efficient optimization algorithms September 18, 2022 postadmin Post in Uncategorized Layout optimization of large-scale pin-jointed frames Matthew Gilbert and Andrew Tyas Department of Civil and Structural Engineering, University of Sheffield, Sheffield, UK Keywords Structural analysis, Linear programming, Optimum design Abstract Computerized layout (or “topology”) optimization was pioneered almost four decades back. However, despite dramatic increases in available computer power and the application of increasingly efficient optimization algorithms, even now only relatively modest sized problems can be tackled using the traditional “ground structure” approach. This is because of the need, in general, for the latter to contain every conceivable member connecting together the nodes in a problem. A simple, but effective solution method capable of tackling problems with large numbers of potential members (e.g. .100,000,000) is presented. Though the method draws on the linear programming technique of “column generation”, since layout optimization specific heuristics are employed it is presented as an iterative “member adding” method. The method requires a ground structure with minimal connectivity to be used in the first iteration; members are then added as required in subsequent iterations until the (provably) optimal solution is found. Introduction Computer-based discrete layout optimization methods for frameworks were first developed almost four decades back by Dorn et al. (1964), who proposed a formulation based on plastic design principles. They found that linear programming (LP) algorithms, which had then only recently been developed, could be used to select the subset of members that should be present in the optimal structure. For plastic layout optimization, the original method was soon extended to allow multiple load cases and self-weight to be included. A comprehensive account of this work has been presented by Hemp (1973). Subsequently, in order to avoid the optimization process leading to impractical Michell type structures (which contain large numbers of joints and very short members), Parkes (1975) proposed a means of including joint costs in the formulation. In fact this method penalises short members rather than the number of joints per se. It is important to note that, for plastic design, inclusion of these additional features does not change the linear nature of the problem. In contrast, for elastic layout optimization with limiting stress criteria it is well known that computational difficulties arise when multiple load cases are involved. This is largely due to the likely existence of singular topologies (Rozvany, 2001; Rozvany and Birker, 1994). In such cases both the mathematical programming and the so-called “optimality criteria” methods (Zhou and Rozvany, 1991) will often fail to identify the true optimal frame The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/researchregister http://www.emeraldinsight.com/0264-4401.htm EC 20,8 1044 Received November 2002 Revised July 2003 Accepted July 2003 Engineering Computations Vol. 20 No. 8, 2003 pp. 1044-1064 q MCB UP Limited 0264-4401 DOI 10.1108/02644400310503017 layout. However, it is also known that for elastic design with multiple load cases, by using the layout from a plastic optimization as the basis for an optimal elastic frame only a relatively small error (e.g. a few per cent) will often ensue (Kirsch, 1990). This might mean that a practical optimization tool might well be based on the plastic method even if limiting (elastic) stress design was the ultimate goal of the optimization process. In fact one of the main problems with both types of layout optimization, elastic and plastic, lies in the fact that a large network of potential members interconnecting specified discrete points, or nodes, in the design domain is required. Such a network of potential members is referred to as a “ground structure” (or “structural universe”), and should, if possible, contain many members since the optimal structure can only be formed from members present initially. Assuming that each node is connected to every other node, for a general 2D or 3D design domain, it can be shown that a total of ðn 2 1Þ þ ðn 2 2Þ þ · · · þ 2 þ 1 [i.e. nðn 2 1Þ=2] members must initially be present, where n is the number of nodes in the problem. This is an upper limit on the number of members present as some workers choose to omit the overlapping members from the ground structure. It should be noted that the latter simplification will, in general, prevent optimal solutions from being obtained, should features such as structural self-weight or joint costs be included in the problem definition (this will invariably be the case for real-world problems). Figure 1 shows a sample design domain and loading and support conditions, together with minimal and full connectivity ground structures. Though using a fully connected ground structure is desirable, unfortunately the number of members present in the problem clearly becomes unmanageably large (because nðn 2 1Þ=2 members are required). Thus, for example, an apparently modest 40 £ 40 node, single load case problem with a fully Figure 1. (a) Design domain (2 units £ 1 unit), loading and support conditions; (b) minimal connectivity ground structure containing 11 bars; and (c) full connectivity ground structure containing 15 bars (includes overlapping bars) Layout optimization 1045 connected ground structure generates several thousand LP constraints and well over 2 million LP problem variables (using the lower bound plastic analysis problem formulation for each physical member a total of 2m LP problem variables are generally required). Such a problem generally requires large amounts of CPU time and memory to solve. Additionally, the fact that the number of members (and LP variables) increases as a function of the square of the number of nodes means that practically large 3D problems are unlikely to be solvable in the foreseeable future using the existing methods. Though it would seem sensible to start with a smaller structure and to add members to this as and when required, the most recent comprehensive review of the available literature (Rozvany et al., 1995) indicates that “no simple methods are available at present for finding the optimal position of additional members”. More recently, Kirsch (1996) has considered expansion processes, but concluded that the “development of a systematic method for adding members. . .is still a challenge”. For plastic optimization problems the method described in this paper appears to overcome this difficulty. Additionally, when only a single load case is involved, the procedure will also provide the optimal structure for the elastic design with stress constraints problem. The proposed member adding scheme described in this paper requires that an initial, minimal connectivity, ground structure is present. The number of members in the ground structure is increased at each step in an iterative process to permit the selection of increasingly efficient structural layouts. Thus, conceptually it may be considered that the method makes use of an “adaptive ground structure” which grows according to the particular design constraints present. Alternatively, since it will be found that the initial ground structure does not influence the optimal structure obtained, as far as the user is concerned the whole notion of a ground structure may be disregarded since it is actually only the extent of the design domain, the specified nodal density and the loading/support conditions which will have any influence on the optimal design. It is shown that by using the method very significant reductions in CPU time and memory requirements can be realised. Although only 2D structures are considered in this paper, the technique is applicable to 3D structures and treatment of quite large, practical, 3D problems should be possible using the method. Classical plastic layout optimization The whole field of plastic layout optimization appears to have been largely ignored for many years. This is despite the fact that many structures are currently designed to meet the requirements of limit state design codes (i.e. codes of practice based on plastic design principles). Additionally, new and increasingly effective LP solution algorithms are becoming available (e.g. the EC 20,8 1046 so-called “interior-point” or “barrier” methods, based on the work by Karmarker (1984)). For these reasons, and with a view to enhance the existing procedures, it was felt that the classical plastic method warranted further study. A brief description of the two classical plastic layout optimization formulations are given in the following sections. The lower bound (primal) formulation The lower bound LP plastic design formulation for the minimum volume problem for a ground structure subjected to a single load case and containing m members and n nodes are stated as follows: min V ¼ qTc ð1Þ subject to Bq ¼ f ð2Þ qþ i ; q2 i $ 0; i ¼ 1; . . .; m where V is the total volume of the structure, qT ¼ {qþ 1 ; 2q2 1 ; qþ 2 ; 2q2 2 ; . . . 2q2 m}; cT ¼ {l1=sþ 1 ; 2l1=s2 1 ; l2=sþ 2 ; 2l2=s2 2 ; . . . 2 lm=sþ m}; B is a suitable ð2n £ 2mÞ equilibrium matrix and fT ¼ { f x 1; f 1y; f x 2; f 2y; . . .f ny}; and li ; qþ i ; q2 i ; sþ i ; s2 i represent, respectively, the length and tensile and compressive member forces and stresses in member i. Finally, f x j ; f y j are x and y direction live load components applied to node j. Using this formulation the LP problem variables are clearly the member forces, qþ i ; q2 i : The upper bound (dual) formulation – obtained using duality principles The upper bound LP plastic design formulation for the same problem may be derived using duality principles from the lower bound formulation (Hemp, 1973), and are possibly stated as follows: max W ¼ fTu ð3Þ subject to: BTu # c ð4Þ where uT ¼ {ux 1; u1y; ux 2; u2y; . . .uny}; and ux j ; ujy are the x and y direction virtual nodal displacement components, i.e. the objective is to maximise the virtual work of the given forces taken over the virtual nodal displacements, subject to limits on the virtual member strains. Note that in the dual LP formulation the problem variables are the nodal displacements uj. Layout optimization 1047 The upper bound (dual) formulation – obtained from Michell’s optimality criteria Alternatively, the constraints required in the dual formulation may be obtained from Michell’s optimality criteria. In a Michell structure, the ground structure effectively comprises an infinite number of members. Michell’s optimality criteria (Michell, 1904) states that in an optimal structure the magnitude of the virtual member strain 1i in each member i with non-zero force is given by: 1i ¼ 1 þsi or 1i ¼ 2 1 2si ð5Þ and also, when (5) does not hold, that: 2 1 2si , 1i , 1 þsi ð6Þ i.e. for all members: 2 1 2si # 1i # 1 þsi ð7Þ Expressed alternatively in terms of member displacement ui we have: 2 1 2si # ui li # 1 þsi ð8Þ If member i connects two nodes j and k, and also assuming that displacements are small, the member displacement can be written in full as: ui ¼ 1 li h i xk 2 xj ux k 2 ux j þ yk 2 yj uky 2 ujy ð9Þ where the original co-ordinates of nodes j and k are denoted, respectively, as ðxj; yjÞ and ðxk; ykÞ; and their x and y direction infinitesimal displacements are, respectively, ux j ; ujy and ux k; uky : Substituting equation (9) into equation (8) gives: 2 1 2si # 1 2li h i ðxk 2 xjÞ ux k 2 ux j þ ð yk 2 yjÞ uky 2 ujy # s1þ i ð10Þ Now, imposing constraint (10) for all m members in the discretized layout opimization problem is exactly equivalent to imposing condition (4), obtained alternatively using duality principles. Thus, for a given ground structure, the dual LP problem constraints are equivalent to Michell’s constraints, and hence are sufficient to enable an optimal solution to be obtained. EC 20,8 1048 Plastic layout optimization: development of a member adding approach The main objective of the present investigation has been to develop an alternative plastic layout optimization procedure in which problem size increases much less rapidly with the number of nodes in the design domain than when using current methods. The subsequent section describes the background to such a method. Solution of large-scale LP problems Over the past few decades, many techniques have been developed for solving the large-scale LP problems. Typically, problems can be so large that it is impractical to store all variables and constraints at any given time. Hence, the so-called “decomposition” procedures are typically employed. “Column generation” describes the process used in decomposition procedures whereby additional variables (or “columns”) are added to a reduced LP problem, the latter usually being termed the “restricted master problem” (RMP). The column generation technique follows on from the analogous technique of “cut generation” applied to the dual problem (Kelley, 1960). In this case, violated constraints (or “cuts”) not present in the reduced LP problem are added. Though originally used in conjunction with simplex based LP algorithms, column generation techniques have been recently considered in the context of interior point methods (Gondzio and Sarkissian, 1997). In order to be effective, column generation techniques are typically used in conjunction with the problem specific heuristics, said to be implemented by a so-called “oracle”. Often the oracle will add just one column (or cut) to the restricted master problem at each iteration though in the method described here typically many columns are added simultaneously. Perhaps surprisingly, decomposition techniques appear to have been rarely applied to structural layout optimization problems. Woo and Schmit (1981) describe a method involving the use of the revised simplex method to solve the restricted master problems of Dantzig Wolfe decomposition, primarily to generate extreme points in the case of problems involving smooth (i.e. non-polyhedral) convex sets (for example, rigid-plastic design using constant stress elements). For the plastic frame layout optimization problem described in this paper the LP variables correspond to the members (i.e. member forces or member forces and areas in an expanded multiple load case formulation). Since the plastic layout optimization specific heuristics are used, the expansion process developed can alternatively be usefully described in structural terms, i.e. as involving “member adding”. Member adding can be viewed as one technique which may be used as part of “expansion”, or “adaptive ground structure”, procedures used in the efficient solution of layout optimization problems. Layout optimization 1049 Initial connectivity heuristic Central to the success of any member adding method is the choice of initial frame. Once chosen this then allows the formulation of the initial RMP. Ideally, the number of members in the initial structure should be as small as possible, to save computational effort. However, the initial frame should be structurally stable (i.e. not a mechanism) to allow a structurally stable optimal structure to be formed from a subset of its members. Furthermore, to reduce the computations required, the initial frame should ideally be as near optimal as possible. The simplest approach is to initially only connect adjacent nodes together (i.e. as in the case of Figure 1(b)). Suitable optimality criteria – checking the virtual member strains The virtual member strain constraints (10) correspond to the member force variables in the dual LP problem. Unless a given member is present in the RMP, these values are not available directly. However, following an initial optimization the virtual member strains for each potential member may readily be calculated and checked using constraint (10). This is straightforward because nodal displacements are included in the dual LP problem formulation; hence the data are available following an optimization and it is trivial, and computationally inexpensive, task to calculate the implied virtual strain in each potential bar linking every pair of unconnected nodes. Additionally, because the data from the dual LP problem are generally available (either throughout or once an optimal solution has been found) it is not necessary to pose the problem in upper bound form. The dual variables corresponding to the lower bound nodal equilibrium constraints represent the nodal displacements required in the calculations above. It is also worth noting that adding a member which initially has zero force/area to the lower bound problem formulation does not lead to violation of any of the lower bound constraints. Thus, the optimal solution from a given iteration will also represent a basic feasible solution for the next iteration; making use of this advanced basis in a computer program can reduce CPU times, particularly when simplex based LP solvers are used. If constraint (10) is not satisfied for any potential member, it follows that the solution is likely to be non-optimal. Hence, in this case one or more new members should be added. Simple example A simple example serves to illustrate how a member adding procedure might work. Suppose the ground structure, loading and support conditions given in Figure 1(b) are used initially (also shown in Figure 2(a)) and that all members are formed from material with a limiting compressive and tensile stress of unity. Following the initial LP optimization process, a minimum volume of 3.36603 is calculated (optimal structure shown in Figure 2(b)). EC 20,8 1050 The virtual nodal displacements associated with the initial optimal solution are required by the member adding method and are presented in Table I. Using the proposed procedure, which is designed to tackle general cases, the virtual strains in all potential members not present in the current ground structure (i.e. AE, AF, BE and BF) would need to be checked to see whether condition (10) is satisfied. However, as potential members AE and BF overlap the existing members and since this particular problem does not include for example, joint costs and/or self-weight, it is clear by inspection that the strains in these particular potential members need not be checked. Then, consider the implied virtual strain in potential member, AF: 1AF ¼ 1 L2 AF ðxF 2 xAÞ ux F 2 ux A þ ð yF 2 yAÞ uy F 2 uy A ¼ 1 5 ½ð1Þð5Þ þ ð22Þð21Þ ¼ 7 5 Figure 2. (a) Initial minimal connectivity ground structure, containing 11 bars; (b) initial optimal structure (volume ¼ 3.36603); and (c) final optimal structure obtained using the member adding procedure (volume ¼ 2.63397) Node x displacement y displacement C 3 21 D 2 0 E* 4* 0* F 5 21 Note: *Because E is not connected by any members in the optimal structure, various displacement values are possible Table I. Example: virtual nodal displacements taken from initial optimal solution Layout optimization 1051 Since 7=5 . 1, condition (10) is violated. Hence, member AF should be considered as a candidate for admission to the ground structure at the next iteration. Additionally, it is evident that prior to the next iteration that if all displacements were scaled down by a factor of 1=ð7=5Þ ¼ 5=7 then all dual constraints would be satisfied. This means that a lower bound on the volume can be calculated to be 3:36603 £ 5=7 ¼ 2:40431: In this example, once member AF is added to the ground structure the form of the final optimal structure may be determined following the next LP iteration. This latter structure has a much reduced weight of 2.63397 and is shown in Figure 2(c). Heuristic criteria for adding a new member It is clear that only those potential members which violate constraint (10) need to be considered as candidates to be added to the ground structure at the next iteration. However, though for a trivial problem a strategy which involves adding to the ground structure all members which fall into this “candidate” category may work well, in the case of larger problems a different approach is required. This is because unless the structure found following the first LP iteration is near optimal, a prohibitively high number of members would fall into the candidate category at this stage and would then have to be added. To counter this, a simple but effective heuristic procedure has been developed. (1) Set iteration count k¼0 and select initial ground structure containing mk members (forms initial RMP). (2) Solve RMP (and its dual). (3) Select value for filter parameter b and set counter of number of candidate members, madd k ¼ 0. (4) For each potential member ði ¼ mk þ 1 . . . mallÞ; compute ki ¼ max 1isþ i ; 21is2 i ; if ðki $ bÞ then madd k ¼ madd k þ 1. (5) If (b ¼ bmin or madd k ¼ am0) then add all madd k members to RMP; else if madd k . am0 sort the madd k candidate members in descending order according to ki and add top am0 members in list to RMP; else k¼k+1 and repeat from step 3. (6) k ¼ k+1. (7) If madd k – 0 then repeat from step 2. The algorithm works by employing a two stage high pass filter. The first stage of the filter, controlled by parameter b, limits the number of candidate members that need to be subsequently sorted into priority order for admission into the frame. In most cases b, which must always be greater than unity, is best set at some fixed proportion of the maximum observed value of ki, denoted k max. The second stage of the filter, controlled by parameter a, governs the EC 20,8 1052 maximum number of members which can be admitted at each iteration (the latter being taken here to be am0, where m0 is the number of members present initially). Sometimes the first stage of the filter will block out too many candidate members, leaving less than am0 members to be added. In such cases, rather than adding only a few members and then solving a RMP which is only slightly different to the previous one, provided b is greater than b min then k is incremented and a new (lower) value of b is tried. Thus, the number of algorithm iterations k may often be greater than the number of LP iterations. As stated earlier, selection of b in step 3 normally requires prior knowledge of k max. To obviate the need to initially loop through potential members to find this maximum value prior to step 3, for the purposes of the work described in this paper, for iterations 1-13, respectively, b has been taken as 1.4142, 1.4, 1.325, 1.2, 1.15, 1.1, 1.075, 1.05, 1.04, 1.03, 1.02, 1.01, 1.005 (subsequently b ¼ bmin ¼ 1:0001Þ: Note that this particular sequence of values is only appropriate for the special case of frames with equal x and y nodal densities and equal compressive and tensile strengths. Finally, whilst for sake of clarity the algorithm shown does not include calculations to bound from below the true optimal volume, this can easily be carried out in step 3 or 4 by dividing the current (upper bound) volume by k max. Additionally, the stopping criteria given in step 7 can readily be modified to alternatively allow termination if the percentage gap between the upper and lower bound volumes is within a specified tolerance. Formal status of the method It is useful to clarify the formal status of both the solutions obtained and the convergence characteristics of the method. Suppose, at the end of a given iteration, it is found that the virtual strain in one or more of the potential members linking nodes in the design domain is greater than the computed strain limit, and that the maximum ratio of virtual to limiting strains is k max. This indicates that the current solution to the restricted upper bound (dual) problem does not represent a feasible solution to the fully connected ground structure problem. Duality theory indicates that the computed volume must represent an upper bound on the optimal volume of the equivalent fully connected ground structure. However, multiplying all virtual displacements by 1/k max both ensures feasibility and reduces the virtual work and hence the computed structural volume by a factor of 1/k max. In this case, duality theory indicates that the scaled down volume must represent a lower bound on the optimal volume of the equivalent fully connected ground structure. Alternatively suppose that, at the end of a given iteration, it is found that the virtual strain in each of the potential members linking nodes in the design domain is less than or equal to the computed strain limit (i.e. k max¼1, or, in practice k max <1). This indicates that the current solution to the restricted Layout optimization 1053 upper bound (dual) problem is a feasible solution to the fully connected ground structure problem. Furthermore, since the current solution is optimal for the reduced structure, it follows that the present solution must also be optimal for the equivalent fully connected ground structure. Finally, since even a fully connected ground structure contains only a finite number of members, any procedure that involves adding (though not removing) members successively, such as the one described here, must only require a finite number of iterations before an optimal solution is obtained (i.e. cycling will not, in general, prevent a solution from being obtained). Though formal proof that the procedure must always find the optimal solution in significantly fewer iterations than mall – m0 has not been established, empirical evidence indicates that a relatively small number of iterations will generally be required (where mall is the number of members in the fully connected ground structure and m0 is the number of members in the initial, minimal connectivity, ground structure used at the start of the member adding scheme). Numerical examples Purpose written software was programmed using an object-oriented approach and the language C++ to initially set up and then modify the LP problem at successive iterations using the member adding procedure described previously. A highly efficient C++ standard template library (STL) sort routine was used in the program to obtain a priority ordered list of members to be added to the frame at a given iteration. The MOSEK (version 2.0) interior point LP solver which uses a homogeneous and self-dual algorithm (www.mosek.com) was used for the numerical studies. The pre-solve feature was switched off and the problem was initially passed to the solver in memory using the supplied subroutine library. Subsequently, it was only necessary to pass changes to the current problem to the solver (rather than the entire revised problem). Problems were run on AMD Athlon XP1900 based PC (running at 1.6 GHz) with 1.5 Gb of RAM, running under Microsoft Windows XP Professional. Test problems Three simple demonstration problems are described here. Problem A is a relatively small problem which can also be solved using the traditional fully connected ground structure approach. Problems B and C are considerably larger, having structural universes containing 13,263,825 and 116,288,875 potential members, respectively. Note that this latter problem is several orders of magnitude larger in size than the “large” truss examples described by Rozvany et al. (1995). In all cases, a unit applied load and equal compressive and tensile strengths of unity were used. Design domains, loading and support conditions are shown in Figure 3. EC 20,8 1054 Figure 3. Numerical test problems: A (353,220 potential members), B (13, 263, 825 potential members) and C (116, 288, 875 potential members) Layout optimization 1055 Influence of initial connectivity For problem A, five alternative initial connection strategies were investigated: (1) connect members to adjacent nodes, omitting all diagonals except backward facing diagonals on the bottom row; (2) connect members to adjacent nodes, omitting forward facing diagonals; (3) connect to adjacent nodes; (4) connect to adjacent nodes and also connect each loaded node to each support node; and (5) fully connected ground structure including overlapping bars. For clarity the five strategies are also illustrated diagrammatically in Figure 4. Figure 4 presents results from the runs. Note that when member adding was employed the number of members added at each iteration was limited to 5 per cent of the number of members present initially. The influence of this parameter will be considered in more detail in the next section. From Figure 4, a number of observations can be made. First, as expected, the computed optimal volume of the frame was found not to be influenced by the initial ground structure used. Secondly, it is clear that all member adding CPU times are significantly shorter than when a fully connected ground structure was used from the outset. Figure 4. Comparative CPU times and LP problem sizes for test problem A (353,220 potential members): influence of initial nodal connectivity EC 20,8 1056 The shortest run time was observed when each node was connected only to immediate adjacent nodes, including diagonals [i.e. (3)]. In this case, the run time was only 8 per cent of that required when the conventional ground structure was used. Perhaps most significantly of all, only a little more than 1 per cent of the number of LP variables were required, reducing memory usage dramatically. Comparing cases (1) and (2) with (3), it is clear that it is counter-productive in terms of CPU time to reduce the level of initial connectivity too much. However, because the peak number of LP variables, and hence memory requirements, are reduced slightly when using reduced connectivity, this may sometimes be a useful strategy. Figure 5 shows all the intermediate optimal structures for case (3). The line thickness on these and other plots presented in the paper were taken to be proportional to the square root of the cross sectional area (for clarity, assuming a circular member cross-section, bars with radius ,0.001 were not plotted. Note that the use of any cut-off would sometime lead to apparently anomalous details on the plots, e.g. a member apparently ending abruptly in space). It is notable from Figure 5 that the near optimal forms (and volumes) are obtained after only a few iterations. It is also useful to examine the performance of the methods with larger problems. Figure 6 shows plots of CPU time against the number of potential members (for clarity a log-log scale is used). To obtain the figure the number of nodes, and hence potential members, in problem A was varied. Figure 6 shows that the member adding procedure is consistently computationally less expensive than the traditional ground structure method. Additionally, the high memory requirements of the latter method meant that the largest initial fully solvable connected ground structure problem contained just 1,225 nodes (749,700 potential members). This compares unfavourably with the size of the largest problem which could be solved when using the member adding method, which in this case contained 18,769 nodes (176,128,296 potential members). It is also noteworthy that the computed volume obtained from the largest problem solvable (i.e. containing 176,128,296 potential members) was just 1.6 per cent lower than the computed volume obtained when using the standard level of nodal density indicated in Figure 3 (i.e. containing 353,220 potential members). Figure 7 shows a plot of the peak number of LP variables in the problem plotted against the number of potential members, for clarity using a log-log scale. The figure indicates that as problem size increases, the ratio of the peak number LP variables required using the initially fully connected ground structure method to those required using the member adding method increases markedly, e.g. with 100,000 potential members the ratio is approximately 18:1 whereas with 100,000,000 potential members the ratio increases to 1080:1. Layout optimization 1057 Influence of member adding selection criteria For problem A, two alternative member adding selection strategies were investigated. The first strategy was to use condition (10) alone to govern whether, at the end of a given iteration, one or more members should be added prior to the next iteration. The second strategy was to limit the total number of members added at any given iteration to fixed proportions of the number of members present in the initial ground structure (i.e. according to the member adding algorithm). For all runs, an initial ground structure which included only connections between the adjacent nodes was used. Results are shown in Table II. Figure 5. Intermediate optimal frames following each LP iteration for test problem A (adjacent nodes initially connected) EC 20,8 1058 It is evident from Table II that the total CPU time expended is not especially sensitive to the percentage of additional members added, provided the latter is not either very small or very large. Of those tried the 5 per cent m0 run was solved most quickly (10.8 s). It is also evident from Table II that in all cases the vast majority of the total CPU time consumed was spent in the LP solution phase, rather than whilst implementing the member adding heuristics (i.e. in the “oracle” phase). Figure 6. CPU times vs no. of potential members: comparison between “ground structure” and “member adding” (5 per cent m0) methods Figure 7. Peak no. of LP variables vs no. of potential members: comparison between “ground structure” and “member adding” (5 per cent m0) methods Layout optimization 1059 Larger test problems B and C Figure 8 shows the output for problem B. The computed optimal volume for problem B was 3.14534. This is 0.12 per cent greater than p, the optimal volume for this type of problem. The CPU time for this problem was 42 min 10 s whilst the peak number of LP variables was 76,847. This is less than 1 per cent of the number of variables which would have been required had an initial fully connected ground structure been used. Figure 9 shows the output for problem C. “Member adding” method: no. of members added at each iteration 1 per cent m0 3 per cent m0 5 per cent m0 10 per cent m0 100 per cent m0 No limit Volume, V 2.43206 2.43206 2.43206 2.43206 2.43206 2.43206 No. of LP iterations 40 18 13 12 6 4 Initial no. of LP variables 6,384 6,384 6,384 6,384 6,384 6,384 Peak no. of LP variables 7,610 7,876 8,092 8,892 16,261 139,556 CPU time to #1.001 £ V(s) 22.7 6.1 5.8 5.8 6.3 53.2 CPU time for LP phase only (s) 28.8 13.0 9.6 10.4 11.2 80.8 Total CPU time (s) 31.7 14.4 10.9 11.7 12.4 82.4 Table II. Comparative CPU times and LP problem sizes for test problem A (353,220 potential members): influence of admission criteria for new members Figure 8. Optimal form for problem B (13,263,825 potential members) EC 20,8 1060 The computed optimal volume for problem C was 4.499827. This is just 0.038 per cent greater than the corresponding Michell structure volume of 4.498115 (obtained for this design problem by Zhou and Rozvany (1991)). The CPU time for this problem was 6 h and 50 min. Following the analysis, it was noted that a computed volume which was within ^15 per cent of the optimal had been obtained in just 20 s and that a computed volume which was within ^5 per cent of the optimal had been obtained in only 9 min. However, during the solution process itself (when the optimal volume was unknown) it took 17 min to obtain a computed volume which was provably within ^15 per cent of the optimal, and 3 h and 40 min to obtain a computed volume which was provably within ^5 per cent of the optimal. The peak number of LP variables for problem C was 215,103. This is less than 0.1 per cent of the number of variables which would have been required had an initial fully connected ground structure been used. Discussion Rather than attempting to solve the design problems with complex design constraints, this paper has focused on solving the simple classical plastic layout optimization problem for large problems (containing up to many millions of potential members). Effective use is made of modern LP algorithms, which are at present extremely well developed. The proposed member adding method uses optimality criteria in conjunction with the heuristic member admission rules. Though the current rules appear very effective, the development of alternative rules may lead to even better Figure 9. Optimal form for problem C (116, 288, 875 potential members) Layout optimization 1061 performance. For example, currently only the addition of members is allowed. However, it is possible to modify the method so that a specified number of non-optimal members can be deleted during a given iteration. This has the advantage that problem size can be more carefully managed and could, for example, allow the analysis of large problems even if the computer memory available is limited (if a limit was placed on the total number of members present during any iteration). Further investigations are required. The proposed method provides major benefits compared with the standard fully-connected ground structure approach (in terms of both analysis time and the size of problem which is tractable). Although, even with the new method, CPU time has been found to rise steeply as the number of nodes increases, it should be borne in mind that real-life design domains are likely to be much more constrained than those considered here, reducing dramatically the number of potential members for a given number of nodes. Additional practical constraints, such as stipulating the maximum length of compression members will also dramatically reduce the number of potential members. Thus, the application of this method raises the possibility of LP-based optimization being used to tackle problems of a size which would be of interest to practicing designers. The use of a dense grid of nodes in the design domain might in many cases obviate the need for more complex optimization methods to be applied (for example, procedures which use flexible though computationally expensive genetic algorithms to add nodes to an initially sparse grid, or move nodes to optimal locations (Kawamura et al., 2002)). Though not considered here, real-world problems will almost always involve the application of several independent load cases. Though efficient procedures which involve solving in succession a series of specially defined single load case problems have been proposed (e.g. for two load cases by several workers, including the work of Hemp (1973)), such methods appear only to apply in the special case of equal compressive and tensile material strengths, ultimately they may not be useful in practice. Alternatively, the lower and upper bound plastic LP formulations can be expanded to tackle multiple load cases (Hemp, 1973). Dual constraints relating to the expanded problem may then be used to determine which members should be added as part of the member adding procedure. This will be demonstrated elsewhere in due course. Though only 2D problems have been considered in the current paper, the extension of the member adding technique to 3D should be relatively straightforward. However, because initially connecting each node to all adjacent nodes, as for 2D problems, may lead to excessive number of members, it is likely that various different initial nodal connectivity strategies will need to be tried before an efficient scheme is found. Additionally, including joint costs in the manner suggested by Parkes (1975) only requires slight modification of constraint (10) in order to be used in conjunction with the member adding technique. EC 20,8 1062 The work described here is an initial step in a larger programme of work concerned with the development of computationally efficient and practically applicable layout optimization tools for use in practice. The significant reduction in computational effort and memory requirements for the optimization of large frameworks should allow the incorporation of more complex constraints, which will enable LP-based optimization methods to be used to design more realistic structural frameworks. Currently, the two areas of investigation by the authors are system stability and member buckling. System stability is an issue because often it is found that the optimal layouts obtained using traditional methods are in unstable equilibrium with the specified applied loads (e.g. as in the case of the optimal solution for problem B – refer to Figure 8). Clearly this is unacceptable. Various methods of overcoming the problem have been suggested. For example, Rozvany (1996) noted the possibility of adding nominal imperfections to the problem formulation. He concluded that the computer time required to solve the expanded problem would be prohibitive. However, because of the availability of the member adding method, this conclusion may no longer be justified. Work is now proceeding in this area. In addition to the system stability problem mentioned above, the problem of individual compression member stability (i.e. buckling) needs to be addressed. In essence the problem is that as the ultimate compressive strength of a member is effectively a function of the geometry of its cross-section (rather than being constant for a member of given length), the optimization problem becomes non-linear. This may be tackled by adopting an iterative LP solution procedure which involves updating member compressive strengths at each iteration (Reinschmidt and Russell, 1974). However, the successive application of LP already used as part of the member adding procedure means that in principle non-linear behaviour may readily be incorporated (though as with all non-linear optimization schemes there is a possibility of arriving at local rather than global optima). This will be discussed elsewhere in due course. Conclusions A conceptually simple, but very effective member adding method has been described. Using the method, layout optimization problems can be tackled which involve far greater number of potential members than has hitherto been possible (e.g. .100,000,000 potential member problems are tractable at present). Though the method requires selection of an initial ground structure, the precise form of the initial connectivity has been shown not to influence the optimal design. Furthermore, bounds on the optimality of intermediate solutions obtained using the method can be computed, with the final solution being provably optimal. Additionally, though the “plastic” design formulation has been used, the solutions are also optimal for the “elastic” with stress constraints design problem provided only a single load case is present. Layout optimization 1063 In the future the method will be applied to large 3D problems and additionally features such as stability, joint costs and member self-weight will be added to the problem formulation, to increase the practical applicability of the output. References Dorn, W.S., Gomory, R.E. and Greenberg, H.J. (1964), “Automatic design of optimal structures”, J. de Mechanique, Vol. 3, pp. 25-52. Gondzio, J. and Sarkissian, R. (1997), “Column generation with a primal-dual method”, in, Logilab Technical Report 96.6, University of Geneva. Hemp, W.S. (1973), Optimum Structures, Clarendon press, Oxford. Karmarker, N.K. (1984), “A new polynomial-time algorithm for linear programming”, Combinatorica, Vol. 4, pp. 373-95. Kawamura, H., Ohmori, H. and Kito, N. (2002), “Truss topology optimization by a modified genetic algorithm”, Struct. Multidisc. Optim., Vol. 23 No. 6, pp. 467-72. Kelley, J.E. (1960), “The cutting-plane method for solving convex programs”, J. S.I.A.M., Vol. 8 No. 4, pp. 703-12. Kirsch, U. (1990), “On singular topologies in optimum structural design”, Struct. Optimization, Vol. 2, pp. 133-42. Kirsch, U. (1996), “Integration of reduction and expansion processes in layout optimization”, Struct. Optimization, Vol. 11, pp. 13-18. Michell, A.G.M. (1904), “The limits of economy of material in frame-structures”, Phil. Mag., Vol. 8, pp. 589-97. Parkes, E.W. (1975), “Joints in optimum frameworks”, Int. J. Solids and Structures, Vol. 11, pp. 1017-22. Reinschmidt, K.F. and Russell, A.D. (1974), “Applications of linear programming in structural layout and optimisation”, Computers and Structures, Vol. 4, pp. 855-69. Rozvany, G.I.N. (1996), “Difficulties in truss topology optimization with stress, local buckling and system stability constraints”, Struct Optimization, Vol. 11 No. 3/4, pp. 213-7. Rozvany, G.I.N. (2001), “On design dependent constraints and singular topologies”, Struct. Multidisc. Optim., Vol. 21 No. 2, pp. 164-72. Rozvany, G.I.N. and Birker, T. (1994), “On singular topologies in exact layout optimization”, Struct. Optimization, Vol. 8, pp. 228-35. Rozvany, G.I.N., Bendsoe, M.P. and Kirsch, U. (1995), “Layout optimization of structures”, Appl. Mech. Review, Vol. 48 No. 2, pp. 41-119. Woo, T.H. and Schmit, L.A. (1981), “Decomposition in optimal plastic design of structures”, Int. J. Solids Structs, Vol. 17 No. 1, pp. 39-56. Zhou, M. and Rozvany, G.I.N. (1991), “The COC algorithm, Part II: topological, geometrical and generalized shape optimizaion”, Comput. Method Appl. Mech. Eng., Vol. 89 No. 1-3, pp. 309-36. Further reading Rozvany, G.I.N. and Zhou, M. (1991), “The COC algorithm, Part I: cross-section optimization or sizing”, Comput. Method Appl. Mech. Eng., Vol. 89 No. 1-3, pp. 281-308. EC 20,8 1064