Design complexity control in truss optimization

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Design complexity control in truss optimization André J. Torii*

·

Rafael H. Lopez

·

Leandro F.F. Miguel

Received: date / Accepted: date

Truss optimization based on the ground structure approach often leads to designs that are too complex for practical purposes. In this paper we present an approach for design complexity control in truss optimization. The approach is based on design complexity measures related to the number of bars (similar to Asadpoure et al (2015)) and a novel complexity measure related to the number of nodes of the structure. Both complexity measures are continuously dierentiable and thus can be used together with gradient based optimization algorithms. The numerical examples show that the proposed approach is able to reduce design complexity, leading to solutions that are more t for engineering practice. Besides, the examples also indicate that in some cases it is possible to signicantly reduce design complexity with little impact on structural performance. Since the complexity measures are non convex, a global gradient based optimization algorithm is employed. Finally, a detailed comparison to a classical approach is presented. Abstract

A.J. Torii* Department of Scientic Computing, Federal University of Paraíba-UFPB, Brazil Tel.: +55-83-32167393 Fax: +55-83-32167393 E-mail: [email protected] Present address: Cidade Universitária, João Pessoa, PB, Brazil, 58051-900

R.H. Lopez Center for Optimization and Reliability in Engineering (CORE), Department of Civil Engineering, Federal University of Santa Catarina-UFSC, Brazil E-mail: [email protected] L.F.F. Miguel Center for Optimization and Reliability in Engineering (CORE), Department of Civil Engineering, Federal University of Santa Catarina-UFSC, Brazil E-mail: [email protected] *corresponding author

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truss optimization · structural optimization · complexity · global optimization Keywords

1 Introduction

Due to its vast range of practical applications, optimization of truss structures has been widely studied by several researchers. Since the pioneer works by Michell (1904); Dorn et al (1964); Fleron (1964); Sved and Ginos (1968); Romstad and Wang (1968); Dobbs and Felton (1969); Pedersen (1970, 1972); Hemp (1973) to the more recent fundamental works by Bendsøe et al (1994); Zhou (1996); Kocvara and Zowe (1996); Cheng and Guo (1997); Ben-Tal et al (2000); Stolpe and Svanberg (2001); Achtziger (2007), to name a few, truss optimization still receives signicant attention from researchers around the world. The reader is referred to the works by Bendsøe et al (1994) and Achtziger (2007) for a time-line of important developments in the eld. The most widespread strategy to tackle the problem was introduced by Dorn et al (1964) and is known in literature as ground structure approach (see also Bendsøe et al (1994)). This approach consists in working with an universal truss with members connecting a set of (possibly all) nodes in a grid. The optimization procedure is then employed to change the cross sectional areas and possibly remove bars in order to optimize some parameter chosen by the designer. The ground structure approach still is the most widely spread technique, even though other strategies have been proposed in the past years (e.g. growth based methods (Martínez et al, 2007; Hagishita and Ohsaki, 2009)). However, a delicate issue may arise when the ground structure approach is applied: the optimum design often becomes too complex for practical purposes. This complexity is generally related to an excessive number of nodes and bars in the optimum structure. Indeed, in several cases, optimum designs provided by truss optimization algorithms are composed by a dense net of bars and nodes that are simply too complex to be used in practice. Take, for example, some of the optimum designs presented by Bendsøe et al (1994), which are reproduced in Fig. 1. These examples show that designs obtained with the ground structure approach often become too complex for practical applications. We then ask ourselves: is it possible to obtain less complex structural designs that are still ecient structural solutions? One approach to overcome this diculty is to limit the design domain of the problem, i.e. to work with a simplied ground structure (see the work by Bendsøe et al (1994)). In practice, one can control design complexity by choosing ground structures of dierent complexity. However, this approach is highly dependent on the designer's experience and intuition. Design complexity was also addressed in the work by He and Gilbert (2015), that describe two interesting approaches to tackle this issue (note that He and Gilbert (2015) refer to the subject as rationalization of truss optimum designs). The rst strategy is to apply geometry optimization together with topology optimization (i.e. also take nodal positions as design variables (Peder-

Design complexity control in truss optimization

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sen, 1972; Bendsøe et al, 1994)), since it has been observed that this approach is able to reduce design complexity in practice. In fact, Achtziger (2007) also concluded that xed nodal positions can lead to structures with many bars, an issue that can be avoided by the use of geometry optimization. However, this approach enforces the use of geometry optimization, what may not be desirable in every case and can lead to some computational diculties (e.g. node melting (Achtziger, 2007)). We also note that this approach actually reduces design complexity to some extent, in the sense that several bars of minor importance may be really removed, but the designer is not able to adjust the level of design complexity control enforced. In other words, one is not able to choose how important design complexity is in comparison to structural performance. The second strategy discussed by He and Gilbert (2015) was presented in the works by Prager (1974) and Parkes (1975) and was originally conceived for the solution of analytical problems. The idea is to penalize bars lengths (i.e. to use increased bars lengths) during volume evaluation, so that shorter bars become less advantageous (see Appendix C). This approach benets from the fact that the optimization problem is only slightly modied (i.e. linearity of the objective function according to cross sectional areas is preserved), what has obvious advantages from both the theoretical and computational point of view. However, He and Gilbert (2015) observed that the approach is not always eective. In this work it has been observed (see the examples) that the approach proposed by Prager (1974) and Parkes (1975) is ineective when the ground structure is composed by bars of similar length (i.e. shorter bars are not so short in comparison to the longer bars), since the approach is based on the penalization of shorter bars. Besides, since modied bars lengths are used to evaluate the volume of the structure, the approach cannot be extended to problems considering volume constraints. Asadpoure et al (2015) also addressed design complexity control, by taking into account fabrications costs related to the number of bars of the structure. In this case, the number of active bars of the design is approximately counted using a regularized Heaviside function. This leads to design complexity reduction as the fabrication costs per bar are increased. Besides, the approximation used for the number of bars is continuously dierentiable, and the resulting problem can be eciently solved using gradient based optimization methods. In this work, an approach to control design complexity in truss optimization is discussed. The main idea is to build continuously dierentiable functions that are able to measure design complexity, in a similar manner to that presented by Asadpoure et al (2015). Here, we discuss two design complexity measures, related to the number of bars and to the number of nodes, respectively. The rst is very similar to that presented by Asadpoure et al (2015), while the second (related to the number of nodes) is a novel design complexity measure. Besides, a global optimization algorithm is employed to solve the resulting structural optimization problem, since the complexity measures (as occurs in the approach proposed by Asadpoure et al (2015)) are non convex. Finally, we present a detailed comparison to the approach presented by Prager (1974) and Parkes (1975) and show that it is not eective in several cases of

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practical interest. This justies the proposal of new strategies to control design complexity in truss optimization. Here we address only the compliance minimization problem with cross sectional areas as design variables. This allows the study to focus only on the proposed design complexity measures, by avoiding several delicate issues that are not of interest here, such as singular solutions (Kirsch, 1990; Cheng and Guo, 1997), correct denition of local buckling (Rozvany and M.Zhou, 1991; Rozvany, 1996; Achtziger, 1999a,b; Torii et al, 2015) and kinematic stability (Bendsøe et al, 1994), to name a few. Extensions of the proposed design complexity measures to other problems regarding truss optimization is straightforward. This paper is organized as follows. In Section 2, we present two design complexity measures based on the number of active bars and nodes in the structure. In Section 3, we rewrite the standard truss compliance minimization problem in order to use these measures for design complexity control. Then, in Section 4, four numerical examples are presented in order to show the eectiveness of the proposed approach. The conclusions of this work are presented in Section 5. The derivatives of the design complexity measures proposed, the global optimization algorithm used here and a brief explanation on Prager (1974) and Parkes (1975) approach are presented in the Appendices. 2 Complexity Measures

2.1 Number of bars The rst step in the construction of the design complexity measures is to dene whether a bar is active or inactive. Here, we consider that a given bar of the structure is inactive when its cross sectional area is smaller than a given threshold, such that it barely aects the structural performance if removed (see the work by Asadpoure et al (2015) for a similar denition). A design complexity measure regarding the number of bars can then be built by counting the number of active bars inside the structure. This approach can be easily employed if discrete optimization algorithms are used. However, this approach cannot be used in the context of continuous optimization, since such a counter does not depend continuously on the design variables of the problem. For this reason, we dene the following complexity measure:

Cb (x) =

m X

c(xi ),

(1)

i=1

where Cb is a design complexity measure related to the number of bars in the structure, m is the number of bars in the ground structure, xi are the cross sectional areas of the bars (taken here as design variables) and c is a function that indicates whether a bar is active or not. The trick is to dene the function c to be continuous and dierentiable, so that continuous optimization algorithms may be employed. In this work, we take c as

Design complexity control in truss optimization

c(t) = 1 −

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1 , (t − ε + 1)a

t ≥ ε,

(2)

with a ≥ 0 and ε ≥ 0. We point out that the resulting complexity measure Cb is very similar to the expression presented by Asadpoure et al (2015). The function c(t) is presented in Fig. 2 for ε = 0.1 and a = 1,2,4. Note that c(ε) = 0 and limt→∞ ci (t) = 1. Consequently, it can be used to indicate if a given bar is active or not in an approximate manner. As the cross sectional area xi is increased, c(xi ) approaches unity and the bar is counted as active in Eq. (1). If the cross sectional area is reduced c(xi ) is reduced to small values and the bar is not counted (actually it is barely counted). Besides, the exponent a can be changed in order to adjust how rapidly c(t) approaches unity when t is increased. In practice, the complexity measure becomes more accurate as a is increased (i.e. only very small cross sectional areas are not counted as active). Finally, we recommend setting the parameter ε equal to the lower bound used for the cross sectional areas, in order to ensure that bars that achieve the lower bound are not counted as active. Of course Cb will not capture the exact number of active bars in the structure. However, this is not the main goal here. What matters is that it is ecient for controlling the complexity of the optimum designs, as already observed by Asadpoure et al (2015). Finally, sensitivity analysis of Cb is very simple, as presented in Appendix A. 2.2 Number of nodes The main novelty of this paper is a complexity measure based on the number of nodes of a given structure. In this case, we dene a node as inactive when only inactive bars are connected to it. That is, if some bar connected to a given node is active, then the node is considered active as well. Again, if discrete optimization algorithms are employed, this measure can be easily evaluated. For the continuous case, we rst dene the sum of the cross sectional areas of the bars connected to a given node i as X bi (x) = xj , (3) j∈Ni

where Ni is the set of bars connected to node i. If some node is inactive, summing the cross sectional areas connected to it in Eq. (3) should give a small value. Consequently, it is possible to approximately evaluate whether some node is active or not by using the same counter function from Eq. (2), i.e. by evaluation of c(bi ). Consequently, the number of active nodes can be approximately counted as

Cn (x) =

q X i=1

c(bi (x)),

(4)

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where Cn is a design complexity measure related to the number of nodes, q is the total number of nodes, and c is the counter function as dened in Eq. (2). We point out that this complexity measure was not presented before. The idea of Eq. (4) is to rst evaluate if the cross sectional areas connected to some node are signicant, by means of bi . Then, we identify whether the node is active or not by evaluating c(bi ). In this case, a node will be considered inactive when the total sum of cross sectional areas connected to it are reduced to small values close to ε. This only occurs if all bars connected to a given node have small cross sectional areas (i.e. become inactive). On the other hand, if the cross sectional areas connected to a given node increase to signicant values, then the total sum from Eq. (3) also increases and the node is counted as active. Again, the number of nodes is counted approximately by Eq. (4), yet the quantity is useful for controlling the complexity of the structural design using a continuous dierentiable function. Sensitivity analysis of this design complexity measure is also described in Appendix A. Finally, in the case of Eq. (4), we recommend setting the parameter ε (used for evaluation of c(t)) equal to the number of bars converging to each node multiplied by the lower bound dened for the cross sectional areas. This ensures that if all bars converging to a given node achieve the lower bound, then the node is not counted as active. This requires setting dierent values of ε for each node, depending on the number of bars that converge to each node. 3 Optimization Problem

The truss optimization problem can be posed in several dierent forms. In this work, we study the case of compliance minimization, in order to focus on the proposed complexity measures. For the same reason, we take only cross sectional areas as design variables. The optimization problem can then be stated as

min W (x, u) = f (x)T u = uT K(x)u

(5)

subject to a volume constraint

g(x) = V (x) − V ≤ 0,

(6)

bounds constraints

xl ≤ xi ≤ xu ,

(i = 1, 2, ..., m)

(7)

and equilibrium constraints

K(x)u = f (x),

(8)

where W is the structural compliance, xi are the design variables (cross sectional areas), K is the stiness matrix, f is the vector of applied forces, u is the

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vector of nodal displacements, V is the structural volume, V is the maximum volume of material to be used, xl and xu are lower and upper bounds on the design variables. Note that we do not remove members with small cross sectional areas, but only set a small lower bound for it, in order to avoid singular stiness matrices related to kinematic unstable structures. The problem given by Eqs. (5)-(8) has been extensively studied in literature. It was demonstrated that it is equivalent to a Linear Programming problem (volume minimization with stress constraints (Hemp, 1973; Bendsøe et al, 1994)) and, consequently, all local minima form a convex set and have the same value of the objective function. In other words, all local minima are also global minimum of the problem and thus global optimization algorithms are not necessary. Besides, if the approach proposed by Prager (1974) and Parkes (1975) is used for design complexity control (see Appendix C), linearity of the volume of material according to the cross sectional areas is preserved. Design complexity can be taken into account in several ways. Here, we propose the penalization of design complexity in the objective function of the original problem, which leads the new objective function

min F (x) =

C(x) W (x) +α , W0 C0

(9)

where α is a penalty parameter, C is some complexity measure (Cb or Cn ), and W0 and C0 are reference values for the compliance and complexity (usually taken as initial values of the original structure being optimized). Normalized values of the compliance and the complexity measures are employed since these quantities can have dierent orders of magnitude. The problem is then solved with the objective function from Eq. (9) with the constraints from Eqs. (6)-(8). Since the objective function was modied, the problem is no longer equivalent to a Linear Programming problem. It is also important to point out that both complexity measures from Eqs. (1) and (4) are non convex functions of x, since c(t) as dened in Eq. (2) is non convex (as is the case of the regularized Heaviside function used by Asadpoure et al (2015)). Consequently, the problem becomes non-convex and local optima may exist. In fact, the authors observed the existence of local optima in some examples studied. This is probably the main drawback of the proposed approach which should be subject of a future work. In this work, global optimization was carried with the algorithm described in Appendix B. Finally, note that other approaches to take into account design complexity would be to include complexity constraints (see Asadpoure et al (2015)) or approach the problem from a multi-objective perspective. Here we simply penalize the original objective function to keep the analysis as simple as possible and to focus on the proposed complexity measures. For this reason the parameter α was not included to have some particular physical meaning (as opposed to the approach presented by Asadpoure et al (2015), where the number of bars is multiplied by a fabrication cost per bar).

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4 Examples

In this section, we analyze four examples in order to show the eectiveness of the proposed approach in controlling design complexity. In all examples, the elastic modulus is E = 1, the applied load is P = 1, xl =1E-6, xu = 200, a = 1, the volume constraint is given by the volume of the original design V = V0 (i.e. we search for structures with the same volume of the original design) and the reference values for C0 and W0 (needed for normalization purposes in Eq. (9)) are obtained from the original structure. Finally, bars with cross sectional areas smaller than 10 × xl are considered as inactive and are not presented in the gures nor counted in the results. In order to obtain the global optimum of the optimization problems we employed the algorithm described in Appendix B. This algorithm is very similar to the restart procedure rst proposed by Luersen et al (2004) and already applied to dierent engineering problems (Luersen and Le Riche, 2004; Luersen et al, 2006; Ritto et al, 2011; Lopez et al, 2014), including truss optimization (Torii et al, 2011). In all examples, k = 40 (the number of restarts of the local search after local convergence) and M = 200 (the number of random trial points generated before each restart, used to choose the next starting point by means of Eq. (19)) were employed (see Appendix B). The local searches were performed using the interior-point algorithm (Nocedal and Wright, 1999; Luenberger and Ye, 2008) available in MATLAB with convergence criteria equal to 1E-6. The global optimization algorithm is not used in the case that complexity measures are not taken into account, since the resulting problem is convex. Finally, when Parkes' approach (as we call the approach proposed by Prager (1974) and Parkes (1975) from now on) is used we search for a structure with minimum volume subjected to stress constraints. In this case the allowable stress is dened so that the optimum design has the same volume as when the proposed approach is applied.

4.1 Example 1 The rst example is presented in Fig. 3. Only neighbor nodes are linked by bars and a 5x5 grid of nodes is used. Consequently, the ground structure is composed by 72 bars and 25 nodes. The initial cross sectional areas are taken as xi = 5, and thus the initial volume of the structure is V0 = 106.568549. The compliance of the original structure is W0 = 0.177478. We search for a structure with the same volume of material and with minimum compliance. The optimum designs obtained with dierent values of α are presented in Fig. 4. We note that the case of α = 0 corresponds to no complexity control (i.e. the original compliance minimization problem with volume constraint). The optimum design obtained using Parkes' approach with s = 1.0 is also presented in Fig. 4. Besides, the structures obtained with α = 20.0 using both Cb and Cn are very similar. More details are presented in Table 1 and Table 2.

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These results indicate that we can control the complexity of the designs by increasing the value of the penalization parameter α. On the other hand, penalization of design complexity by increasing values of α leads to more exible designs. In others words, one is able to reduce design complexity, but this reduction likely comes at the cost of a more exible optimum structure. Note that the structures obtained with α = 1.0, however, are much less complex than and almost as sti as the one obtained without complexity control (this observation does not hold for the structures obtained with α = 20.0). This indicates that reduction of design complexity does not necessarily leads to severe reduction of structural performance. Finally, note that the designs obtained with Cn are, in general, dierent from those obtained with Cb . For comparison purposes, this problem was also solved using Parkes' approach, by taking s = 0.1, 1.0, 10.0 (see Appendix C). All three optimum designs obtained with Parkes' approach are almost identical and thus only the one obtained with s = 1.0 is presented in Fig. 4. More details are presented in Table 3. From these results we note that Parkes' approach was not able to reduce design complexity in this case. In fact, the nal design is somewhat insensitive on the parameter s and similar to the one obtained without design complexity control. This happens because, in this example, design complexity cannot be reduced simply by avoiding shorter bars, since most bars have similar lengths. We also conclude that, in some cases, avoidance of shorter bars does not necessarily reduces design complexity.

4.2 Example 2 The second example is presented in Fig. 5. Only neighbor nodes are linked by bars. Consequently, the ground structure is composed by 198 bars and 64 nodes. The initial cross sectional areas are taken as xi = 5, and thus the initial volume of the structure is V0 = 392.131300. The compliance of the original structure is W0 = 6.098322. We search for a structure with the same volume of material and with minimum compliance. The optimum designs obtained with Cb and Cn (α = 0.1), no complexity control (α = 0) and Parkes' approach (s = 0.1) are presented in Fig. 6. More details are presented in Table 4. We note that the solution obtained using Parkes' approach with other values of s (e.g. s = 1.0) are very similar to the one obtained with s = 0.1. Again, the proposed approach was able to eectively reduce design complexity. The design obtained with Parkes' approach is also very similar to the one obtained with α = 0. We also point out that the design obtained with Cn is dierent from that obtained with Cb . Besides, the structural compliance of the designs obtained with α = 0.1 is almost identical to the one obtained with α = 0. This example clearly indicates that design complexity control does not necessarily reduce structural performance.

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4.3 Example 3 In this example a simply supported beam is studied. Because of symmetry, only half structure is modeled, as shown in Fig. 7. All nodes are linked to one another, resulting in a total of 28 nodes and 378 bars. The initial cross sectional areas are taken as xi = 1, and thus the initial volume of the structure is V0 = 374.814003. The compliance of the original structure is W0 = 0.645194. The solutions obtained using Cb and Parkes' approach are presented in Fig. 8. More details are presented in Tables 5 and 6. We note that the solutions obtained with Cn (α = 0.1, 1.0, 10.0) are very similar to the ones obtained with Cb (α = 0.1, 0.5, 5.0) (i.e. have the same topology and similar compliance). For this reason, only the structures obtained with Cb are presented. We point out that the optimum design obtained without design complexity control (i.e. α = 0) has several superposed bars. This is the reason why 41 bars are counted as active in Table 5 while a lower number of bars appear in Fig. 8. Other local optima without the superposed bars and with the same compliance can also be found (see Section 3). However, the original compliance minimization problem given by Eqs. (5)-(8) do not distinguish between local optima containing or not superposed bars. The design obtained with α = 0.1, on the other hand, is almost identical to the one obtained without complexity control (i.e. have similar topology and the same compliance), but do not have superposed bars. Again, we see that by increasing the value of the penalization parameter α one is able to reduce design complexity. Besides, the structural compliance of the design obtained using α = 5.0 is about 6.5% higher in comparison to the one obtained with α = 0 (no complexity control), but the design is much simpler. In fact, it is a classical solution often obtained with coarser grids. Again, this indicates that it is possible to reduce design complexity and still obtain an ecient solution. Finally, we note that Parkes' approach is also eective in this example. 4.4 Example 4 The last example is shown in Fig. 9. All nodes are linked to one another, resulting in a total of 35 nodes and 595 bars. The initial cross sectional areas are taken as xi = 1, and thus the initial volume of the structure is V0 = 475.053763. The compliance of the original structure is W0 = 0.213581. The designs obtained using Cb are presented in Fig. 10. The designs obtained using Cn are very similar to the ones obtained with Cb with the same values of α and thus are not presented. More details are presented in Table 7. From these results, we note that the proposed approach was once again able to reduce the complexity of the optimum designs. Besides, note that the structures obtained with α = 5.0 are composed by only 4 bars/nodes and have compliance only 5.5% higher than the structure obtained with α = 0. This indicates that the structures obtained considering design complexity are

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still ecient in comparison to the one obtained without complexity control. We also note that, as occurred in the previous example, the optimum design obtained without complexity control has several superposed bars. The designs obtained with Parkes' approach are presented in Fig. 11 and Table 8. Note that the designs obtained with s = 0.01, 0.2, 0.5 are very similar to the ones obtained with the proposed approach. However, the design obtained with s = 0.10 has the same number of bars as the one obtained with s = 0.01, but has a higher structural compliance. This shows that avoidance of shorter bars (i.e. Parkes' approach) dot not necessarily enforce design complexity reduction. From this point of view we can say that Parkes' approach may not be consistent for design complexity control in some cases, as already observed by He and Gilbert (2015).

5 Conclusions

In this paper, we presented an approach for design complexity control in truss optimization. Two design complexity measures were built as continuous differentiable approximations for the total number of nodes and bars. These measures were then included in the optimization problem using penalization of the objective function. The complexity measure related to the number of bars nd a similar counterpart in the recently published paper by Asadpoure et al (2015), while the complexity measure related to the number of nodes is a novel concept. The numerical examples show that the proposed approach is able to reduce design complexity in truss optimization, leading to solutions that are more t for engineering practice. The rst two examples presented show that the proposed approach is eective for controlling design complexity even in cases where Parkes' approach fails. Besides, the designer is able to enforce different levels of design complexity control by adjusting the penalization factor. We also point out that the two complexity measures presented can lead to dierent optimum designs. The third and fourth examples demonstrate that the proposed approach is also able to reproduce the solutions obtained with Parkes' approach (when this approach is eective). Note that taking into account design complexity can lead to optimum designs with higher structural compliance. Consequently, the designer must choose the most advantageous compromise between design complexity and structural eciency. However, the examples presented show that reduction of design complexity does not necessarily lead to signicant reduction of structural performance (measured here using the structural compliance). In other words, it is possible to signicantly reduce design complexity with little impact on structural performance in some cases. The impact on the structural performance, of course, is expected to depend on the value of α. In the examples we observed that α in the range 0.01 − 0.1 leads, in general, to small impacts on structural compliance. However, the choice of appropriate values for the

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penalization parameter depends on the problem being addressed and we thus recommend testing for dierent values of α. The main drawback of the proposed approach is that the design complexity measures adopted are not convex. In the case of compliance minimization with cross sectional areas as design variables, this is a serious drawback, since the original problem is very well posed (i.e. it is equivalent to a Linear Programming problem). However, in several problems regarding truss optimization (e.g. shape optimization and some problems with nonlinear constraints) the original problem may be already non-convex. In these cases, global optimization schemes are already necessary and non-convexity of the complexity measures becomes less important. Still, the search for better behaved design complexity measures is recommended as subject for future works. Some other aspects that deserve further investigation are related to: building of other complexity measures (e.g. related to bars superpositions and bar intersections); enforcing design complexity control in a dierent way (e.g. use of design complexity constraints or a multi-objective approach); applying the approach to other problems concerning truss and frame optimization (e.g. shape optimization, local and global stability constraints). The authors would like to thank CNPq and CAPES for the nancial support of this research. Acknowledgements

References

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Hagishita T, Ohsaki M (2009) Topology optimization of trusses by growing ground structure method. Struct Multidisc Optim 37(4):377393 He L, Gilbert M (2015) Rationalization of trusses generated via layout optimization. Struct Multidisc Optim 52(4):677694 Hemp W (1973) Optimum structures. Clarendon Press, Oxford Kirsch U (1990) On singular topologies in optimum structural design. Struct Optim 2(3):133142 Kocvara M, Zowe J (1996) How mathematics can help in design of mechanical structures. In: Griths D, Watson G (eds) Numerical analysis, Longman Scientic and Technical, Harlow, pp 7693 Lopez RH, Ritto TG, Sampaio R, Cursi JESD (2014) A new algorithm for the robust optimization of rotor-bearing systems. Eng Optim 46(8):11231138 Luenberger DG, Ye Y (2008) Linear and Nonlinear Programming, 3rd edn. Springer, New York Luersen MA, Le Riche R (2004) Globalized nelder-mead method for engineering optimization. Comput Struct 82(23-26):22512260 Luersen MA, Riche R, Guyon F (2004) A constrained, globalized and bounded nelder-mead method for engineering optimization. Struct Multidisc Optim 27:4354 Luersen MA, Riche RL, Lemosse D, Maître OL (2006) A computationally ecient approach to swimming monon optimization. Struct Multidisc Optim 31(6):488496 Martínez P, Martín P, OMQuerin (2007) Growth method for size, topology, and geometry optimization of truss structures. Struct Multidisc Optim 33(1):1326 Michell A (1904) The limits of economy of material in frame-structures. Phil Mag 8(47):589597 Nocedal J, Wright SJ (1999) Numerical Optimization. Springer, New York, Parkes E (1975) Joints in optimum frameworks. Int J Solids Struct 11(9):1017 1022 Pedersen P (1970) On the minimum mass layout of trusses. In: AGARD Conference Proceedings n. 36 Pedersen P (1972) On the optimal layout of multi-purpose trusses. Comput Struct 10:803805 Prager W (1974) A note on discretized michell structures. Comput Methods Appl Mech Eng 3(3):349?355 Ritto TG, Lopez RH, Sampaio R, Cursi JESD (2011) Robust optimization of a exible rotor-bearing system using the campbell diagram. Eng Optim 43(1):7796 Romstad K, Wang C (1968) Optimum design of framed structures. J Struct Div ASCE 94:28172845 Rozvany G (1996) Diculties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optim 11(3-4):213217 Rozvany G, MZhou (1991) A note on truss design for stress and displacement constraints by optimality criteria methods. Struct Optim 3(1):4550

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Appendix A: Derivatives of Complexity Measures

The derivative of Cb according to a given design variable is given by

∂c(xi ) ∂Cb , = ∂xi ∂t

(10)

where

∂c a = , ∂t (t − ε + 1)1+a

t ≥ ε.

(11)

The derivative of Cn according to a given design variable is given by n

X ∂c ∂bj ∂Cn = . ∂xi ∂bj ∂xi j=1

(12)

The derivative ∂c/∂bj is as presented in Eq. (11). The derivative ∂bj /∂xi , on the other hand, is simply

∂bj = ∂xi



1, if i ∈ Nj 0, if i ∈ / Nj

(13)

Note that both derivatives are very simple to evaluate and should not lead to signicant increases in the computational eort required for sensitivity analysis.

Design complexity control in truss optimization

15

Appendix B: Global Optimization Algorithm

The global optimization algorithm used in this paper is based on starting local searches from dierent initial design vectors. The algorithm is very similar to the restart procedure rst proposed by Luersen et al (2004) and already applied for truss optimization by Torii et al (2011). We rst start the local search with an initial design vector x0 to obtain an optimum design x∗0 . Both x0 and x∗0 are then included in a set of design vectors (14)

S0 = {x0 , x∗0 },

that will store design vectors where we already started/ended local searches. The next step is to randomly generate a set of M trial design vectors yi (15)

T = {y1 , y2 , ..., yM }

satisfying the bounds on the design variables, but not necessarily satisfying the nonlinear constraints (check of nonlinear constraints are avoided because of the computational costs involved). The average distance from all trial vectors yi ∈ T to all vectors in S0 is then evaluated, i.e.,

d (yi , S0 ) =

X 1 kyi − xk, card(S0 )

i = 1, 2, ..., M,

(16)

x∈S0

where card(S0 ) stands for cardinality of S0 (i.e. number of elements in S0 ) and k.k is some vector norm (in this work Euclidean norm is used). We then take the next starting point as the one in T with maximum average distance from the vectors in S0 , i.e.,

x1 = arg max

y∈T

d(y, S0 ).

(17)

In this way we choose a starting point that is, in the average sense, the most distant from the ones in S0 . A new local search is then started with x1 to obtain an optimum design x∗1 and both vectors are included in a new set S1 (together with x0 and x∗0 ). A new set of trial vectors T is randomly generated and the process is repeated. After k restarts (k + 1 local searches) we have the set of starting/found design vectors

Sk = {x0 , x∗0 , x1 , x∗1 , ..., xk , x∗k },

(18)

we randomly generate a set of trial vectors T and choose the new starting design vector as the one that maximizes

xk+1 = arg max

y∈T d(y, Sk ).

(19)

The parameters required to use this algorithm are the number of local searches k to be made and the number of trial vectors M generated before each restart. The local searches can be made using any optimization algorithm.

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André J. Torii* et al.

In this paper an interior point algorithm is used (Nocedal and Wright, 1999; Luenberger and Ye, 2008). The only dierence between this algorithm and the one proposed by Luersen et al (2004) is that here we take the trial vector yi ∈ T that maximizes the average distance from the ones in Sk . In the original approach the new starting design vector was chosen as the one that minimizes a normal multidimensional probability density function, representing the probability of arriving at some point from Sk when starting from some trial vector yi ∈ T .

Appendix C: Prager-Parkes' approach

The approach presented by Prager (1974) and Parkes (1975) was originally conceived for problems based on volume minimization. The idea is to replace Pm the real volume of the structure, given by V = i=1 li xi (where li are the bars lengths), by the modied volume

Vp =

m X (li + s)xi ,

(20)

i=1

where s > 0 is a parameter named by Parkes (1975) as joint radius. In this way, shorter bars are penalized in comparison to longer ones. In order to see this fact, it is interesting to rewrite Vp as

Vp =

m X

φ(li )li xi ,

(21)

  s . 1+ li

(22)

i=1

with

φ(li ) =

This expression puts in evidence that the approach is the same as multiplying the volume of each bar li xi by a unitary cost per volume φ(li ). This cost per volume, however, is smaller the longer the bar is (i.e. limli →∞ φ(li ) = 1). On the other hand, for shorter bars we have limli →0 φ(li ) = ∞, i.e. shorter bars have increased unitary costs. In this way, the optimization algorithm will try to avoid shorter bars, that are comparatively more expensive than longer ones. It is important to point out that this approach was originally developed in order to tackle analytical solutions.

Design complexity control in truss optimization

Fig. 1

17

Examples of complex optimum designs (reproduced from Bendsøe et al (1994))

a=4

a=2 a=1

c(t)

t Fig. 2

Counter function ci (t) for dierent values of a

Fig. 3

Example 1: Ground structure

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André J. Torii* et al.

α=0

Cb α=1

Cb α = 20

Parkes s=1

Cn α=1

Cn α = 20

Fig. 4

Example 1: Optimum designs

Fig. 5

Example 2: Ground structure

Design complexity control in truss optimization

19

α=0

Cb α = 0.1

Parkes s = 0.1

Cn α = 0.1

Fig. 6

Example 2: Optimum designs

Fig. 7

Example 3: Ground structure

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André J. Torii* et al.

α=0

α = 0.1

s = 0.01

α = 0.5

s = 0.1

α=5

s = 0.15

Fig. 8

Example 3: Optimum designs obtained using Cb (left) and Parkes' approach (right)

Fig. 9

Example 4: Ground structure

Design complexity control in truss optimization

Fig. 10

Fig. 11

21

α=0

α = 0.1

α = 5.0

α = 20.0

Example 4: Optimum designs obtained with Cb

s = 0.01

s = 0.10

s = 0.20

s = 0.50

Example 4: Optimum designs obtained with Parkes' approach

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André J. Torii* et al.

Table 1

Example 1: Comparison between optimum solutions obtained with Cb α

0.0 1.0 20.0 Table 2

α

s

Case

Cn

17.009933 13.671849 8.851052

max xi

31.498986 37.909666 49.566286

W

0.084453 0.084453 0.084453

V

106.568545 106.568549 106.568549

N. bars 34 34 34

max xi

31.217755 31.068528 31.002430

1.723911 1.723912 1.723913 1.723915

N. bars 69 69 36 36

Cb

60.646917 60.647503 34.677292 34.676988

N. nodes 41 41 35 34

Cn

40.045867 40.045483 34.346062 33.370697

W

0.161529 0.161529 0.163711 0.170752

N. bars 41 12 6 4

Cb

32.452465 11.497119 5.883587 3.938796

max xi

53.856443 72.286627 95.723812 93.790096

Example 3: Comparison between optimum solutions obtained with Parkes' aps

0.00 0.01 0.10 0.15 Table 7

N. nodes 18 14 9

Example 3: Comparison between optimum solutions obtained with Cb α

proach

W

0.084453 0.084463 0.100608

W

0.0 0.1 0.5 5.0 Table 6

max xi

31.498986 35.574247 47.851242

Example 2: Comparison between optimum designs

α=0 Parkes (s = 0.1) Cb (α = 0.1) Cn (α = 0.1) Table 5

Cb

27.638409 11.584124 8.762439

Example 1: Optimum designs obtained with Parkes' approach 0.1 1.0 10.0

Table 4

N. bars 34 12 9

Example 1: Comparison between optimum solutions obtained with Cn 0.0 1.0 20.0

Table 3

W

0.084453 0.084453 0.099279

W

0.161529 0.161529 0.161924 0.166076

N. bars 41 12 10 4

Cb

32.452465 11.497442 9.687106 3.967747

max xi

53.856443 72.256063 72.256065 96.341421

Example 4: Comparison between optimum solutions obtained with Cb α

0.0 0.1 5.0 20.0

W

0.045516 0.045516 0.048032 0.055290

N. bars 19 8 4 2

Cb

18.278127 7.883254 3.961506 1.986563

max xi

112.326697 112.333502 153.875227 155.487243

Design complexity control in truss optimization Table 8

proach

23

Example 4: Comparison between optimum solutions obtained with Parkes' aps

0.00 0.01 0.10 0.20 0.50

W

0.045516 0.045516 0.045883 0.046757 0.050165

N. bars 19 8 8 4 2

Cb

18.278127 7.882623 7.845039 3.961944 1.987209

max xi

112.326697 112.326700 122.857328 157.959421 171.331049