Structural Optimization of Grid Shells: Design Parameters and Combined Strategies Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Ernesto Grande1; Maura Imbimbo2; and Valentina Tomei3 Abstract: The optimization of structures is a tricky process that involves strategies and mathematical algorithms in which the design parameters—introduced in terms of variables, constraint conditions, optimization functions, penalty conditions, and so forth—play very important roles. Their selection and introduction in the optimization process could inﬂuence the characteristics and the level of optimization of the derived solutions. This paper describes the roles of design parameters used in different optimization strategies. Moreover, an efﬁcient optimization approach, which combines form-ﬁnding (FF), sizing-optimization (SO), and topologic-optimization (TO) strategies in a multilevel process for which design variables and constraint conditions are opportunely selected, is proposed. Numerical analyses referring to some canopy case studies derived from the current literature are also presented in the paper. The comparisons among optimization approaches, featuring FF, SO, and TO optimization strategies performed singularly or in combination throughout a simple sequence of phases, emphasize the effectiveness of the proposed approach for obtaining light structural solutions for grid shells. DOI: 10.1061/(ASCE)AE.1943-5568.0000286. © 2017 American Society of Civil Engineers. Introduction Nowadays, grid-shell structures are an important structural typology widely chosen in many recent applications thanks to their capacity to cover large spaces with free form and light solutions. In the design of grid shells, the shape and the structure cannot be separated because they are the same thing. For this reason, a grid shell should have a shape that respects architectural requirements and is structurally efﬁcient; this requirement means that the architectural aspects are treated together with the structural ones, and to pursue an optimal design, both aspects should be optimized at the same time. Numerous examples of grid-shell structures based on optimized shapes and involving different structural materials are described in the current literature (Pone et al. 2013; Williams 2014). In some cases, they were created through innovative design approaches that explicitly adopt form-ﬁnding (FF) and optimization strategies as efﬁcient support tools for the entire design process. An interesting and recent example in Italy is the prototype of a timber grid shell covering an area of 75 m2 built in the Faculty of Architecture courtyard at the University Federico II in Naples (Pone et al. 2013). For the design of this prototype, a design procedure based on the particle-spring model, followed by a ﬁnite- element analysis, was developed for the FF phases and structural analysis. Other fascinating examples of grid-shell structures based on FF or structural-optimization design processes are present in Europe. In particular, two examples that are representative of the different 1 Associate Professor, Dept. of Sustainability Engineering, Guglielmo Marconi Univ., Rome, Italy (corresponding author). E-mail: e.grande@ unicas.it 2 Associate Professor, Dept. of Civil and Mechanical Engineering, Univ. of Cassino and Southern Lazio, Cassino, Italy. 3 Assistant Professor, Dept. of Civil and Mechanical Engineering, Univ. of Cassino and Southern Lazio, Cassino, Italy. Note. This manuscript was submitted on March 28, 2017; approved on July 26, 2017; published online on October 26, 2017. Discussion period open until March 26, 2018; separate discussions must be submitted for individual papers. This paper is part of the Journal of Architectural Engineering, © ASCE, ISSN 1076-0431. © ASCE design strategies and construction techniques are the Mannheim Multihalle in Germany and the British Museum canopy in England. Indeed, for the Mannheim Multihalle, which is a standalone multilayered grid system, the designer, Frei Otto, created the optimal shape by using the physical-hanging model (Linkwitz and Veenendaal 2014; Williams 2014). In the case of the British Museum canopy, designed by Foster and Partners, the original shape was determined from a nonlinear analysis on the basis of the dynamic relaxation method (Williams 2014). In the context of function and design, it is then evident that FF and structural-optimization techniques can have key roles as efﬁcient supports for the design of grid-shell structures. Indeed, the growing number of studies concerning grid shells (Richardson et al. 2013, Bouhaya et al 2014; Adriaenssens et al. 2014; D’Amico et al. 2014; Dimcic 2011; Dini et al. 2013; Douthe et al. 2006) testify to the interest of the scientiﬁc community, and at the same time, the need to create efﬁcient and robust design tools involving innovative design strategies. The present paper has the two-fold aim of examining the inﬂuences of design parameters on the optimization process of gridshell structures and an efﬁcient structural-optimization approach that combines three different strategies: FF, sizing optimization (SO), and topologic optimization (TO). The main peculiarity of the proposed approach is that each strategy was considered a phase of a global-optimization process in which the design parameters and structural requirements were opportunely selected for each phase and converted into variables and constraint conditions of the optimization problem. Before presenting the proposed approach, the selected optimization strategies were considered both individually and in combination throughout a simple sequence of optimization phases that were all characterized by the same set of constraints in terms of displacements and utilization ratios (i.e., the ratio between the stress and strength of the members). In particular, considering some canopy case studies derived from the current literature, the optimization strategies were applied by accounting for different design parameters (boundary restraints, grid densities, ranges of cross-section diameter) and considering speciﬁc criteria for introducing these parameters into the optimization process. The results that emerged 04017027-1 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. from these analyses are reported in the ﬁrst part of the paper. The analyses were ﬁnalized to obtain information concerning the inﬂuence of the design parameters and structural requirements on the performance of the selected optimization strategy. The evidence that emerged from this group of preliminary analyses also accounted for the degree of effectiveness of the proposed approach. By considering the same canopy case studies, the results derived by applying the proposed approach, which combines the optimization strategies created by opportunely selecting the constraint conditions, are presented in detail in the second part of the paper. Then, the results of these analyses are compared with the ones derived from the preliminary ones. In comparison to the approaches in which FF, SO, and TO optimization strategies are taken singularly or combined together throughout a simple sequence of phases, all characterized by the same set of design constraints, the proposed approach allows for lighter structural solutions for grid shells. The structure of the paper is quite schematic and describes stepby-step the constraints and the objective functions adopted in each of the analyzed strategies. In fact, after the “Introduction,” the subsequent paragraphs describe the case studies, the traditional optimization strategies, and the proposed approach by speciﬁcally explaining the underlying roles of the design parameters. In particular, for both the traditional optimization strategies and the proposed approach, the constraints and objective functions are clearly stated. Indeed, the efﬁciency of the proposed approach becomes clearer as the paper presents the work because of the advantage gained through comparisons of it with the basic strategies. Case Studies The case studies analyzed in the paper come from recent research carried out by Richardson et al. (2013), who presented some interesting results concerning a coupled FF and grid-optimization approach for single-layer grid shells. In particular, the case studies consist of grid-shell canopies that cover a surface area of 24 24 m, with a geometry of a 6 6, ﬂat, square grid composed of nodes equally spaced by 4 4 m (Fig. 1). Three schemes characterized by three different boundary conditions were considered: The ﬁrst is pin supported on all four sides of the canopy (R4); the second is pin supported on three sides of the canopy (R3); the third is pin supported on two opposite sides of the canopy (R2). The three described canopy schemes present different values for the peak height (Richardson et al. 2013): 5, 5.85, and 4.728 m for R4, R3, and R2, respectively. In addition, all the described schemes are characterized by hollow circular-steel cross sections with a S355-steel-grade material: The yield strength, fyd, was 355 MPa and Young’s modulus, Es, was 210 GPa. A distributed gravity load equal to 3 kN/m2 was considered for evaluating the nodal vertical forces determined during the numerical analyses. Optimization Strategies and Role of Design Parameters The ﬁrst part of the paper is mainly devoted to showing the analysis of the inﬂuence of some design parameters on the optimization process by FF, SO, and TO strategies. In particular the following design parameters were considered: • Boundary constraint schemes R2, R3, and R4; © ASCE Fig. 1. Grids described in the paper: (a) 6 6; (b) 3 3; (c) 12 12 Density of the grid composing the canopy with 6 6 [as in Richardson et al. (2013)], 3 3, and 12 12 grid schemes (Figure 1); • Diameters used during the optimization process, including a single diameter (i.e., same cross section) for all the members of the canopy (solutions in terms of d1), and three different diameters used in the canopy (solutions in terms of d3); and • Criteria for assigning the three diameters among the members composing the canopy during the optimization process: a criterion based on the orientation of the members, horizontal elements, vertical elements, diagonals (solution denoted d3a); a criterion based on the utilization ratio U of members, U < 33%, 33% ≤ U ≤ 66%, U > 66% (denoted in the following d3b). All these parameters were then combined together leading to different structural solutions to consider during the optimization process. • Form Finding FF was the ﬁrst strategy considered for the shape derivation of the described canopies. In particular, to derive an optimal shape for the canopies, the shape corresponding to the hanging model was determined by considering the three different grid densities for each of the three different restraint schemes. These shapes were derived using two different computer codes and selecting different modeling and numerical strategies. The ﬁrst-used approach refers to the dynamic relaxation method (Schek 1974; Day 1965), which is implemented in Kangaroo. In this case, the optimal shape was set by the level of internal prestress and on the basis of the boundary supports. The second approach was performed by using a commercial code in SAP2000. In this case, the canopies were modeled by using cable elements instead of truss elements. Second-order incremental analyses were performed by introducing a large-displacements option. Applications of the different approaches and computer codes provided the same results in terms of the canopy shapes (Fig. 2). 04017027-2 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 2. Shapes derived after the FF approach by grid scheme: (a) 3 3; (b) 6 6; (c) 12 12 Nevertheless, more expensive computational time and convergence drawbacks emerged in the case of the cable-modeling approach. Furthermore, it was observed that the shapes of the analyzed canopies were signiﬁcantly inﬂuenced both by the restraint boundary conditions and, moreover, by the grid density. As emphasized in Tomei (2017) and Grande et al. (2016), the shape based on the hanging model allows for particular reductions in the structural weight while preserving good levels of global stiffness. For these reasons, the canopy shapes derived from this phase were adopted for performing the other optimization strategies. Sizing Optimization Taking into account the shapes of canopies derived from the FF, the SO was performed by considering the different boundary restraint schemes (R2, R3, and R4), the different grid densities (3 3, 6 6, and 12 12), and moreover, the range of diameters (d1, d3) and the criteria for assigning the diameter to canopy members during the optimization process (d3a, d3b). The numerical analyses were developed through the commercial software Karamba, a ﬁnite-element plug-in developed for Grasshopper and fully embedded in the three-dimensional (3D) modeling software, Rhinoceros. Karamba allows for interactive calculations on the responses of 3D structures while considering the parametric environment of Grasshopper, and then, it shows all the potentialities in terms of geometric modeling and structural optimization. Moreover, it introduces user subroutines for both the preprocessing and postprocessing phases. A mono-objective genetic algorithm strategy was used for developing the SO process according to the following parameters: population size of 50, crossover rate of 0.8, and mutation rate of 0.1. In particular, constraints in terms of maximum displacement (Dmax ≤ 0.12 m) and the maximum utilization ratio of members (Umax ≤ 1, which takes into account both the strength and local buckling), were both introduced. In addition, the structural weight, W, was considered the objective function to minimize during the SO (Fig. 3). The results derived from the SO strategy are presented in Figs. 4 and 5 in terms of weight and of maximum displacement normalized with respect to the limit value (Dmax), respectively. The ﬁgures show that the SO provides better results in terms of weight when the three diameters were considered and assigned according to the criterion that was based on the utilization ratio (solutions in d3b). This © ASCE Fig. 3. Flowchart of the optimization strategies evidence is common to all the restraint schemes and to all the mesh densities. All the schemes were characterized by a maximum displacement value lower than the admissible one. This ﬁnding means that the constraint condition inﬂuencing the SO was the utilization ratio of the canopy members. The results also reveal that both the mesh density and the restraint scheme inﬂuenced the weight and the stiffness of the solution derived at the end of the SO. In particular, although it was observed that an increase in the mesh reﬁnement provided a reduction in the weight, the inﬂuence of the boundary restraints depended on both the number of restrained sides of the canopy and the conﬁguration of the restraints. Indeed, the solutions for the restraint conﬁguration R3 (three sides restrained) were characterized by greater values of structural weight than were the solutions for R2 (two sides restrained); however, the solutions corresponding to the restraint conﬁguration R4 (four sides restrained) showed lower values of structural weight than either the R2 or R3 scheme conﬁgurations did. This evidence is common to both criteria selected for assigning the diameters to the members. 04017027-3 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 4. Weight derived from application of only the SO strategy by diameter criteria: (a) d1; (b) d3a; (c) d3b Sizing-Optimization and Topologic-Optimization Approach Considering the previously obtained solutions of the canopies derived through the FF and the subsequent SO strategy, an additional optimization strategy was analyzed. In particular, a TO based on the removal of some of the diagonals of the grid shell was performed by considering the same computer code and parameters, objective function, and constraints of the mono-objective genetic algorithm as were used for the SO strategy. For this optimization, a sequence composed of a preliminary FF phase and two subsequent optimization phases, where both SO and TO were based on the same objective function and constraint conditions, were used to derive a ﬁnal solution that is characterized by a reduced number of © ASCE Fig. 5. Displacements derived from application of only the SO strategy by diameter criteria: (a) d1; (b) d3a; (c) d3b diagonals than either the solution derived from the FF or from the SO. For the sake of brevity, the proposed strategy that was applied to the 6 6 mesh for the three restraint schemes is the only one presented. The results obtained from the TO approach are reported in Fig. 6 in terms of structural weight: The obtained solutions were compared against the corresponding solutions derived by applying only the SO. Moreover, Fig. 7 shows the grid conﬁgurations obtained at the end of the optimization process (i.e., after the TO phase). From Figs. 6 and 7, it can be observed that, despite a reduction in the number of diagonals composing the grid, which depended on both the restraint scheme and the range of diameters and criteria of assignments used for the SO, the introduction of the TO led to a 04017027-4 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 7. Grid conﬁgurations at the end of the SO followed by the TO approach: (a) SO(d1)-TO; (b) SO(d3a)-TO; (c) SO(d3b)-TO Fig. 6. Comparison of solutions derived from the singular SO strategy and from the SO strategy followed by the TO strategy by speciﬁc criteria: (a) same cross-section sizes for all the members; (b) orientation of members; (c) utilization ratio of members slight reduction in the structural weight. This effect particularly depended on the parameters considered for developing the SO: The lowest reduction of the weight corresponded to the d3b case, for which the SO depended on a range of three diameters and a criterion that was based on the utilization ratio of members. Mixed Sizing-Optimization and Topologic-Optimization Approach Because the introduction of the TO did not cause a signiﬁcant reduction in the structural weight with respect to the approach that © ASCE was based only on the SO, another approach was used that combines the FF, SO, and TO approaches and is referred to as mixed SO and TO approach; it is composed of four phases (Fig. 3): a preliminary FF phase for deriving the shape; a SO phase based on the same cross-section size for each member (d1), which differs from the sizes used in the previous approach; a TO phase; and a ﬁnal SO phase developed by considering only the d3a and d3b criteria. In particular, all phases of the process that were based on the SO and the TO were developed by considering the same objective function (i.e., the structural weight, W) and the same constraint conditions (i.e., the maximum displacement, Dmax, and the maximum utilization ratio, Umax) as introduced in the mono-objective genetic algorithm. The results derived at the end of the SO phase, introduced after the TO phase, are reported in Fig. 8 in terms of weight and in comparison to the other analyzed approaches. From the comparisons, it is clear that, despite some cases in which the mixed SO and TO approach led to lower structural weights, in some cases, the mixed SO and TO approach led to solutions of greater structural weight than the other approaches. Considerations At the end of this, the ﬁrst part of the paper, the roles of some design parameters in the context of different optimization strategies and approaches were described. In particular, it was observed that both the restraint scheme and the grid density affected the hangingmodel shape derived from the FF phase. Moreover, these parameters, together with additional ones introduced in the optimization process in terms of variables and constraint conditions, also inﬂuenced the solutions derived from both the SO and the TO strategies. Another important aspect that emerged from the ﬁrst part of the study was the low level of effectiveness in reducing the structural 04017027-5 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 8. Comparison of the solutions derived from the SO, SO followed by the TO, and mixed SO and TO approaches by speciﬁc criteria: (a) orientation of members; (b) utilization ratio of members weight when a process based on a simple sequence of optimization strategies was used. This result seems to suggest the use of SO and TO strategies separately—without combining them—is the better approach. Nevertheless, it is important to emphasize that the approaches combining the SO and TO, presented in this ﬁrst part of the paper, were based on a simple sequence of SO and TO phases from which the solution was derived from an optimization phase that was used for developing the subsequent solution. All these phases were, indeed, developed by considering the same constraint conditions. As shown in the following part of the paper, the peculiarity of the presented approaches is the main reason for their low effectiveness in reducing the structural weight of the canopies when the SO and TO strategies are combined together. Proposed Approach In remainder of the paper, an optimization approach that draws only from the combination of different optimization strategies is proposed. The proposed approach is similar to the mixed SO and TO approach presented in the section with of the same title. It is based on the same sequence of phases but different constraint conditions that were adopted for each of the optimization phases composing the entire process. In particular (Fig. 3): © ASCE Fig. 9. Comparison of solutions derived from the SO, the SO followed by TO, the mixed SO and TO, and the proposed approaches by speciﬁc criteria: (a) orientation of members; (b) utilization ratio of members Fig. 10. Grid conﬁgurations at the end of the proposed approach 1. The FF strategy was preliminary considered for deriving the shape of the canopy according to the hanging model. 2. The SO strategy was applied according to the solution derived from the FF: The same cross-section size was used for all the members (d1); constraint conditions were introduced on both the displacements and the utilization ratios; and an objective function, in terms of structural weight, was adopted (i.e., the same as in the mixed SO and TO approach). 3. The TO strategy was applied by considering the solution derived from the previous SO strategy solution. Different from the mixed SO and TO approach, the constraint conditions only for displacements were introduced. An objective function in terms of structural weight was adopted. 4. The SO strategy was applied by using the solution derived from the previous TO strategy. Three different cross-section sizes and two criteria for assigning them (d3a and d3b) were considered. Different from the mixed SO and TO approach, the constraint conditions that concern only the utilization ratios 04017027-6 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 11. Comparison of solutions derived from the described strategies and approaches © ASCE 04017027-7 J. Archit. Eng., 2018, 24(1): 04017027 J. Archit. Eng. were introduced, and an objective function in terms of structural weight was used. Then, the differences with respect to the mixed SO and TO approach that concern only the type of constraints were introduced into the two last optimization phases. Nevertheless, despite these few differences, the results presented in Fig. 9 in terms of structural weight and in Fig. 10 in terms of grid conﬁguration show that the proposed approach led to a signiﬁcant reduction in the number of diagonals after the TO phase, and it also led to a signiﬁcant reduction in the structural weight compared to the other approaches considered in the study. Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved. Conclusions The optimization of structures involves strategies and mathematical algorithms for which the design parameters, introduced in terms of variables, constraint conditions, optimization functions, penalty conditions, and so forth, play a very important role. Their selection and introduction in the optimization process can inﬂuence the characteristics and the level of optimization of the derived solutions. In this study, the roles of the design parameters in the context of different optimization strategies were analyzed. Moreover, an efﬁcient optimization approach that combines the FF, SO, and TO strategies into a multilevel process, with design variables and constraint conditions opportunely selected, was proposed and assessed with reference to case studies derived from the literature. When compared with the usual optimization approaches, in which the FF, SO and TO optimization strategies are performed singularly or combined together throughout a simple sequence of phases and all characterized by the same set of design constraints (stress, displacement, buckling, etc.), the proposed approach is shown to obtain lighter structural solutions for grid shells (Fig. 11). Indeed, it was observed that the introduction of constraints that concerned only the displacements into the TO phase led to a substantial reduction in the number of diagonal members composing the structural scheme. Furthermore, the subsequent SO phase, which used the solution that emerged from the TO phase and for which constraint conditions were introduced that only concerned the utilization ratio of members, led to a greater reduction of the global structural weight of canopies than the other approaches. The numerical analyses presented in the paper clearly highlight the role of the selected design parameters on the structural design optimization process. For these parameters, the role of the grid density, which inﬂuenced both the FF process and the SO and TO optimization solution, was investigated. Indeed, this parameter represented the discretization of the shell surface of the canopy so it necessarily accounts for the derivation of the optimal shape. Moreover, it also represents a parameter inﬂuencing the length and slope of the members composing the structural solution better than the mixed TO and SO process did. Furthermore, the parameters described in the paper were concerned with the chosen number of diameters as variables of the problem, and, in particular, the criterion for distributing them during the SO process. Indeed, it was observed that an increase in the range of diameters and the selection based on the stress ratio of members allowed for lighter solutions than were found with use of the other criteria analyzed in the study. The paper shows the potential for combining different optimization strategies. Nevertheless, it also revealed the importance in managing the design parameters in terms of variables, constraints, and objective functions of the mathematical optimization problem, particularly when different optimization strategies are combined. © ASCE Indeed, although the solutions derived at the end of each phase by using genetic algorithms mathematically respect the constraints and represent a minimum for the objective function, the ﬁnal solution at the end of the entire process does not represent the most optimized structural design solution. The proposed approach, based on both a combination of different optimization strategies and on a selection of constraints, improved the level of optimization of canopies in terms of structural weight with respect to the other approaches. Finally, although they are not considered in the present research, the inﬂuences of the joints and global buckling, which represent additional aspects, could play an important role in the optimization process of grid shells. Indeed, the mechanical behavior, dimensions, and construction aspects of member-to-member connections can signiﬁcantly affect the selection of members and the construction sequences and the shape of the grid shell. Indeed, some curvatures are unfeasible because of the rigidity and the limited strength of the connections. The proneness to global buckling of grid shell solutions derived from only considering the local buckling phenomena of members requires the introduction of modiﬁcations, generally in terms of the cross-sectional areas of the canopy members. Aspects of both joints and the global buckling are the subjects of recent research carried out by the authors [Grande et al. (2017), “Role of global buckling in the optimization process of grid shells: design strategies,” submitted to Engineering Structure by Elsevier]. References Adriaenssens, S., Block, P., Veenendaal, D., and Williams, C. (2014). Shell structure for architecture: Form ﬁnding and optimization, Routledge, New York. Bouhaya, L., Baverel, O., and Caron, J.-F. (2014). “Optimization of gridshell bar orientation using a simpliﬁed genetic approach.” Struct. Multidiscip. Optim., 50(5), 839–848. D’Amico, B., Kermani, A., and Zhang, H. (2014). “Form ﬁnding and structural analysis of actively bent timber grid shells.” Eng. 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