Abstract
In this study, we present a novel non-parametric shape-topology optimization method for frame structures with multi-materials, aiming at designing more light and stiff frame structures. The sum of squared error norm for achieving the target displacements on the specified members is minimized under the volume constraints of multi-materials as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the material gradient functions for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The generalized solid isotropic material with penalization (GSIMP) method is employed to classify into the multi-materials in topology optimization. The gradient functions derived are applied to the unified H1 gradient method for frame structures to determine the optimal shape and material variations. With this method, the optimal free-form and topology for arbitrary large-degree of design freedom frame structures with multi-materials can be obtained without any shape and topology parameterization. Numerical examples with various materials and different boundary conditions are demonstrated and the results are discussed.
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References
Ahrari A, Atai AA (2013) Fully stressed design evolution strategy for shape and size optimization of truss structures. Comput Struct 123:58–67. https://doi.org/10.1016/j.compstruc.2013.04.013
Al-Ketan O, Rowshan R, Al-Rub RK (2018) Topology-mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials. Addit Manuf 19:167–183. https://doi.org/10.1016/j.addma.2017.12.006
Amir O, Mass Y (2017) Topology optimization for additive manufacturing: accounting for overhang limitations using a virtual skeleton. Addit Manuf 58:58–73. https://doi.org/10.1016/j.addma.2017.08.001
Azegami H, Kaizu S, Takeuchi K (2011) Regular solution to topology optimization problems of continua. Japan Soc Ind Appl Math 3:1–4. https://doi.org/10.14495/jsiaml.3.1
Azegami H, Wu ZC (1996) Domain optimization analysis in linear elastic problems: approach using traction method. JSME Int J Ser A, Mech Mat Eng 39(2):272–278. https://doi.org/10.1299/jsmea1993.39.2_272
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654. https://doi.org/10.1007/s004190050248
Bruns TE (2005) A reevaluation of the SIMP method with filtering and an alternative formulation for solid-void topology optimization. Struct Multidiscip Optim 30:428–436. https://doi.org/10.1007/s00158-005-0537-x
Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization 1. Springer
Chao D, Hamed S, Shilin D, Yi MX (2017) A new node-shifting method for shape optimization of reticulated spatial structures. Eng Struct 152:727–735. https://doi.org/10.1016/j.engstruct.2017.09.051
Dede T (2014) Application of teaching-learning-based-optimization algorithm for the discrete optimization of truss structures. KSCE J Civ Eng 18:1759–1767. https://doi.org/10.1007/s12205-014-0553-8
Fernández-Cabán PL, Masters FJ (2018) Hybridizing particle swarm and big bang-big crunch optimization methods to explore then exploit the design domain of large planar frame structures. Comput Struct 202:1–14. https://doi.org/10.1016/j.compstruc.2018.02.014
Gholizadeh S, Danesh M, Gheyratmand C (2020) A new Newton metaheuristic algorithm for discrete performance-based design optimization of steel moment frames. Comput Struct 234:106250. https://doi.org/10.1016/j.compstruc.2020.106250
Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Dover Publications
Hvejsel CF, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43:811–825. https://doi.org/10.1007/s00158-011-0625-z
Kaminakis NT, Stavroukajus GE, Drosopoulos GA (2012) Topology optimization for compliant mechanisms, using evolutionary-hybrid algorithms and application to the design of auxetic materials. Compos Part B 43(6):2655–2688. https://doi.org/10.1016/j.compositesb.2012.03.018
Kaveh A, Khayatazad M (2013) Ray optimization for size and shape optimization of truss structures. Comput Struct 117:82–94. https://doi.org/10.1016/j.compstruc.2012.12.010
Kociecki M, Adeli H (2015) Shape optimization of free-form steel space-frame roof structures with complex geometries using evolutionary computing. Eng Appl Artif Intell 38:168–182. https://doi.org/10.1016/j.engappai.2014.10.012
Larsen SD, Sigmund O, Groen JP (2018) Optimal truss and frame design from projected homogenization-based topology optimization. Struct Multidiscip Optim 57(4):1641–1474. https://doi.org/10.1007/s00158-018-1948-9
Lieu QX, Do DTT, Lee J (2018) An adaptive hybrid evolutionary firefly algorithm for shape and size optimization of truss structures with frequency constraints. Comput Struct 195:99–112. https://doi.org/10.1016/j.compstruc.2017.06.016
Lopez C, Burggraeve S, Stroobants J (2018) A preliminary approach to multi-material topology optimization considering cost estimation. WIT Trans Built Environ 175(11):121–131. https://doi.org/10.2495/HPSM180131
Paz J, Diaz J, Romera L, Costas M (2015) Size and shape optimization of aluminum tubes with GFRP honeycomb reinforcements for crashworthy aircraft structures. Compos Struct 133:499–507. https://doi.org/10.1016/j.compstruct.2015.07.077
Shimoda M, Liu Y (2014) A non-parametric free-form optimization method for shell structures. Struct Multidiscip Optim 50:409–423. https://doi.org/10.1007/s00158-014-1059-1
Shimoda M, Liu Y, Morimoto T (2014) Non-parametric free-form optimization method for frame structures. Struct Multidiscip Optim 50(1):129–146. https://doi.org/10.1007/s00158-013-1037-z
Shimoda M, Kameyama K, Shi JX (2017) Tailoring static deformation of frame structures based on a non-parametric shape-size optimization method. Int J Solids Struct 223:143–154. https://doi.org/10.1016/j.ijsolstr.2017.02.011
Shimoda M, Liu Y, Wakasa M (2020) Free-form optimization method for frame structures aiming at controlling time-dependent responses. Struct Multidiscip Optim (on line). https://doi.org/10.1007/s00158-020-02708-y
Wang D (2007) Optimal shape design of a frame structure for minimization of maximum bending moment. Eng Struct 29(8):1824–1832. https://doi.org/10.1016/j.engstruct.2006.10.004
Zhao ZL, ZhouS CK, Xie YM (2020) A direct approach to controlling the topology in structural optimization. Comput Struct 227:106141. https://doi.org/10.1016/j.compstruc.2019.106141
Zuo W, Saitou K (2017) Multi-material topology optimization using ordered SIMP interpolation. Struct Multidiscip Optim 55(2):477–491. https://doi.org/10.1007/s00158-016-1513-3
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A part of this work was supported by JKA and its promotion funds from AUTORACE (KEIRIN RACE).
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The optimization system developed consists of in-house C programs and MSC NASTRAN for FE analyses. Their executions are controlled with “Batch program” on Windows OS until the convergence. For the benchmark calculation by readers, we will provide the NASTRAN model data used in this paper.
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Shimoda, M., Tani, S. Simultaneous shape and topology optimization method for frame structures with multi-materials. Struct Multidisc Optim 64, 699–720 (2021). https://doi.org/10.1007/s00158-021-02871-w
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DOI: https://doi.org/10.1007/s00158-021-02871-w