The ESO method revisited

Noname manusript No. (will be inserted by the editor) The ESO method revisited Kazem Ghabraie the date of reeipt and aeptane should be inserted later Abstrat This paper examines the evolutionary strutural optimisation (ESO) method and it shortomings. By proposing a problem statement for ESO followed by an aurate sensitivity analysis a framework is presented in whih ESO is mathematially justifiable. It is shown that when using a suffiiently aurate sensitivity analysis, ESO method is not prone to the problem proposed by Zhou and Rozvany (200). A omplementary disussion on previous ommuniations about ESO and strategies to overome the Zhou-Rozvany problem is also presented. Finally it is shown that even the proposed rigorous ESO approah an result in highly ineffiient loal optima. It is disussed that the reason behind this shortoming is ESO s inherent unidiretional approah. It is thus onluded that the ESO method should only be used on a very limited lass of optimisation problems where the problem s onstraints demand a unidiretional approah to final solutions. It is also disussed that the Bidiretional ESO (BESO) method is not prone to this shortoming and it is suggested that the two methods be onsidered as ompletely separate optimisation tehniques. Keywords evolutionary strutural optimization (ESO) sensitivity analysis ompliane design bidiretional evolutionary strutural optimization (BESO) sequential element rejetion and admission (SERA) hard-kill methods Introdution The evolutionary strutural optimization (ESO) method was initially introdued by Xie and Steven (993) based Shool of Civil Engineering and Surveying, University of Southern Queensland, Toowoomba QLD 4350, Australia E-mail: kazem.ghabraie@usq.edu.au on the simple idea of gradually removing ineffiient parts of strutures to reah optimised designs. A bidiretional version of ESO (BESO) was proposed later by Querin et al (998) and Yang et al (999) whih also permits adding new elements to the effiient parts of strutures. The term SERA (sequential element rejetion and admission) was later proposed by Rozvany and Querin (2002) for this method to distinguish it from Darwinian evolutionary-based methods. In ESO/BESO terminology, the measure of an element s effiieny is its sensitivity number. In earlier works on ESO/BESO, not enough attention was paid to mathematially stating the optimisation problem and the sensitivity numbers used to be defined rather intuitively. For example, to obtain a fully stressed struture, a stress invariant of elements suh as von Mises stress was onsidered as sensitivity number (Xie and Steven 993). In more reent works on BESO, some researhers have adopted a rigorous approah of learly stating the optimisation problem and defining sensitivity numbers based on sensitivity analysis of the objetive funtion. For example, one an refer to Huang and Xie (2007, 2009, 200a,b), Ghabraie (2009), Ghabraie et al (200), and Nguyen et al (204). Despite the apparent similarity between ESO and BESO, however, to the best knowledge of the author, no satisfying mathematial problem statement has been proposed For ESO. As explained by Rozvany (2009), the problem statements proposed earlier (e.g. by Tanskanen 2002) are not justifiable. Another serious ritiism on ESO/BESO methods was proposed by Zhou and Rozvany (200). Through a simple example, Zhou and Rozvany showed that ESO method an lead to non-optimal solutions. Suggestions for hanges in ESO algorithm to answer this ritiism

2 Kazem Ghabraie have been proposed by a number of authors. Rozvany (2009) reviewed some of the most important proposed disussions and solutions, inluding Tanskanen (2002), Rozvany and Querin (2002), Rozvany et al (2004), Edwards et al (2007), and Huang and Xie (2008). Rozvany disusses that some of these proposed solutions (e.g. Edwards et al 2007) are unjustifiable. For other suggestions, suh as mesh refinement and freezing ritial boundary elements (proposed by Huang and Xie 2008), he disusses that the solution is not always reliable. The problem an be fixed if a soft-kill BESO approah is employed. In soft-kill BESO, also known as the virtual material approah (e.g. in Rozvany and Querin 2002), the ineffiient elements are weakened by using a very soft material rather than being ompletely removed from the design domain (hard-kill). However, as Rozvany orretly summarises, none of the proposed solutions ompletely retifies this problem for the ESO method. He thus onludes that ESO is presently fully heuristi, omputationally rather ineffiient, methodologially laking rationality, oasionally unreliable, with highly haoti onvergene urves. This paper presents a framework in whih the ESO approah an be mathematially justifiable. We start by proposing a problem statement for ESO ( 2) followed by an aurate sensitivity analysis whih overomes the Zhou-Rozvany problem ( 2.2). A omplementary disussion on some other proposed strategies to overome the Zhou-Rozvany problem and relevant ommuniations is also presented ( 4). It is disussed that the proposed aurate sensitivity analysis an ensure that ESO always reahes a loal optimum. Finally we will show that even this rigorous ESO approah an result in highly ineffiient loal optima ( 5). The reasons behind this behaviour are disussed. It is onluded that the ESO method should only be used on a very limited lass of optimisation problems. It is also disussed that the BESO method is not prone to this problem. 2 A problem statement for ESO In order to investigate the theoretial bakground of the ESO method, it is neessary to know what sort of optimisation problems are atually solved when implementing this method. Consider a design domain whih is disretised using finite element method into N elements. To be able to modify the topology of the design domain, onsider the following linear material interpolation sheme, ( K e (x e ) = K (0) e + x e K () e K (0) e ), e =,..., N () where K e is the stiffness matrix of element e and x e is the design variable of this element whih an vary ontinuously between 0 and. K (0) e and K () e are the stiffness matries of the same element if it was made of two different materials, namely, material 0 and material respetively. For single material-void designs, K (0) e is a zero matrix of the same size as K () e. In this ase, () simplifies to K e = x e K () e. To simplify the matters hereafter we limit our study to ompliane-based ESO. Starting from the full design domain, in eah step ESO removes one or more elements whih are onsidered to be ineffiient. Throughout this proedure the funtion whih is onstantly redued is the volume of the struture. It is thus reasonable to think of volume as the objetive funtion for ESO. Based on this fat, we propose the following problem statement for the ompliane-based ESO method: min V = x,...,x N suh that N v e x e e= and x e {0, }, e =,..., N (2) where V is the total volume of the design domain, v e is the volume of element e, and is the mean ompliane of the struture defined as = f T u (3) with f and u representing the nodal fore and displaement vetors respetively. is a predefined upper limit for the mean ompliane. It should be noted that the only allowable values of design variables in problem (2) are the binary values of 0 and. Based on the fat that removal of any element will inrease the value of mean ompliane ( < 0), it is obvious that the problem stated in (2) does not always have a solution. Solutions an only exist if is not smaller than the ompliane of the full struture. Also, it is obvious that the optimal solution requires the ompliane to attain its maximum feasible value beause otherwise it is possible to obtain a better solution by removing more elements. 2. Sensitivity numbers in ESO The ESO proedure approahes the optimal point from one side. In order to find a solution for problem (2), in eah step ESO needs to remove the element (or elements) whih results in the smallest inrement to. The proedure should be stopped when the ondition beomes ative (i.e. = ) or when even the smallest inrement to aused by removal of the least effiient element violates the ondition.

The ESO method revisited 3 The sensitivity number defined for eah element in this problem should thus reflet the effet of removing that element on the mean ompliane of the system. Considering a single material-void design, the sensitivity number used by ESO for this problem is defined as α e = uk () e u (4) This sensitivity number is onsistent with the results of first order sensitivity analysis. Using Taylor s series, the hange in due to a hange in the value of x e an be expressed as = x e + 2 e 2! x 2 x e 2 e + + m m! x m x e m + (5) e In ESO, like nearly all other gradient-based topology optimisation methods, only the first term in the above series is onsidered and it is assumed that e = x e (6) Beause the elements an only be totally removed in ESO, we always have x e = 0 =. Hene, we have = = uk () e u = α e e The problem in the above sensitivity analysis is the assumption made in (6). Looking at (5), for x e, the higher powers of x e an be negleted and the hange in is approximately equivalent to the first term. In this ase, assumption (6) is valid. In the ESO method, however, x e = and the higher powers of x e an not be negleted. In fat, in some ases, when x e ± the ontribution of higher order terms beomes quite onsiderable. This effet is demonstrated in 3.. 2.2 Aurate sensitivity numbers As disussed above, from (5), it is lear that when x e ±, for obtaining an aurate estimate of e, one has to onsider the higher order terms. In order to alulate m x m e, we differentiate (3) several times. Noting that the fore vetor does not depend on design variables, we have m x m = f T m u (7) e x m e To alulate m u m, we need to use the equilibrium equation. In linear systems, this equation an be expressed as Ku = f (8) where K is the global stiffness matrix of the system whih an be obtained by assembling the elements stiffness matries in the following form: K = N K e (x e ) (9) e= Rearranging the equilibrium equation and differentiating it several times, noting that the fore vetor is not dependent on design variables, we obtain m u x m = m K e x m f (0) e On the other hand, using the definition KK = I and differentiating, we have K K = K K () The term K is alulable from (). Noting that a linear relationship is used in (), m K x m e = 0 for any m >. Now by further differentiation of () we an obtain the following general expression m K m = m! ( K K ) m K (2) Substituting from (2) into (0) and then into (7), we an write m ( ) m x m = m! f T K K K f (3) e whih an be written in terms of displaement as m x m = ( )m m! u T K ( K e K ) m u (4) Noting that x e =, the m-th term in the Taylor series (5), is δe m = u T K ( K K ) m u, m N (5) Noting the positive-definiteness of K and K, it is lear that for any m the above term is positive. This means that in ESO, using the first order sensitivity It should be noted that K is only invertible when the omponents orresponding to the restrained degrees of freedom are eliminated from it. Hereafter, by K we are atually referring to this redued matrix.

4 Kazem Ghabraie number always results in an underestimation of the atual hange in. Moreover, it is also lear that the variation of with respet to any single design variable is ompletely monotoni and thus the Taylor s series is onverging. In mathematial terms we have 2 δ m e > δ m+n e > 0, e {,..., N}, m, n N (6) The onvergene of the series in (5) will be also demonstrated numerially in 3.. Due to onvergene of the Taylor s series, it is lear that by using enough terms in the Taylor s series, one an approah the most aurate sensitivity numbers with any arbitrary tolerane. 3 The Zhou-Rozvany problem The Zhou-Rozvany problem is shown in Fig.. As noted by Zhou and Rozvany (200), when applied on this problem, both stress-based and ompliane-based ESO approahes will eliminate the top-most element resulting in a non-optimal solution (Fig. 2). Beause ESO annot reintrodue the removed elements, obviously the problem annot be fixed by further iterations. Fig. : The Zhou-Rozvany problem. Fig. 2: The solution obtained by ESO for the Zhou- Rozvany problem after removing element. Zhou and Rozvany (200) orretly mentioned that in ESO, suh a failure an our if the sensitivity for the rejeted element inreases signifiantly as its normalized density (t i ) [(here x i )] varies from to zero. It is also shown in the previous setion that in ESO the first order sensitivity analysis may not be suffiiently aurate. In the next setion, we study the effets of onsidering higher order terms in sensitivity analysis of some of the elements in the Zhou-Rozvany problem. 2 In general we have δ m δ m+n, but the equal sign is only appliable when the whole struture is removed resulting in. 2 3. Sensitivity numbers of some of the elements in the Zhou-Rozvany problem To observe the effets of higher order terms in (5), in this setion we study the variation of the mean ompliane of the Zhou-Rozvany system due to gradual hanges in design variable of some elements. Three elements are onsidered individually and their design variable is gradually redued from. For eah value of the design variable the atual value of the mean ompliane is alulated using finite element analysis. For eah value, we also estimated the value of using the Taylor s series (5) with different number of terms. Using M terms, for example, we have the following approximation = 0 + M δe m m= where 0 is the mean ompliane of the full design where x e =, e {,..., N}. The results are shown in Fig. 3. The 00th element is the top most element whih is removed by ESO in its first iteration. It an be seen in Fig. 3 that for this element the value of shows a signifiantly steep inrease when the design variable approahes zero. Only after onsidering 3000 terms in the Taylor s series, the relative error between the predited and atual values of x00=0.00 beomes negligible. For x 00 < 0.00, even more terms need to be onsidered. The values of the first 0 terms of the Taylor s series (5) for the three onsidered elements are reported in Table. Considering the first term only, the 00th element shows the lowest value (and hene is onsidered the least effiient by ESO if only first order sensitivity numbers are onsidered). From this table, and also from Fig. 3, it is obvious that even seond-order sensitivities (as suggested by Rozvany 2009) would not retify ESO s problem. It is only after onsidering 7 terms, that it is revealed that the 95th element is in fat the orret hoie for removal. Despite the fat that several thousand terms are needed to orretly alulate 00 from (5), it should be noted that no further alulations is required one the Taylor s series for at least one of the elements is onverged. For example, onsidering a relative error tolerane of % for Taylor s series onvergene, by alulating 0 terms of the series one already ensures that element 95 is the suitable andidate for removal. Solutions obtained by using ESO with an aurate sensitivity numbers on the Zhou-Rozvany problem are reported in 5.. Before presenting those results, however, in the next setion some omments are made on

The ESO method revisited 5 420 45 40 405 Atual values M = M = 5 M = 5 M = 35 395 394 393 392 Atual values M = M = 3 M = 9,200,000 800 Atual values M = M = 50 M = 50 M = 500 M = 200 M = 3000 400 395 39 390 600 390 389 400 0 0.2 0.4 0.6 0.8 x 94 0 0.2 0.4 0.6 0.8 x 95 0 3 0 2 0 0 0 x 00 Fig. 3: Variation of the atual and predited values of using different number of terms (M) in (5) with respet to hanges in x e for e {94, 95, 00} in the Zhou-Rozvany problem. The onsidered elements are highlighted at the top of the legends. For e = 00, due to extremely steep hanges, the variations are only plotted for x 00 [0.00, ] and with logarithmi sale. Table : The values of first 0 terms of the series (5) for elements 94, 95, and 00. e = 94 e = 95 e = 00 M δ M 94 M m= δm 94 δm 95 M m= δm 95 δm 00 M m= δm 00 4.65 4.65 2.894 2.894 0.996 0.996 2 3.329 7.494.560 4.453 0.995.992 3 2.85 0.309 0.864 5.37 0.994 2.985 4 2.437 2.746 0.485 5.803 0.993 3.978 5 2.3 4.877 0.275 6.078 0.99 4.969 6.873 6.750 0.56 6.234 0.990 5.960 7.649 8.399 0.089 6.323 0.989 6.948 8.454 9.853 0.050 6.374 0.988 7.936 9.283 2.37 0.029 6.403 0.986 8.923 0.33 22.270 0.07 6.420 0.985 9.908 some of the other treatments suggested for this problem. 4 Complementary omments on treatments suggested for ESO failure in the Zhou-Rozvany problem overed many aspets of the suggested treatments and disussions on ESO, this author identifies some important points whih need to be addressed about these ommuniations. 4. ESO breakdown in statially determinate problems Rozvany (2009) reviewed some of the most important suggested treatments for ESO failure in the Zhou-Rozvany problem. Apart form using soft-kill BESO, whih is atually solving the problem using a different optimisation method, the other key suggested solutions involve using aurate sensitivity numbers (as was followed herein) and mesh refinement. Although Rozvany The treatments suggested by Huang and Xie (2008) are based on monitoring the boundary onditions to detet breaking of supports. In their paper, Huang and Xie (2008) stated that failure of ESO may our when a presribed boundary support is broken for a statially indeterminate struture. When a boundary support is broken, the strutural system ould be om-

6 Kazem Ghabraie pletely hanged from the one originally defined in the initial design and even BESO would not be able to retify the nonoptimal design. To avoid this problem, it is imperative that the presribed boundary onditions for the struture be heked and maintained at eah iteration during the optimization proess. Presumably, based on the nature of the Zhou-Rozvany problem, it seems to be believed that the ESO breakdown an only happen in similar statially indeterminate problems. Here we show that ESO failure an also happen in statially determinate problems. The finite element model of the Zhou-Rozvany problem an be onsidered to be a half domain of a symmetri statially determinate problem shown in Fig. 4a. Obviously the same problem will our if ESO is tried on this problem. Figs. 4b, illustrate other statially determinate problems in whih ESO will fail. 2 4.2 Deteting ESO failure by omparing predited hange to atual hange Restating the onlusions from Rozvany and Querin (2002), Rozvany (2009) argues that ESO would give a orret iteration-wise optimal element hange, if for all rejeted elements of that iteration the relevant sensitivities did not hange signifiantly as their thikness varies from unity to zero. Similar point was also mentioned by Zhou and Rozvany (200). At least for omplianebased ESO, the orretness of this statement is apparent from (5) and (6). If, after removing an element, the atual hange in the mean ompliane is not signifiantly different from the first order sensitivity number used by ESO, it means that the higher order terms in (5) were in fat negligible. Based on this, Rozvany suggests that this differene is heked in eah iteration by omparing the sensitivity value with the atual hange aused by a unit hange in the density of the rejeted elements. If the differene is large, the orresponding elements ould be stopped from being eliminated. Two points need to be mentioned here about this proposal: (a) (b) 3 2 2 2. Although this approah is overall reasonable, it is arguable that how an one ensure that the differene between the sensitivities and atual hange in the objetive funtion is signifiant enough to onsider the elimination as erroneous? Is there any guarantee that a ertain threshold value whih works for one speifi problem an work well on other problems as well? 2. Moreover one an think of some problems in whih this approah fails to work (or prevents ESO to proeed). Consider, for example, the problem depited in Fig. 5b. In this problem elimination of any element will result in a signifiant variation between the predited and atual values of the objetive funtion. Nevertheless, one an obtain a design with smaller volume if a suffiiently large upper limit for ompliane is adopted. () Fig. 4: Some statially determinate problems in whih ESO fails. (a) S 0 (b) S Clearly the tehnique suggested by Huang and Xie (2008) annot detet the failure of ESO in these statially determinate problems. Fig. 5: Examples of problems in whih the first order sensitivity of ompliane in all the elements are signifiantly smaller than the atual hange aused by their removal (S problems).

The ESO method revisited 7 We will ome aross this lass of problems again in 5.. In these problems, removing any element will hange the onnetivity of the system. We denote this lass of problems by S. The S problems an be divided further into two sublasses. In some S problems, removal of any element results in instability of the system. In other words the mean ompliane of the system approahes infinity by removing any element from it. We denote this sublass by S 0. The other sublass, denoted hereafter by S, inludes all S designs whih do not belong to S 0. Fig. 5 illustrates examples of both sublasses. It is lear that one ESO reahes a S 0, no further solutions an be obtained. However, if a S design is reahed, depending on the ondition of the problem, it is possible to obtain further solutions by removing elements until reahing a S 0. The problem with the approah suggested by Rozvany and Querin (2002) and Rozvany (2009) is that if the initial problem is a S design, no solution an be found using this approah. 4.3 Non-optimal or loal optimal solution? In an interesting observation, Huang and Xie (200b) noted that the solution obtained by ESO for the Zhou- Rozvany problem after eliminating four elements (Fig. 6) is a highly ineffiient loal optimum rather than a non-optimal solution. Fig. 6: The solution obtained by ESO for the Zhou- Rozvany problem after removing 4 elements. To prove their statement, Huang and Xie (200b) used the Solid Isotropi Mirostruture with Penalisation (SIMP) method (Bendsøe 989; Rozvany and Zhou 99; Rozvany et al 992) to solve the Zhou-Rozvany problem starting from the initial design shown in Fig. 6 with x i = for all elements in the horizontal beam and x i = x min = 0.00 for the four elements in the vertial tie. They report that with a penalty fator of p 3. the SIMP method annot improve this initial design any further and then onluded that beause the SIMP method with ontinuous design variables guarantees that its solution should be at least a loal optimum this design is a loal minimum. There are a number of points whih need to be mentioned about this approah and onlusion:. Although the design shown in Fig. 6 is in fat a loal minimum (as will be demonstrated soon), the approah used by the authors to prove this is arguable. The results obtained by the SIMP method depend on its algorithmi parameters; most importantly the penalty fator (p), the minimum allowable value of design variables (x min ), the move limit (m), and the regulating power (η) (here we adopted the notation used by Sigmund 200). By playing with these parameters, one an fore the SIMP method to lok on many learly non-optimal initial designs. 2. More importantly, the problems solved by SIMP and ESO are different from eah other. Even if a partiular design is a loal minimum in SIMP, one annot readily onlude that it is a loal minimum for ESO. 3. As will be soon demonstrated, some of the solutions obtained by the ESO method annot be onsidered as loal minimum. Thus the onlusion made by Huang and Xie (200b) annot be generalised. In the following we elaborate more on points 2 and 3 above. 4.3. Loal minima for problems with ontinuous variables A ompliane minimisation problem with ontinuous variables an be expressed as follows. min x,...,x N suh that V V and x e [0, ], e =,..., N (7) where V is a predefined upper limit on the volume of the struture. The typial problem solved by the SIMP method is a penalised version of the above form with a typial power-law interpolation sheme in the form E e (x e ) = x p ee, p > where E e is the Young s modulus of element e, and E is the Young s modulus of a base material. A feasible neighbourhood with radius ɛ > 0 about a feasible point like x for problem (7) takes the following simple form. N ɛ (x) = { x + x [0, ] N x < ɛ, } N x e = 0 e= (8) This set defines an open hyper-disk 3 formed by intersetion of a hyper-ball with radius ɛ and a hyper-plane 3 A hyper-ball in (N )-dimensional spae

8 Kazem Ghabraie in an N-dimensional spae. A point x (with ( x) = and V ( x) = V ) is a loal minimum of problem (7) if ɛ > 0 : ( x N ɛ ( x) : (x) ) (9) 4.3.2 Loal minima for ESO The big differene between problems (7) and (2) is in the last ondition whih hanges from a ontinuous boxing ondition in SIMP to a binary ondition in ESO. The feasible domain of ESO problem is not ontinuous. Thus the onept of neighbourhood needs to be onsidered arefully. Consider the following problem whih is the binary form of problem (7). min x,...,x N suh that V V and x e {0, }, e =,..., N (20) A feasible neighbourhood about a feasible point x for this problem an be defined based on (8) as M ɛ (x) = N ɛ (x) {0, } N (2) whih only ontains a finite number of points of the aforementioned hyper-disk. Beause of its disrete nature, trivially the smallest neighbourhood around eah point ontains only that point (0 < ɛ < 2). Obviously we annot aept this neighbourhood when assessing whether a point is a loal minimum or not. Negleting this trivial ase, the next smallest neighbourhood around eah point is obtained when 2 < ɛ < 2, i.e. when only two omponents of x are non-zero 4. For simpliity we show this neighbourhood by M(x). Noting that this is the smallest non-trivial feasible neighbourhood, we an all a point x a loal minimum for problem 20 if x M( x) : (x) (22) It an be easily shown that any loal minimum point x of problem 20 is also a loal minimum point of problem 2. Otherwise, problem 2 has a different loal minimum x in the smallest non-trivial feasible neighbourhood with V < V and. Now if we add an element to x to inrease its volume (up to V ), its ompliane will derease and we obtain a solution for problem 20 with V < V V and < in M( x) whih ontradits our assumption that x is a loal minimum of problem 20. 4 Due to the binary nature of design variables, the only feasible non-zero values in x are ±. Keeping the volume onstant requires the sum of the omponents in x to vanish. Thus the smallest feasible positive value of x is 2. Now we an argue that a point x is a loal minimum for ESO if (22) holds. In simple words, an ESO solution is a loal minimum if it yields the minimum value of among all designs obtained by swithing one solid element to void and one void element to solid. It is now lear that the solution depited in Fig. 6 is in fat a loal minimum for ESO. Beause removing any of the beam elements, inreases the mean ompliane to the extent that turning none of the tie elements into solid an suffiiently derease it down to its initial value. It should be noted however, that not all the solutions obtained by ESO are loal minimum. For example, it is lear that the solution obtained after removal of the first element from the Zhou-Rozvany problem (Fig. 2) is not a loal minimum and is in fat a non-optimal solution. Based on this disussion, we an also readily onlude that the solutions obtained by ESO through using aurate sensitivity analysis are always loally minimum. Although, as we will demonstrate in the next setion, they may be highly ineffiient. 5 Another shortoming of ESO 5. Solution to the Zhou-Rozvany problem using aurate sensitivity numbers The results obtained for the Zhou-Rozvany problem after applying the ESO method with high-order sensitivity analysis are illustrated in Fig. 7. The result obtained after removing element expetedly mathes the global optimum for V = 99 (or for = 395.3) as reported by Stolpe and Bendsøe (20). After that ESO results slightly deviate from the global optima. Again, after removing 3 elements (Fig. 7e) the ESO method reahes a global optimum at V = 69, = 579.93. The next point at V = 68, = 608.85 (Fig. 7f) also mathes a global optimum. This solution is learly a S design. By further removal of elements, ESO jumps to a S 0 design at V = 65, = 4747.5. Beyond this point, any further element removal will result in an unstable system. The designs obtained after the S design (Figs. 7g h) are highly ineffiient. In fat the intuitively suggested design by Zhou and Rozvany (200) depited in Fig. 8 provides a muh better result of V = 40 at = 7 5. Noting that aurate sensitivity numbers have been used to obtain these results, it is obvious that this time 5 The differene between this number and the value of = 2 reported for this design by Zhou and Rozvany (200) is due to using analytially integrated stiffness matries for finite elements in this paper. Also see Stolpe and Bendsøe (20).

The ESO method revisited 9 V = 99, = 395.3 V = 40, = 7. (a) V = 98, = 402.24 (b) V = 95, = 420.69 () V = 85, = 48.20 (d) V = 69, = 579.93 (e) V = 68, = 608.85 (f) V = 67, = 4747.3 (g) V = 65, = 4747.5 (h) Fig. 7: Solutions to the Zhou-Rozvany problem using aurate sensitivity numbers with different ompliane limits. Fig. 8: An intuitively suggested solution by Zhou and Rozvany (200). ESO s problem is not due to using inorret or inaurate sensitivity numbers. 5.2 The reason behind this shortoming of ESO As seen in the previous setion, ESO method an lead to highly ineffiient (loally optimum) solutions even if aurate sensitivity numbers are used. Perhaps this shortoming of ESO is even more serious than the problem aused by using inaurate sensitivity numbers. This shortoming is due to the fat that ESO is restrited to move in only one diretion. In fat, ESO modifies the problem as it proeeds. So, for example, the solution obtained by removing 32 elements (Fig. 7f) is atually a solution to the problem with the initial design depited in Fig. 7e, and likewise Fig. 7g is a solution to Fig. 7f. Using aurate sensitivity analysis, one an ensure that the solution obtained in iteration i+ is the optimal solution to the problem with the initial design equivalent to the solution obtained in iteration i. But this is not enough to ensure that the solution obtained is an optimal solution to the initial problem. Although Fig. 7g is the optimal solution to a problem with the design domain depited in Fig. 7f, it is not an effiient solution to the Zhou-Rozvany problem. Even for statially determinate problems, ESO s solution may be far from optimum. Consider for example, the statially determinate problem shown in Fig. 4b. After removing 32 elements, for this problem ESO reahes the design illustrated in Fig. 9. As this is a S 0 design, no further elements an be removed by ESO. This is a good solution for = 66 (yielding the volume of 68 elements) but it is a highly ineffiient solution, for example, for = 2000. For this value of, an intuitive design similar to the one depited in Fig. 8 (without the top rollers) would yield a onsiderably lower volume of V = 40. 5.3 Unreliable behaviours of ESO Noting that ESO hanges the problem as it proeeds, it an be onluded that the solutions obtained by ESO are only (loally) optimal in one branh of possible solutions. It an then be expeted that the overall behaviour

0 Kazem Ghabraie V = 68, = 65.3 Fig. 9: The solution obtained by ESO for the problem depited in Fig. 4b for all 65.3. To illustrate this a short antilever beam is onsidered whih is disretised into a finite element mesh of 30 20 square 9-node elements (Fig. ). Some of the solutions obtained by ESO using first order sensitivity numbers and aurate sensitivity numbers are shown in Fig. 2. of ESO is extremely problem-dependent. In the following we mention two interesting observations. 5.3. A more restrited version of the problem may lead to better results It is possible to obtain a better solution for the same problem if we fore ESO to stik to another branh of solutions. For example, if we start with the initial design depited in Fig. 0a (whih is a subset of the Zhou- Rozvany problem), we an obtain the result shown in Fig. 0b with V = 54, = 93.8 whih is obviously better than Fig. 7g and Fig. 7h. This simple example shows that allowing more elements in the initial design domain does not neessarily mean a better solution an be obtained by ESO. V = 76, = 693.90 (a) V = 54, = 93.8 (b) Fig. 0: A modified version of the Zhou-Rozvany problem: a) initial design, and b) a solution found by ESO using aurate sensitivity numbers. Fig. : A short antilever beam problem. When the ondition < = 40 is imposed, the results obtained using aurate sensitivity numbers are better than the ones obtained using first order sensitivity numbers. But the first order sensitivity numbers yield better results when the ondition is hanged to < = 40. The graph in Fig. 3 shows the relationship between volume (V ) and ompliane () for the solutions obtained using the two sets of sensitivity numbers. It an be seen that for almost all values of > 43 the first order sensitivity numbers lead to better results. As explained before, suh a behaviour an be expeted beause ESO hanges the problem as it proeeds. In the earlier stages, the problem is not modified very muh so the higher order sensitivity numbers work better. But there is no guarantee that staying in the branh followed by the aurate sensitivity numbers always lead to better results. Based on these unreliable behaviours, even when aurate sensitivity numbers are employed, perhaps one an argue that ESO method should be generally avoided. 5.3.2 Using aurate sensitivity analysis might lead to worse results Due to the above observation, it is expeted that at some point, an inaurate first-order sensitivity analysis leads to a better solution ompared to more aurate higher-order sensitivity analyses. In other words there is no guarantee that a more aurate approah yields a better result. 6 What about BESO? It should be noted here that BESO is essentially different from its predeessor. By allowing elements to be added as well, the initial problem is not modified at least for soft-kill BESO. 6 It is thus possible for BESO 6 In ontrast to what seems to be generally believed, it an be shown that the same an be true even for hard-kill BESO. This is however beyond the sope of this work and the author wishes to address this matter in a separate ommuniation.

The ESO method revisited 40 first order sensitivities aurate sensitivities 20 00 80 V = 298, = 36.65 (a) First order sensitivities with < 40. 60 40 20 0 00 200 300 400 500 600 Fig. 3: A omparison between ESO results obtained using first and higher order sensitivity numbers. V V = 280, = 39.79 (b) High order sensitivities with < 40. V = 4, = 39.52 () First order sensitivities with < 40. to move aross different branhes of solutions and the method is not prone to this shortoming of ESO. The other important differene between the two methods is the range of problems that an be solved with them. Due to its nature, ESO an only minimise the volume (weight) of a struture but BESO (like SIMP) an be formulated to minimise a wide range of objetive funtions. Based on this disussion, this author suggests that ESO and BESO methods are treated as ompletely separate and distint methods despite their historial relationship. 6. Is there any problem for whih using ESO is preferred? V = 32, = 34.6 (d) High order sensitivities with < 40. Fig. 2: ESO solutions obtained for the short antilever beam problem for different values of with first and higher order sensitivity numbers. Generally for all problems using a bidiretional method suh as BESO (or SIMP) is preferable. Nevertheless, there are some speifi types of problems where the limitations of the problem justify a unidiretional approah. A good example is the problem of finding the next piee of ground to be removed in an exavation projet. In this ase, one one part of the domain is removed it annot be physially reintrodued so the unidiretional approah of ESO fits well to this problem (Ghabraie et al 2008). For the urrent disussion, it is enough to aept that soft-kill BESO will not suffer from this shortoming of ESO.

2 Kazem Ghabraie 7 Conlusion In this paper the ESO method and its shortomings are studied. A problem statement is proposed for ESO. An aurate sensitivity analysis is also proposed and aurate sensitivity numbers are alulated for omplianebased ESO. It is shown that the proposed approah an solve the Zhou-Rozvany problem. It is then demonstrated that due to its unidiretional approah to optimal points, even when using aurate sensitivity numbers, the ESO method an lead to highly ineffiient solutions. Based on the observations and disussions, it is onluded that ESO should only be used in problems where the limitations justify a unidiretional approah to the solution. A distintion should be made between BESO and ESO, and these two methods should be onsidered separately. Referenes Bendsøe MP (989) Optimal shape design as a material distribution problem. 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