Skip to main content
SpringerLink
Log in

Improved discrete-continuous parameterization method for concurrent topology optimization of structures and continuous material orientations

基于改进离散-连续参数化的结构和连续材料方向协同拓扑优化

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Concurrent topology optimization of structures and material orientations is a hot topic over the past decades. However, how to avoid the local optima of such problems is quite challenging. To handle this issue, a method combining the discrete material optimization method and continuous fiber orientation optimization method is proposed in our previous work, referred to as discrete-continuous parameterization (DCP), which takes advantage of the global search capability of discrete methods and the full design space of continuous methods. However, the DCP method requires too many design variables, resulting in a huge computational burden. Hence, we provide an improved DCP method to reduce the number of design variables and at the same time without sacrificing the convexity of the optimization problem in this work. In the proposed method, an extended multimaterial interpolation is firstly developed, which is capable of reducing the number of design variables greatly. Then, we integrate the proposed interpolation into the DCP method, generating an improved DCP method for the concurrent optimization of structural topology and fiber orientation. Several benchmark optimization examples show that the proposed method can greatly reduce the risk of falling into local optima with much fewer design variables.

摘要

摘要结构和材料方向的并行拓扑优化是过去几十年来的一个热门话题. 然而, 如何避免此类问题掉入局部最优相当具有挑战性. 为 了解决这个问题, 我们在以前的工作中提出了一种将离散材料优化方法和连续纤维取向优化方法相结合的方法, 称之为离散-连续参数 化(DCP), 该方法同时继承了离散方法的全局搜索能力和连续方法的全设计空间. 然而, DCP方法需要太多设计变量, 导致了巨大的计 算负担. 因此, 我们在本工作中提出了一种改进的DCP方法, 以减少设计变 量的数量, 同时又不牺牲优化问题的凸性. 在该方法中, 我们 提出了一种扩展的多材料插值方法, 该方法能够大大减少设计变量的数量然后, 我们将所提出的插值方法集成到DCP方法中, 最终建 立了结构拓扑和纤维取向协同优化的改进DCP方法. 几个基准优化实例表明, 该方法可以在能大大降低陷入局部最优的风险的同时, 使 用更少的设计变量.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. O. Sigmund, and K. Maute, Topology optimization approaches, Struct. Multidisc. Optim. 48, 1031 (2013).

    Article  MathSciNet  Google Scholar 

  2. Y. Luo, O. Sigmund, Q. Li, and S. Liu, Topology optimization of structures with infill-supported enclosed voids for additive manufacturing, Addit. Manuf. 55, 102795 (2022).

    Google Scholar 

  3. Y. Luo, O. Sigmund, Q. Li, and S. Liu, Additive manufacturing oriented topology optimization of structures with self-supported enclosed voids, Comput. Methods Appl. Mech. Eng. 372, 113385 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  4. Y. Luo, Q. Li, and S. Liu, Topology optimization of shell-infill structures using an erosion-based interface identification method, Comput. Methods Appl. Mech. Eng. 355, 94 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Zhu, H. Zhou, C. Wang, L. Zhou, S. Yuan, and W. Zhang, A review of topology optimization for additive manufacturing: Status and challenges, Chin. J. Aeronaut. 34, 91 (2021).

    Article  Google Scholar 

  6. Y. Xu, J. Zhu, Z. Wu, Y. Cao, Y. Zhao, and W. Zhang, A review on the design of laminated composite structures: Constant and variable stiffness design and topology optimization, Adv. Compos. Hybrid. Mater. 1, 460 (2018).

    Article  Google Scholar 

  7. H. Ghiasi, D. Pasini, and L. Lessard, Optimum stacking sequence design of composite materials, Part I: Constant stiffness design, Compos. Struct. 90, 1 (2009).

    Article  Google Scholar 

  8. H. Ghiasi, K. Fayazbakhsh, D. Pasini, and L. Lessard, Optimum stacking sequence design of composite materials, Part II: Variable stiffness design, Compos. Struct. 93, 1 (2010).

    Article  Google Scholar 

  9. S. A. Emam, and D. J. Inman, A review on bistable composite laminates for morphing and energy harvesting, Appl. Mech. Rev. 67, 060803 (2015).

    Article  Google Scholar 

  10. J. Chen, Y. Tang, R. Ge, Q. An, and X. Guo, Reliability design optimization of composite structures based on PSO together with FEA, Chin. J. Aeronaut. 26, 343 (2013).

    Article  Google Scholar 

  11. X. Wang, Z. Meng, B. Yang, C. Cheng, K. Long, and J. Li, Reliability-based design optimization of material orientation and structural topology of fiber-reinforced composite structures under load uncertainty, Compos. Struct. 291, 115537 (2022).

    Article  Google Scholar 

  12. L. Esposito, A. Cutolo, M. Barile, L. Lecce, G. Mensitieri, E. Sacco, and M. Fraldi, Topology optimization-guided stiffening of composites realized through automated fiber placement, Compos. Part B-Eng. 164, 309 (2019).

    Article  Google Scholar 

  13. K. Suzuki, and N. Kikuchi, A homogenization method for shape and topology optimization, Comput. Methods Appl. Mech. Eng. 93, 291 (1991).

    Article  MATH  Google Scholar 

  14. J. P. Groen, and O. Sigmund, Homogenization-based topology optimization for high-resolution manufacturable microstructures, Int. J. Numer. Meth. Eng. 113, 1148 (2018).

    Article  MathSciNet  Google Scholar 

  15. J. P. Groen, J. Wu, and O. Sigmund, Homogenization-based stiffness optimization and projection of 2D coated structures with orthotropic infill, Comput. Methods Appl. Mech. Eng. 349, 722 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Boddeti, Z. Ding, S. Kaijima, K. Maute, and M. L. Dunn, Simultaneous digital design and additive manufacture of structures and materials, Sci. Rep. 8, 15560 (2018).

    Article  Google Scholar 

  17. J. Liu, and A. C. To, Deposition path planning-integrated structural topology optimization for 3D additive manufacturing subject to self-support constraint, Comput.-Aided Des. 91, 27 (2017).

    Article  Google Scholar 

  18. W. Wu, S. Li, X. Qin, W. Liu, X. Cui, H. Li, M. Shi, and H. Liu, Effects of fiber orientation on tool wear evolution and wear mechanism when cutting carbon fiber reinforced plastics, Chin. J. Aeronaut. 36, 549 (2023).

    Article  Google Scholar 

  19. N. Boddeti, D. W. Rosen, K. Maute, and M. L. Dunn, Multiscale optimal design and fabrication of laminated composites, Compos. Struct. 228, 111366 (2019).

    Article  Google Scholar 

  20. Y. Luo, W. Chen, S. Liu, Q. Li, and Y. Ma, A discrete-continuous parameterization (DCP) for concurrent optimization of structural topologies and continuous material orientations, Compos. Struct. 236, 111900 (2020).

    Article  Google Scholar 

  21. T. Nomura, E. M. Dede, J. Lee, S. Yamasaki, T. Matsumori, A. Kawamoto, and N. Kikuchi, General topology optimization method with continuous and discrete orientation design using isoparametric projection, Int. J. Numer. Meth. Eng. 101, 571 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Pedersen, On optimal orientation of orthotropic materials, Struct. Optim. 1, 101 (1989).

    Article  Google Scholar 

  23. P. Pedersen, Bounds on elastic energy in solids of orthotropic materials, Struct. Optim. 2, 55 (1990).

    Article  Google Scholar 

  24. P. Pedersen, On thickness and orientational design with orthotropic materials, Struct. Optim. 3, 69 (1991).

    Article  Google Scholar 

  25. A. R. Díaz, and M. P. Bendsøe, Shape optimization of structures for multiple loading conditions using a homogenization method, Struct. Optim. 4, 17 (1992).

    Article  Google Scholar 

  26. H. C. Cheng, N. Kikuchi, and Z. D. Ma, An improved approach for determining the optimal orientation of orthotropic material, Struct. Optim. 8, 101 (1994).

    Article  Google Scholar 

  27. J. H. Luo, and H. C. Gea, Optimal bead orientation of 3D shell/plate structures, Finite Elem. Anal. Des. 31, 55 (1998).

    Article  MATH  Google Scholar 

  28. J. H. Luo, and H. C. Gea, Optimal orientation of orthotropic materials using an energy based method, Struct. Optim. 15, 230 (1998).

    Article  Google Scholar 

  29. X. Yan, Q. Xu, D. Huang, Y. Zhong, and X. Huang, Concurrent topology design of structures and materials with optimal material orientation, Compos. Struct. 220, 473 (2019).

    Article  Google Scholar 

  30. M. Miki, and Y. Sugiyama, Optimum design of laminated composite plates using lamination parameters, AIAA J. 31, 921 (1993).

    Article  Google Scholar 

  31. J. Foldager, J. S. Hansen, and N. Olhoff, A general approach forcing convexity of ply angle optimization in composite laminates, Struct. Optim. 16, 201 (1998).

    Article  Google Scholar 

  32. J. P. Foldager, J. S. Hansen, and N. Olhoff, Optimization of the buckling load for composite structures taking thermal effects into account, Struct. Multidisc. Optim. 21, 14 (2001).

    Article  Google Scholar 

  33. D. Peeters, D. van Baalen, and M. Abdallah, Combining topology and lamination parameter optimisation, Struct. Multidisc. Optim. 52, 105 (2015).

    Article  MathSciNet  Google Scholar 

  34. Y. Zhou, T. Nomura, and K. Saitou, Multi-component topology and material orientation design of composite structures (MTO-C), Comput. Methods Appl. Mech. Eng. 342, 438 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Lee, D. Kim, T. Nomura, E. M. Dede, and J. Yoo, Topology optimization for continuous and discrete orientation design of functionally graded fiber-reinforced composite structures, Compos. Struct. 201, 217 (2018).

    Article  Google Scholar 

  36. T. Nomura, A. Kawamoto, T. Kondoh, E. M. Dede, J. Lee, Y. Song, and N. Kikuchi, Inverse design of structure and fiber orientation by means of topology optimization with tensor field variables, Compos. Part B-Eng. 176, 107187 (2019).

    Article  Google Scholar 

  37. R. A. Salas, F. J. Ramirez-Gil, W. Montealegre-Rubio, E. C. N. Silva, and J. N. Reddy, Optimized dynamic design of laminated piezo-composite multi-entry actuators considering fiber orientation, Comput. Methods Appl. Mech. Eng. 335, 223 (2018).

    Article  MATH  Google Scholar 

  38. J. Stegmann, and E. Lund, Discrete material optimization of general composite shell structures, Int. J. Numer. Meth. Eng. 62, 2009 (2005).

    Article  MATH  Google Scholar 

  39. C. F. Hvejsel, and E. Lund, Material interpolation schemes for unified topology and multi-material optimization, Struct. Multidisc. Optim. 43, 811 (2011).

    Article  MATH  Google Scholar 

  40. S. R. Henrichsen, E. Lindgaard, and E. Lund, Robust buckling optimization of laminated composite structures using discrete material optimization considering “worst” shape imperfections, Thin-Walled Struct. 94, 624 (2015).

    Article  Google Scholar 

  41. Y. Xu, Y. Gao, C. Wu, J. Fang, and Q. Li, Robust topology optimization for multiple fiber-reinforced plastic (FRP) composites under loading uncertainties, Struct. Multidisc. Optim. 59, 695 (2019).

    Article  Google Scholar 

  42. M. Bruyneel, SFP—a new parameterization based on shape functions for optimal material selection: application to conventional composite plies, Struct. Multidisc. Optim. 43, 17 (2010).

    Article  Google Scholar 

  43. Y. Zhang, Y. Hou, and S. Liu, A new method ofdiscrete optimization for cross-section selection of truss structures, Eng. Optim. 46, 1052 (2013).

    Article  Google Scholar 

  44. T. Gao, W. H. Zhang, and P. Duysinx, Simultaneous design of structural layout and discrete fiber orientation using bi-value coding parameterization and volume constraint, Struct. Multidisc. Optim. 48, 1075 (2013).

    Article  MathSciNet  Google Scholar 

  45. T. Gao, W. Zhang, and P. Duysinx, A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate, Int. J. Numer. Meth. Eng. 91, 98 (2012).

    Article  MATH  Google Scholar 

  46. H. Ding, and B. Xu, Optimal design of vibrating composite plate considering discrete-continuous parameterization model and resonant peak constraint, Int. J. Mech. Mater. Des. 17, 679 (2021).

    Article  Google Scholar 

  47. Z. Qiu, Q. Li, Y. Luo, and S. Liu, Concurrent topology and fiber orientation optimization method for fiber-reinforced composites based on composite additive manufacturing, Comput. Methods Appl. Mech. Eng. 395, 114962 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  48. H. Ding, and B. Xu, A novel discrete-continuous material orientation optimization model for stiffness-based concurrent design of fiber composite, Compos. Struct. 273, 114288 (2021).

    Article  Google Scholar 

  49. M. P. Bendsøe, and O. Sigmund, Material interpolation schemes in topology optimization, Arch. Appl. Mech. 69, 635 (1999).

    Article  MATH  Google Scholar 

  50. L. Zhang, L. Guo, P. Sun, J. Yan, and K. Long, A generalized discrete fiber angle optimization method for composite structures: Bipartite interpolation optimization, Numer. Meth. Eng. 124, 1211 (2023).

    Article  MathSciNet  Google Scholar 

  51. F. Wang, B. S. Lazarov, and O. Sigmund, On projection methods, convergence and robust formulations in topology optimization, Struct. Multidisc. Optim. 43, 767 (2010).

    Article  MATH  Google Scholar 

  52. K. Svanberg, The method of moving asymptotes—a new method for structural optimization, Int. J. Numer. Meth. Eng. 24, 359 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  53. P. Wang, B. Zou, S. Ding, L. Li, and C. Huang, Effects of FDM-3D printing parameters on mechanical properties and microstructure of CF/PEEK and GF/PEEK, Chin. J. Aeronaut. 34, 236 (2021).

    Article  Google Scholar 

  54. X. Yan, M. Lai, D. Huang, Y. Zhang, and X. Huang, Manufacturing-oriented topological design of CFRC structures with variable fiber volume and orientation, Compos. Struct. 310, 116779 (2023).

    Article  Google Scholar 

Download references

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 12202154, 12272076, and 52188102), the China Postdoctoral Science Foundation (Grant No. 2022M711249), and the Natural Science Foundation of Hubei Province (Grant No. 2020CFA028).

Author information

Authors and Affiliations

  1. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    Yunfeng Luo  (罗云锋) & YongAn Huang  (黄永安)

  2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Shutian Liu  (刘书田), Zheng Qiu  (仇政) & Yaohui Ma  (麻耀辉)

Authors
  1. Yunfeng Luo  (罗云锋)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shutian Liu  (刘书田)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zheng Qiu  (仇政)
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yaohui Ma  (麻耀辉)
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. YongAn Huang  (黄永安)
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Author contributions Yunfeng Luo designed the research, set up the experiment, and wrote the first draft of the manuscript. Shutian Liu helped organize and revise the manuscript, and provided supervision. Zheng Qiu and Yaohui Ma reviewed the manuscript. YongAn Huang helped organize the manuscript and provided supervision.

Corresponding authors

Correspondence to Shutian Liu  (刘书田) or YongAn Huang  (黄永安).

Ethics declarations

Conflict of interestOn behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, Y., Liu, S., Qiu, Z. et al. Improved discrete-continuous parameterization method for concurrent topology optimization of structures and continuous material orientations. Acta Mech. Sin. 40, 422496 (2024). https://doi.org/10.1007/s10409-023-22496-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-023-22496-x

Search

Navigation