Combination of the phase field method and BESO method for topology optimization

  • 173 Accesses


In this paper, the phase field method and the BESO (bidirectional evolutionary structural optimization) are combined to solve the topology optimization problems. The phase field function is used to represent the structure, and a time-dependent reaction diffusion equation called the Allen–Cahn equation is used to update the phase field function. In the sensitivity of Lagrange function, the Lagrange multiplier is replaced by the product of the Lagrange multiplier and the phase field function for fairing. The material removal scheme of the BESO which is easy to implement is employed to nucleate holes in the phase field method–based topology optimization. For a given target volume in each iterative step, a threshold of sensitivity is used to determine which elements should be removed; then, the structure is updated to match the target volume. Several numerical examples based on a two-dimensional minimum compliance problem are studied to demonstrate the effectiveness of this method.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

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
Fig. 21
Fig. 22
Fig. 23


  1. Allaire G, De Gournay F, Jouve F, Toader AM (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34(1):59–80

  2. Bourdin B, Chambolle A (2006) The phase-field method in optimal design. Solid Mech Its Appl 137:207–215

  3. Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362

  4. Challis VJ (2010) A discrete level-set topology optimization code written in MATLAB. Structural & Multidisciplinary Optimization 41(3):453–464

  5. Da D, Xia L, Li G, Huang X (2017) Evolutionary topology optimization of continuum structures with smooth boundary representation. Struct Multidiscip Optim 57:2143–2159

  6. De Faria J, Novotny A, Feijoo R, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Methods Appl Mech Eng 192(7):803–829

  7. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38

  8. Gain AL, Paulino GH (2012) Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation. Struct Multidiscip Optim 46(3):327–342

  9. Hu X, Yixin L, Hangjie J (2018) A nodal finite element approximation of a phase field model for shape and topology optimization. Appl Math Comput 339:675–684

  10. Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43(14):1039–1049

  11. Huang X, Xie YM (2010a) A further review of ESO type methods for topology optimization. Struct Multidiscip Optim 41(5):671–683

  12. Huang X, Xie YM (2010b) Evolutionary topology optimization of continuum structures: methods and applications. John Wiley & Sons, Hoboken

  13. Jeong SH, Yoon GH, Takezawa A, Choi DH (2014) Development of a novel phase-field method for local stress-based shape and topology optimization. Comput Struct 132:84–98

  14. Jia H et al (2011) Evolutionary level set method for structural topology optimization. Comput Struct 89(5–6):445–454

  15. Jiang L, Chen S, Jiao X (2017) Parametric shape & topology optimization: a new level set approach based on cardinal basis functions. Int J Numer Methods Eng 114: 66–87

  16. Lee K, Ahn K, Yoo J (2016) A novel p-norm correction method for lightweight topology optimization under maximum stress constraints. Comput Struct 171:18–30

  17. Li H, Gao L, Xiao M, Gao J, Chen H, Zhang F (2016) Topological shape optimization design of continuum structures via an effective level set method. Cogent Eng 3:1–14

  18. Munk D, Steven G, Vio G (2015) Topology and shape optimization methods using evolutionary algorithms: a review. Springer-Verlag New York, Inc., New York

  19. Querin OM, Steven GP (1998) Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comput 15(8):1031–1048

  20. Querin OM, Steven GP, Xie YM (2000) Evolutionary structural optimisation using an additive algorithm. Finite Elem Anal Des 34(3):291–308

  21. Seong HK, Yoo J (2017) Probability distribution function inspired structural optimization for frequency response problems. Comput Methods Appl Mech Eng 318:783–802

  22. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055

  23. Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718

  24. Takezawa A, Yoon GH, Jeong SH, Kobashi M, Kitamura M (2014) Structural topology optimization with strength and heat conduction constraints. Comput Methods Appl Mech Eng 276:341–361

  25. Tavakoli, Rouhollah (2014) Multimaterial topology optimization by volume constrained Allen–Cahn system and regularized projected steepest descent method. Comput Methods Appl Mech Eng 276:534–565

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

  27. Wang SY, Wang MY (2006b) Structural shape and topology optimization using an implicit free boundary parametrization method. CMES - Comput Model Eng Sci 13(2):119–147

  28. Wang Y, Luo Z, Kang Z, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570–1586

  29. Wei P, Li Z, Li X, Wang MY (2018) An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Struct Multidiscip Optim 58:831–849

  30. Xia L, Xia Q, Huang X, Xie YM (2016) Bi-directional evolutionary structural optimization on advanced structures and materials: a comprehensive review. Arch Comput Methods Eng 25:437–478

  31. Xia L, Zhang L, Xia Q, Shi T (2018a) Stress-based topology optimization using bi-directional evolutionary structural optimization method. Comput Methods Appl Mech Eng 333:356–370

  32. Xia Q, Shi T, Xia L (2018b) Topology optimization for heat conduction by combining level set method and BESO method. Int J Heat Mass Transf 127:200–209

  33. Xia Q, Shi T, Xia L (2019) Stable hole nucleation in level set based topology optimization by using the material removal scheme of BESO. Comput Methods Appl Mech Eng 343:438–452

  34. Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896

  35. Zhou S, Wang MY (2007) Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct Multidiscip Optim 33(2):89

  36. Zhu B, Zhang X, Fatikow S, Wang N (2015) Bi-directional evolutionary level set method for topology optimization. Eng Optim 47(3):17

Download references

Author information

Correspondence to Jiawen Gao.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

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

Responsible Editor: Hai Huang

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gao, J., Song, B. & Mao, Z. Combination of the phase field method and BESO method for topology optimization. Struct Multidisc Optim 61, 225–237 (2020).

Download citation


  • Phase field method
  • BESO
  • Topology optimization
  • Allen–Cahn equation
  • Nucleate holes