99, Schoofs et al., 997, Smpson et al., 997, Toropov et al., 999c). Sacks et al. (989) proposed a methodology to appromate determnstc responses based on nterpolaton models. The dea s that, as opposed to classcal desgn of eperments, replcated runs at the same settngs wth the same nputs wll be dentcal. The appromatons are found by krgng models (Lews, 998) evaluated wth Latn hypercube samplng. Applcatons are revewed n the papers by Sacks et al. (989), Booker (998), Bates et al. (998), Torczon and Trosset (998) and Gunta and Watson (998) among others. gure 2. llustrates the dfference between determnstc and nondetermnstc curve fttng (Smpson et al., 997). Determnstc Non-determnstc gure 2. Determnstc and non-determnstc curve fttng
Revew of appromaton technques 2 2.3.3 Neural network Many applcatons use a neural network to realse self-learnng models. It works by adjustng weghts between pars of nput/output. Once traned, the neural network can replace comple analyss procedures. Therefore, neural networks present an alternatve approach to global functon appromaton. The major dsadvantage of neural models s that nput/output s appromated by a black bo approach and no understandng of the underlyng relatonshp can be ganed. 2.3.4 Genetc Programmng Genetc Programmng (Koza, 992) uses the same prncples as genetc algorthms (Goldberg, 989) wth a dfferent representaton of the solutons n the form of computer programmes. The ablty of genetc programmng to evolve symbolc solutons s nvestgated n ths thess for the selecton of the structure of global appromatons. Chapter 4 descrbes the methodology of genetc programmng and Chapters 5 and 6 show applcatons to smulated and epermental responses. 2.4 Md-range appromatons Global appromatons allow the constructon of eplct appromatons vald n the entre desgn space, but as the number of desgn varables grows, they requre too many functon evaluatons to buld.
Revew of appromaton technques 3 An alternatve approach based on varatons of local appromatons has been suggested wth nformaton about objectve functon and constrants calculated at more than one pont. The purpose s to gve an enhanced accuracy and epanded applcablty. Such applcatons can be classfed as md-range appromatons. Early work n multpont appromatons concentrated n constructng appromatons along a lne search defned by values of the constrants (Haftka and Gurdal, 993). urther developments used data from several optmzaton teratons to fnd appromatons n the entre desgn space. Usually they are based on two or three ponts (Barthelemy and Haftka, 993). A dfferent approach combnes response surface methodology (RSM) wth multpont appromatons. In ths way, the appromatons are obtaned by leastsquares procedures nstead of nterpolaton between desgn ponts. Toropov replaced the orgnal optmzaton problem by a successon of md-range appromatons of the correspondng orgnal functons. Ths technque employs move lmts to defne a new sub-regon around the current optmum pont. Each appromaton s evaluated by a weghted least-squares method usng the orgnal functon values (and ther dervatves, when avalable (Toropov et al., 993)) at several ponts of the desgn varable space. Applcatons are dscussed n Toropov and Carlsen (994), van Keulen and Toropov (997,998) and Markne (999). Recent developments are presented n Toropov et al. (999c), ncludng an mplementaton n a parallel computng envronment (van Keulen and Toropov, 999).
Revew of appromaton technques 4 Other work on multpont appromatons can be found n adel et al. (99), Etman (997), Schoofs et al. (997) and Xu and Grandh (999). An etensve lterature revew s gven by Venter (998). 2.5 Concluson The basc appromaton methodologes used n desgn optmzaton have been revewed. Local appromatons lack a general perspectve of the desgn space, whch has several mplcatons, e.g. the rsk of convergence to local optma or an nsuffcent overvew of the problem. Also, the convergence can be serously affected by the presence of numercal nose. Generally, global and md-range appromatons are preferred. The man dffculty n the current global appromaton technques based on RSM s the necessty to specfy a structure for the appromaton functon. In contrast, the genetc programmng methodology s the only technque that does not assume the structure of the model n advance, but suggests a soluton for both the structure and the coeffcents of the model. Therefore, t s proposed to nvestgate the use of the genetc programmng methodology to obtan hgh qualty appromatons based on RSM. RSM wll be descrbed n Chapter 3 and genetc programmng n Chapter 4.