Skip to main content
SpringerLink
Log in

A novel discrete adjoint-based level set topology optimization method in B-spline space

  • Research Article
  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

This paper presents a novel computational scheme for sensitivity analysis of the velocity field in the level set method using the discrete adjoint method. The velocity field is represented in B-spline space, and the adjoint equations are constructed based on the discretized governing equations. The key contribution of this work is the demonstration that the velocity field in the level set method can be entirely obtained from the discrete adjoint method. This eliminates the need for shape sensitivity analysis, which is commonly used in standard level set methods. The results demonstrate the effectiveness of the approach in producing optimized results for stress and linearized buckling problems. Overall, the proposed method has the potential to simplify the way in which topology optimization problems using level set methods are solved, and has significant implications for the design of a broad range of engineering applications.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  • Adalsteinsson D, Sethian JA (1999) The fast construction of extension velocities in level set methods. J Comput Phys 148(1):2–22

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    Article  MathSciNet  MATH  Google Scholar 

  • Amir O (2021) Efficient stress-constrained topology optimization using inexact design sensitivities. Int J Numer Meth Eng 122(13):3241–3272

    Article  MathSciNet  Google Scholar 

  • Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, Berlin

    MATH  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  MathSciNet  MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654

    MATH  Google Scholar 

  • Dunning PD, Ovtchinnikov E, Scott J, Kim HA (2016) Level-set topology optimization with many linear buckling constraints using an efficient and robust eigensolver. Int J Numer Meth Eng 107(12):1029–1053

    Article  MathSciNet  MATH  Google Scholar 

  • Ferrari F, Sigmund O (2020) Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses. Comput Methods Appl Mech Eng 363:112911

    Article  MathSciNet  MATH  Google Scholar 

  • Ferrari F, Sigmund O, Guest JK (2021a) Topology optimization with linearized buckling criteria in 250 lines of Matlab. Struct Multidiscip Optim 63:3045–3066

    Article  MathSciNet  Google Scholar 

  • Ferrari F, Sigmund O, Guest JK (2021b) Topology optimization with linearized buckling criteria in 250 lines of Matlab. Struct Multidiscip Optim 63(6):3045–3066

    Article  MathSciNet  Google Scholar 

  • Gao X, Ma H (2015) Topology optimization of continuum structures under buckling constraints. Comput Struct 157:142–152

    Article  Google Scholar 

  • Giles MB, Duta MC, Muller J-D, Pierce NA (2003) Algorithm developments for discrete adjoint methods. AIAA J 41(2):198–205

    Article  Google Scholar 

  • Guirguis D, Hamza K, Aly M, Hegazi H, Saitou K (2015) Multi-objective topology optimization of multi-component continuum structures via a Kriging-interpolated level set approach. Struct Multidiscip Optim 51(3):733–748

    Article  MathSciNet  Google Scholar 

  • Gunzburger MD (2002) Perspectives in flow control and optimization. SIAM, Philadelphia

    Book  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech. https://doi.org/10.1115/1.4027609

    Article  Google Scholar 

  • Huang X, Xie M (2010) Evolutionary topology optimization of continuum structures: methods and applications. Wiley, Hoboken

    Book  MATH  Google Scholar 

  • Jiang L, Chen S (2017) Parametric structural shape & topology optimization with a variational distance-regularized level set method. Comput Methods Appl Mech Eng 321:316–336

    Article  MathSciNet  MATH  Google Scholar 

  • Kambampati S, Chung H, Kim HA (2021) A discrete adjoint based level set topology optimization method for stress constraints. Comput Methods Appl Mech Eng 377:113563

    Article  MathSciNet  MATH  Google Scholar 

  • Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620

    Article  Google Scholar 

  • Luo Z, Wang MY, Wang S, Wei P (2008) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Meth Eng 76(1):1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Norato J, Haber R, Tortorelli D, Bendsøe MP (2004) A geometry projection method for shape optimization. Int J Numer Meth Eng 60(14):2289–2312

    Article  MathSciNet  MATH  Google Scholar 

  • Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comput Methods Appl Mech Eng 293:306–327. https://doi.org/10.1016/j.cma.2015.05.005

    Article  MathSciNet  MATH  Google Scholar 

  • Picelli R, Townsend S, Brampton C, Norato J, Kim HA (2018) Stress-based shape and topology optimization with the level set method. Comput Methods Appl Mech Eng 329:1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Poon NM, Martins JR (2007) An adaptive approach to constraint aggregation using adjoint sensitivity analysis. Struct Multidiscip Optim 34(1):61–73

    Article  Google Scholar 

  • Sarcar M, Rao KM, Narayan KL (2008) Computer aided design and manufacturing. PHI Learning Pvt. Ltd., New Delhi

    Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Meth Eng 24(2):359–373

    Article  MathSciNet  MATH  Google Scholar 

  • Van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013a) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472

    Article  MathSciNet  Google Scholar 

  • Van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013b) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48:437–472

    Article  MathSciNet  Google Scholar 

  • Wang Y, Kang Z (2018) A velocity field level set method for shape and topology optimization. Int J Numer Meth Eng 115(11):1315–1336

    Article  MathSciNet  Google Scholar 

  • Wang Y, Kang Z (2021) MATLAB implementations of velocity field level set method for topology optimization: an 80-line code for 2D and a 100-line code for 3D problems. Struct Multidiscip Optim 64(6):4325–4342

    Article  Google Scholar 

  • Wang S, Wang MY (2006) Radial basis functions and level set method for structural topology optimization. Int J Numer Meth Eng 65(12):2060–2090

    Article  MathSciNet  MATH  Google Scholar 

  • Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Kang Z, Liu P (2019) Velocity field level-set method for topological shape optimization using freely distributed design variables. Int J Numer Meth Eng 120(13):1411–1427

    Article  MathSciNet  Google Scholar 

  • Wang Y, Yang H, Kang Z (2022a) Velocity field level set method incorporating topological derivatives for topology optimization. J Appl Mech 89(6):061002

    Article  Google Scholar 

  • Wang C, Xie YM, Lin X, Zhou S (2022b) A reaction diffusion-based B-spline level set (RDBLS) method for structural topology optimization. Comput Methods Appl Mech Eng 398:115252

    Article  MathSciNet  MATH  Google Scholar 

  • Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Meth Eng 78(4):379–402

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang S, Gain AL, Norato JA (2017b) Stress-based topology optimization with discrete geometric components. Comput Methods Appl Mech Eng 325:1–21

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Struct Multidiscip Optim 53(6):1243–1260. https://doi.org/10.1007/s00158-015-1372-3

    Article  MathSciNet  Google Scholar 

  • Zhang W, Yang W, Zhou J, Li D, Guo X (2017a) Structural topology optimization through explicit boundary evolution. J Appl Mech 84(1):011011

    Article  Google Scholar 

  • Zhang W et al (2017) Explicit three dimensional topology optimization via Moving Morphable Void (MMV) approach. Comput Methods Appl Mech Eng 322:590–614. https://doi.org/10.1016/j.cma.2017.05.002

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to Prof. Kazu Saitou for editing the manuscript.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    Hao Deng

Authors
  1. Hao Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Deng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, H. A novel discrete adjoint-based level set topology optimization method in B-spline space. Optim Eng (2023). https://doi.org/10.1007/s11081-023-09851-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11081-023-09851-7

Keywords

Search

Navigation