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Evolutionary structural optimization for connection topology design of multi-component systems Qing Li Department of Aeronautical Engineering, The University of Sydney, Sydney, New South Wales, Australia

Grant P. Steven School of Engineering, University of Durham, Durham, UK, and

Y.M. Xie Faculty of Engineering and Science, Victoria University of Technology, Melbourne, Victoria, Australia Keywords Finite element analysis, Structural optimization, Design, Components Abstract Most engineering products contain more than one component or structural element, a consideration that needs to be appreciated during the design process and beyond, to manufacturing, transportation, storage and maintenance. The allocation and design of component interconnections (such as bolts, rivets, or springs, spot-welds, adhesives, others) usually play a crucial role in the design of the entire multi-component system. This paper extends the evolutionary structural optimization method to the generic design problems of connection topology. The proposed approach consists of a simple cycle of a finite element analysis followed by a rule-driven element removal process. To make the interconnection elements carry as close to uniform a load as possible, a ``fully stressed’’ design criterion is adopted. To determine the presence and absence of the interconnection elements, the usage efficiencies of fastener elements are estimated in terms of their relative stress levels. This avoids the use of gradient-based optimization algorithms and allows designers to readily seek an optimization of connection topology, which can be implemented in their familiar CAD/CAE design platforms. To demonstrate the capabilities of the proposed procedure, a number of design examples are presented in this paper.

1. Introduction In recent years, structural optimization has been exhaustively studied for systems made of single components (Bendsùe, 1995; Rozvany et al., 1995; Xie and Steven, 1997; Steven et al., 1998; Hassani and Hinton, 1999), where various efficient and reliable design tools are becoming accepted. In real life, however, most engineering structures consist of more than one component or structural part. Pin or rigid joint trusses, frames of car bodies, machine tools, computer

Engineering Computations, Vol. 18 No. 3/4, 2001, pp. 460-479. # MCB University Press, 0264-4401

In the past years of our research in structural optimization, Professor Ernie Hinton has given his encouragement and support to us. He and his co-researchers at Swansea have made significant contributions to the development of structural optimization techniques. The authors dedicate the work described in this paper in honor of Ernie Hinton, a great engineer and good friend.

cases and aircraft structures are some practical examples in various engineering applications. Traditionally, multi-component systems are usually analyzed at the level of individual components in order to simplify the modelling and design process. In such analyses, the interconnections are simply treated as some form of sub-boundaries with appropriate load transfer and kinematic constraints. The location of the support points and the magnitude of the loads need to be carefully determined in advance for single component analyses to be of any value. Obviously, such a procedure is convincing only when the connection patterns between the components can be definitely identified. Therefore, one could doubt a single component design is really an optimum if the fastener locations and sizes have been decided at the outset of the analysis. To design connection pattern, convention, experience and manufacturing factors play a dominant role in determining the fastener type, positioning and proportioning. However, this may not always guarantee an optimum in a structural sense. Design engineers may be interested to know what design patterns and how many fasteners would achieve a best possible structural performance within the manufacturing constraints. For this reason, optimization of interconnection is of great practical significance and is also capable of providing a more reliable solution to the design of the entire multicomponent system. If this connection optimization can be done in conjunction with, and as part of, the component topology optimization then the whole assembly benefits. It is noted that the problem of connection optimization has attracted some attention from a manufacturing point of view (Menassa and DeVries, 1989, 1991; Cai et al., 1996), where finite element analysis (FEA) and mathematical programming techniques were adopted to achieve an optimum. In this work, the number of points to be located in a fixture design is limited to six, due to restrictions of work-piece positioning. Although this research work does consider certain aspects of fastener location design, a generalized methodology for connection optimization is still needed. The great success of various general-purpose structural optimization techniques in 1990s has aroused interest in adapting them to solve connection optimization problems. Chirehdast and Jiang (1996) extended the concept of topology optimization method to the design of spot-weld and adhesive bond patterns. Later, they proposed a theoretical framework for generalized connection optimization (Jiang and Chirehdast, 1997). To solve the coupled problem of connection and component design, Chickermane and Gea (1997) considered the multi-component system as a whole, in which the optimization of interconnections and components were achieved simultaneously. This avoided the potential problem of detecting a limited or local optimum due to the separation of both design processes. In their solutions, a stiffness maximization criterion was adopted. Later, a constraint on the maximum fastener load was integrated into the optimization process so as to avoid excessive loads in connection elements. The common points behind the above-mentioned

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solutions were to model the various interconnections as linear springs, whereby different connection types were represented by different spring stiffnesses. Spring stiffness was treated as a continuous design variable and the structural mean compliance as the objective function. In their procedure, the structural response was evaluated using FEA and the optimum was achieved using mathematical programming techniques. Similarly to other continuous variable based optimization approaches (Bendsùe, 1995; Gea, 1996; Hassani and Hinton, 1999), the intermediate stiffness of a spring needs to be penalized, in some appropriate form, such that the connection stiffness eventually goes to either of two possible extremes, i.e. full or null stiffness. Evolutionary structural optimization (ESO) has been proposed and developed to provide the engineering design community with an alternative optimization strategy whereby traditional mathematical programming based optimal processes are replaced by a simple heuristic approach. The physical concept behind this method is intuitive and simple. By progressively removing a certain amount of under-utilized material and/or adding some material to over-utilized regions, the structure evolves towards an optimum. It has been found that the ESO approach offers significant simplicity in its computer implementation; usually only adding a few lines into finite element preprocessing and post processing batch files. A great number of numerical examples in a wide range of engineering and physical disciplines have demonstrated the ESO method to be very effective and robust (Xie and Steven, 1993, 1994, 1997; Hinton and Sienz, 1995; Hinton et al., 1998; Rosko, 1995; Li, 2000). Similarly to other structural optimization methods, previous studies using ESO have been limited to structures and physical fields made of a single component. Being an engineering oriented method, it is clearly necessary to extend its application to multi-component structural systems. In this paper, a stress based approach is proposed to design a fastener layout/topology that achieves an almost uniform stress level in each interconnection element. The fundamental idea is to model the connection between components using nonadjacent discrete brick elements for every possible candidate fastener location. The von Mises stress levels of these interconnection elements are treated as an indicator to determine whether its presence is required. With the ESO method, the connection element itself, rather than its associated physical parameters, such as stiffness, is treated as the design variable. This paper develops a systematic procedure for the positioning of fasteners within the connection design space. At the same time as considering fastener location, the conventional ESO process can be applied to the components being connected, thereby producing an overall approach to the topology optimization of a multi-component structure. A number of practical examples are presented to demonstrate the capabilities and efficacy of the proposed methodology.

2. Relative efficiencies of interconnection elements In the proposed procedure, brick elements are employed to model candidate interconnections. To reflect the nature of discrete connections, such brick elements are isolated from each other as shown in Figure 1. The distance ¢, between two adjacent elements is usually set up to be greater than, or equal to, some minimum manufacturing and/or assembly dimension ¢m in . In bolted connections, for example, the distance ¢m in may need to consider the nut size and the operational space for a spanner. To comply with these requirements, a certain initial form of finite element mesh would be needed to ensure the appropriate connection arrangement. It is considered that use of brick elements, rather than beams, makes the FE modelling of the connection mechanics and entire structural behaviour closer to practical situations since any type of fasteners would have an actual size for their cross section. With the different material properties of elements, different connection topologies can be represented that are closer to physical situations. In a multi-component system, it is frequently found that failure occurs either at the connection itself or around the attachment regions in the connected components (Chickermane and Gea, 1999). This may be caused by an inappropriate allocation of interconnections, e.g. some locations may require more connections than others under a certain load. A reliable sign of connection failure is excessive stress or strain. Alternatively, a low stress or strain level may reflect an inefficient use of the connection material. Ideally, the stress levels in all connections in a multi-component system should be near the same safe stress level. To represent the relative stress levels of the connection elements, a dimensionless factor is formulated by comparing the von Mises stress ¼Ci (Orthwein, 1986; Rosko, 1995; Young et al., 1999) of a candidate interconnection i with the highest one ¼Cmaxt (or allowable stress ¼Callowt of material of connection elements, but for convenience, the following formulation uses only the maximum) as ¬i ˆ

¼Ci ; ¼Cmaxt


…1†

Figure 1. Modeling for multiple component system

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where, subscript i denotes the number of the connection element, sub-subscript t means element i located in the tth connection region and superscript C means connection elements. From the viewpoint of structural strength, the factor reflects a relative efficiency of the interconnection usage. At a more complex but realistic level, where the multi-component system may be operated under circumstances involving multiple load cases, as governed by the finite element equation: Ku k ˆ p k …k ˆ 1; 2; ¢ ¢ ¢ ; LCN †;

…2†

the relative efficiency under all the load cases can be estimated by a weighted average scheme as: Á !¿ LCN LCN X X ¼Ci …p k † …3† ¬i ˆ w…p k † C w…p k † ¼maxi …p k † kˆ1 kˆ1 or, by an extreme value scheme as: ¬ˆ

Mt max iˆ1

LCN max kˆ1 n

¼Ci …p k †

LCN max kˆ1

o ¼Ci …p k †

…4†

where, LCN denotes the total number of load cases, pk (k = 1, 2, ¢ ¢ ¢, LCN) the vector of the kth load case, ¼Ci (pk ) the von Mises stress of the ith connection element under the kth load case, w(pk ) gives the weighting factor for the kth C load case, LCN max ¼i …p k †calculates n the highest o stress level of the ith connection kˆ1 Mt LCN C under all load cases and max max ¼i …p k † calculates the highest stress level at iˆ1

kˆ1

all load cases LCN over all interconnections Mt in the tth connection region. For convenience, the former is termed as the overall efficiency and the latter is named as the extreme efficiency. The efficiency factor provides a criterion to justify which connection elements should remain and which ones should be eliminated. The formulations in equation (1)-(4), could treat various connection regions differently. This means that the goal of equal efficiency or iso-strength joints can be sought in individual connection regions. This provides a way of dealing with different connection types and sizes. On other hand, if the desired connection type and size among all connection regions are the same, the global maximum stress of all regions n o T ¼C …5† ¼Cmax ˆ max maxt ; tˆ1

can be adopted as the reference criterion (¼Cmaxt ), where T denotes the total number of candidate connection regions. In this sense, an iso-strength design can be achieved over all connection regions.

3. Evolutionary optimization procedure In a traditional ESO procedure (Xie and Steven, 1993, 1994, 1997; Hinton and Sienz, 1995), the optimization starts from a more conservative design, where the fasteners are initially allocated over all possible positions. To improve the efficiency of interconnections, those less efficiently utilized elements are gradually removed. As a result, the relative efficiencies (or stress levels) of the remaining interconnections become more uniform than before. To remove under-utilized interconnections, the ESO algorithm introduces a simple rejection mechanism. If the efficiency of a connection element is lower than a threshold level or a so-called rejection ratio (RR) (Xie and Steven, 1993), i.e.: ¬i µ RRSS …i ˆ 1; 2; ¢ ¢ ¢ ; Mt ; t ˆ 1; 2; ¢ ¢ ¢ ; T†

…6†

then this connection element is considered to be relatively lowly stressed or structurally less efficient, and as a result, should be removed from the specific connection region t. Based on this idea, finite element analysis, plus an element assessment/removal cycle is repeated using the same value of RRSS , until there are no more interconnections that can be removed. This means that an ESO steady state (SS) has been reached, by which the lowest efficiency within a specific connection region has become higher than a certain level or percentage RRSS . To advance such an optimization process, an evolutionary rate (ER) is introduced and added to RR as RRSS‡1 ˆ RRSS ‡ ER

…7†

with the increased threshold efficiency or rejection ratio, the iterations take place again until a new steady state is attained. For clarification, a detailed optimization procedure can be organized as follows for the design of connections or whole multi-component system: Step 1: Discretize the multi-component system using an appropriate dense FE mesh, assign the property type of connection and/or component elements to a number greater than 1, define ESO parameter ER, RR0 and set SS = 0. Step 2: Perform a FEA to determine the relative efficiency factor of all candidate connection elements as equations (1)-(5). Step 3: For all candidate elements, if their relative efficiency satisfies equation (6), then assign their property type to zero (i.e. remove from the system). Step 4: If a steady state is reached, increase RRSS by ER, as equation (7) and set SS = SS + 1, repeat Step 3; otherwise, repeat Steps 2 to 3 until an optimum is achieved. The optimization procedure can be terminated at a point where some form of optimum is attained. To indicate the overall structural performance, Querin et al. (1998, 1999) and Liang et al. (1999) proposed two performance indices (PI) respectively. For many typical topology optimization problems, these two PIs


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can give an approximate indicator quantitatively. Taking Querin et al.’s PI as an example, the performance of surviving interconnection elements can be formulated as: ¿ Mt X …8† PI ˆ ¼i Vi …P £ l†; iˆ1

where P is a nominal load and l a nominal length. The lower the PI, the better the connection system. In order to avoid a big jump in structural response during connection element removal, the number of connection elements to be eliminated at each stage should be relatively small (Hinton and Sienz, 1995). In other words the initial RR0 and the ER should be small, e.g. less than 1 percent of the maximum connection element stress value. The ESO procedure presented herein only allows elements to be removed from an oversized interconnection region. It is worth noting that the recent progress made by Querin et al. (1998), Yang et al. (1999) and Young et al. (1999) allows the system to gradually grow from an undersized structure into an optimum by both adding elements where the stresses are highest (over-utilized) and taking elements away where stresses are lowest (under-utilized). It is shown that the bi-directional ESO (BESO) approach can produce almost identical solutions to those by the ESO. For simplicity, only the ESO approach to multiple component systems is presented in this paper. 4. Design examples To demonstrate the capabilities of the proposed procedure, several typical connection designs are investigated. In all the design cases, two or three thick plates can be connected in an overlapping manner by spot welds, rivets, pins or threaded fasteners. The overlapping part (i.e. connection region) is meshed with a 15 £ 15 grid as illustrated in Figure 2, where the nominal dimensions a and l are equal to 150mm and 300mm respectively. Within this grid, 7 £ 7 candidate interconnections are modelled by eight-node bricks with 30 percent thickness and half the Young’s modulus of the plates. Assume that the distance of 10mm

Figure 2. Initial finite element model of straight overlapped connection

between two adjacent connection elements and between connections and edges is greater than the minimum requested space of manufacturing and assembly. For each design case, two point load cases are applied. In all the evolutionary optimization processes below, an initial rejection ratio of RR0 = 0 and an evolution rate of ER = 1 per cent are adopted.

Evolutionary structural optimization

4.1 Straight connection case A cantilever type system consisting of two thick plates A and B is shown in Figure 2, in which two point load cases are applied in the upper and lower corners of the free end. Figure 3 shows the optimal connection patterns at several ESO steady states near the end of the evolution process. As the rejection ratio (RR) increases, more and more, relatively inefficient connection elements are eliminated from candidate locations. It can be found that, by the end of the evolutionary process (SS = 17), the last survived connection elements locate at the four outmost corners of the overlapping square as shown in Figure 3(c). Such an obvious result could be argued to be some form of validation of the ESO process used in connection optimization problems. In the example, two load cases are applied. It is found that the results produced from the extreme efficiency and the overall efficiency schemes are almost the same. This indicates the surviving interconnections have both higher extreme efficiency and higher overall efficiency for these two load cases. To identify an optimal design, the evolutionary process can be monitored by plotting the highest stress, the lowest stress and the performance index, Figure 4. It can be seen that the deviation between the highest and the lowest stress in the surviving connection elements becomes smaller and smaller with the evolution process, which reflects a path approaching a fully stressed design. Also, from Figure 4, one can clearly identify the lowest point of the performance index, which corresponds to the four connection elements located near the corner positions, as shown in Figure 3(c). This provides further evidence to validate the ESO solution.

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4.2 Crossover connection case Crossover joints are one of the most representative junction types in structural engineering, in which one plate is attached to another at right angles. As shown in Figure 5, the entire structure is fully supported on the both ends of the horizontal plate and two load cases are applied at the ends of the vertical plate. In this design case, the last four surviving interconnections also locate at the

Figure 3. Connection patterns in the straight joint

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Figure 5. Initial finite element model of cross overlapped connection

corner positions (Figure 6(d)), as in the straight connection case. This can be validated both intuitively and by evolution histories of the stress deviation as well as through the performance index as in Figure 7. It is found that the pattern of the four outmost interconnections has the smallest stress deviation (zero) and the lowest performance index. Such a solution is in good agreement with a conventional design. To deal with multiple load cases, the overall efficiency scheme and extreme efficiency scheme are also considered separately. Unlike the preceding straight joint case, in the crossover joint, these two schemes may result in different design patterns at some iterations. Taking the eight interconnection elements as an example, the overall efficiency scheme produces a pattern as in Figure 6(a)


Figure 6. Connection patterns in the cross joint

Figure 7. Evolution history of PI and stress levels for the cross joint optimization

and the extreme efficiency scheme yields one as in Figure 6(b). The former reflects an average use efficiency of connection elements, while the latter emphasizes the highest efficiency under different load cases. However, it is found that both the schemes produce the same final pattern for M = 4. This means that the pattern in Figure 6(d) is an optimum to both the overall efficiency scheme and the extreme efficiency one. In addition, it is interesting to note that the last eight remaining interconnections, as in Figure 6(b), are not evenly distributed around the edges of the connection domain as in Figure 6(c). By

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comparing the ratio of highest extreme efficiency to the lowest one, it is found that the design in Figure 6(b) is only 79 percent of that in Figure 6(c) (1.75 vs. 2.22). This means that the uniform allocation of eight elements around the edges of the connection domain does not offer the optimum in the sense of iso-strength or equal efficiency.

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4.3 T-shape connection case For the third design study the modelled region reflects a transverse support joint that can be typical for a frame structure, which engineers frequently incorporate into designs. Different from the crossover joint case as shown before, the T-joint has only one symmetric axis in a vertical direction. For this reason, two situations of load-support can be considered in the T-joint design. Design A is loaded at the both ends (CD and EF) of the long horizontal plate and supported at the lower end (GH) of the short vertical plate, as illustrated in Figure 8. Design B has a reverse load-support arrangement, i.e. horizontally loaded at points G and H and supported on edges CD and EF respectively. From the evolutionary patterns at the last several iterations as in Figure 9(a-c), it can be seen that the final connection pattern (M = 4) for Design A is similar to that of the cross joint case, i.e. locating at the four corners as in Figure 9(c). This appears very intuitive for structural engineers. In this case, the bending loads play a dominating role in the connection region. But for Design B, the last four remaining connections (M = 4) are located near left and right lower corners as in

Figure 8. Initial model for the T-shape overlapped connection (Design A)

Figure 9. ESO solution for T-connection patterns (Design A)

Figure 10(c). If the connection pattern of Design A (M = 4, at the four outmost corners) is adopted in Design B (also M = 4), it is found that the ratio of the highest to the lowest efficiency (or stress level) would increase by 53 percent (1.14 vs. 2.21). This means that, for Design B, the lower corner areas need more connection elements in order to more uniformly carry both the bending and shear loads. From the evolution histories of stress deviation and performance index, one can identify the optimal connection patterns. For Design A, the pattern of four interconnections as in Figure 9(c) has the lowest performance index from Figure 11(a). This is in good agreement with the intuitive or practical design. But for Design B, the pattern of six interconnections as in Figure 10(b) has the lowest PI from Figure 11(b). This implies that, although the fourinterconnection pattern (Figure 10(c)) is of more uniform stress distribution, the increase in the stress level exceeds the decrease in connection element number. From a structural strength point of view, the design with four-interconnections gives a poorer performance. As in the case of the crossover connection, the overall efficiency scheme and extreme efficiency one also produce different solutions to the connection patterns at some iterations. Taking M = 6 in Design A as an example, the overall efficiency scheme has the connections 1 and 2 (black squares, as labeled in Figure 9(b)) to locate at positions 3 and 4 (circled in Figure 9(b)). Once again, however, these two schemes share the same pattern for the four-interconnection design as in Figure 9(c).


4.4 L-shaped connection case By simple adaptation of the mesh in the previous two examples, a L-shaped joint can be constructed as in Figure 12, in which two load cases are applied at the free end of the system. Figure 13 shows the evolution results at the last few ESO iterations. It is found the optimal distribution of the fasteners is somewhat different from the intuitive design of the cross and T-joints. Taking the case of M = 4 as an example, the even allocation at the four corner position may not give an optimum under the fully stressed criterion. This can be concluded from the comparison of the ratios of the highest to the lowest efficiency between the ESO pattern and even allocation one, where the former (1.69) is only 55 percent of the latter (3.13). That is to say that the stress distribution of the former is much more uniform than that of the latter.

Figure 10. ESO solution for T-connection patterns (Design B)

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Figure 11. Evolution history for T-joint optimization

The evolution history curves also provide the quantitative indication of the design optimization. From Figure 14, one can readily find out that the lowest point occurs at M = 3 from the performance index curve. This explores that the connection pattern of M = 3 as in Figure 13(c) is of the best strength-efficiency performance. It is interesting to observe that the performance of M = 6 (Figure 13(a)) appears to be better than those of M = 5 and M = 4 (Figure 13(b)). This means that not only the location of connections, also their number should be considered in order to achieve the best design. 4.5 U-shaped connection case This example presents a three-component system, in which two connection regions (t1 and t2 ) are optimized simultaneously. In this example, the same


Figure 12. Initial model for the L-shape overlapped connection

Figure 13. ESO solution for L-joint pattern optimization

Figure 14. Evolution history for L-joint optimization

number of candidate connection elements are initially allocated to each region (M1 = M2 = 7 £ 7) as in Figure 15. To simplify the calculation of elemental efficiency factor, the highest stress level of both regions is taken as n o C C C ¼max ˆ max ¼max1 ; ¼max2 , (equation (5)). That is to say that these two domains are considered as a whole in the design process.

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474 Figure 15. Initial model for the U-shape overlapped connection of three components

The optimal patterns of the joint elements in the example seem to be an identical pattern to that of the L-shape design as shown in Figure 13. In this example, although the structural system is non-symmetric, an even distribution of two group fasteners can be observed. It can be known that the bending moments in both connection regions are the same, and this leads to the symmetry of the von Mises stress fields in these two connection regions. For the design of four connection elements, similarly, one can compare the stress distribution of the ESO pattern with that of the even corner pattern, as in L-shape design. Once again, it is found that the ratio of the highest to the lowest efficiency of the ESO solution is notably lower than that of the even allocation solution (1.68 vs. 3.13). This implies that the loads carried by the ESO pattern are more uniform than those carried by the even allocation one (Figure 16). 4.6 Case of multi-component system design In addition to the location design of connections, the topology of component material can also be optimized at the same time. This means that element elimination could be carried out in the connection regions and the components concurrently dependant on whether the dual evolution criteria, i.e.: ( ¬i µ RRSS …i ˆ 1; 2; . . . ; Mt ; t ˆ 1; 2; . . . ; T† …9† ; ¬j µ RRSS …j ˆ 1; 2; . . . ; Ns ; s ˆ 1; 2; . . . ; S† are respectively satisfied or not, where ¬j ˆ ¼Pj =¼Pmaxs denotes the relative usage efficiency of component material at the jth element. In (9) T is the total number of interconnections, each with M elements, and S is the total number of components being connected, each with N elements. To show this design feature, the optimization of the entire ``cross connection’’ system, as illustrated in Figure 5, is performed again but elements are removed from both connection and component regions of the model. At some iteration, there could be both connection elements and component elements removed at


Figure 16. ESO solution for U-joint optimization

one cycle. But at other iterations, only connection elements or only component elements could be removed from the system. From Figures 17 and 18 and comparing with the connection-only design in section 4.2, it is interesting to find that, whether the component topologies are changed or not, and whether the extreme efficiency scheme or overall efficiency scheme is used, the last four surviving connection elements are located at the outmost corner positions of the connection region. This appears to be in appropriate agreement with the conventional design. From this point, it also indicates that such a traditional design may be of the most uniform stress distribution.

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Figure 17. Connection system design with the overall efficiency criteria (V/Vo = 40 per cent)

It is worth noting that the overall efficiency scheme and extreme efficiency scheme produce different topologies of the attached components, compare Figure 17(b) with Figure 18(b). It seems that the overall scheme gives more complex topologies than the extreme scheme, in particular near the overlapped connection region. Seemingly, such a coupled optimization procedure gives a more meaningful methodology to practical design issues. It not only reduces the design time (two design cycles down to one cycle), but also avoids the problem of a conditional or local optimum due to the separation of both design processes.


Figure 18. Connection system design with extreme efficiency criteria (V/Vo = 40 per cent)

5. Concluding remarks This paper extends the evolutionary structural optimization method to the design of multi-component systems. To have the interconnection elements carry as uniform a stress as possible, stress levels of all candidate connection elements are employed to estimate the relative efficiency. With this concept, the absence and presence of an interconnection is determined in terms of its relative efficiency. In the optimization process, less efficient connection elements are gradually removed from the structural system by following a rule based evolutionary procedure. This significantly simplifies the optimization process and makes the algorithm easy to be implemented into various design platforms.

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In addition to achieving even utilization of the fasteners the ESO technique can simultaneously be applied to determine the optimum topology of the connected components as well as the connection topology. Such a simultaneous topology optimization, when fully implemented into commercial optimization software, offers designers the opportunity to more fully achieve optimum designs for realistic engineering objects. To deal with multiple load cases, the overall efficiency scheme and the extreme efficiency scheme are presented in the paper. The former emphasizes the average extent of use under various load cases, while the latter reflects the highest stress or lowest strength in different load cases. These two schemes may lead to different connection patterns or component topologies. The choices of the schemes depend on the design requirements. Under some circumstances, designers should compare and analyze both solutions to determine the most suitable one. The definition of the connection can be as broad as possible herein, which can include various fasteners such as pins, rivets, bolts, spot-welds, adhesive bonds, that depends on the areas of application. Although eight-node brick elements are employed to model the interconnection characteristics of the structural system, it is easy to be replaced by other element types when necessary. This work develops a very promising application area and useful methodology for design engineers. References and further reading Bendsùe, M.P. (1995), Optimization of Structural Topology, Shape, and Material, Springer-Verlag, Berlin. Cai, W., Hu, S.J. and Yuan, J.X. (1996), ``Deformable sheet metal fixturing: principles, algorithms and simulations’’, ASME Transactions: Journal of Manufacturing Science & Engineering, Vol. 118, pp. 318-24. Chickermane, H. and Gea, H.C. (1997), ``Design of multi-component structural systems for optimal layout topology and joint locations’’, Engineering with Computers, Vol. 13, pp. 235-43. Chickermane, H., Gea, H.C., Yang, R.J. and Chung, C.H. (1999), `Òptimal fastener pattern design considering bearing loads’’, Structural Optimization, Vol. 17, pp. 140-6. Chirehdast, M. and Jiang, T. (1996), `Òptimal design of spot-weld and adhesive bond patterns’’, International Congress & Exposition 1996, SAE Paper No. 960812. Gea, H.C. (1996), ``Topology optimization: a new micro-structure based design domain method’’, Computers & Structures, Vol. 61, pp. 781-8. Hassani, B. and Hinton, E. (1999), Homogenization and Structural Topology Optimization, Springer-Verlag, Berlin. Hinton, E. and Owen, D.R.J. (1979), An Introduction to Finite Element Computation, Pineridge Press, Swansea. Hinton, E. and Sienz, J. (1995), ``Fully stressed topological design of structure using an evolutionary procedure’’, Engineering Computations, Vol. 12, pp. 229-44. Hinton, E., Sienz, J., Bulman, S. and Lee, S.J. (1998), ``Fully integrated design optimization of engineering structures using adaptive finite element models’’, in Steven, G.P., Querin, O.M., Guan, G. and Xie, Y.M. (Eds), Proceedings of the First Australasian Conference on Structural Optimization, Keynote paper, pp. 73-8.

Jiang, T. and Chirehdast, M. (1997), `À system approach to structural topology optimization: designing optimal connections’’, ASME Transactions: Journal of Mechanical Design, Vol. 119, pp. 40-7. Li, Q. (2000), `Èvolutionary structural optimization for mechanical and thermal problems’’, PhD dissertation, The University of Sydney, Sydney, p. 314. Li, Q., Steven, G.P. and Xie, Y.M. (1997), ``Connection topology design using the evolutionary structural optimization method’’, Research Progress Report, Finite Element Research Centre, The University of Sydney, Sydney. Li, Q., Steven, G.P. and Xie, Y.M. (2000), `À simple checkerboard suppression alogorithm for evolutionary structural optimization’’, Structural Optimization, in press. Liang, Q., Xie, Y.M. and Steven, G.P. (1999), `Òptimal selection of topologies for the minimumweight design of continuum structures with stress constraints’’, to appear in Journal of Mechanical Engineering Science: Part C of Proceedings of the Institution of Mechanical Engineers. Menassa, R.J. and DeVries, W.R. (1989), ``Locating point synthesis in fixture design’’, Annals of the CIRP, Vol. 38, pp. 165-9. Menassa, R.J. and DeVries, W.R. (1991), `Òptimization methods applied to selecting support positions in fixture design’’, ASME Transactions: Journal of Engineering for Industry, Vol. 113, pp. 412-18. Orthwein, W.C. (1986), ``Finding principal stresses in three dimensions’’, Computers in Mechanical Engineering, pp. 51-8. Papadrakakis, M., Tsompanakis, Y., Hinton, E. and Sienz, J. (1996), `Àdvanced solution methods in topology optimization and shape sensitivity analysis’’, Engineering Computations, Vol. 13, pp. 57-90. Querin, O.M., Steven, G.P. and Xie, Y.M. (1998), `Èvolutionary structural optimization using a bidirectional algorithm’’, Engineering Computations, Vol. 15, pp. 1031-48. Querin, O.M., Steven, G.P. and Xie, Y.M. (1999), ``Development of a performance indicator for structural topology optimization’’, to appear in Structural Optimization. Rosko, P. (1995), ``Three-dimensional topology design of structures using crystal models’’, Computers & Structures, Vol. 55, pp. 1077-83. Rozvany, G.I.N., Bendsùe, M.P. and Kirsch, U. (1995), ``Layout optimization of structures’’, Applied Mechanics Review, Vol. 48, pp. 41-118. Steven, G.P., Li, Q. and Xie, Y.M. (1999), ``Connection topology optimization by evolutionary methods’’, Proceedings of NAFEMS World Congress ’99, Vol. 2, pp. 901-11. Steven, G.P., Querin, O.M., Guan, H. and Xie, Y.M. (1998), Structural Optimization, Oxbridge Press, Melbourne. Xie, Y.M. and Steven, G.P. (1993), `À simple evolutionary procedure for structural optimization’’, Computers & Structures, Vol. 49, pp. 885-96. Xie, Y.M. and Steven, G.P. (1994), `Òptimal design of multiple load case structures using an evolutionary procedure’’, Engineering Computations, Vol. 11, pp. 295-302. Xie, Y.M. and Steven, G.P. (1997), Evolutionary Structural Optimization, Springer-Verlag, Berlin. Yang, X.Y., Xie, Y.M., Steven, G.P. and Querin, O.M. (1999), ``Bidirectional evolutionary method for stiffness optimization’’, AIAA Journal, Vol. 37, pp. 1483-8. Young, V., Querin, O.M., Steven, G.P. and Xie, Y.M. (1999), ``3D and multiple load case bidirectional evolutionary structural optimization (BESO)’’, Structural Optimization, Vol. 18, pp. 183-92.
