Phenotype Building Blocks and Geometric Crossover in Structural Optimisation

Transcription

1 Paper 5 Cvl-Comp Press, 22 Proceedngs of the Eghth Internatonal Conference on Engneerng Computatonal Technology, B.H.V. Toppng, (Edtor), Cvl-Comp Press, Strlngshre, Scotland Phenotype Buldng Blocks and Geometrc Crossover n Structural Optmsaton A. Maher, T. Macquart, D. Safar 2 and M.. Maher 2 School of Computng, Engneerng and Informaton Scences Northumbra Unversty, Newcastle upon Tyne, Unted Kngdom 2 Department of Cvl Engneerng, Shraz Unversty, Iran Abstract Macro or phenotype buldng blocks (PBBs) contan nformaton at phenotype level. PBBs are ether the components of a mult-component system, or dfferent parts of a contnuous system wth dfferent desgn qualtes and/or evaluaton measures. Usng PBBs can lead to enhancement of the search effcency by utlsng problem specfc search operators and heurstcs appled on the PBBs. In order to preserve, propagate and recombne good buldng blocks effcently, buldng blocks should have a low probablty of beng dsrupted by crossover. Therefore, the crossover operaton should be desgned at phenotype level. Geometrc crossover (GCO) s appled on the phenotype rather than the genotype. Identfyng PBBs and usng GCO, partal ftness can be defned and employed to mprove the performance of the search algorthm. Another advantage of usng GCO s as a result of ts ease of applcaton to nonfxed-length and varable-length chromosomes. Varable-length chromosomes are a common feature of topology optmsaton problems when genetc algorthms are employed. GCO can be easly appled to the structural optmsaton problems ncludng, topology optmsaton wthout predefnng a grd. These two advantages have been demonstrated by mplementng GCO n genetc algorthms employed for optmsaton of plane trusses supportng dstrbuted loads, two-dmensonal steel frames and wnd turbne blades. Keywords: geometrc crossover, phenotype buldng block, partal ftness, topology optmsaton, genetc algorthm, structural optmsaton. Introducton The standard explanaton of how genetc algorthms (GAs) operate, s often referred as the buldng block hypothess []. Accordng to ths hypothess, GAs operate by combnng small buldng blocks nto larger buldng blocks. The ntutve dea behnd recombnaton s that combnng features (or buldng blocks) from two good parents wll often produce better chldren.

2 It s well recognsed that the effcency of a genetc algorthm n exploraton and explotaton of the soluton space can be mproved by ncorporatng doman-specfc knowledge nto the algorthm. In many real-world applcatons, the physcs of the problem suggests heurstcs that can be ncorporated nto the search and selecton procedures. Whenever GAs are appled to such problems, knowledge about the doman of applcaton should be consdered n the desgn of the reproducton operators as well as the representaton and selecton. Doman knowledge has been broadly ncorporated n selecton and reproducton operators. Heurstcs or knowledge-augmented operators are talored for ndvdual applcatons. The prme consderaton n defnng these operators s how to mplement heurstcs n a reproducton operator wthout jeopardsng the exploratory nature of the search. 2 Phenotype buldng blocks, partal ftness and geometrc crossover In ths paper, a phenotype buldng block refers to a part of soluton that contans nterpretable nformaton; hence t can be assessed by ts own mert based on a set of measurable assessment crtera. Assessment crtera of a PBB can be a subset of the assessment crtera of the ndvdual tself, or alternatvely, there s at least a known qualtatve relatonshp between the PBB assessment crtera and the ndvdual assessment crtera. A PBB can be a combnaton of components of a mult-component or dscrete system or dfferent parts of a contnuous system. A partal ftness s assgned to each PBB. Partal ftness can be defned usng a nonconflctng subset of the assessment crtera of the PBB. In the context of engneerng desgn problems: f j ( x, y) =, ( j =,, q ), n whch y = { y, y2,..., yq} s the set of desgn qualtes and x = { x, x,..., x p} s the set of desgn varables, the set of assessment crtera refers to the set of desgn qualtes. Desgn qualtes can be treated as objectves or constrants of the correspondng optmsaton problem. In order to preserve, propagate and recombne good buldng blocks effcently, buldng blocks should have a low probablty of beng dsrupted by crossover. Therefore, ether the length of the mportant buldng blocks must be short, or alternatvely the crossover operator must be talored for the problem at hand wsely reducng the probablty of breakng down good buldng blocks. In fact, the former s the Holland s orgnal assumpton that the mportant buldng blocks are of short length, whch s true f segment-based crossover operators such as one- or twopont crossover for btwse bnary strngs s used. Usng unform crossover, on the other hand, makes t dffcult to propagate good buldng blocks once they are found. Usng PBBs to ncorporate doman-knowledge, the crossover operaton should be desgned n such a way that dsrupton of good buldng blocks s avoded once they are found. PBBs are not of short length n genotype space. Therefore, the tradtonal crossover operatons that operate on the genotype are not sutable for combnng the PBBs. Geometrc crossover, on the other hand, s appled on the phenotype (real desgn space) rather than the genotype to avod dsrupton of good buldng blocks. 2

3 Generally, a crossover operator, dependng on the type of the chromosome encodng and the mechansm of the crossover, may lead to one or a combnaton of the followngs: Arthmetc recombnaton of parents, n whch chldren are n an Eucldean dstance of the parents Attrbute swap between parents, n whch the chldren have common attrbutes wth ther parents. A desgn attrbute may be dentfed by one or more desgn varables. Attrbute recombnaton of parents, n whch chldren may have common attrbutes wth ther parents or new attrbutes resulted by mxng the parents attrbutes. Geometrc recombnaton, n whch chldren have common PBBs wth ther parents. Dependng on the type of the buldng blocks, a GCO may lead to arthmetc recombnaton (e.g. new span n the wnd turbne blade example of Secton 4), attrbute swap (e.g. constructng a chld by the axes of one parent and bays of another n the plane frame example of Secton 3), and attrbute recombnaton (e.g. weght of chld n the truss example of Secton 5). The applcaton of GCO s not lmted to the structural optmsaton. Cut and splce s a type of crossover whch s appled on the real desgn space. Ths type of crossover has been employed for pattern recognton and structural optmsaton of atomc clusters problems. Cross et al [2] and later Myers and Hancock [3] appled a smlar concept for pattern recognton problems. They combned the solutons by physcally dvdng the graphs nto two dsjont subgraphs. Ther results show that ther crossover operaton mproves the search convergence speed. More recently, Nazm and Erkoc [4] and Froltsov and euter [5] appled smlar concept n structural optmsaton of atomc clusters. In ther genetc algorthm a new structure s generated by cuttng exstng cluster geometres nto two halves and then recombnng the halves of dfferent confguratons. It should be noted that n none of these works the concept of partal ftness has been used n the process of parent selecton. 3 Applcaton to planner frames Fgure, shows a typcal m n two-dmensonal structural frame made of n c = ( m +) n standard columns and n b = mn standard beams. Each column and beam can be dentfed by an ndex, referrng to ts standard code. Usng an ndexed codng, the chromosome of ths frame can be defned by a strng of length n c + nb. For ths chromosome representaton, beams and columns are genotypes. However, one can defne phenotypes buldng blocks as axs (four columns), bay (four beams) and storey (four columns and three beams). Each of these three PBBs can be assessed based on a subset of the frame assessment crtera (e.g. weght, lateral storey drft, beam deflectons, beam and column maxmum stresses, column crtcal buckng load, etc). Each frame can be made of storeys or combnaton of axes and bays. 3

4 Fgure : A typcal m n plane frame and ts PBBs. Fgure 2: Geometry crossover of the two-dmensonal frame problem usng PBBs: (a) homologous and (b) non-homologous. Fgure 2 shows the PBB exchange n a GCO of the frame of Fgure. Snce there s more than one type of each PBBs, both homologous and non-homologous crossover can be appled to two parent frames. In order to nvestgate the effect of mplementng PBB, partal ftness and GCO n the desgn of frames, one of the desgn case studes of reference [6] s adopted n whch the optmum allowable stress desgn of steel frames subjected to varous loadngs and load combnatons under constrants of AISC ASD specfcaton s 4

5 desred. Desgn varables are the element secton szes from the avalable W shapes of a standard lst. For a steel frame consstng of N members that are collected n N d desgn groups (number of desgn varables), the objectve s to fnd a vector of nteger values I (Eq. ()), representng the sequence numbers of steel sectons assgned to N member groups: d m T I = I, I,..., I ] () [ 2 Nd for mnmzng the weght of the frame, expressed as: W N = d Nt A = j= γ L, (2) j subject to a seres of constrants, as formulated and detaled n [6], on combnaton of axal and flexural stress, column Euler stress, column slenderness rato, shear stress, ductlty, beam deflecton servceablty, storey drft servceablty and constructablty. In Equaton 2, A and γ, respectvely, represent the area and the unt weght of N t stands for the total number of L s the length of the member j whch belongs to group. the steel secton adopted for the member group, members n group and j Fgure 3: Search hstory usng (a) btwse crossover converged at generaton number 2 to.583 and (b) geometrc crossover converged at generaton number 9 to.568 In order to nvestgate the performance of employng GCO, PBB and partal ftness n optmsaton of frames, startng wth the same ntal populaton, two types of crossover s used: () GCO wth partal ftness PF = weght for all types of PBBs, and () conventonal btwse crossover. Fgure 3 shows the search hstores for the two runs. Evdently, when usng GCO, the search converges faster (generaton number 9 versus 4) to a better soluton (penalsed objectve functon of.568 versus.583). 5

6 4 Applcaton to wnd turbne blades Aerodynamc desgn of wnd turbne blades ncludes optmsaton of the topology of the blade. Parameters such as rotor radus and the span-wse dstrbutons of the chord length, pretwst and aerofols defne the topology of the blade and are treated as desgn varables. A wde range of parameters construct the set of desgn qualtes (objectves/constrants). Normally, the average annual power yeld or the power coeffcent at the desgn wnd speed are treated as optmsaton objectves, whle constrants are appled on the aerodynamc loads, blade nternal forces, blade weght, fatgue lfe, etc. Hub Inner secton Outer secton Tp Structural demand Aerodynamc demand Chord Chord n Pretwst Pretwst n Aerofol Aerofol n Blade span Genotype chromosome Inner secton Outer secton Phenotype buldng blocks Fgure 4: Wnd turbne blade and ts PBBs Fgure 4 shows a wnd turbne blade. Usng n desgn (precson) ponts for the dstrbuted desgn varables (chord length, pretwst and the aerofol code), the total number of desgn varables wll be3 n +. Employng a mxed real-number/ndexed encodng, each ndvdual can be defned by a chromosome of length3 n +. The assessment crtera for the blade can be defned as the weght of the blade, maxmum flap stress and the average power producton correspondng to a gven wnd probablty densty functon. Blade s a contnuous structure, but ts nner and outer sectons have dfferent functons and can be defned as two PBBs. The outer secton of the blade s aerodynamcally more effcent. That s, the power s mostly produced by the outer secton of the blade. Whlst the nner secton experences greatest nternal forces and the desgn s drven based on the structural demand. Snce the nner and outer sectons have dfferent functons they can be assessed separately. The nner secton can be assessed based on, for example, the maxmum flap stress or the weght of the blade, whle the outer secton can be assessed based on the produced average power. In contrast to the frame example, there s no sharp boundary between the wnd turbne blade PBBs. 6

7 The genotype chromosome s a strng of real numbers as well as ndexes, as shown n Fgure 5.a. The desgn varables can be ether dstrbuted (chord, pretwst, aerofol thckness and aerofol code) or sngle value (blade span). The desgn varables can also be categorsed as ether contnuous (blade span, chord, pretwst and aerofol thckness) or dscrete (aerofol code). Fgure 5.b demonstrates the GCO, assumng that parents P and P 2, respectvely, have better performance n terms of desgn qualtes correspondng to the nner and outer segments of the blade. The cut pont s a randomly selected real number between and the span of the blade. Snce there s only one of each type of PBBs, only homologous crossover s possble. Dstrbuted desgn varables (n desgn ponts per each dstrbuted desgn varables dstrbuted from hub to tp) Sngle desgn varables (a) Chord Pretwst Aerofol code Span Dscrete desgn varable Contnuous (real number) desgn varable Parents genotype chromosome P: better performance n terms of desgn qualtes correspondng to the nner segment Hub Tp P2: better desgn qualtes correspondng to outer segment (b) Chord (P) Chord (P2) Pretwst (P) Pretwst (P2) Af code (P) Af code (P2) Span new Chld genotype chromosome: Geometrc crossover Fgure 5: Wnd turbne blade (a) Genotype chromosome, (b) GCO Havng the radal coordnate normalsed by span length ( r = r / span ), the cut pont s a randomly selected precson pont r. The cut dvdes each parent blade nto two parts. The dstrbuted desgn varables of the chld blades are formed by those of the left and rght hand sdes of each parent blade. A repar operaton s also requred to retan the contnuty of the dstrbuted desgn varable. Fgure 6 llustrates the process of formng a dstrbuted desgn varable (here the pretwst dstrbuton β ) of a chld from a par of parents. The repared pretwst s c 7

8 obtaned by multplyng the unrepared pretwst by the left and rght multplers M ( r L ) and M ( r ). [ ( r )] [ β ( r )] M ( r ) f r r P L c β = (3) C, [ β ( r )] M ( r ) f r < r P2 c where, subscrpts C, P, P2 and stand for chld, parent, parent 2 and repared, respectvely. M ( r L ) and M ( r ) are the left and rght segments of a multpler curve. The multpler curve for chld s a lnear curve between at r = and [ β, c ] C, [ β, c ] P [ β ], c C, at the cut pont; and [ β, c ] P2 Fgure 6.c. The pretwst at the cut pont at the cut pont r and at r = as shown n c rc s denoted by β, c. The repared pretwst at the cut pont s a combnaton of the left and rght values proportonal to the length of the left and rght segments respectvely. That s, the repar process has less effect on the segment wth longer length. Pre-Twst (deg) (a) Parent Parent cut pont r Pre-Twst (deg) (b) Chld before repar Chld after repar r Multpler (c) r Fgure 6: Pretwst formaton of a chld blade; chld s formed based on the left segment of parent and the rght segment of parent 2. 8

9 Sngle value desgn varables, for example the blade span of chld,, s the combnaton of those of parent blades n a weghtng sense. chld rc parent + ( rc ) parent 2 = (4) Interchangng ndces and 2 n the above equatons, the second chld wll be formed. In order to nvestgate the performance of employng GCO, PBB and partal ftness n optmsaton of wnd turbne blades, blade of AWT27, a 2-bladed stall regulated wnd turbne, s selected to be optmsed for the pretwst and chord dstrbutons. An ntal populaton of sze s generated randomly. Blades of the ntal populaton have the same span and aerofol dstrbuton as the baselne. Usng the same ntal populaton, two types of crossover s used: () GCO wth partal ftnesses PF = Pav for the outer secton and PF2 = weght for the nner secton; and () arthmetc crossover. For GCO the pretwst of the produced chld s obtaned by [ ( r )] [ ϕ( r )] P M ( ) L r f r rc ϕ C =, (5) [ ϕ( r )] P M ( r ) f r < r 2 c n whch, ϕ represents pretwst β and chord c. In the case of arthmetc crossover the pretwst and chord dstrbutons of the chld s gven by: ϕ chld = λϕ (6) parent + λ) ( ϕ parent 2 n whch, agan,ϕ represents pretwst β and chord c and λ s a random number. In performng arthmetc crossover, a roulette wheel constructed based on the ftness ftness = Pav s employed, whle n the case of GCO, two roulette wheels are constructed based on the partal ftnesses defned above. These roulette wheels are employed to select the parents. Startng wth the same ntal populaton, 2 crossover operatons of each type s carred out. esults presented n Fgure 7 show an mprovement of.6% n the average power of the populaton when GCO s used versus an mprovement of 8.6% when arthmetc crossover s used. esults also show that the populaton qualty n terms of the weght mproves slghtly more when usng GCO (2.2% versus.7%). 9

10 Fgure 7: Percent mprovement n the qualtes of the populaton: geometrc versus arthmetc crossover 5 Applcaton to trusses supportng dstrbuted loads Fgure 8 shows a two dmensonal truss, constructed of a strng of trangular panels and subjected to a dstrbuted load q (x). The topology of ths truss, ncludng the number of panels, s to be optmsed wth the objectve of mnmsng the weght of the truss, subject to a constrant on the load converson expedency (LCE), a parameter defned based on the structural propertes of the load doman [7]. Assumng that members are made of the same materal and that ther cross sectonal areas are proportonal to ther nternal forces, mnmsaton of the weght of the truss s equvalent to mnmsng the objectve functon h, defned as subject to the constrant h = m l = ( X ) P ( X ) ; {,2,...,m} (7) LCE LCE c (8) where, LCEc s the smallest permssble LCE, l and P ( P ) represent the length and the nternal force of member respectvely and m s the number of members n the truss. In the case of zero members ( P = ), P wll be replaced wth the smallest nternal force n the truss members. In the above equaton X = { n, x j, y j }, ( j {,2,...,n} ), stands for the vector of desgn varables, parameter n s the total number of nodes, and x j and y j are the coordnates of node j respectvely. More detals on the formulaton of the problem can be found n reference [7]. Usng a mxed nteger-real number encodng, the genotype chromosome of ths truss wll have a varable length of 2n 3, where n s the total number of nodes.

11 y q(x) Dstrbuted Load h /2 ght secton h /2 Left secton x Free node Load-bearng node Support node n x l Truss Doman l h x n-2 y y n-2 Genotype chromosome Left secton ght secton Phenotype buldng blocks Fgure 8: Plane truss and ts PBBs In the frst two examples, the PBBs were ether, confguratonally, easy to dentfy as n the case of the frame or had dfferent functons as n the case of the wnd turbne blade. However, none of these condtons are necessarly requred to be vald for PBBs. Ths truss can be vewed as a dscrete system smlar to the frame example. However, snce t s to be optmsed for the topology, t does not have a fxed topology through optmsaton process. As the PBBs are fxed, the bggest possble PBB s a trangular panel. Obvously, the smaller the PBB, the closer t becomes to the genotype form. Ths s not algned wth the orgnal phlosophy behnd usng PBBs for ncorporatng knowledge nto the search process. An alternatve approach s to treat the truss as a contnuous system smlar to the wnd turbne blade. In ths way each truss can be defned based on two PBBs: ts left and rght sectons. In the wnd turbne blade problem, the phonotype buldng blocks are dentfed based on ther dfferent functons and have dfferent assessment crtera. However, n the truss problem, the functons of the left and rght sectons of the truss are the same; hence, ther assessment crtera wll be the same.

12 n x x n-2 y y n-2 (a) Parent P genotype chromosome; length= 2n 3 m x x m-2 y y m-2 Parent P2 genotype chromosome; length= 2m 3 (b) r=p+q x x r-2 y y r-2 Chld genotype chromosome (p nodes of parent P are located n the left hand sde of the cut pont and q nodes of parent P2 are located n the rght hand sde of the cut pont) Parent Cut pont l x (c) Chld x Parent Parents and chld load-bearng node strngs Fgure 9: Plane truss (a) parents genotype chromosome, (b) chld genotype chromosome after GCO, (c) llustratve example showng the change n the number of panels due to GCO (only the x-coordnate of the load-bearng nodes s shown) l l x The GCO of the truss problem s llustrated n Fgure 9. Fgures 9.a and 9.b show the genotype chromosome of two parents and the offsprng made by a GCO operaton. A geometrc cut dvdes each parent truss nto two parts. Assumng that the left secton of parent P s better than the left secton of parent P 2 and the rght secton of parent 2 s better than the rght secton of parent, the left secton of the frst parent s combned wth the rght secton of the second parent to form a chld truss. A repar operaton also mght be requred at the cut pont to make the chld truss knematcally stable. The locaton of the cut pont and the topology of the parents nfluence the topology of the offsprng. Generally, the length of these three chromosomes can be dfferent. That s, the offsprng can have a dfferent topology from ts parents. The length of the offsprng chromosome s r = p + q, where p and q are the number of nodes on the left secton of the frst parent and the rght secton of the second parent, respectvely. Fgure 9.c shows the x-coordnate of the loadbearng nodes of two parents and ther offsprng. Ths fgure llustrates how two chromosomes wth dfferent lengths are combned and make a new chromosome wth a dfferent length. 2

13 Fgure shows the search hstores of the above optmsaton problem once usng GCO and once applyng conventonal arthmetc crossover. Evdently, employng GCO and partal ftness mproves the search convergence rate. The ftness s defned as the product of the recprocal of the weght of the truss and a penalty functon: The penalty functon p s defned as: ftness = p (9) h LCE LCE p = mn, LCEc LCE () where, LCE s the mnmum possble value for LCE. The partal ftnesses used for the left and rght hand sdes of the truss are defned as the recprocal of the weght of the left and rght hand sdes of the truss respectvely. Fgure : Search hstores n the qualtes of the populaton: geometrc versus arthmetc crossover 6 Concluson By dentfyng PBBs and usng GCO, partal ftness can be defned and employed to mprove the performance of the search algorthm. It s shown that these concepts can be appled to a wde range of structural optmsaton problems wth dfferent characterstcs. These concepts can be appled to both, dscrete and contnuous structures. PBBs may or may not have sharp boundares. PBB can be defned as a combnaton of genotype buldng blocks of a dscrete system or t can be dentfed based on dfferent functons of dfferent segments of a contnuous system. Partal ftness can be defned based on ether the ndvdual assessment crtera or some new assessment crtera. There s no restrcton on the type of the desgn varables 3

14 nvolved n the problem (ndexed, real value, nteger, dstrbuted, sngle value, contnuous and dscrete). Another advantage of usng GCO s due to ts ease of applcaton to non-fxed-length and varable-length chromosomes. Varable-length chromosomes are a common feature of topology optmsaton problems when usng genetc algorthms. Hence, GCO can be easly appled to structural optmsaton problems ncludng topology optmsaton wthout predefnng a grd. eferences [] Handbook of Evolutonary Computaton Edted by Thomas Baeck, D.B Fogel, Z Mchalewcz, Taylor & Francs, 2 [2] A.D.J. Cross,.C. Wlson and E.. Hancock, Inexact Graph Matchng Usng Genetc Search, Pattern ecognton, 3(6), , 997. [3]. Myers, E.. Hancock, Least-commtment graph matchng wth genetc algorthms, Pattern ecognton, 34, , 2. [4] N. Nazm Dugan, S. Erkoc, Genetc algorthm Monte Carlo hybrd geometry optmzaton method for atomc clusters, Computatonal Materals Scence, 45, 27 32, 29. [5] V.A. Froltsov, K. euter, obustness of cut and splce genetc algorthms n the structural optmzaton of atomc clusters, Chemcal Physcs Letters, 473, , 29. [6] D. Safar, M.. Maher and A. Maher, Optmum Desgn of Steel Frames Usng a Multple-Deme GA wth Improved eproducton Operators, Constructonal Steel esearch, 67 (8), , 2. [7] A. Maher and M.. Maher, A robust Method for Integrated Desgn of Trusses Supportng Dstrbuted Loads, Engneerng Structures, 4, , 22. 4