MULTI-OBJECTIVE OPTIMUM DESIGN OF 3D STRUCTURES UNDER STATIC AND SEISMIC LOADING CONDITIONS

Transcription

1 4 th GRACM Congress on Computatonal Mechancs GRACM 2002 Patra, June, 2002 GRACM MULTI-OBJECTIVE OPTIMUM DESIGN OF 3D STRUCTURES UNDER STATIC AND SEISMIC LOADING CONDITIONS Manols Papadrakaks, Nkos D. Lagaros and Vagels Plevrs Insttute of Structural Analyss & Sesmc Research (ISASR) Natonal Techncal Unversty of Athens Zografou Campus, Athens, Greece Emal: {mpapadra, web page: Greek Assocaton of Computatonal Mechancs Keywords: Mult-obectve optmzaton, evoluton strateges, sesmc loadng. Abstract. Almost every real world problem nvolves smultaneous optmzaton of several ncommensurable and often competng obectves whch consttutes a mult-obectve optmzaton problem. In mult-obectve optmzaton problems the optmal soluton s not unque as n sngle-obectve optmzaton problems. Ths paper s concerned wth large-scale structural optmzaton of skeletal structures such, as space frames and trusses, under statc and/or sesmc loadng condtons wth multple obectves. Combnatoral optmzaton methods and n partcular algorthms based on evoluton strateges are mplemented for the soluton of ths type of problems. In treatng sesmc loadng condtons a number of accelerograms are produced from the elastc desgn response spectrum of the regon. These accelerograms consttute the multple loadng condtons under whch the structures are optmally desgned. Ths approach for treatng sesmc loadng s compared wth an approxmate desgn approach, based on smplfcatons adopted by the sesmc codes, n the framework of multobectve optmzaton. 1 Introducton In sngle-obectve optmzaton problems the optmal soluton s usually clearly defned snce t s the mnmum one, ths does not hold n real world problems havng multple and conflctng obectves. Instead of a sngle optmal soluton, there s usually a set of alternatve solutons, generally denoted as the set of Pareto optmal solutons. These solutons are optmal n the wder sense snce no other soluton n the search space s superor to them when all obectves are consdered. In the absence of preference nformaton, none of the correspondng trade-offs can be sad to be better than the others. On the other hand, the search space can be too large and too complex, whch s the usual case of real world problems, to be solved by the conventonal determnstc optmzers. Thus, effcent optmzaton strateges are requred able to deal wth the multple obectves and the complexty of the search space. Evolutonary Algorthms (EA) have several characterstcs that are desrable for ths knd of problems and most frequently outperform the determnstc optmzers such as gradent based optmzaton algorthms. There are some classcal methods for dealng wth the mult-obectve optmzaton problems, such as the lnear weghtng method, the dstance functon method and the constrant method, that have to be combned wth the optmzaton algorthm. The mplementaton of gradent based optmzers for ths type of problems becomes even more cumbersome. The applcaton of EA n mult-obectve optmzaton problems has receved consderable attenton n the last fve years due to ths dffculty of conventonal optmzaton technques, to be extended to mult-obectve optmzaton problems [1]. EA optmzers employ multple ndvduals that can search smultaneously for multple solutons. Usng some modfcatons on the operators used by the EA optmzers the search process can be drven to a famly of solutons representng the set of Pareto optmal solutons. The performance of the proposed method for handlng optmzaton problems wth multple obectves s examned n one space frame. For ths test example both the rgorous approach and the smplfed one wth respect to the loadng condton are mplemented and ther effcency s compared n the framework of fndng the optmum desgn of a structure under multple obectves. In the context of the rgorous approach a number of artfcal accelerograms are produced from the desgn response spectrum of the regon for elastc structural response.

2 Manols Papadrakaks and Nkos D. Lagaros 2 Sngle-obectve Structural Optmzaton In szng optmzaton problems the am s to mnmze a sngle-obectve functon, usually the weght of the structure, under certan behavoral constrants on stress and dsplacements. The desgn varables are most frequently chosen to be dmensons of the cross-sectonal areas of the members of the structure. Due to fabrcaton lmtatons the desgn varables are not contnuous but dscrete snce cross-sectons belong to a certan set. A dscrete structural optmzaton problem can be formulated n the followng form mn f (s) subect to g (s) 0 =1,...,k (1) d s R, =1,...,n where R d s a gven set of dscrete values and the desgn varables s (=1,...,n) can take values only from ths set. In the optmal desgn of frames the constrants are the member stresses and nodal dsplacements or nterstorey drfts. For rgd frames wth Ι-shapes, the stress constrants, under allowable stress desgn requrements specfed by Eurocode 3 [2], are expressed by the followng formula N M sd y,sd Mz,sd (2) Af y / γm1 Wpl,yf y/ γm1 Wpl,zf y / γm1 where N sd, M sd,y, M sd,z are the stress resultants, W pl,y,w pl,z are the plastc frst moment of nerta, and f y s the yeld stress. The safety factor γ M1 s a Eurocode 1 [3] box value usually taken as Mult-obectve Structural Optmzaton In practcal applcatons of structural optmzaton of 3D frames and trusses the materal weght rarely gves a representatve measure of the performance of the structure. In fact, several conflctng and ncommensurable crtera usually exst n real-lfe desgn problems that have to be dealt wth smultaneously. Ths stuaton forces the desgner to look for a good compromse desgn between the conflctng requrements. These knds of problems are called optmzaton problems wth many obectves. The consderaton of mult-obectve optmzaton n ts present sense orgnated towards the end of the 19 th century when Pareto presented the optmalty concept n economc problems wth several competng crtera. Snce then, although many technques have been developed n order to deal wth mult-obectve optmzaton problems the correspondng applcatons were confned to mathematcal functons. The frst applcatons n the feld of structural optmzaton wth multple obectves appeared at the end of the seventes. 3.1 Crtera and conflct The desgner lookng for the optmum desgn of a structure s faced wth the queston of selectng the most sutable crtera for measurng the economy, the strength, the servceablty or any other factor that affects the performance of a structure. Any quantty that has a drect nfluence on the performance of the structure can be consdered as a crteron. On the other hand, those quanttes that must satsfy only some mposed requrements are not crtera but they can be treated as constrants. Most of the structural optmzaton problems are treated wth one sngle-obectve usually the weght of the structure, subected to some strength constrants. These constrants are set as equalty or nequalty constrants usng some upper and lower lmts. When there s a dffculty n selectng these lmts, then these parameters are better treated as crtera. One mportant basc property n the multcrteron formulaton s the conflct that may or may not exst between the crtera. Only those quanttes that are competng should be treated as ndependent crtera whereas the others can be combned nto a sngle crteron to represent the whole group. The local conflct between two crtera can be defned as follows: The functons f and f are called locally collnear wth no conflct at pont s f there s c > 0 such that f (s)=c f (s). Otherwse, the functons are called locally conflctng at s. Accordng to ths defnton any two crtera are locally conflctng at a pont of the desgn space f ther maxmum mprovement s acheved n dfferent drectons. The global conflct between two crtera can be defned as follows: The functons f and f are called globally conflctng n the feasble regon F of the desgn space when the two optmzaton problems mn s F f (s) and mn s F f (s) have dfferent optmal solutons. 3.2 Formulaton of a multple obectve optmzaton problem In formulatng an optmzaton problem the choce of the desgn varables, crtera and constrants represents undoubtedly the most mportant decson made by the engneer. In general the mathematcal formulaton of a mult-obectve problem ncludes a set of n desgn varables, a set of m obectve functons and a set of k constrant functons and can be defned as follows:

3 Manols Papadrakaks and Nkos D. Lagaros mn s F [f 1(s),f 2(s),,f m(s)] subect to g (s) 0 =1,...,k d s R, =1,...,n where the vector s = [s 1 s 2... s n ] T represents a desgn varable vector and F s the feasble set n desgn space R n. It s defned as the set of desgn varables that satsfy the constrant functons g(s) n the form: F = { s R n g(s) 0} (4) Usually there exsts no unque pont whch would gve an optmum for all m crtera smultaneously. Thus the common optmalty condton used n sngle-obectve optmzaton must be replaced by a new concept the so called Pareto optmum: A desgn vector s* F s Pareto optmal for the problem of eq. (4) f and only f there exsts no other desgn vector s F such that f (s) f (s * ) for =1,2,...,m wth (5) f (s) < f (s * ) for at least one obectve. 3.3 Solvng the mult-obectve optmzaton problem Classcal methods for generatng the Pareto optmal set combne the obectves nto a sngle, parameterzed obectve functon. Bascally, ths procedure s ndependent of the underlyng optmzaton algorthm. Three prevously used methods n the lterature [4-6] are brefly dscussed and are compared n ths study wth the proposed modfed ES n terms of computatonal tme and effcency for treatng multobectve optmzaton problems Lnear weghtng method The frst method called the lnear weghtng method [6] combnes all the obectves nto a sngle scalar parameterzed obectve functon by usng weghtng coeffcents. If w, =1,2,...,m are the weghtng coeffcents the problem of eq. (5) can be wrtten as follows: m mns F w f (s) (7) = 1 wth no loss of generalty the followng normalzaton of the weghtng coeffcents s employed m w = 1 (8) = 1 By varyng these weghts t s now possble to generate the set of Pareto optma solutons for problem of eq. (5). The values of the weghtng coeffcents are adusted accordng to the mportance of each crteron. Every combnaton of those weghtng coeffcents correspond to a sngle Pareto optmal soluton, thus, performng a set of optmzaton processes usng dfferent weghtng coeffcents t s possble to generate the full set of Pareto optmal solutons. In real world problems dfferent unts correspond to dfferent obectves leadng to varatons of some orders of magntude between the values of the obectves. It s therefore advsable that the obectves should be normalzed accordng the followng expresson: f (s) f mn f (s) = f max f (9) mn where the normalzed obectves f () s [0,1], = 1,2,...,m, use the same desgn space wth the non normalzed ones, whle f mn and f max are the mnmum and maxmum values of the obectve functon Dstance functon method The dstance methods [4] are based on the mnmzaton of the dstance between the set of the obectve functon values and some chosen reference ponts belongng to the so called crteron space. Where as crteron space s defned the set of the obectve functon values that correspond to desgn vectors of the feasble doman. The resultng scalar problem s: mn s F d p (s) (10) where the dstance functon can be wrtten as follows: m 1/ p p d(s) p = w[ f(s) z] (11) = 1 T (3)

4 Manols Papadrakaks and Nkos D. Lagaros and p s an nteger number. The reference pont z d R m that s selected by the desgner s also called deal or utopan pont. A reference pont that s frequently used s the followng: z d = [ f 1 mn f 2 mn... f m mn ] T (12) where f mn s the optmum soluton of the sngle-obectve optmzaton problem where the -th obectve functon s treated as the unque obectve. The normalzaton functon eq. (8) for the weghtng factors w s also used. In the case that p = eq. (10) s transformed to the mnmax problem: mn max[ w f (x) ], = 1, 2,..., m (13) s F In the case of p=1 the formulaton of the dstance method s equvalent to the lnear method when the reference pont used s the zero ẑ = 0, whle the case of p=2 the method s called weghted quadratc method Constrant method Accordng to ths method the orgnal multcrteron problem s replaced by a scalar problem where one crteron f k s chosen as the obectve functon and all the other crtera are removed nto the constrants [2]. By ntroducng parameters ε nto these new constrants an addtonal feasble set s obtaned: F k ( ε ) = { s R n f (s) ε, = 1, 2,..., m µε k } (14) If the resultng feasble set s denoted by Fk = F F k the parameterzed scalar problem can be expressed as: mn f s k (s) F (15) k The constrant method gves the opportunty to obtan the full doman of optmum solutons, n the horzontal or vertcal drecton usng one crteron as obectve functon and the other as constrant Modfed Evoluton strateges for multobectve optmzaton The three above mentoned methods for mult obectve optmzaton have been used n the past wth mathematcal programmng optmzaton algorthms where at each optmzaton step one desgn pont was examned as an optmum desgn canddate. In order to locate the set of Pareto optmum solutons a famly of optmzaton runs have to be executed. On the other hand, evolutonary algorthms work smultaneously wth a populaton of desgn ponts, nstead of a sngle desgn pont, whch consttute a populaton of optmum desgn canddates, n the space of desgn varables. Due to ths characterstc, evolutonary algorthms have a great potental n fndng multple optma, n a sngle optmzaton run, whch s very useful n Pareto optmzaton problems. Snce the early nnetes a number of researchers have suggested the use of evolutonary algorthms n multobectve optmzaton problems [1]. In our study the method of Evoluton Strateges (ES) s appled for the frst tme for structural mult-obectve optmzaton problems. To ths purpose some changes have to be made n the random operators n order to gude the convergence to a populaton that represent the set of Pareto optmal solutons. These changes refer to () the selecton of the parent populaton at each generaton that has to be modfed n order to gude the search procedure towards the set of Pareto optmum solutons, and () the presentaton from convergence to a sngle desgn pont, and preserve dversty n the populaton n every generaton step. The frst demand s possble to be fulflled usng random selecton of the obectve accordng to whch the ndvdual wll be chosen for reproducton. Whle n order to preserve dversty n the populaton and fulfl the second requrement, the ftness sharng s mplemented. The dea behnd sharng s to degrade those ndvduals that are represented n hgher percentage n the populaton. The modfed obectves after sharng are the followng: f (s) f (s) = sh(d(s,h)) (16) h where the sharng functon used n the current study s the followng: d(s,h) α 1 f d(s,h) <σshare sh(d(s, h)) = σshare (17) 0 otherwse The dstance functon used s n the obectve space: d(s,h) = f(s) f(h) (18) 4 Structural desgn under sesmc loadng The equatons of equlbrum for a fnte element system n moton can be wrtten n the usual form M(s)u + C(s)u + K(s)u = R (19) t t t t

5 Manols Papadrakaks and Nkos D. Lagaros where M(s ), C(s ), and K(s ) are the mass, dampng and stffness matrces for the -th desgn vector s ; R t s the external load vector, whle u,u and u are the dsplacement, velocty, and acceleraton vectors, respectvely of the fnte element assemblage. The soluton methods of response spectrum modal analyss, whch s based on the mode superposton approach and drect ntegraton of the equatons of moton wll be consdered n ths work. 5 Soluton of the optmzaton problem There are three types of algorthms belongng to the class of evolutonary computaton that mtate nature by usng bologcal methodologes n order to fnd the optmum soluton of a problem: () evolutonary programmng (EP), () genetc algorthms (GAs) and () evoluton strateges (ESs). Ther man dfference s that GAs deal wth bt-strngs of fxed szes, ES wth real vectors and EP wth fnte state automata. GAs basc assumpton s that the optmal soluton can be found by assemblng buldng blocks,.e. partal peces of solutons, whle ESs and EP smply ensure the emergence of the best solutons. The most mportant consequence of ths dfferent approach s related to the recombnaton operator, vewed as essental for GA, as potentally useful for ES and as possbly harmful for EP. The modern tendences seem to follow combnatons of the two approaches, snce GA users have turned to real number representatons when dealng wth real numbers followng expermental results or heurstc demonstratons, whereas ES users have ncluded recombnaton as a standard operator, and have desgned specal operators for non real-valued problems. 5.1 ES n mult-obectve structural optmzaton problems The applcaton of evolutonary algorthms n mult-obectve optmzaton problems has attracted the nterest of a number of researchers n the last fve years due to the dffculty of conventonal optmzaton technques, such as gradent based methods, to be extended to mult-obectve optmzaton problems. EA, however, have been recognzed to be more approprate to mult-obectve optmzaton problems snce early n ther development. Multple ndvduals can search for multple solutons smultaneously, takng advantage of any smlartes avalable n the famly of possble solutons to the problem. In the frst mplementaton where the classcal methods are used, the optmzaton procedure, n order to generate a set of Pareto optmal solutons, ntates wth a set of parent desgn vectors needed by the ES optmzer and a set of weghtng coeffcents for the combnaton of all obectves nto a sngle scalar parameterzed obectve functon. These weghtng coeffcents are not set by the desgner but are beng systematcally vared by the optmzer after a Pareto optmal soluton has been acheved. There s an outer loop whch systematcally vares the parameters of the parameterzed obectve functon, and s called decson makng loop. The nner loop s the classcal ES process, startng wth a set of parent vectors. If any of these parent vectors gves an nfeasble desgn then ths parent vector s modfed untl t becomes feasble. Subsequently, the offsprngs are generated and checked f they are n the feasble regon. Accordng to (µ+λ) selecton scheme n every generaton the values of the obectve functon of the parent and the offsprng vectors are compared and the worst vectors are reected, whle the remanng ones are consdered to be the parent vectors of the new generaton. On the other hand, accordng to (µ,λ) selecton scheme only the offsprng vectors of each generaton are used to produce the new generaton. Ths procedure s repeated untl the chosen termnaton crteron s satsfed. The number of parents and offsprngs nvolved affects the computatonal effcency of the mult-membered ES dscussed n ths work. It has been observed that when the values of µ and λ are equal to the number of the desgn varables produce better results. Two ES algorthms for mult-obectve structural optmzaton applcatons under sesmc loadng are compared and tested n the subsequent secton: () The ES algorthm combned wth the classcal methods whch can be stated as follows: Outer loop - Decson makng loop Set the parameters w of the parameterzed obectve functon Inner loop - ES loop 1. Selecton step : selecton of s ( = 1,2,...,µ) parent vectors of the desgn varables 2. Analyss step : solve M(s)u + C(s)u + K(s)u = R(t) (=1,2,...,µ) 3. Evaluaton of parameterzed obectve functon 4. Constrants check : all parent vectors become feasble 5. Offsprng generaton : generate s, (=1,2,...,λ) offsprng vectors of the desgn varables 6. Analyss step : solve M(s )u + C(s )u + K(s )u = R(t) (=1,2,...,λ) 7. Evaluaton of the parameterzed obectve functon 8. Constrants check : f satsfed contnue, else change s and go to step 5

6 Manols Papadrakaks and Nkos D. Lagaros 9. Selecton step : selecton of the next generaton parents accordng to (µ+λ) or (µ,λ) selecton schemes 10. Convergence check : If satsfed stop, else go to step 5 End of Inner loop End of Outer loop () The modfed ES algorthm (ESMO) as descrbed n secton whch can be stated as follows: 1. Selecton step : selecton of s ( = 1,2,...,µ) parent vectors of the desgn varables 2. Analyss step : solve M(s)u + C(s)u + K(s)u = R(t) (=1,2,...,µ) 3. Evaluaton of the obectve functons 4. Constrants check : all parent vectors become feasble 5. Offsprng generaton : generate s, (=1,2,...,λ) offsprng vectors of the desgn varables 6. Analyss step : solve M(s )u + C(s )u + K(s )u = R(t) (=1,2,...,λ) 7. Evaluaton of the obectve functons 8. Constrants check : f satsfed contnue, else change s and go to step 5 9. Selecton step : random selecton of the potental obectve for the each ndvdual and selecton of the next generaton parents accordng to (µ+λ) or (µ,λ) selecton schemes 10. Ftness sharng 11. Convergence check : If satsfed stop, else go to step 5 6 Numercal results The performance of the mult-obectve optmzaton methods dscussed n ths paper s nvestgated n two benchmark test examples: A sx storey space frame and a mult-layered space truss. The followng abbrevatons are used n ths secton: DTI refers to the Newmark Drect tme Integraton method. RSMA refers to the Response Spectrum Modal Analyss. LWM refers to the Lnear Weghtng method for treatng mult-obectve optmzaton problems. DFM refers to the Dstance Functon method for treatng mult-obectve optmzaton problems. CM refers to the Constrant method for treatng mult-obectve optmzaton problems. ESMO refers to the proposed Evoluton Strateges for treatng Mult-obectve Optmzaton problems. The obectve functons consdered for ths problem are the weght of the structure and the maxmum dsplacement. The constrants are mposed on the nter-storey drfts and for each element group on the maxmum stress constrant of Eqs. (2) under a combnaton of axal force and bendng moments. The space frame conssts of 63 elements wth 180 degrees of freedom as shown n Fgure 2. The length of the beams and the columns are L 1 =7.32 m and the columns L 2 =3.66 m, respectvely. The structure s loaded wth a kpa gravty load on all floor levels and a statc lateral load of 109 kn appled at each node n the front elevaton along the z drecton. The element members are dvded nto 5 groups, each one havng two desgn varables resultng n ten total desgn varables. The cross secton of each member s assumed to be a Ι-shape and for each member two desgn varables are consdered as shown n Fgure 1. The modulus of elastcty s 200 GPa and the yeld stress s σ y =250 MPa. b h Fgure 1: I-shape cross secton

7 Manols Papadrakaks and Nkos D. Lagaros y z x Fgure 2: Sx storey space frame The Pareto optmal set of solutons was frst computed wth the LWM. The performance of ths method for the case of seekng the smultaneous mnmzaton of weght and maxmum dsplacement s depcted n Fgures 3 and 4 for both statc and sesmc loadng condtons. In Fgures 3 and 4 the performance of the DFM and ESMO methods are also presented. For the case of the DFM the zero (0) pont was consdered as the utopan pont, whle four dfferent schemes of the DFM were examned. p=1: equvalent to the LWM, p=2: called quadratc LWM and p=8: equvalent to the p= LWM (Statc) ESMO (Statc) LWM (Dynamc-DTI) ESMO (Dynamc-DTI) DFM-p=2 (Statc) DFM-p=8 (Statc) DFM-p=2 (Dynamc-DTI) DFM-p=8 (Dynamc-DTI) 1.2 Max Dpslacement (cm) Weght (kn) Fgure 3: Sx Storey frame: Performance of the methods for statc and combned statc and sesmc loadng condtons 1.45 LWM (RSMA) DFM-p=2 (RSMA) ESMO (RSMA) DFM-p=8 (RSMA) LWM (DTI) DFM-p=2 (DTI) ESMO (DTI) DFM-p=8 (DTI) 1.2 Max Dpslacement (cm) Weght (kn) Fgure 4: Sx Storey frame: Performance of the methods for combned statc and sesmc loadng condtons

8 Manols Papadrakaks and Nkos D. Lagaros 6 Conclusons Evoluton Strateges can be consdered as an effcent tool for mult-obectve desgn optmzaton of structural problems such as space frames under statc and sesmc loadng condtons. The ESMO method compared to the lnear weghng method and the constrant method appears to be robust and relable for treatng mult-obectve structural optmzaton problems gvng almost dentcal results. The generalzaton of the lnear weghng method for p>1 called the dstance functon method s also examned n ths study. The results obtaned by the dstance functon method were somewhat dfferent than those taken by the other two methods, whle for large values of p t produces ether too close or dsperse ponts n the Pareto sets. References [1] Fonseca C.M. and Flemng P.J., An Overvew of Evolutonary Algorthms n Multobectve Optmzaton, Evolutonary Computatons, 3(1), pp. 1-16, [2] Eurocode 1, Actons on structures, Part 1.1: Denstes, self-weght and mposed loads for buldngs, Part 1.3: Snow loads, Part 1.4: Wnd loads, EN /EN /EN [3] Eurocode 3, Desgn of steel structures, Part 1.1: General rules, Part 1.5: Strength and stablty of planar plated structures wthout transverse loadng, EN /EN [4] Zeleny M., Multple Crtera Decson Makng, McGraw-Hll, New York, [5] Hames Y.Y. and Hall W.A., Mult-obectves n water resource systems analyss: The surrogate worth trade off method, Water Resources Res., 10, pp , [6] Cohon J.L., Multobectve Programmng and Plannng, Academc Press, New York, 1978.