The ESO method revisited
|
|
- Randolph Heath
- 6 years ago
- Views:
Transcription
1 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 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 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.
3 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 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 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 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
5 The ESO method revisited Atual values M = M = 5 M = 5 M = Atual values M = M = 3 M = 9,200, Atual values M = M = 50 M = 50 M = 500 M = 200 M = x x 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 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 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 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) 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).
7 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 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 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) 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 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, = The next point at V = 68, = (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, = 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).
9 The ESO method revisited 9 V = 99, = V = 40, = 7. (a) V = 98, = (b) V = 95, = () V = 85, = (d) V = 69, = (e) V = 68, = (f) V = 67, = (g) V = 65, = (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 = 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 = 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
10 0 Kazem Ghabraie V = 68, = 65.3 Fig. 9: The solution obtained by ESO for the problem depited in Fig. 4b for all To illustrate this a short antilever beam is onsidered whih is disretised into a finite element mesh of 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 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, = (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 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.
11 The ESO method revisited 40 first order sensitivities aurate sensitivities V = 298, = (a) First order sensitivities with < Fig. 3: A omparison between ESO results obtained using first and higher order sensitivity numbers. V V = 280, = (b) High order sensitivities with < 40. V = 4, = () 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.
12 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. Strutural Optimization (4):93 202, DOI 0.007/BF Edwards CS, Kim HA, Budd CJ (2007) An evaluative study on ESO and SIMP for optimising a antilever tie beam. Strutural and Multidisiplinary Optimization 34(5):403 44, DOI 0.007/s x Ghabraie K (2009) Exploring topology and shape optimisation tehniques in underground exavations. PhD thesis, Shool of Civil, Environmental and Chemial Engineering Siene, RMIT University, Melbourne, Australia Ghabraie K, Xie YM, Huang X (2008) Shape optimization of underground exavation using ESO method. In: Xie YM, Patnaikuni I (eds) Innovations in Strutural Engineering and Constrution: Proeedings of the 4th International Strutural Engineering and ConstrutiomConferene (ISEC-4), Sep. 2007, Melbourne, Australia, Taylor and Franis, London, pp Ghabraie K, Chan R, Huang X, Xie YM (200) Shape optimization of metalli yielding devies for passive mitigation of seismi energy. Engineering Strutures 32(8): , DOI 0.06/j.engstrut Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-diretional evolutionary strutural optimization method. Finite Elements in Analysis and Design 43(4): , DOI 0.06/j.finel Huang X, Xie YM (2008) A new look at ESO and BESO optimization methods. Strutural and Multidisiplinary Optimization 35():89 92, DOI 0.007/s Huang X, Xie YM (2009) Bi-diretional evolutionary topology optimization of ontinuum strutures with one or multiple materials. Computational Mehanis 43(3):393 40, DOI 0.007/s Huang X, Xie YM (200a) Evolutionary Topology Optimization of Continuum Strutures: Methods and Appliations. John Wiley & Sons, Chihester, England Huang X, Xie YM (200b) A further review of ESO type methods for topology optimization. Strutural and Multidisiplinary Optimization 4(5):67 683, DOI 0.007/s Nguyen T, Ghabraie K, Tran-Cong T (204) Applying bidiretional evolutionary strutural optimisation method for tunnel reinforement design onsidering nonlinear material behaviour. Computers and Geotehnis 55:57 66, DOI 0.06/j.ompgeo Querin OM, Steven GP, Xie YM (998) Evolutionary strutural optimisation (ESO) using a bidiretional algorithm. Engineering Computations 5(8):03 048, DOI 0.08/ Rozvany GIN (2009) A ritial review of established methods of strutural topology optimization. Strutural and Multidisiplinary Optimization 37(3):27 237, DOI 0.007/s Rozvany GIN, Querin OM (2002) Theoretial foundations of sequential element rejetions and admissions (SERA) methods and their omputational implementations in topology optimisation. In: Proeedings of the 9th AIAA/ISSMO Symposium on Multidis. Anal and Optim.(held in Atlanta, Georgia), AIAA, Reston, VA, DOI 0.254/ Rozvany GIN, Zhou M (99) Appliations of the COC algorithm in layout optimization. In: Eshenauer HA, Matthek C, Olhoff N (eds) Engineering optimization in design proesses. Proeedings of the International Conferene (Karlsruhe Nulear Researh Center, Germany September 34, 990), Springer, Berlin, pp Rozvany GIN, Zhou M, Birker T (992) Generalized shape optimization without homogenization. Strutural Optimization 4(3-4): , DOI 0.007/BF Rozvany GIN, Querin OM, Logo J (2004) Sequential element rejetions and admissions (SERA) method: appliations to multionstraint problems. In: Proeedings of the 0th AIAA/ISSMO Multidis. Anal. Optim. Conferene, Albany, NY, DOI 0.254/ Sigmund O (200) A 99 line topology optimization ode written in matlab. Strutural and Multidisiplinary Optimization 2(2):20 27, DOI 0.007/s Stolpe M, Bendsøe MP (20) Global optima for the Zhou Rozvany problem. Strutural and Multidisiplinary Optimization 43(2):5 64, DOI 0.007/s y Tanskanen P (2002) The evolutionary strutural optimization method: theoretial aspets. Computer Methods in Applied Mehanis and Engineering 9(47-48): , DOI 0.06/S (02) Xie YM, Steven GP (993) A simple evolutionary proedure for strutural optimization. Computers & Strutures 49(5): , DOI 0.06/ (93)90035-C Yang XY, Xie YM, Steven GP, Querin OM (999) Bidiretional evolutionary method for stiffness optimization. AIAA Journal 37(): Zhou M, Rozvany GIN (200) On the validity of ESO type methods in topology optimization. Strutural and Multidisiplinary Optimization 2():80 83, DOI 0.007/s
Millennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More informationThe Hanging Chain. John McCuan. January 19, 2006
The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a
More informationSimplified Buckling Analysis of Skeletal Structures
Simplified Bukling Analysis of Skeletal Strutures B.A. Izzuddin 1 ABSRAC A simplified approah is proposed for bukling analysis of skeletal strutures, whih employs a rotational spring analogy for the formulation
More informationComplexity of Regularization RBF Networks
Complexity of Regularization RBF Networks Mark A Kon Department of Mathematis and Statistis Boston University Boston, MA 02215 mkon@buedu Leszek Plaskota Institute of Applied Mathematis University of Warsaw
More informationFig Review of Granta-gravel
0 Conlusion 0. Sope We have introdued the new ritial state onept among older onepts of lassial soil mehanis, but it would be wrong to leave any impression at the end of this book that the new onept merely
More informationThe Effectiveness of the Linear Hull Effect
The Effetiveness of the Linear Hull Effet S. Murphy Tehnial Report RHUL MA 009 9 6 Otober 009 Department of Mathematis Royal Holloway, University of London Egham, Surrey TW0 0EX, England http://www.rhul.a.uk/mathematis/tehreports
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationEvaluation of effect of blade internal modes on sensitivity of Advanced LIGO
Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple
More informationControl Theory association of mathematics and engineering
Control Theory assoiation of mathematis and engineering Wojieh Mitkowski Krzysztof Oprzedkiewiz Department of Automatis AGH Univ. of Siene & Tehnology, Craow, Poland, Abstrat In this paper a methodology
More informationMaximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More informationA Spatiotemporal Approach to Passive Sound Source Localization
A Spatiotemporal Approah Passive Sound Soure Loalization Pasi Pertilä, Mikko Parviainen, Teemu Korhonen and Ari Visa Institute of Signal Proessing Tampere University of Tehnology, P.O.Box 553, FIN-330,
More informationA model for measurement of the states in a coupled-dot qubit
A model for measurement of the states in a oupled-dot qubit H B Sun and H M Wiseman Centre for Quantum Computer Tehnology Centre for Quantum Dynamis Griffith University Brisbane 4 QLD Australia E-mail:
More informationNonreversibility of Multiple Unicast Networks
Nonreversibility of Multiple Uniast Networks Randall Dougherty and Kenneth Zeger September 27, 2005 Abstrat We prove that for any finite direted ayli network, there exists a orresponding multiple uniast
More informationHankel Optimal Model Order Reduction 1
Massahusetts Institute of Tehnology Department of Eletrial Engineering and Computer Siene 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Hankel Optimal Model Order Redution 1 This leture overs both
More informationSingular Event Detection
Singular Event Detetion Rafael S. Garía Eletrial Engineering University of Puerto Rio at Mayagüez Rafael.Garia@ee.uprm.edu Faulty Mentor: S. Shankar Sastry Researh Supervisor: Jonathan Sprinkle Graduate
More informationDevelopment of a user element in ABAQUS for modelling of cohesive laws in composite structures
Downloaded from orbit.dtu.dk on: Jan 19, 2019 Development of a user element in ABAQUS for modelling of ohesive laws in omposite strutures Feih, Stefanie Publiation date: 2006 Doument Version Publisher's
More informationAssessing the Performance of a BCI: A Task-Oriented Approach
Assessing the Performane of a BCI: A Task-Oriented Approah B. Dal Seno, L. Mainardi 2, M. Matteui Department of Eletronis and Information, IIT-Unit, Politenio di Milano, Italy 2 Department of Bioengineering,
More informationFrequency Domain Analysis of Concrete Gravity Dam-Reservoir Systems by Wavenumber Approach
Frequeny Domain Analysis of Conrete Gravity Dam-Reservoir Systems by Wavenumber Approah V. Lotfi & A. Samii Department of Civil and Environmental Engineering, Amirkabir University of Tehnology, Tehran,
More informationIMPEDANCE EFFECTS OF LEFT TURNERS FROM THE MAJOR STREET AT A TWSC INTERSECTION
09-1289 Citation: Brilon, W. (2009): Impedane Effets of Left Turners from the Major Street at A TWSC Intersetion. Transportation Researh Reord Nr. 2130, pp. 2-8 IMPEDANCE EFFECTS OF LEFT TURNERS FROM THE
More informationMODELLING THE POSTPEAK STRESS DISPLACEMENT RELATIONSHIP OF CONCRETE IN UNIAXIAL COMPRESSION
VIII International Conferene on Frature Mehanis of Conrete and Conrete Strutures FraMCoS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang Eds) MODELLING THE POSTPEAK STRESS DISPLACEMENT RELATIONSHIP
More informationLightpath routing for maximum reliability in optical mesh networks
Vol. 7, No. 5 / May 2008 / JOURNAL OF OPTICAL NETWORKING 449 Lightpath routing for maximum reliability in optial mesh networks Shengli Yuan, 1, * Saket Varma, 2 and Jason P. Jue 2 1 Department of Computer
More informationLOAD-RATIO DEPENDENCE ON FATIGUE LIFE OF COMPOSITES
LOAD-RATIO DEPENDENCE ON FATIGUE LIFE OF COMPOSITES Joakim Shön 1 and Anders F. Blom 1, 1 Strutures Department, The Aeronautial Researh Institute of Sweden Box 1101, SE-161 11 Bromma, Sweden Department
More informationAdvances in Radio Science
Advanes in adio Siene 2003) 1: 99 104 Copernius GmbH 2003 Advanes in adio Siene A hybrid method ombining the FDTD and a time domain boundary-integral equation marhing-on-in-time algorithm A Beker and V
More informationINTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 4, 2012
INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume, No 4, 01 Copyright 010 All rights reserved Integrated Publishing servies Researh artile ISSN 0976 4399 Strutural Modelling of Stability
More informationA Functional Representation of Fuzzy Preferences
Theoretial Eonomis Letters, 017, 7, 13- http://wwwsirporg/journal/tel ISSN Online: 16-086 ISSN Print: 16-078 A Funtional Representation of Fuzzy Preferenes Susheng Wang Department of Eonomis, Hong Kong
More informationAn improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases
Noname manuscript No. (will be inserted by the editor) An improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases Kazem Ghabraie the date of receipt and acceptance
More informationPhysical Laws, Absolutes, Relative Absolutes and Relativistic Time Phenomena
Page 1 of 10 Physial Laws, Absolutes, Relative Absolutes and Relativisti Time Phenomena Antonio Ruggeri modexp@iafria.om Sine in the field of knowledge we deal with absolutes, there are absolute laws that
More informationMODE I FATIGUE DELAMINATION GROWTH ONSET IN FIBRE REINFORCED COMPOSITES: EXPERIMENTAL AND NUMERICAL ANALYSIS
21 st International Conferene on Composite Materials Xi an, 20-25 th August 2017 MODE I FATIUE DELAMINATION ROWTH ONSET IN FIBRE REINFORCED COMPOSITES: EXPERIMENTAL AND NUMERICAL ANALYSIS Man Zhu 1,3,
More informationA Time-Dependent Model For Predicting The Response Of A Horizontally Loaded Pile Embedded In A Layered Transversely Isotropic Saturated Soil
IOSR Journal of Mehanial and Civil Engineering (IOSR-JMCE) e-issn: 2278-1684,p-ISSN: 232-334X, Volume 16, Issue 2 Ser. I (Mar. - Apr. 219), PP 48-53 www.iosrjournals.org A Time-Dependent Model For Prediting
More informationWavetech, LLC. Ultrafast Pulses and GVD. John O Hara Created: Dec. 6, 2013
Ultrafast Pulses and GVD John O Hara Created: De. 6, 3 Introdution This doument overs the basi onepts of group veloity dispersion (GVD) and ultrafast pulse propagation in an optial fiber. Neessarily, it
More informationUTC. Engineering 329. Proportional Controller Design. Speed System. John Beverly. Green Team. John Beverly Keith Skiles John Barker.
UTC Engineering 329 Proportional Controller Design for Speed System By John Beverly Green Team John Beverly Keith Skiles John Barker 24 Mar 2006 Introdution This experiment is intended test the variable
More informationThe Laws of Acceleration
The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the
More information3 Tidal systems modelling: ASMITA model
3 Tidal systems modelling: ASMITA model 3.1 Introdution For many pratial appliations, simulation and predition of oastal behaviour (morphologial development of shorefae, beahes and dunes) at a ertain level
More informationBreakdown of the Special Theory of Relativity as Proven by Synchronization of Clocks
Breakdown of the Speial Theory of Relativity as Proven by Synhronization of Cloks Koshun Suto Koshun_suto19@mbr.nifty.om Abstrat In this paper, a hypothetial preferred frame of referene is presumed, and
More informationA Queueing Model for Call Blending in Call Centers
A Queueing Model for Call Blending in Call Centers Sandjai Bhulai and Ger Koole Vrije Universiteit Amsterdam Faulty of Sienes De Boelelaan 1081a 1081 HV Amsterdam The Netherlands E-mail: {sbhulai, koole}@s.vu.nl
More informationSensitivity Analysis in Markov Networks
Sensitivity Analysis in Markov Networks Hei Chan and Adnan Darwihe Computer Siene Department University of California, Los Angeles Los Angeles, CA 90095 {hei,darwihe}@s.ula.edu Abstrat This paper explores
More informationOptimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach
Amerian Journal of heoretial and Applied tatistis 6; 5(-): -8 Published online January 7, 6 (http://www.sienepublishinggroup.om/j/ajtas) doi:.648/j.ajtas.s.65.4 IN: 36-8999 (Print); IN: 36-96 (Online)
More informationA Characterization of Wavelet Convergence in Sobolev Spaces
A Charaterization of Wavelet Convergene in Sobolev Spaes Mark A. Kon 1 oston University Louise Arakelian Raphael Howard University Dediated to Prof. Robert Carroll on the oasion of his 70th birthday. Abstrat
More informationVolume 29, Issue 3. On the definition of nonessentiality. Udo Ebert University of Oldenburg
Volume 9, Issue 3 On the definition of nonessentiality Udo Ebert University of Oldenburg Abstrat Nonessentiality of a good is often used in welfare eonomis, ost-benefit analysis and applied work. Various
More informationA NETWORK SIMPLEX ALGORITHM FOR THE MINIMUM COST-BENEFIT NETWORK FLOW PROBLEM
NETWORK SIMPLEX LGORITHM FOR THE MINIMUM COST-BENEFIT NETWORK FLOW PROBLEM Cen Çalışan, Utah Valley University, 800 W. University Parway, Orem, UT 84058, 801-863-6487, en.alisan@uvu.edu BSTRCT The minimum
More informationA NEW FLEXIBLE BODY DYNAMIC FORMULATION FOR BEAM STRUCTURES UNDERGOING LARGE OVERALL MOTION IIE THREE-DIMENSIONAL CASE. W. J.
A NEW FLEXIBLE BODY DYNAMIC FORMULATION FOR BEAM STRUCTURES UNDERGOING LARGE OVERALL MOTION IIE THREE-DIMENSIONAL CASE W. J. Haering* Senior Projet Engineer General Motors Corporation Warren, Mihigan R.
More informationA simple expression for radial distribution functions of pure fluids and mixtures
A simple expression for radial distribution funtions of pure fluids and mixtures Enrio Matteoli a) Istituto di Chimia Quantistia ed Energetia Moleolare, CNR, Via Risorgimento, 35, 56126 Pisa, Italy G.
More informationDIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS
CHAPTER 4 DIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS 4.1 INTRODUCTION Around the world, environmental and ost onsiousness are foring utilities to install
More informationUPPER-TRUNCATED POWER LAW DISTRIBUTIONS
Fratals, Vol. 9, No. (00) 09 World Sientifi Publishing Company UPPER-TRUNCATED POWER LAW DISTRIBUTIONS STEPHEN M. BURROUGHS and SARAH F. TEBBENS College of Marine Siene, University of South Florida, St.
More informationNormative and descriptive approaches to multiattribute decision making
De. 009, Volume 8, No. (Serial No.78) China-USA Business Review, ISSN 57-54, USA Normative and desriptive approahes to multiattribute deision making Milan Terek (Department of Statistis, University of
More informationStress triaxiality to evaluate the effective distance in the volumetric approach in fracture mechanics
IOSR Journal of ehanial and Civil Engineering (IOSR-JCE) e-issn: 78-1684,p-ISSN: 30-334X, Volume 11, Issue 6 Ver. IV (Nov- De. 014), PP 1-6 Stress triaxiality to evaluate the effetive distane in the volumetri
More informationDevelopment of Fuzzy Extreme Value Theory. Populations
Applied Mathematial Sienes, Vol. 6, 0, no. 7, 58 5834 Development of Fuzzy Extreme Value Theory Control Charts Using α -uts for Sewed Populations Rungsarit Intaramo Department of Mathematis, Faulty of
More informationFailure Assessment Diagram Analysis of Creep Crack Initiation in 316H Stainless Steel
Failure Assessment Diagram Analysis of Creep Crak Initiation in 316H Stainless Steel C. M. Davies *, N. P. O Dowd, D. W. Dean, K. M. Nikbin, R. A. Ainsworth Department of Mehanial Engineering, Imperial
More informationCALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS
International Journal of Modern Physis A Vol. 24, No. 5 (2009) 974 986 World Sientifi Publishing Company CALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS PAVEL SNOPOK, MARTIN
More informationEinstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk
Einstein s Three Mistakes in Speial Relativity Revealed Copyright Joseph A. Rybzyk Abstrat When the evidene supported priniples of eletromagneti propagation are properly applied, the derived theory is
More informationSTUDY OF INTERFACIAL BEHAVIOR OF CNT/POLYMER COMPOSITE BY CFE METHOD
THE 19TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS STUDY OF INTERFACIAL BEHAVIOR OF CNT/POLYMER COMPOSITE BY CFE METHOD Q. S. Yang*, X. Liu, L. D. Su Department of Engineering Mehanis, Beijing University
More informationFIBER/MATRIX DEBONDING CRITERIONS IN SIC/TI COMPOSITE. NUMERICAL AND EXPERIMENTAL ANALYSIS
FIBER/MATRIX DEBONDING CRITERIONS IN SIC/TI COMPOSITE. NUMERICAL AND EXPERIMENTAL ANALYSIS A. Thionnet 1, J. Renard 1 1 Eole Nationale Supérieure des Mines de Paris - Centre des Matériaux P. M. Fourt BP
More informationRelative Maxima and Minima sections 4.3
Relative Maxima and Minima setions 4.3 Definition. By a ritial point of a funtion f we mean a point x 0 in the domain at whih either the derivative is zero or it does not exists. So, geometrially, one
More informationNUMERICAL SIMULATION OF ATOMIZATION WITH ADAPTIVE JET REFINEMENT
Paper ID ILASS8--7 ILASS 28 Sep. 8-, 28, Como Lake, Italy A44 NUMERICAL SIMULATION OF ATOMIZATION WITH ADAPTIVE JET REFINEMENT Anne Bagué, Daniel Fuster, Stéphane Popinet + & Stéphane Zaleski Université
More information7 Max-Flow Problems. Business Computing and Operations Research 608
7 Max-Flow Problems Business Computing and Operations Researh 68 7. Max-Flow Problems In what follows, we onsider a somewhat modified problem onstellation Instead of osts of transmission, vetor now indiates
More informationAn Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems
An Integrated Arhiteture of Adaptive Neural Network Control for Dynami Systems Robert L. Tokar 2 Brian D.MVey2 'Center for Nonlinear Studies, 2Applied Theoretial Physis Division Los Alamos National Laboratory,
More informationThe gravitational phenomena without the curved spacetime
The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,
More informationLecture 3 - Lorentz Transformations
Leture - Lorentz Transformations A Puzzle... Example A ruler is positioned perpendiular to a wall. A stik of length L flies by at speed v. It travels in front of the ruler, so that it obsures part of the
More informationLikelihood-confidence intervals for quantiles in Extreme Value Distributions
Likelihood-onfidene intervals for quantiles in Extreme Value Distributions A. Bolívar, E. Díaz-Franés, J. Ortega, and E. Vilhis. Centro de Investigaión en Matemátias; A.P. 42, Guanajuato, Gto. 36; Méxio
More informationModeling of Threading Dislocation Density Reduction in Heteroepitaxial Layers
A. E. Romanov et al.: Threading Disloation Density Redution in Layers (II) 33 phys. stat. sol. (b) 99, 33 (997) Subjet lassifiation: 6.72.C; 68.55.Ln; S5.; S5.2; S7.; S7.2 Modeling of Threading Disloation
More informationarxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Aug 2004
Computational omplexity and fundamental limitations to fermioni quantum Monte Carlo simulations arxiv:ond-mat/0408370v1 [ond-mat.stat-meh] 16 Aug 2004 Matthias Troyer, 1 Uwe-Jens Wiese 2 1 Theoretishe
More informationSufficient Conditions for a Flexible Manufacturing System to be Deadlocked
Paper 0, INT 0 Suffiient Conditions for a Flexile Manufaturing System to e Deadloked Paul E Deering, PhD Department of Engineering Tehnology and Management Ohio University deering@ohioedu Astrat In reent
More informationTunnel Reinforcement Optimization for Nonlinear Material
November 25-27, 2012, Gold Coast, Australia www.iccm-2012.org Tunnel Reinforcement Optimization for Nonlinear Material T. Nguyen* 1,2, K. Ghabraie 1,2, T. Tran-Cong 1,2 1 Computational Engineering and
More information10.2 The Occurrence of Critical Flow; Controls
10. The Ourrene of Critial Flow; Controls In addition to the type of problem in whih both q and E are initially presribed; there is a problem whih is of pratial interest: Given a value of q, what fators
More informationAverage Rate Speed Scaling
Average Rate Speed Saling Nikhil Bansal David P. Bunde Ho-Leung Chan Kirk Pruhs May 2, 2008 Abstrat Speed saling is a power management tehnique that involves dynamially hanging the speed of a proessor.
More informationEE 321 Project Spring 2018
EE 21 Projet Spring 2018 This ourse projet is intended to be an individual effort projet. The student is required to omplete the work individually, without help from anyone else. (The student may, however,
More informationThe Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge
The Conept of Mass as Interfering Photons, and the Originating Mehanism of Gravitation D.T. Froedge V04 Formerly Auburn University Phys-dtfroedge@glasgow-ky.om Abstrat For most purposes in physis the onept
More informationAnalysis of discretization in the direct simulation Monte Carlo
PHYSICS OF FLUIDS VOLUME 1, UMBER 1 OCTOBER Analysis of disretization in the diret simulation Monte Carlo iolas G. Hadjionstantinou a) Department of Mehanial Engineering, Massahusetts Institute of Tehnology,
More informationVariation Based Online Travel Time Prediction Using Clustered Neural Networks
Variation Based Online Travel Time Predition Using lustered Neural Networks Jie Yu, Gang-Len hang, H.W. Ho and Yue Liu Abstrat-This paper proposes a variation-based online travel time predition approah
More informationEvaluation of a Dual-Load Nondestructive Testing System To Better Discriminate Near-Surface Layer Moduli
52 TRANSPORTATION RESEARCH RECORD 1355 Evaluation of a Dual-Load Nondestrutive Testing System To Better Disriminate Near-Surfae Layer Moduli REYNALDO ROQUE, PEDRO ROMERO, AND BYRON E. RUTH Theoretial analyses
More informationDeveloping Excel Macros for Solving Heat Diffusion Problems
Session 50 Developing Exel Maros for Solving Heat Diffusion Problems N. N. Sarker and M. A. Ketkar Department of Engineering Tehnology Prairie View A&M University Prairie View, TX 77446 Abstrat This paper
More informationStructural Integrity of Composite Laminates with Embedded Microsensors
Strutural Integrity of Composite Laminates with Embedded Mirosensors Yi Huang, Sia Nemat-Nasser Department of Mehanial and Aerospae Engineering, Center of Exellene for Advaned Materials, University of
More informationMethods of evaluating tests
Methods of evaluating tests Let X,, 1 Xn be i.i.d. Bernoulli( p ). Then 5 j= 1 j ( 5, ) T = X Binomial p. We test 1 H : p vs. 1 1 H : p>. We saw that a LRT is 1 if t k* φ ( x ) =. otherwise (t is the observed
More informationAn Adaptive Optimization Approach to Active Cancellation of Repeated Transient Vibration Disturbances
An aptive Optimization Approah to Ative Canellation of Repeated Transient Vibration Disturbanes David L. Bowen RH Lyon Corp / Aenteh, 33 Moulton St., Cambridge, MA 138, U.S.A., owen@lyonorp.om J. Gregory
More informationOn Component Order Edge Reliability and the Existence of Uniformly Most Reliable Unicycles
Daniel Gross, Lakshmi Iswara, L. William Kazmierzak, Kristi Luttrell, John T. Saoman, Charles Suffel On Component Order Edge Reliability and the Existene of Uniformly Most Reliable Uniyles DANIEL GROSS
More informationEFFECTS OF COUPLE STRESSES ON PURE SQUEEZE EHL MOTION OF CIRCULAR CONTACTS
-Tehnial Note- EFFECTS OF COUPLE STRESSES ON PURE SQUEEZE EHL MOTION OF CIRCULAR CONTACTS H.-M. Chu * W.-L. Li ** Department of Mehanial Engineering Yung-Ta Institute of Tehnology & Commere Ping-Tung,
More informationTHEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE?
THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine beause they are fusing hydrogen into helium in their entral parts. In this problem we use onepts of
More informationADHESION MEASURES OF ELASTO-PLASTIC THIN FILM VIA BUCKLE-DRIVEN DELAMINATION
ADHESION MEASURES OF ELASTO-PLASTIC THIN FILM VIA BUCKLE-DRIVEN DELAMINATION Yu Shouwen and Li Qunyang Department of Engineering Mehanis, Tsinghua University, Beijing 184, China Yusw@mail.tsinghua.edu.n
More informationMath 151 Introduction to Eigenvectors
Math 151 Introdution to Eigenvetors The motivating example we used to desrie matrixes was landsape hange and vegetation suession. We hose the simple example of Bare Soil (B), eing replaed y Grasses (G)
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 4
Advaned Computational Fluid Dynamis AA5A Leture 4 Antony Jameson Winter Quarter,, Stanford, CA Abstrat Leture 4 overs analysis of the equations of gas dynamis Contents Analysis of the equations of gas
More informationReliability Guaranteed Energy-Aware Frame-Based Task Set Execution Strategy for Hard Real-Time Systems
Reliability Guaranteed Energy-Aware Frame-Based ask Set Exeution Strategy for Hard Real-ime Systems Zheng Li a, Li Wang a, Shuhui Li a, Shangping Ren a, Gang Quan b a Illinois Institute of ehnology, Chiago,
More informationChapter 8 Hypothesis Testing
Leture 5 for BST 63: Statistial Theory II Kui Zhang, Spring Chapter 8 Hypothesis Testing Setion 8 Introdution Definition 8 A hypothesis is a statement about a population parameter Definition 8 The two
More informationThe coefficients a and b are expressed in terms of three other parameters. b = exp
T73S04 Session 34: elaxation & Elasti Follow-Up Last Update: 5/4/2015 elates to Knowledge & Skills items 1.22, 1.28, 1.29, 1.30, 1.31 Evaluation of relaxation: integration of forward reep and limitations
More informationJAST 2015 M.U.C. Women s College, Burdwan ISSN a peer reviewed multidisciplinary research journal Vol.-01, Issue- 01
JAST 05 M.U.C. Women s College, Burdwan ISSN 395-353 -a peer reviewed multidisiplinary researh journal Vol.-0, Issue- 0 On Type II Fuzzy Parameterized Soft Sets Pinaki Majumdar Department of Mathematis,
More informationPacking Plane Spanning Trees into a Point Set
Paking Plane Spanning Trees into a Point Set Ahmad Biniaz Alfredo Garía Abstrat Let P be a set of n points in the plane in general position. We show that at least n/3 plane spanning trees an be paked into
More informationRemark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the function V ( x ) to be positive definite.
Leture Remark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the funtion V ( x ) to be positive definite. ost often, our interest will be to show that x( t) as t. For that we will need
More informationParticle-wave symmetry in Quantum Mechanics And Special Relativity Theory
Partile-wave symmetry in Quantum Mehanis And Speial Relativity Theory Author one: XiaoLin Li,Chongqing,China,hidebrain@hotmail.om Corresponding author: XiaoLin Li, Chongqing,China,hidebrain@hotmail.om
More informationRobust Flight Control Design for a Turn Coordination System with Parameter Uncertainties
Amerian Journal of Applied Sienes 4 (7): 496-501, 007 ISSN 1546-939 007 Siene Publiations Robust Flight ontrol Design for a urn oordination System with Parameter Unertainties 1 Ari Legowo and Hiroshi Okubo
More informationAharonov-Bohm effect. Dan Solomon.
Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that
More informationQCLAS Sensor for Purity Monitoring in Medical Gas Supply Lines
DOI.56/sensoren6/P3. QLAS Sensor for Purity Monitoring in Medial Gas Supply Lines Henrik Zimmermann, Mathias Wiese, Alessandro Ragnoni neoplas ontrol GmbH, Walther-Rathenau-Str. 49a, 7489 Greifswald, Germany
More informationGeneralized Dimensional Analysis
#HUTP-92/A036 7/92 Generalized Dimensional Analysis arxiv:hep-ph/9207278v1 31 Jul 1992 Howard Georgi Lyman Laboratory of Physis Harvard University Cambridge, MA 02138 Abstrat I desribe a version of so-alled
More informationRelativity in Classical Physics
Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of
More informationmax min z i i=1 x j k s.t. j=1 x j j:i T j
AM 221: Advaned Optimization Spring 2016 Prof. Yaron Singer Leture 22 April 18th 1 Overview In this leture, we will study the pipage rounding tehnique whih is a deterministi rounding proedure that an be
More informationThe experimental plan of displacement- and frequency-noise free laser interferometer
7th Edoardo Amaldi Conferene on Gravitational Waves (Amaldi7) Journal of Physis: Conferene Series 122 (2008) 012022 The experimental plan of displaement- and frequeny-noise free laser interferometer K
More informationChapter 2 Linear Elastic Fracture Mechanics
Chapter 2 Linear Elasti Frature Mehanis 2.1 Introdution Beginning with the fabriation of stone-age axes, instint and experiene about the strength of various materials (as well as appearane, ost, availability
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')
22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966),
More informationSINCE Zadeh s compositional rule of fuzzy inference
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006 709 Error Estimation of Perturbations Under CRI Guosheng Cheng Yuxi Fu Abstrat The analysis of stability robustness of fuzzy reasoning
More informationMetric of Universe The Causes of Red Shift.
Metri of Universe The Causes of Red Shift. ELKIN IGOR. ielkin@yande.ru Annotation Poinare and Einstein supposed that it is pratially impossible to determine one-way speed of light, that s why speed of
More informationChapter 13, Chemical Equilibrium
Chapter 13, Chemial Equilibrium You may have gotten the impression that when 2 reatants mix, the ensuing rxn goes to ompletion. In other words, reatants are onverted ompletely to produts. We will now learn
More informationEXPERIMENTAL STUDY ON BOTTOM BOUNDARY LAYER BENEATH SOLITARY WAVE
VOL. 11, NO. 8, APRIL 16 ISSN 1819-668 6-16 Asian Researh Publishing Network (ARPN). All rights reserved. EXPERIMENTAL STUDY ON BOTTOM BOUNDARY LAYER BENEATH SOLITARY WAVE Bambang Winarta 1, Nadiatul Adilah
More information