Abstract
Truss lattices are potential candidates for the design of innovative aerostructures, thanks to their high stiffness-to-weight ratio, modularity, and aeroelastic properties. However, when designing ultralight structures, multiple mechanical constraints, such as maximum internal stress or local buckling constraints, must be taken into account since the early design phase. In response to this, a volume minimization problem for 3D structures, subject to multiple load cases, maximum stress, and topological buckling constraints, is formulated in this work. The optimization is solved using a two-step optimization strategy. First, a relaxed formulation is solved by a Sequential Linear Programming algorithm and is used to explore the vast design space of the optimization. During this phase, a heuristic is proposed to reduce the influence of the starting point on the optimized structure. The solution is refined in a second optimization step in which the full non-linear problem is solved using IPOPT, making sure that all the mechanical constraints are respected. The proposed method is validated on multiple two-dimensional classical benchmarks, showing robust behavior with respect to random starting point initializations. Later, the three-dimensional wingbox of the Common Research Model subject to multiple load cases is optimized. The results show that the proposed method can deal with real-sized structures with thousands of candidate members, all while being computationally efficient, optimizing the structure in minutes on a consumer notebook.
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Data availability
The full numeric results of all the load cases presented in the paper can be found at Stragiotti (2023), together with the ground structures and the starting points.
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ES contributed toward conceptualization, methodology, software, validation, formal analysis, writing-original draft, review & editing, and visualization. F-XI contributed toward conceptualization, validation, writing-review & editing, and supervision. CJ contributed toward conceptualization, validation, writing-review & editing, and supervision. JM contributed toward validation, writing-review & editing, and supervision.
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The method is open and transparent, and all information necessary for the reproducibility of results has been mentioned in the article. All software used is open source (except for linear solver PARDISO, distributed with an academic license). The random initialization points are generated with the Numpy random function, setting the generator seed to 12, and are provided in the reference data set (Stragiotti 2023).
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Appendices
Appendix 1: Optimization parameters
Table 7 provides the values of the parameters used for the SLP and NLP steps of the current study to generate the examples shown in Sect. 4. The objective function is scaled so that the initial volume is 1000, the areas are in the interval [0, 1000], the initial forces in [0, 1000], and the displacement in [0, 1000] for the SLP and the NLP.
Several additional parameters are used in the NLP step for cyipopt and IPOPT:
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mu_strategy is set to adaptive
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grad_f_constant is set to yes
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hessian_constant is set to yes
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alpha_for_y is set to min-dual-infeas
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linear_solver is set to pardiso
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expect_infeasible_problem is set to yes
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bound_push is set to 1e-12
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constr_viol_tol is set to 1e-6
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nlp_scaling_method is set to user-scaling.
Numerical results
In this annex, we detail the numerical results of the optimized 2D cantilever beam and the multiple load cases ten-bar truss presented in Sects. 4.2 and 4.3. In Table 8, the member forces, areas, and volumes of all the active members at the solution of the 2D cantilever beam are presented. This design shows a 9 % lower volume compared to the one found by Achtziger (1999b). In Table 9, we show the member forces of the two load cases, the areas, and the volumes of the active members of the optimized ten-bar truss with multiple load cases. The full numeric results of all the load cases can be found at Stragiotti (2023).
Iteration history
In this Section, we plot the objective function and constraint violation history curves of the multiple load cases ten-bar truss presented in Sect. 4.3 and the CRM presented in Sect. 4.4.
Iteration history of the ten-bar truss with multiple load cases example solved with the 2S-1R algorithm. a objective function history for the SLP and NLP step. The sharp increase in the objective function during the SLP step corresponds to the reinitialization call. b constraint violation for the NLP step
Iteration history of the CRM-315 example solved with the 2S-5R algorithm. a objective function history for the SLP and NLP step. The sharp increases in the objective function during the SLP step correspond to the reinitialization calls. b constraint violation for the NLP step
Iteration history of the CRM-2370 example solved with the 2S-5R algorithm. a objective function history for the SLP and NLP step. The sharp increases in the objective function during the SLP step correspond to the reinitialization calls. b constraint violation for the NLP step
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Stragiotti, E., Irisarri, FX., Julien, C. et al. Efficient 3D truss topology optimization for aeronautical structures. Struct Multidisc Optim 67, 42 (2024). https://doi.org/10.1007/s00158-024-03739-5
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DOI: https://doi.org/10.1007/s00158-024-03739-5