Abstract
The present work introduces a novel concurrent optimization formulation to meet the requirements of lightweight design and various constraints simultaneously. Nodal displacement of macrostructure and effective thermal conductivity of microstructure are regarded as the constraint functions, which means taking into account both the load-carrying capabilities and the thermal insulation properties. The effective properties of porous material derived from numerical homogenization are used for macrostructural analysis. Meanwhile, displacement vectors of macrostructures from original and adjoint load cases are used for sensitivity analysis of the microstructure. Design variables in the form of reciprocal functions of relative densities are introduced and used for linearization of the constraint function. The objective function of total mass is approximately expressed by the second order Taylor series expansion. Then, the proposed concurrent optimization problem is solved using a sequential quadratic programming algorithm, by splitting into a series of sub-problems in the form of the quadratic program. Finally, several numerical examples are presented to validate the effectiveness of the proposed optimization method. The various effects including initial designs, prescribed limits of nodal displacement, and effective thermal conductivity on optimized designs are also investigated. An amount of optimized macrostructures and their corresponding microstructures are achieved.
Similar content being viewed by others
References
Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988)
Bendsøe, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1, 193–202 (1989)
Zhou, M., Rozvany, G.I.N.: The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89, 309–336 (1991)
Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49, 885–896 (1993)
Huang, X., Xie, Y.M.: Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem. Anal. Des. 43, 1039–1049 (2007)
Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003)
Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163, 489–528 (2000)
Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)
Zhou, S., Wang, M.Y.: Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct. Multidiscip. Optim. 33, 89 (2007)
Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometricallyła new moving morphable components based framework. J. Appl. Mech. 81, 081009 (2014)
Guo, X., Zhang, W., Zhang, J., et al.: Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Comput. Methods Appl. Mech. Eng. 310, 711–748 (2016)
Zhang, W., Zhang, J., Guo, X.: Lagrangian description based topology optimization—a revival of shape optimization. J. Appl. Mech. 83, 041010 (2016)
Zhang, W., Yang, W., Zhou, J., et al.: Structural topology optimization through explicit boundary evolution. J. Appl. Mech. 84, 011011 (2016)
Zhang, W., Chen, J., Zhu, X., et al.: Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Comput. Methods Appl. Mech. Eng. 322, 590–614 (2017)
Guo, X., Zhou, J., Zhang, W., et al.: Self-supporting structure design in additive manufacturing through explicit topology optimization. Comput. Methods Appl. Mech. Eng. 323, 27–63 (2017)
Eschenauer, H.A., Olhoff, N.: Topology optimization of continuum structures: a review. J. Appl. Mech. Appl. Mech. Rev. 54, 331–390 (2001)
Rozvany, G.I.N.: A critical review of established methods of structural topology optimization. Struct. Multidiscip. Optim. 37, 217–237 (2009)
Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)
Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31, 2313–2329 (1994)
Sigmund, O.: Tailoring materials with prescribed elastic properties. Mech. Mater. 20, 351–368 (1995)
Clausen, A., Wang, F., Jensen, J.S., et al.: Topology optimized architectures with programmable Poisson’s ratio over large deformations. Adv. Mater. 27, 5523–5527 (2015)
Xie, Y.M., Yang, X., Shen, J., et al.: Designing orthotropic materials for negative or zero compressibility. Int. J. Solids Struct. 51, 4038–4051 (2014)
Wang, X., Xu, S., Zhou, S., et al.: Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: a review. Biomaterials 83, 127–141 (2016)
Rodrigues, H., Guedes, J.M., Bendsoe, M.P.: Hierarchical optimization of material and structure. Struct. Multidiscip. Optim. 24, 1–10 (2002)
Coelho, P.G., Fernandes, P.R., Guedes, J.M., et al.: A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct. Multidiscip. Optim. 35, 107–115 (2008)
Liu, L., Yan, J., Cheng, G.: Optimum structure with homogeneous optimum truss-like material. Comput. Struct. 86, 1417–1425 (2008)
Niu, B., Yan, J., Cheng, G.: Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct. Multidiscip. Optim. 39, 115–132 (2009)
Deng, J., Yan, J., Cheng, G.: Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Struct. Multidiscip. Optim. 47, 583–597 (2013)
Guo, X., Zhao, X., Zhang, W., et al.: Multi-scale robust design and optimization considering load uncertainties. Comput. Methods Appl. Mech. Eng. 283, 994–1009 (2015)
Huang, X., Zhou, S.W., Xie, Y.M.: Topology optimization of microstructures of cellular materials and composites for macrostructures. Comput. Mater. Sci. 67, 397–407 (2013)
Yan, X., Huang, X., Sun, G., et al.: Two-scale optimal design of structures with thermal insulation materials. Compos. Struct. 120, 358–365 (2015)
Liu, Q., Chan, R., Huang, X.: Concurrent topology optimization of macrostructures and material microstructures for natural frequency. Mater. Des. 106, 380–390 (2016)
Xu, B., Jiang, J.S., Xie, Y.M.: Concurrent design of composite macrostructure and multi-phase material microstructure for minimum dynamic compliance. Compos. Struct. 128, 221–233 (2015)
Xu, B., Xie, Y.M.: Concurrent design of composite macrostructure and cellular microstructure under random excitations. Compos. Struct. 123, 65–77 (2015)
Xu, B., Huang, X., Xie, Y.M.: Two-scale dynamic optimal design of composite structures in the time domain using equivalent static loads. Compos. Struct. 142, 335–345 (2016)
Zhang, W., Sun, S.: Scale-related topology optimization of cellular materials and structures. Int. J. Numer. Methods Eng. 68, 993–1011 (2006)
Xia, L., Breitkopf, P.: Concurrent topology optimization design of material and structure within FE\(_2\) nonlinear multiscale analysis framework. Comput. Methods Appl. Mech. Eng. 278, 524–542 (2014)
Xia, L., Breitkopf, P.: Recent advances on topology optimization of multiscale nonlinear structures. Arch. Comput. Methods Eng. 24, 227–249 (2016)
Jia, J., Cheng, W., Long, K., et al.: Hierarchical design of structures and multiphase material cells. Comput. Struct. 165, 136–144 (2016)
Long, K., Han, D., Gu, X.: Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct Poisson’s ratios for maximum frequency. Comput. Mater. Sci. 129, 194–201 (2017)
Chen, W., Tong, L., Liu, S.: Concurrent topology design of structure and material using a two-scale topology optimization. Comput. Struct. 178, 119–128 (2017)
Sui, Y., Peng, X.: The ICM method with objective function transformed by variable discrete condition for continuum structure. Acta Mech. Sin. 22, 68–75 (2006)
Sui, Y., Yang, D.: A new method for structural topological optimization based on the concept of independent continuous variables and smooth model. Acta Mech. Sin. 14, 179–185 (1998)
Sui, Y.: Modelling, Transformation and Optimizationł New Developments of Structural Synthesis Method. Dalian University of Technology Press, Dalian (1996)
Andreassen, E., Andreasen, C.S.: How to determine composite material properties using numerical homogenization. Comput. Mater. Sci. 83, 488–495 (2014)
Zuo, Z.H., Xie, Y.M.: Evolutionary topology optimization of continuum structures with a global displacement control. Comput. Aided Des. 56, 58–67 (2014)
Lazarov, B.S., Sigmund, O.: Filters in topology optimization based on Helmholtz-type differential equations. Int. J. Numer. Methods Eng. 86, 765–781 (2011)
Amstutz, S., Giusti, S.M., Novotny, A.A., et al.: Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int. J. Numer. Methods Eng. 84, 733–756 (2010)
Svanberg, K.: The method of moving asymptotes-a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 359–373 (1987)
Acknowledgements
The project was supported by the National Natural Science Foundation of China (Grants 11202078, 51405123) and the Fundamental Research Funds for the Central Universities (Grant 2017MS077). We are thankful for Professor Krister Svanberg for MMA program made freely available for research purposes.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Long, K., Wang, X. & Gu, X. Concurrent topology optimization for minimization of total mass considering load-carrying capabilities and thermal insulation simultaneously. Acta Mech. Sin. 34, 315–326 (2018). https://doi.org/10.1007/s10409-017-0708-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-017-0708-1