A fast method for binary programming using first-order derivatives, with application to topology optimization with buckling constraints

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2012) Publshed onlne n Wley Onlne Lbrary (wleyonlnelbrary.com) A fast method for bnary programmng usng frst-order dervatves, wth applcaton to topology optmzaton wth bucklng constrants P. A. Browne 1, *,, C. Budd 1,N.I.M.Gould 2,H.A.Km 3 and J. A. Scott 2 1 Department of Mathematcal Scences, Unversty of Bath, Bath, BA2 7AY, UK 2 Numercal Analyss Group, STFC Rutherford Appleton Laboratory, Oxfordshre, OX11 0QX, UK 3 Department of Mechancal Engneerng, Unversty of Bath, Bath, BA2 7AY, UK SUMMARY We present a method for fndng solutons of large-scale bnary programmng problems where the calculaton of dervatves s very expensve. We then apply ths method to a topology optmzaton problem of weght mnmzaton subject to complance and bucklng constrants. We derve an analytc expresson for the dervatve of the stress stffness matrx wth respect to the densty of an element n the fnte-element settng. Results are presented for a number of two-dmensonal test problems. Copyrght 2012 John Wley & Sons, Ltd. Receved 17 November 2011; Revsed 1 May 2012; Accepted 9 May 2012 KEY WORDS: topology optmzaton; bucklng; egenvalue; structural optmzaton; bnary programmng 1. INTRODUCTION The problem consdered n ths paper s how to mnmze the weght of an elastc structure whlst mantanng the ntegrty of the structure by prescrbng two constrants. The frst ensures that the structure has a prescrbed level of stffness, and the second that the structure s not prone to bucklng. The purpose of ths paper s to present an algorthm that can provde a soluton for problems such as these n a reasonable computng tme. To formulate the problem mathematcally, the desgn space (regon n whch materal s placed) s dscretzed usng fnte elements. The goal of optmzaton s to determne whch elements should contan materal and whch should be vod of materal. Ths s usually represented by a bnary varable where a value of 1 s assgned to an element wth materal and a value of 0 to an element contanng no materal, thus resultng n a bnary programmng problem. Fndng a global soluton to bnary programmng problems s notorously dffcult. The methods for fndng such mnma can be broadly put nto three categores: mplct enumeraton, branchand-bound and cuttng-plane methods. The most popular mplementatons nvolve hybrds of branch-and-bound and cuttng-plane methods. For a comprehensve descrpton of these bnary programmng methods, see, for example, Wolsey [1]. These methods were popular for structural optmzaton from the late 1960s through the early 1990s. In 1994, Arora and Huang [2] revewed the methods for solvng structural optmzaton problems dscretely. In 1968, Toakley [3] appled a combnaton of cuttng-plane methods and branch-and-bound to solve truss optmzaton problems. Usng what s now known as the branch-and-cut method, *Correspondence to: P. A. Browne, Department of Mathematcal Scences, Unversty of Bath, Bath, BA2 7AY, U.K. E-mal: P.A.Browne@bath.ac.uk Copyrght 2012 John Wley & Sons, Ltd.

2 P. A. BROWNE ET AL. ths method was resurged n 2010 by Stolpe and Bendsøe [4] to fnd the global soluton to a mnmzaton of complance problem, subject to a constrant on the volume of the structure. In 1980, Farkas and Szabo [5] appled an mplct enumeraton technque to the desgn of beams and frames. Branch-and-bound methods have been used by, amongst others, John et al. [6], Sandgren [7, 8] and Salajegheh and Vanderplaats [9] for structural optmzaton problems. In the latest of these papers, the number of varables n the consdered problem was 100 and n some cases took over 1 week of CPU tme on a modern server to compute the soluton. Whlst these methods do fnd global mnma, they suffer from exponental growth n the computaton tme as the number of varables ncreases. Beckers [10] used a dual method to fnd dscrete solutons to structural optmzaton problems. In 2003, Stolpe and Svanberg [11] formulated a topology optmzaton problem as a mxed 0 1 programme and solved t usng branch-and-bound methods. Achtzger and Stolpe [12 14] have studed n detal the topology optmzaton of truss structures usng branch-and-bound methods, and have been able to fnd global solutons to problems wth over 700 bars n the ground structure. To avod the computatonal ssues assocated wth bnary programmng, the tradtonal approach to topology optmzaton has been to relax the bnary constrant and to look for a soluton that vares contnuously n R n. Ths s known as contnuous relaxaton of the problem. Physcally, ths relaxed varable can correspond to the stffness of the materal n the element. Nested analyss and desgn s then performed, meanng that the structural analyss of the current structure s carred out and approprate dervatves calculated. These values are then fed to an optmzaton routne that updates the structure. The analyss s performed agan and ths process terates untl an optmum s attaned. By far, the most popular optmzaton method to update the structure s the Method of Movng Asymptotes (MMA) [15]. MMA requres the functon values and values of the dervatves at the current teraton n order to progress to a new teraton wth lower objectve functon. There are many examples of MMA beng very effcent at solvng topology optmzaton problems wth dfferent objectves/constrants (e.g. [16]). The bucklng load of a structure s found as the soluton to an egenvalue problem, and a structure s sad to buckle at the lowest postve egenvalue, referred to as the crtcal load. Dervatves of ths crtcal load are only well defned f there s a sngle modeshape correspondng to the crtcal load, that s, we have a smple egenvalue. Hence, a drect bound on the crtcal load that cannot be used as a constrant wth MMA as the dervatve s not well defned. Semdefnte programmng methods have been developed specfcally to deal wth such eventualtes. Kocvara [17], and n conjuncton wth Stngl [18], has appled such methods to topology optmzaton problems. More recently, along wth Bogan [19], they have appled an adapted verson of ther semdefnte codes to fnd non-nteger solutons to bucklng problems. Ths made use of a reformulaton of a semdefnte constrant usng the ndefnte Cholesky factorzaton of the matrx, and solvng a resultng nonlnear programmng problem wth an adapted verson of MMA. Wth these technques, they were able to solve a non-dscrete problem wth 5000 varables n about 35 mn on a standard PC. When a contnuous relaxaton approach s used n problems nvolvng calculatng the bucklng modes (or harmonc modes) of a structure, unwanted numercal effects are ntroduced. Tenek and Hagwara [20], Pedersen [21] and Neves et al. [22] all noted that spurous bucklng (or harmonc) modes would be computed n whch the bucklng s confned to regons where the densty of materal s less than 10%. Whlst assgnng zero stress stffness (or mass n the harmonc analyss case) contrbutons from these elements can eradcate these spurous modes; ths s not consstent wth the underlyng model of the structure. Indeed, f one were to consder a structure where a small fracton (less than 10%) of materal was equdstrbuted throughout the desgn doman, the stress stffness matrx would be the zero matrx, and as a result, the crtcal load of the structure would be computed as nfnte. In ths paper, we ntroduce an effcent method for bnary programmng and apply t to topology optmzaton problems wth a bucklng constrant. In dong so, we avod the problem of spurous bucklng modes and can fnd solutons to large two-dmensonal problems (O.10 5 / varables). Because of the dmensonalty of the problems, and the complexty of dervatve-free methods for bnary programmes, we wll use dervatve nformaton to reduce ths complexty. The effcency

3 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES of topology optmzaton methods nvolvng a bucklng constrant s severely hndered by the calculaton of the dervatves of the bucklng constrant. Ths calculaton typcally takes an order of magntude more tme than the lnear elastcty analyss. Wth ths n mnd, the proposed fast bnary descent method we ntroduce wll try to reduce the number of dervatve calculatons requred. The remander of ths paper s organzed as follows. In Secton 2, we formulate the topology optmzaton problem to nclude a bucklng constrant. Secton 3 motvates and states the new method, whch we use to solve the optmzaton problem. Secton 4 then contans mplementaton detals and results for a number of two-dmensonal test problems. Fnally n Secton 5, we draw conclusons about the proposed algorthm. 2. FORMULATION OF TOPOLOGY OPTIMIZATION TO INCLUDE A BUCKLING CONSTRAINT Let be the desgn doman contanng the elastc structure that s dscretzed usng a fnte-element mesh T.Aloadf s appled to, and ths nduces dsplacements u, whch are the soluton to the equlbrum equatons of lnear elastcty Ku D f (1) where K s the fnte-element stffness matrx. Complance, defned as the product f T u, s a measure of external work done on the structure and addng an upper bound c max ensures that the structure remans stff. The mnmzaton of weght subject to a complance constrant s a well-studed problem, for example, [16]. However, t has long been observed that structures optmzed for mnmum weght or complance are prone to bucklng [23]. The crtcal bucklng load of a structure s defned by the smallest postve value of correspondng to a nonzero v for whch.k C K /v D 0. (2) In ths equaton, K s the symmetrc fnte-element stffness matrx, and K s the symmetrc stress stffness matrx.., v/ s an egenpar of the generalzed egenvalue problem (2). s referred to as the egenvalue and v 0 the correspondng egenvector (or mode shape). The crtcal load s tmes the appled load f. Gven a safety factor parameter c s >0, a bound of the form > c s s equvalent to the semdefnte constrant K C c s K 0. Ths means that all the egenvalues of the system.k C c s K / are non-negatve. Ths happens only f P M D1 vt.k C c sk /v > 0 where v are the M bucklng modes that solve.k C K /v D 0. If we let x 2¹0, 1º n represent the densty of materal n each of the elements of the mesh, wth x D 0 correspondng to an absence of materal n element and x j D 1 correspondng to element j beng flled wth materal, the problem to be solved becomes X mn xj x (3a) subject to c 1.x/ WD c max f T u.x/ > 0 (3b) MX c 2.x/ WD v.x/ T.K.x/ C c s K.x//v.x/ > 0 (3c) D1 x 2¹0, 1º n (3d) K.x/u.x/ D f (3e) ŒK.x/ C.x/K.x/ v.x/ D 0. 8 D 1, :::, M (3f)

4 P. A. BROWNE ET AL Dervatve calculatons To use the bnary descent method (dscussed n Secton 3) we need an effcent way of calculatng the dervatve of the constrants wth respect to the varables x. As wll be seen n Secton 4, the computaton of dervatves of the bucklng constrant (3c) s the bottleneck n our optmzaton algorthm, so t s mperatve that we have an analytc expresson for ths. To calculate the dervatves, the bnary constrants on the varables are relaxed and assume that the followng holds K.x/ D X` x`k`, where K` s the local element stffness matrx. The dervatve of ths wth respect to the densty of an element x s gven D K. Calculatng the dervatve of the bucklng constrant requres the dervaton of an expresson Ths quantty s nontrval to compute, unlke the dervatve of a mass matrx, whch would be n place of the stress stffness matrx n structural optmzaton nvolvng harmonc modes. The stress feld ` on an element ` s a 3 3 tensor wth sx DOFs. Ths can be wrtten n three dmensons as follows: whch n two dmensons reduces to ` D ` D D x`e`b`u, 7 5 ` D x`e`b`u, 12 where u are the nodal dsplacements of the element, E` s a constant matrx of materal propertes and B` contans geometrc nformaton about the element. The ndces 1, 2 and 3 refer to the coordnate drectons of the system. We consder the two-dmensonal case and note that all the followng steps have a drect analogue n three dmensons. We wrte the stress stffness matrx gven n (2) as follows. K D Z `D1 G T` ` ` G`dV`, (4) where G` s a matrx contanng dervatves of the bass functons that relates the dsplacements of an element ` to the nodal DOFs [24], and n s the total number of elements n the fnte-element mesh T. Now, defne a map W R 3 7! R 44 by ˇ 5A 6 WD ˇ ˇ

5 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES Note that s a lnear operator. Usng ths, (4) becomes Z K D G T`.x`E`B`u/G` dv` D `D1 Z G`./ T.x`E`B`./u/G`./ dv` `D1 X! j G`. j / T.x`E`B`. j /u/g`. j / (5) `D1 j where! j are the weghts assocated wth the approprate Gauss ponts j that mplement a chosen quadrature rule to approxmate the ntegral. Dfferentatng the equlbrum equaton (1) wth respect to the densty x u C D 0 and D K u. Now consder the dervatve of the operator wth respect to x.snce D x`e`b`u.x / D ı `E`B`. j /u C x`e`b`. j where ı ` s the Kronecker Delta. Applyng the chan rule to (5), X! j G`. j / j /u/ j / ld1 ld1 j X! j G`. j / T ı `E`B`. j /u x`e`b`. j /K u j /, (6) j where the approxmaton s due to the error n the quadrature rule used. Ths matrx can now be used to fnd the dervatve of the bucklng constrant that we requre. For each varable x D 1, :::, n, (6) must be computed. As (6) contans a sum over D 1, :::, n, t can be seen that has computatonal complexty of O.n/ for each and hence computng (6) for all varables has complexty of O.n 2 /. 3. FAST BINARY DESCENT METHOD In ths secton, we motvate and descrbe the new method that we propose for solvng the bnary programmng problem. If we solve the state equatons (3e) and (3f), then problem (3) takes the general form mn x et x (7a) subject to c.x/ > 0 (7b) x 2¹0, 1º (7c)

6 P. A. BROWNE ET AL. wth x 2 R n, c 2 R m and e D Œ1, 1, :::, 1 T 2 R n and note that problem (3) s of ths form. Typcally m wll be small (less than 10) andm<<n. We also assume that x 0 D e s an ntal feasble pont of (7). Let k denote the current teraton, and x k s the value of x on the kth teraton. The objectve functon e T x s a lnear functon of x that can be optmzed by successvely reducng the number of nonzero terms n x, and we need not worry about errors n approxmatng ths. However, the constrants are nonlnear functons of x and ensurng that (7b) holds s dffcult. Accordngly, we now descrbe how a careful lnearzaton of the constrant equatons can lead to a feasble algorthm. Taylor s theorem can be used to approxmate c.x k / / c x kc1 D c.x k / C k / x kc1 x k C hgher-order terms s determned usng the explct dervatve results of the prevous secton. The method wll take dscrete steps so that x kc1 x k 2¹ 1, 0, 1º 8 D 1, :::, n, and so we must assume that the hgher-order terms wll be small, but later, a strategy wll be ntroduced to cope wth ths when they are not. Consder now varables x k such that x k D 1 that we wsh to change to x kc1 x kc1 x k D 1, for the dfference n the lnearzed constrant functons c x kc1 c.x k / D k / x kc1 x k D 0. Because to be mnmal, all the terms / need to be as small as possble. However, because there are multple constrants, the varables for whch the gradent of one constrant s small may have a large gradent for another constrant. Assumng a feasble pont such that c.x k />0and gnorng the hgher order terms, c x kc1 D c.x k / C We have to ensure c x kc1 >0,so k / x kc1 x k. (8) or equvalently 1 C D1 c.x k / C k T.xk / =c j.x k / x kc1 x kc1 x k x k >0 >0 8j D 1, :::, m. If x kc1 x k then each normalzed constrant c j.x k / s changed j.x k / =c j.x k /. Defne the senstvty of varable to be s.x k / D max j j.x k / ± = max c j.x k /, 10 (9) where s the machne epslon that guards aganst round off errors. For each varable, s.x k / s the most conservatve estmate of how the constrants wll vary f the value of the varable s changed. In one varable, ths has the form shown n Fgure 1. Fgure 1(a) shows the absolute values of the lnear approxmatons to the constrants based on ther values and correspondng dervatves. Fgure 1(b)

7 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES (a) (b) Fgure 1. Senstvty calculaton n one varable for the case when m D 2. (a) Lnear approxmatons to the constrants c x k n the case where m = 2. In ths stuaton x k D 1. (b) Senstvty calculaton n one varable. Here, s.x k / D max¹a, bºdb. shows the calculaton that we make based on normalzng these approxmatons to compute whch of the constrants would decrease the most f the varable x k were changed. ˇj s the pont at whch the lne assocated wth the constrant c j crosses the y-axs and so ˇj D j.x k / =c j.x k /.The amount that the normalzed constrant c j would change f the varable x k were changed s then gven by 1 ˇj j.x k / =c j.x k /. In ths case, the dervatves ndcate that f the varable x k were to be decreased, the second constrant s affected relatvely more than the frst constrant (as max¹a, bº Db), and hence, the senstvty assocated wth ths varable x s gven the value s.x k / 2.x k / =c 2.x k /. Ths senstvty measure also provdes an orderng so that f we choose to update varables n ncreasng order of ther senstvty, the changes n the constrant values are mnmzed. Now, for ease of notaton, let us assume that the varables are ordered so that s 1 6 s 2 6 :::6 s p 8s s.t. x1 k, xk 2, :::, xk p D 1 (10) s pc1 > s pc2 > :::> s n 8s s.t. x k n, xk n 1, :::, xk pc1 D 0 (11) To be cautous, nstead of requrng c.x kc1 / > 0, we allow for the effects of the nonlnear terms and so are content f nstead c.x kc1 / >.1 /c.x k / for some 2.0, 1/. Ths mples that c.x T.x k / / C x kc1 x k >.1 /c.x k /, D1 or equvalently c.x k / C T.x k / x kc1 x k > 0. To update the current soluton, we consder the varables ordered so that (10) and (11) hold and fnd for some > 0 `X L WD max ` s.t. c j.x j.x k / / >0 for all j 2 1, :::, m. (12) 16`6p D1

8 P. A. BROWNE ET AL. Then we decrease from 1 to 0 those varables x1 k, :::, xk L so as to reduce the objectve functon by a value of L. However, there s the possblty that ncreasng varables from 0 to 1 could further reduce the objectve functon by reducng yet more varables from 1 to 0. Ths s tested by fndng (or attemptng to fnd) J>0such that J WD max 06`6.p L/=2 ` s.t. `X j.x k pc X2` j.x k LC > 0 for all j 2 1, :::, m. (13) So, the varables correspondng to the terms n the frst sum are ncreased from 0 to 1, butfor each of these two varables are decreased from 1 to 0, correspondng to the terms n the second summaton. As there are more terms n the second summaton, the objectve functon mproves whlst remanng a feasble soluton. Hence, the varables xlc1 k, :::, xk LC2J are decreased from 1 to 0, and the varables xpc1 k, :::, xk pcj are ncreased from 0 to 1. Note that n (12) and (13) the equatons have to hold for each of the constrants j D 1, :::, m. The coeffcent s a measure of how well the lnear gradent nformaton s predctng the change n the constrants. If the problem becomes nfeasble, then the method has taken too large a step, so s reduced n order to take a smaller step. However, recall that the goal of ths method s to compute the gradents as few tmes as possble, and so we wsh to take steps that are as large as possble. If the step has been accepted for the prevous two teratons wthout reducng then s ncreased to try and take larger steps and thus speed up the algorthm. Note that f s too large and the soluton becomes nfeasble, then s reduced and a smaller step s taken wthout recomputng the dervatves. Hence, ncreasng by too much s not too detrmental to the performance of the algorthm. Based on experence, s reset to 0.7 when the soluton becomes nfeasble and s set to 1.5 when we want to ncrease t. These values appear stable and gve good performance for most problems. To ensure that at least one varable s updated, must be larger than a crtcal value c gven by! j.x k / c D max =c j.x k /. j k 1 Ths guarantees that L > 1 and at least one varable s updated. The upper bound 6 1 must also be enforced so that c.x kc1 / > 0. If we cannot make any further progress wth ths algorthm, we stop. Makng further progress would be far too expensve as we would have to swtch to a dfferent nteger programmng strategy, and the curse of dmensonalty for the problems that we wsh to consder prohbts ths. However, we beleve the computed soluton s good because f we try and mprove the objectve functon by changng the varable for whch the constrants are nfntesmally least senstve, the soluton becomes nfeasble. The fast bnary descent algorthm s thus presented n Algorthm IMPLEMENTATION AND RESULTS We consder optmzng sotropc structures wth Young s modulus 1.0 and Posson s rato 0.3. The desgn spaces are dscretzed usng square blnear elements on a unform mesh. The fast bnary descent method has been mplemented n Fortran90 usng the HSL mathematcal software lbrary [25] and appled to a seres of two-dmensonal structural problems. The lnear solve for the calculaton of dsplacements (1) used HSL_MA87 [26], a DAG based drect solver desgned for shared memory systems. For the sze of problems consdered, HSL_MA87 has been found to be very effcent and stable. The frst sx bucklng modes of the system (2) were computed as these were suffcent to ensure all correspondng egenvectors of the crtcal load were found. These egenpars were calculated usng HSL_EA19 [27], a subspace teraton code, precondtoned by the Cholesky factorzaton already computed by HSL_MA87. The senstvtes were passed through a

9 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES Fgure 2. Desgn doman of a centrally loaded cantlevered beam. The aspect rato of the desgn space s 1.6, and a unt load s appled vertcally from the centre of the rght hand sde of the doman. standard low-pass flter [28] wth radus 2.5h where h s the wdth of an element and ordered usng HSL_KB22, a heapsort [29] algorthm. The codes were executed on a desktop wth an Intel Core 2 Duo CPU 2.83 Ghz wth 2 GB RAM runnng a 32-bt Lnux OS and were compled wth the gfortran compler n double precson. All reported tmes are wall-clock tmes measured usng system_clock Short cantlevered beam We consder a clamped beam wth a vertcal unt external force appled to the free sde as shown n Fgure 2. Fgures 3 5 refer to the solutons found wth the same desgn doman and materal propertes, but wth bucklng and complance constrants.

10 P. A. BROWNE ET AL. Fgure 3. Soluton found on mesh of elements. The bucklng constrant s set to c s D 0.9 and the complance constrant c max D 35. Avolumeof s attaned. The bucklng constrant c 2 s actve, and the complance constrant c 1 s not. Fgure 4. Soluton found on mesh of elements. The bucklng constrant s set to c s D 0.9 and the complance constrant c max D 60. Avolumeof s attaned. Here, the bucklng constrant c 2 s actve, and the complance constrant c 1 s not. Fgure 5. Soluton found on mesh of elements. The bucklng constrant s set to c s D 0.1 and the complance constrant c max D 30.Avolumeof0.692 s attaned. Here, the complance constrant c 1 s actve, and the bucklng constrant c 2 s not. Fgure 3 s the computed soluton to the problem wth parameters c s D 0.9 n (3c) and c max D 35 n (3b). In ths case, the complance constrant c 1.x 0 / s large ntally but the bucklng constrant c 2.x 0 / s small ntally. We see that the method has produced a typcal optmum grllage structure wth four bars under compresson and only three bars under tenson. Note that n the upper bar near the pont of loadng, there s a dstnct corner n the computed soluton. Ths type of formaton attracts hgh concentratons of stran energy, and so, f the problem were mnmzaton of complance, then an optmzaton method would wsh to avod such stuatons. However, n ths case, optmzaton of ths regon s prmarly domnated by the bucklng constrant, and the complance s not the crtcal constrant. Fgure 4 s the computed soluton to a problem wth the same bucklng constrant as n Fgure 3 (c s D 0.9) but s allowed to be more flexble wth c max D 60 (.e., the complance constrant s not as restrctve). Ths results n a clear asymmetry n the computed soluton n whch the lower bar s much thcker than the upper bar. Ths lower bar s under compresson wth ths loadng, and hence would be prone to bucklng. Thus optmzaton renforced the lower bar to meet the bucklng constrant.

11 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES Fgure 5 was obtaned as the soluton for a problem wth c s D 0.1 and c max D 30. In ths case, the ntal value of c 1.x 0 / s close to 0. The computed soluton has only the complance constrant actve, and hence, the computed soluton s more symmetrcal than the solutons shown n Fgures 3 and 4. From Fgures 3 5, t s possble to see a clear dfference n the topology of the resultng soluton dependng on the parameters c s and c max. Note that whlst one constrant may be volated f the updatng process were to proceed, the other constrants have been utlzed throughout the computaton and have affected the path taken and resultng soluton of the algorthm. The hstory of the algorthm when appled to the problem solved n Fgure 3 where c max D 35 and c s D 0.9 s dsplayed n Fgures 6 8. The plot of the objectve functon aganst teraton number shown n Fgure 6 s monotoncally decreasng and so shows that the method as descrbed n Secton 3 s ndeed a descent method. Note that n the ntal stages of the computaton, large steps are made and ths vares as the computaton progresses. Untl teraton 4 large steps have been made and thus the objectve functon s swftly decreasng. When gong to teraton 5 takng a large step would make the current soluton nfeasble so the method automatcally decreases the step sze and hence the decrease n the objectve functon s reduced. Fgure 7 shows that the complance constrant s nactve at the soluton of ths problem. Note that at all ponts the complance of the structure s below the maxmum complance c max, and so, the soluton s feasble at all ponts wth respect to c 1. If ths plot s compared wth Fgure 6, then the large changes n complance can be seen to occur where there are large reductons n volume and smlarly when there s a small change n the volume, the change n complance s also small. Fgure 6. Volume teratons of the fast bnary descent method. Fgure 7. Complance teratons of the fast bnary descent method.

12 P. A. BROWNE ET AL. Fgure 8. Egenvalues teratons of the fast bnary descent method. Note that on the 20th teraton, the egenvalue constrant s volated; thus, the computed soluton s at the 19th teraton. Fgure 8 shows the lowest sx egenvalues of the system as the bnary descent method progresses. We see that on the 20th teraton, the lowest egenvalue s below the constrant c s,andso,the computed soluton s at teraton 19. At teratons 5 and 8 we see that the egenvalue constrant s close to beng volated. The ncrease n the lowest egenvalue at the subsequent steps corresponds to a local thckenng of the structure around the place where the bucklng s most concentrated. Ths shows that the method has re-ntroduced materal n order to move away from the constrant boundary. The nonlnearty n c 2.x/ s clear from the non-monotonc behavour seen n Fgure 8. Generally, we do see the egenvalues convergng and that supports the ntutve optmalty crtera of concdental egenvalues. Fgure 8, when vewed n combnaton wth Fgure 7 shows that for the hstory of the algorthm, the solutons are all feasble Sde loaded column In ths secton, we consder a tall desgn space fxed completely at the bottom carryng a vertcal load appled at the top corner of the desgn space. The desgn space s shown n Fgure 9(a), and the computed solutons to ths problem wth dfferng constrants are shown n Fgures 9(b) and (c). The problem solved n Fgure 9(b) has c s D and c max D The problem solved n Fgure 9(c) has c s D and c max D 60. In Fgure 9(c), as the constrants are relaxed compared wth the problem n Fgure 9(b), the computed soluton has a sgnfcantly lower objectve functon. However, t follows the same structural confguraton where the man compressve column drectly under the load ressts the bucklng and the slender column on the sde provdes addtonal support n tenson to reduce bendng. In both of these structures the path of the optmzaton s drven by the frst bucklng mode Centrally loaded column We consder a square desgn doman (Fgure 10). A unt load s appled vertcally downwards at the centre of the top of the desgn doman, and the base s fxed. Fgures present results for a mesh of elements for a range of values of the constrants. Fgures 11 and 12 have c s D 0.5 wth c max D 5 and c max D 5.5, respectvely. Ths small change n the complance constrant results n two dstnct confguratons. Fgure 12 wth the hgher complance constrant acheves a lower volume and has the bucklng constrant actve as opposed to the complance constrant whch s actve n Fgure 11.

13 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES (a) (b) (c) Fgure 9. A column loaded at the sde. (a) Desgn doman wth wdth to heght rato 3 W 10. (b) Optmal desgn on a mesh wth c s D and c max D Here, c 2 s actve and c 1 s not. (c) Optmal desgn on a mesh wth c s D and c max D 60. Here, c 1 s actve and c 2 s not. Fgure 10. Desgn doman of model column problem. Ths s a square doman of sde length 1 wth a unt load actng vertcally at the mdpont of the upper boundary of the space. Dstnct ƒ-lke structures have been found n Fgures 13 and 14. These problems share the parameter c max D 8, but vary n that they have c s D 0.4 and c s D 0.1, respectvely. The hgher bucklng constrant of Fgure 13 leads to the development of thck regons n the centre of the supportng legs. These regons help to resst the frst-order bucklng mode of the ndvdual legs and are not seen n Fgure 14 as the bucklng constrant s lower. Fgure 15 s the soluton to a problem wth the same parameters as the problem consdered n Fgure 14, but s solved on a much fner mesh.

14 P. A. BROWNE ET AL. Fgure 11. Soluton computed on a mesh of elements. The bucklng constrant s set to c s D 0.5 and the complance constrant c max D 5. Here, the complance constrant s actve and the bucklng constrant s nactve. Fgure 12. Soluton computed on a mesh of elements. The bucklng constrant s set to c s D 0.5 and the complance constrant c max D 5.5. In ths case, compared wth Fgure 11, the hgher complance constrant has led to a soluton where ths constrant s nactve and the bucklng constrant s now actve. Fgure 13. Soluton computed on a mesh of elements. The bucklng constrant s set to c s D 0.4 and the complance constrant c max D 8. Avolumeof0.276 s attaned.

15 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES Fgure 14. Soluton computed on a mesh of elements. The bucklng constrant s set to c s D 0.1 and the complance constrant c max D 8. Avolumeof0.183 s attaned. Fgure 15. Soluton computed on a mesh of elements. The bucklng constrant s set to c s D 0.1 and the complance constrant c max D 8. Avolumeof s attaned. Compare wth Fgure 14. From Fgures 11 15, we see that the symmetry of the problem s not present n the computed soluton. As Stolpe [30] and Rozvany [31] have shown, because we do not have contnuous varables, we do not necessarly expect the optmal soluton to these bnary programmng problems to be symmetrc. The asymmetry n the computed solutons arse from (12) and (13) as only a subset of elements wth precsely the same senstvty values may be chosen to be updated and so the symmetry may be lost. Table I summarzes the results obtaned when solvng the problem consdered n Fgures 14 and 15, but wth varyng mesh szes. Note the problem sze that the fast bnary method has been able to solve. A computaton on a two-dmensonal mesh of elements took less than 8 hona modest desktop and elements took around 12 h. Ths speed s attaned because the number of dervatve calculatons appears to be not dependent on the number of varables. Fgure 16 shows a log log plot of the number of optmzaton varables aganst the wall-clock tme taken to compute a soluton. As the plot appears to have a gradent close to 2, ths ndcates that the tme to compute a soluton s O.n 2 /. A detaled examnaton of the computatonal cost ndcates that the vast majorty of the computatonal cost s n the computaton of the dervatve of the bucklng constrant (see the fnal column of Table I). A massvely parallel mplementaton of ths step s possble, and t s antcpated

16 P. A. BROWNE ET AL. Table I. Table of results for the centrally loaded column. Proporton Problem sze Dervatve Tme (mn) of tme on n Objectve calculatons Analyses to 3 2 =@x D D D D D D D D D D D D D D D D Fgure 16. Log log plot of tme aganst the number of optmzaton varables. The gradent of ths plot appears to be 2, suggestng that the tme to compute the soluton to a problem wth n varables s O.n 2 /. that t should acheve near optmal speedup as no nformaton transfer s requred for the calculaton of the dervatve wth respect to the ndvdual varables. 5. CONCLUSIONS The man computatonal cost assocated wth topology optmzaton nvolvng bucklng s the calculaton of the dervatves of the bucklng load. We have employed an analytc formula for ths, but t stll remans the most expensve part of the algorthm. To reduce the computatonal cost, we have developed an algorthm that ams to mnmze the number of these computatons that are requred. The method s a descent method that enforces feasblty at each step and thus could be termnated early and would stll result n a feasble structure. We have numercally shown that the algorthm scales quadratcally wth the number of elements n the fnte-element mesh of the desgn space. Ths corresponds to the analytcal result that the dervatve of the stress-stffness matrx wth respect to each of the desgn varables s an O.n 2 / operaton. The numercal experments demonstrate the effcency of the method for bnary topology optmzaton usng complance and bucklng constrants.

17 BINARY PROGRAMMING USING FIRST-ORDER DERIVATIVES ACKNOWLEDGEMENTS Ths paper was funded by the EPSRC wth contract grant no. EP/E053351/1 and CASE award MCA /2009. REFERENCES 1. Wolsey L. Integer programmng, Wley-Interscence seres n dscrete mathematcs and optmzaton. Wley: New York, USA, Avalable from: 2. Arora J, Huang M. Methods for optmzaton of nonlnear problems wth dscrete varables: a revew. Structural Optmzaton 1994; 8: Toakley A, Optmum desgn usng avalable sectons. Journal of the Structural Dvson: proceedngs of the Amercan Socety of Cvl Engneers 1968; 94: Stolpe M, Bendsøe MP. Global optma for the Zhou Rozvany problem. Structural and Multdscplnary Optmzaton 2010; 43(2): Farkas J, Szabo L. Optmum desgn of beams and frames of welded I-sectons by means of backtrack programmng. Acta Technca Academae Scentarum Hungarcae 1980; 91(1): John K, Ramakrshna C, Sharma K. Optmum desgn of trusses from avalable sectons use of sequental lnear programmng wth branch and bound algorthm. Engneerng Optmzaton 1988; 13(2): Sandgren E. Nonlnear nteger and dscrete programmng for topologcal decson makng n engneerng desgn. Journal of Mechancal Desgn 1990; 112(1): DOI: / Avalable from: lnk/?jmd/112/118/1 8. Sandgren E. Nonlnear nteger and dscrete programmng n mechancal desgn optmzaton. Journal of Mechancal Desgn 1990; 112(2): DOI: / Avalable from: 9. Salajegheh E, Vanderplaats G. Optmum desgn of trusses wth dscrete szng and shape varables. Structural Optmzaton 1993; 6: Beckers M. Dual methods for dscrete structural optmzaton problems. Internatonal Journal for Numercal Methods n Engneerng 2000; 48(12): Stolpe M, Svanberg K. Modellng topology optmzaton problems as lnear mxed 0 1 programs. Internatonal Journal for Numercal Methods n Engneerng June 2003; 57(5): Avalable from: Achtzger W, Stolpe M. Truss topology optmzaton wth dscrete desgn varables guaranteed global optmalty and benchmark examples. Structural and Multdscplnary Optmzaton December 2007; 34(1): DOI: /s Avalable from: Achtzger W, Stolpe M. Global optmzaton of truss topology wth dscrete bar areas Part I: theory of relaxed problems. Computatonal Optmzaton and Applcatons November 2008; 40(2): DOI: /s Avalable from: Achtzger W, Stolpe M. Global optmzaton of truss topology wth dscrete bar areas Part II: mplementaton and numercal results. Computatonal Optmzaton and Applcatons 2009; 44(2): DOI: /s Svanberg K. The method of movng asymptotes a new method for structural optmzaton. Internatonal Journal For Numercal Methods n Engneerng 1987; 24: Bendsøe MP, Sgmund O. Topology Optmzaton: Theory, Methods and Applcatons. Sprnger: Berln, Germany, Kočvara M. On the modellng and solvng of the truss desgn problem wth global stablty constrants. Structural and Multdscplnary Optmzaton Aprl 2002; 23(3): DOI: /s Avalable from: Kočvara M, Stngl M. Solvng nonconvex SDP problems of structural optmzaton wth stablty control. Optmzaton Methods and Software October 2004; 19(5): DOI: / Avalable from: crossref D404A21C5BB053405B1A640AFFD44AE3 19. Bogan C, Kočvara M, Stngl M. A new approach to the soluton of the VTS problem wth vbraton and bucklng constrants. 8th World Congress on Structural and Multdscplnary Optmzaton, Lsbon, Portugal, 2009; Tenek LH, Hagwara I. Egenfrequency maxmzaton of plates by optmzaton of topology usng homogenzaton and mathematcal programmng. JSME Internatonal Journal Seres C 1994; 37(4): Pedersen N. Maxmzaton of egenvalues usng topology optmzaton. Structural and Multdscplnary Optmzaton 2000; 20(1): Neves MM, Sgmund O, Bendsøe MP. Topology optmzaton of perodc mcrostructures wth a penalzaton of hghly localzed bucklng modes. Internatonal Journal for Numercal Methods n Engneerng June 2002; 54(6): Avalable from: Hunt G, Thompson J. A General Theory of Elastc Stablty. Wley-Interscence: London, UK, Cook RD, Malkus DS, Plesha M. Concepts and Applcatons of Fnte Element Analyss. John Wley and Sons: London, UK, 1989.

18 P. A. BROWNE ET AL. 25. HSL(2011). A collecton of Fortran codes for large scale scentfc computaton. Avalable from: ac.uk 26. Hogg J, Red J, Scott J. Desgn of a multcore sparse cholesky factorzaton usng DAGs. SISC 2010; 32: Ovtchnnkov E, Red J. A precondtoned block conjugate gradent algorthm for computng extreme egenpars of symmetrc and Hermtan problems. Techncal Report RAL-TR , STFC Rutherford Appleton Laboratory, Ddcot, UK, Huang X, Xe YM. Evolutonary Topology Optmzaton of Contnuum Structures. John Wley & Sons Ltd: Chchester, UK, Wllams J. Algorthm 232: heapsort. Communcatons of the ACM 1964; 7(6): Stolpe M. On some fundamental propertes of structural topology optmzaton problems. Structural and Multdscplnary Optmzaton 2010; 41(5): Rozvany GIN. Authors reply to a dscusson by Gengdong Cheng and Xaofeng Lu of the revew artcle On symmetry and non-unqueness n exact topology optmzatoni by George I. N. Rozvany (2011, Struct Multdsc Optm 43: ). Structural and Multdscplnary Optmzaton September 2011; 44(5): DOI: /s Avalable from: