SUMMARY. Topology optimization, buckling, eigenvalue, derivative, structural optimization 1. INTRODUCTION

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 0000; 00:1 20 Publshed onlne n Wley InterScence ( A fast method for solvng bnary programmng problems usng frst order dervatves, wth specfc applcaton to topology optmzaton wth bucklng constrants P. A. Browne 1, Prof. C. Budd 1, Prof. N. I. M. Gould 2, Dr H. A. Km 3 and Dr J. A. Scott 2 1 Department of Mathematcal Scences, Unversty of Bath, UK 2 Numercal Analyss Group, Rutherford Appleton Laboratory, STFC, UK 3 Department of Mechancal Engneerng, Unversty of Bath, 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 mnmsaton of weght 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 c 0000 John Wley & Sons, Ltd. Receved... KEY WORDS: Topology optmzaton, bucklng, egenvalue, dervatve, structural optmzaton 1. INTRODUCTION Topology optmzaton can be thought of n many ways. To an engneer, t may be thought of as placng materal somewhere n a desgn space n order to attan an optmal value of a functon that s mportant to the system. Mathematcally t can be posed as fndng an optmal doman on whch a functon of the soluton of the underlyng PDE s mnmal. We are nterested n a conceptually straghtforward problem: to mnmze the weght of an elastc structure. However, we also want to mantan the ntegrty of the structure, and we do ths 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 dscretsed usng fnte elements. Our goal s to determne whch elements should contan materal Correspondence to: Department of Mathematcal Scences, Unversty of Bath, Bath, UK. E-mal: [email protected] Copyrght c 0000 John Wley & Sons, Ltd. [Verson: 2010/05/13 v3.00]

2 2 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT and whch should be vod of materal. If we assocate a value of 1 to an element wth materal and a value of 0 to an element contanng no materal, ths topology optmzaton problem s 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 to the early 1990s. In 1994, Arora & 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, ths method was resurged n 2010 by Stolpe and Bendsøe [4] to fnd the global soluton to a mnmsaton 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 & Vanderplaats [9] for structural optmzaton problems. In the latest of these papers, the number of varables n the consdered problem was 100. Whlst these methods do fnd global mnma of the problems, they suffer from exponental growth n the computaton tme as the number of varables ncreases. Beckers 2000 [10] uses a dual method to fnd dscrete solutons to structural optmzaton problems. In 2003, Stolpe and Svanberg [11] formulate a topology optmzaton problem as a mxed 0-1 program and solve usng branch-and-bound methods. Achtzger and Stolpe [12, 13, 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 densty of the materal n the element or the thckness of the materal n the element (f the problem s 2 dmensonal). 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 optma s attaned. By far the most popular optmzaton method to update the structure s the Method of Movng Asymptotes (MMA) [15]. To functon, MMA needs only the functon values and values of the dervatves at that pont. There are many examples of MMA beng very effcent at solvng topology optmzaton problems wth dfferent objectves/constrants. 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 only one modeshape correspondng to the crtcal load. Mathematcally, ths means we have a smple egenvalue. Hence a drect bound on the crtcal load cannot be used as a constrant wth MMA as the dervatve s not well defned.

3 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 3 Semdefnte programmng methods have been developed specfcally to deal wth such eventualtes. Kocvara [16], and n conjuncton wth Stngl [17], has appled such methods to topology optmzaton problems. More recently, along wth Bogan [18], they have appled an adapted verson of ther semdefnte codes to fnd nonnteger solutons to bucklng problems. Ths made use of a reformulaton of a semdefnte constrant usng the ndefnte Cholesky factorsaton of the matrx, and solvng a resultng nonlnear programmng problem wth an adapted verson of MMA. 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 [19], Pedersen [20] and Neves et al. [21] 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 ther proposed soluton of havng no 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 that s able to fnd a local mnma of the topology optmzaton problem wth bucklng constrants. In dong so, we avod the problem of spurous bucklng modes and can fnd solutons to large two-dmensonal problems (O(10 5 ) varables). However, the method does not guarantee the computed soluton s a global mnmser. Due to the dmensonalty of the problems we wsh to solve, and the complexty of dervatvefree methods for bnary programs, we wll use dervatve nformaton to reduce ths complexty. The effcency 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 structural analyss. Wth ths n mnd, the bnary descent method we ntroduce wll try to reduce the number of dervatve calculatons made to mprove effcency. The remander of ths paper s organsed as follows. In Secton 2 we formulate the topology optmzaton problem to nclude the 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 BUCKLING CONSTRAINTS Let Ω be the desgn doman contanng the elastc structure that we dscretse usng a fnte-element mesh T. We apply a load f to Ω, and ths nduces dsplacements u, whch are the soluton to the equlbrum equatons of lnear elastcty Ku = f (1)

4 4 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT where K s the fnte element stffness matrx. To prescrbe stffness of the system, we bound a quantty known as complance. Complance s defned as the product f T u, whch s a measure of external work done on the structure. We want to gve ths an upper bound c max so that our structure remans stff. The mnmsaton of weght subject to a complance constrant s a well-studed problem, see for example [22]. However, t has long been observed that structures optmzed for mnmum weght or complance are prone to bucklng [23]. The crtcal load of a structure s defned by the smallest postve value of λ correspondng to a nonzero egenvector v for whch (K + λk σ )v = 0. (2) In ths equaton, K s the symmetrc fnte-element stffness matrx and K σ s the symmetrc stress stffness matrx. (λ, v) s an egenpar of the generalsed egenvalue problem (2). λ s refered to as the egenvalue and v 0 the correspondng egenvector (or modeshape). The crtcal load s then λ tmes the appled load f. Gven a safety factor parameter c s > 0, a bound of the form λ c s s equvalent to the semdefnte constrant K + c s K σ 0. Ths means that all the egenvalues of the system (K + c s K σ ) are non-negatve. Ths happens only f M =1 vt (K + c sk σ )v 0 where v are the M bucklng modes that solve (K + λk σ )v = 0. If we let x {0, 1} n represent the densty of materal n each of the elements of our mesh, wth x = 0 correspondng to an absence of materal n element and x j = 1 correspondng to element j beng flled wth materal, the problem we wsh to solve becomes: mn x x (3a) subject to c 1 (x) := c max f T u(x) 0 (3b) M c 2 (x) := v (x) T (K(x) + c s K σ (x))v (x) 0 (3c) =1 x {0, 1} n K(x)u(x) = f [K(x) + λ(x)k σ (x)]v(x) = 0. (3d) (3e) (3f) 2.1. Dervatve calculatons To use the bnary descent method that we wll explan n Secton 3 we need an effcent way of calculatng the dervatve of the constrants wth respect to the varables x. As we wll see later n the results secton, the computaton of dervatves of the bucklng constrant (3c) s the bottleneck of our optmzaton algorthm, so t s mpertve that we have an analytc expresson for ths. To calculate the dervatves, we relax the bnary constrants on our varables and assume the followng holds

5 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 5 K(x) = l x l K l, where K l s the local element stffness matrx. The dervatve of ths wth respect to the densty of an element x s gven by K (x) = K. Calculatng the dervatve of the bucklng constrant requres the dervaton of an expresson for K σ. 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 σ l on an element l s a 3 3 tensor wth 6 degrees of freedom. Ths can be wrtten n three dmensons as whch n two dmensons reduces to σ l = σ l = σ 11 σ 22 σ 33 σ 12 σ 13 σ 23 σ 11 σ 22 l = x l E l B l u, = x l E l B l u, σ 12 l where u are the nodal dsplacements of the element, E l s a constant matrx of materal propertes and B l 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. n K σ = l=1 G T l σ 11 σ σ 12 σ G l dv l, (4) 0 0 σ 11 σ σ 12 σ 22 where G l s a matrx contanng dervatves of the bass functons that relates the dsplacements of an element l to the nodal degrees of freedom [24] and n s the total number of elements n the fnte-element mesh T. Now defne a map Θ : R 3 R 4 4 by α γ 0 0 α γ β 0 0 Θ( β ) := 0 0 α γ. γ 0 0 γ β l

6 6 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT Note that Θ s a lnear operator. Usng ths, (4) becomes K σ = = n G T l Θ(x l E l B l u)g l dv l l=1 n G l (ξ) T Θ(x l E l B l (ξ)u)g l (ξ) dv l l=1 n ω j G l (ξ j ) T Θ(x l E l B l (ξ j )u)g l (ξ j ) (5) l=1 j where ω j are the weghts assocated wth the approprate Gauss ponts ξ j that mplement a chosen quadrature rule to approxmate the ntegral. Dfferentatng the equlbrum equaton (1) wth respect to the densty x yelds and hence K u + K u = 0 u 1 K = K u. Now consder the dervatve of the operator Θ wth respect to x. Snce Θ s lnear Θ(x l E l B l u) where δ l s the Kronecker Delta. ( ) = Θ x l E l B l u(x ) ( = Θ Applyng the chan rule to (5) we obtan δ l E l B l (ξ j )u + x l E l B l (ξ j ) u ) K σ K σ n ω j G l (ξ j ) T Θ(x le l B l (ξ j )u) G l (ξ j ) l=1 l=1 j n ω j G l (ξ j ) T 1 K Θ(δ l E l B l (ξ j )u x l E l B l (ξ j )K u)g l (ξ 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 whch we requre. For each varable x = 1,..., n, (6) must be computed. As (6) contans a sum over = 1,..., n one can see that computng Kσ has computatonal complexty of O(n) for each and hence computng (6) for all varables has complexty of O(n 2 ). 3. BINARY DESCENT METHOD In ths secton, we motvate and descrbe the new method whch we propose for solvng the bnary programmng problem. If we solve the state equatons (3e) and (3f) then problem (3) takes the

7 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 7 general form mn x e T x (7a) subject to c(x) 0 (7b) x {0, 1} wth x R n, c R m and e = [1, 1,..., 1] T R n and note that problem (3) s of ths form. Typcally m wll be small (less than 10) and m << n. We also assume that x 0 = e s an ntal feasble pont of (7). From now on we let k denote the current teraton, and x k the value x on the k-th teraton. The objectve functon e T x s a purely lnear functon of x ths 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 lnearsaton of the constrant equatons can lead to a feasble algorthm. Taylor s theorem can then be used to approxmate c(x k ) c(x k+1 ) = c(x k ) + n =1 c(x k ) (x k+1 x k ) + hgher order terms where c(xk ) s determned usng the explct dervatve results of the prevous secton. Our method wll take dscrete steps so that x k+1 x k { 1, 0, 1} = 1,..., n, and so we must assume that the hgher order terms wll be small, but later we wll ntroduce a strategy to cope wth when they are not. If we now consder varables x k such that xk x k = 1 and so for the dfference n the lnearsed constrant functons x k+1 c(x k+1 ) c(x k ) = = 1 whch we wsh to change to xk+1 n =1 c(x k ) (x k+1 x k ) (7c) = 0. Then to be mnmzed, we want all the terms of c(xk ) to be as small as possble. However, we have multple constrants, so the varables for whch the gradent of one constrant s small may have a large gradent for another constrant. Assume we are at a feasble pont so that c(x k ) > 0. Ignorng the hgher order terms, we can wrte c(x k+1 ) = c(x k ) + n =1 c(x k ) (x k+1 x k ). (8)

8 8 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT We have to ensure c(x k+1 ) > 0, so 1 + c(x k ) + n =1 n =1 c(x k ) (x k+1 x k ) > 0 c T j (xk ) /c j (x k )(x k+1 x k ) > 0 j = 1,..., m. If x k+1 x k then each normalsed constrant c j(x k ) s changed by ± cj(xk ) /c j (x k ). If we defne a senstvty for each varable as s (x k ) = c j (x k ) max /c j (x k ) (9) j=1,...,m ths gves us a quantty for each varable whch s the most conservatve estmate of how the constrants wll vary f we change the value of the varable. In one varable, ths has the form shown n Fgure 1. Fgure 1a shows the absolute values of the lnear approxmatons to the constrants based on ther values and correspondng dervatves. Fgure 1b shows the calculaton that we make based on normalsng these approxmatons to compute whch of the constrants would decrease the most f the varable x k were changed. c(x) c 2 (x k ) c 1 (x k ) a 1 b c 1 (x ) c 2 (x ) 0 0 (a) Lnear approxmatons to the constrants c(x k ) n the case where m = 2. 1 x k 0 0 (b) Senstvty calculaton n one varable. Here s (x k ) = max{a, b} = b. 1 x k Fgure 1. Senstvty calculaton n one varable for the case when m = 2. Ths senstvty measure also provdes an orderng so that f we choose to update varables n ncreasng order of ths quantty, the change n the constrant values are mnmsed. Now for ease of notaton, let us assume that the varables are ordered so that s 1 s 2... s p s s.t. x k 1, x k 2,..., x k p = 1 s p s p+1... s n s s.t. x k n, x k n 1,..., x k p+1 = 0

9 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 9 To be cautous, nstead of requrng c(x k+1 ) 0, we allow for the effects of the nonlnear terms and so would content f nstead c(x k+1 ) (1 α)c(x k ). Ths mples that.e. we requre c(x k ) + n =1 αc(x k ) + c T (x k ) (x k+1 x k ) (1 α)c(x k ), n =1 To update the current soluton we can fnd L := max l s.t. αc(x k ) c T (x k ) (x k+1 x k ) 0. l j=1 x k j =1 c(x k ) x j > 0 (10) where the lower summaton condton x k j = 1 means we only consder those gradents correspondng to varables wth the value 1. Then we change from 1 to 0 those varables x k 1,..., xk L so as to reduce the objectve functon by a value of L. However, there s the possblty that changng varables from 0 1 could allow us to further reduce the objectve functon by changng yet more varables from 1 0. We test for ths stuaton by fndng (or attemptng to fnd) J > 0 such that J := max j s.t. j =1 x k =0 c(x k ) 2j =1 x k =1 c(x k ) x L+ 0 (11) So we can change varables the varables correspondng to the terms n the frst sum from 0 1 but we can then fnd more varables to change from 1 0, correspondng to the terms n the second summaton, so that the objectve functon mproves whlst remanng a feasble soluton. The coeffcent α s a measure of how well the lnear gradent nformaton s predctng the change n the constrants as we go from one teraton to another. If the problem becomes nfeasble, then we know we have taken too bg a step, so we must reduce α n order to take a smaller step. However, recall the goal of ths method s to compute the gradents as few tmes as possble, and so we wsh to take steps whch are as large as possble. If the step has been accepted for the prevous 2 teratons wthout reducng α then we choose to ncrease α n order to attempt to take larger steps and thus speed up the algorthm. Note that f α s too large and the step we take becomes nfeasble then we can reduce α and try a smaller step all wthout recomputng the dervatves. Hence ncreasng α by too much would not be too detrmental to the performance of the algorthm. After expermentng wth dfferent values to ncrease and decrease α n these cases, we have chosen to reset α to 0.7α when the problem becomes nfeasble and we reset α to 1.5α when we want to ncrease t. These values appear stable and gve good performance for most problems.

10 10 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT To ensure that we try and update at least one varable, α must be larger than a crtcal value α c gven by α c = max {( c j(x k ) )/c j (x k )}. j=1,...,m x k 1 Ths guarantees that L 1 and so we update at least 1 varable. If we cannot make any further progress wth ths algorthm, we stop. When we have nonlnear constrants we cannot say for certan that we are at an optmal pont, however makng further progress would be far too expensve as we would have to swtch to a dfferent nteger programmng strategy and the dmenson of the problems that we wsh to consder prohbts ths. we now present the algorthm: Algorthm 1 Fast bnary descent method 1: Intalse x 0 and α. Compute objectve functon (7a) and constrants (7b) 2: f x 0 not feasble then 3: Stop 4: else 5: Compute dervatves c(xk ) 6: Sort s (9) 7: Compute values L (10) and J (11) 8: Update, based on L and J, the varables x k 9: f no varables updated then 10: {We cannot change any varables} 11: return wth optmal soluton 12: end f 13: Compute objectve functon and constrants. 14: f not feasble then 15: {Reject update step} 16: Reduce α. 17: GO TO 7 18: else 19: {Accept update step} 20: Increase α f desred 21: k = k : GO TO 5 23: end f 24: end f 4. IMPLEMENTATION AND RESULTS In our experments, we consder optmsng an sotropc materal wth Young s modulus 1.0 and Posson rato 0.3. We dscretse the desgn spaces usng square blnear elements on a unform mesh.

11 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 11 Algorthm 1 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 6 bucklng modes of the system (2) were computed as these were suffcent to ensure we found all correspondng egenvectors of the crtcal load. These egenpars were calculated usng HSL EA19 [27], a subspace teraton code, precondtoned by the Cholesky factorsaton already computed by HSL MA87. The senstvtes were ordered usng HSL KB22, a heapsort algorthm. The codes were executed on a desktop wth an Intel R Core TM 2 Duo CPU 2.83Ghz wth 2GB RAM runnng a 32-bt Lnux OS and were compled wth the gfortran compler n double precson Short cantlevered beam We consder a clamped beam wth a vertcal external force appled to the free sde as shown n Fgure 2. Fgure 2. Desgn doman of a centrally loaded cantlevered beam. Here we have a wdth to heght rato of 8 : 5 and a unt load actng vertcally from the centre of the rght hand sde of the doman. Fgure 3. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.9 and the complance constrant c max = 35. A volume of s attaned. Smlarly to Fgure 4 the bucklng constrant c 2 s actve and the complance constrant c 1 s not.

12 12 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT Fgure 3 s the computed soluton to a problem that s allowed to be reasonably flexble (the complance constrant c 1 (x 0 ) s large ntally) but the bucklng constrant s reasonably tght (c 2 (x 0 ) s small ntally). We see that the method has produced a structure wth 4 bars under compresson and only 3 bars under tenson. Fgure 4. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.9 and the complance constrant c max = 60. A volume of s attaned. Here the bucklng constrant c 2 s actve and the complance constrant c 1 s not. Fgure 4 s the computed soluton to a problem wth the same bucklng constrant as n Fgure 3 but s allowed to be more flexble (the complance constrant s not as restrctve). There s 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. The method has automatcally desgned the structure wth more materal n ths compressed bar n order to keep the bucklng load of the structure above the constrant. Fgure 5. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.1 and the complance constrant c max = 30. A volume of s attaned. Here the complance constrant c 1 s actve and the bucklng constrant c 2 s not. Fgure 5 has a very small bucklng constrant but a very tght complance constrant (c 1 (x 0 ) s close to 0). The method has computed a soluton that only has the complance constrant actve. In ths calculaton, the complance constrant has been the major nfluencng factor and hence

13 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 13 the computed soluton s much more symmetrcal than one when the bucklng constrant s more promnent. Fgures 3 to 5 refer to the solutons found wth the same desgn doman and materal propertes but wth dfferent values for the bucklng and complance constrants. We can see that there s a clear dfference n the topology of the resultng soluton dependng on these qualtes. We now dsplay some of the hstory of the algorthm when appled to the problem solved n Fgure 3 where c max = 35 and c s = Volume Iteraton Fgure 6. Volume - teratons of Bnary method 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 compuaton progresses. 35 c max = Complance Iteraton Fgure 7. Complance - teratons of the Bnary method

14 14 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT Fgure 7 shows how 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. 2 Lowest 6 Egenvalues c s = Iteraton Fgure 8. Egenvalues - teratons of the Bnary method Fgure 8 shows the lowest 6 egenvalues of the system as bnary descent method progresses. We see that on the 20-th teraton the lowest egenvalue s below the constrant c s and so the computed soluton s at teraton 19. The nonlnearty n c 2 (x) s clear to see from Fgure 8 as we can see that t s not behavng monotoncally. When vewed n combnaton wth Fgure 7 we see 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. Ths type of loadng would typcally be representatve of that experenced by a cross secton of a wall supportng a roof. The desgn space s shown n Fgure 9a and the computed solutons to ths problem wth dfferng constrants are shown n Fgures 9b and 9c. We can see from Fgure 9b because of the complance constrant we have a thck structure vertcally underneath the appled load whch carres the force drectly to the ground. The bucklng constrant has ntroduced the slender support n the opposte sde of the desgn doman to the load akn to a buttress. We can see that ths s carryng comparatvely less load but s helpng the man support to resst bucklng. In Fgure 9c as the constrants are relaxed compared wth the problem n Fgure 9b, the computed soluton has a sgnfcantly lower objectve functon, but we stll see the same thck support drectly below the load and the more slender support to the sde to help resst bucklng.

15 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 15 (a) Desgn doman wth wdth to heght rato 3 : 10. (b) Optmal desgn on a mesh wth c s = and c max = Here c 2 s actve and c 1 s not. (c) Optmal desgn on a mesh wth c s = and c max = 60. Here c 1 s actve and c 2 s not. Fgure 9. A column loaded at the sde 4.3. Centrally loaded column As shown n Fgure 10 we have a square desgn doman. The loadng s vertcally downwards at the top of the desgn doman and the base s fxed completely. Fgure 10. Desgn doman of model column problem. Ths s a square doman of sde length 1 wth a unt load actng vertcally on the space at the mdpont of the upper boundary of the space. Fgure 11 through to Fgure 14 show results of ths problem for a specfc mesh sze wth varyng values of the dfferent constrants. Fgure 16 s the resultng soluton when solved on the mesh wth 200 elements.

16 16 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT Fgure 11. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.5 and the complance constrant c max = 5. Here, the complance constrant t actve and the bucklng constrant s nactve. Fgure 12. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.5 and the complance constrant c max = 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 found on mesh of elements. The bucklng constrant s set to c s = 0.4 and the complance constrant c max = 8. A volume of s attaned. Fgure 14. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.1 and the complance constrant c max = 8. A volume of s attaned. In all the solutons to ths test problem shown n Fgures 11 to 16 we can see that the symmetry of the problem s not present n the computed soluton. As Rozvany [28] has shown, we do not necessarly expect the optmal soluton to these bnary programmng problems to be symmetrc. The asymmetry n the computed solutons shown n Fgures 11 to 16 arse from (10) and (11). Only a subset of elements wth precsely the same senstvty values may be chosen to be updated and so the symmetry may be lost. The frst thng to note about the results s the problem sze whch the fast bnary method has been able to solve n reasonable tme. A computaton on a two-dmensonal mesh of elements

17 BINARY TOPOLOGY OPTIMIZATION WITH BUCKLING CONSTRAINTS 17 Problem sze n = = = = = = = = = = = = = = = = Objectve Dervatve calculatons Analyses Tme (mns) to 3 s.f. Proporton on c Table I. Table of results for the centrally loaded column n less than 8 hours on a modest desktop s a speed whch would be useful to a structural engneer. Ths speed s attaned because the number of dervatve calculatons appears to not be dependent on the number of varables. However, f we look at the fnal column, we see that the vast majorty of the work s spent calculatng the dervatve of the stablty constrant. However, we note that t s possble to parallelse ths step. A massvely parallel mplementaton should acheve near optmal speedup as no nformaton transfer s requred for the calculaton of the dervatve wth respect to the ndvdual varables. A soluton to a problem wth 10 5 varables was found usng the algorthm descrbed wthn 3 days whch shows the O(n 2 ) behavour and that larger problems can be solved f the user s tme constrants allow.

18 18 P. A. BROWNE, C. BUDD, N. I. M. GOULD, H. A. KIM, J. A. SCOTT 10 3 Tme (Mnutes) n Fgure 15. 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 ). Fgure 16. Soluton found on mesh of elements. The bucklng constrant s set to c s = 0.1 and the complance constrant c max = 8. A volume of s attaned. Compare wth Fgure CONCLUSIONS The man computatonal cost assocated wth topology optmzaton problems nvolvng bucklng s the calculaton of the dervatves of the bucklng load. We have wrtten an analytc formula for ths but t remans stll the most expensve part of an algorthm. To reduce the computatonal cost we have developed an algorthm whch tres to mnmse the number of these computatons that have to be made. Ths method does not guarantee fndng the global mnmum, but nstead wll fnd a pont

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