Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Structural topology optimization with design-dependent pressure loads

  • 936 Accesses

  • 11 Citations


One way to solve topology optimization of continuum structures under design-dependent pressure loads is to recover the loading surface at each step of the minimization process. During the topology evolution, the intermediate topologies obtained by using the SIMP (Solid Isotropic Material with Penalization) method actually can be regarded as gray scale images, for which the paper proposes a new material boundary identification scheme based on image segmentation technique. The Distance Regularized Level Set Evolution (DRLSE) method proposed by Li et al., IEEE Trans Image Process 19(12):3243–3254 (2010) is utilized to detect the image edge. Then the pressure boundary is represented as the zero level contour of a level set function (LSF). Inheriting the merits of the level set method, the current scheme can handle the topological change of the pressure boundary efficiently and be easily extended to the three-dimensional problems. In addition, the scheme is more stable compared with the conventional loading surface searching methods since it works well for the intermediate topologies with local scattered densities. A new optimization framework is also proposed to avoid the load sensitivity analysis. Four numerical examples are presented to show the validity and advantages of the proposed scheme.

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

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


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

  2. Aubert G, Kornprobst P (2006) Mathematical problems in image processing: partial differential equations and the calculus of variations, applied mathematical sciences, vol 147. Springer, New York

  3. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):192–202

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

  5. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635–654

  6. Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM: Control Optim Calc Var 9:19–48

  7. Bruggi M, Cinquini C (2009) An alternative truly-mixed formulation to solve pressure load problems in topology optimization. Comput Methods Appl Mech Eng 198(17-20):1500–1512. doi:10.1016/j.cma.2008.12.009

  8. Chen BC, Kikuchi N (2001) Topology optimization with design-dependent loads. Finite Elem Anal Des 37 (1):57–70. doi:10.1016/S0168-874X(00)00021-4

  9. Cook RD, Malkus DS, Plesha ME, Witt RJ (2007) Concepts and applications of finite element analysis. Wiley

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

  11. Du J, Olhoff N (2004a) Struct Multidiscip Optim 27(3):151–165. doi:10.1007/s00158-004-0379-y

  12. Du J, Olhoff N (2004b) Topological optimization of continuum structures with design-dependent surface loading — part ii: algorithm and examples for 3d problems. Struct Multidiscip Optim 27(3):166–177. doi:10.1007/s00158-004-0380-5

  13. Hammer V, Olhoff N (2000) Topology optimization of continuum structures subjected to pressure loading. Struct Multidiscip Optim 19(2):85–92. doi:10.1007/s001580050088

  14. Hammer V, Olhoff N (2001) Topology optimization of 3D structures with design dependent loads. In: Proceedings of the 4th WCSMO. Liaoning Electronic Press

  15. Lee E, Martins JR (2012) Structural topology optimization with design-dependent pressure loads. Comput Methods Appl Mech Eng 233-236:40–48. doi:10.1016/j.cma.2012.04.007

  16. Li C (2010) Matlab code for drlse. available from. http://www.engr.uconn.edu/cmli/

  17. Li C, Xu C, Gui C, Fox MD (2010) Distance regularized level set evolution and its application to image segmentation. IEEE Trans Image Process 19(12):3243–3254

  18. Michaleris P, Tortorelli DA, Vidal CA (1994) Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int J Numer Methods Eng 37(14):2471–2499

  19. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations. J Comput Phys 79(1):12–49

  20. Querin O, Steven G, Xie Y (1998) Evolutionary structural optimisation (eso) using a bidirectional algorithm. Eng Comput 15(8):1031–1048

  21. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524. doi:10.1080/08905459708945415

  22. Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21 (2):120–127

  23. Sigmund O, Clausen P (2007) Topology optimization using a mixed formulation: an alternative way to solve pressure load problems. Comput Methods Appl Mech Eng 196(13-16):1874–1889. doi:10.1016/j.cma.2006.09.021

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

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

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

  27. Zhang H, Zhang X, Liu S (2008) A new boundary search scheme for topology optimization of continuum structures with design-dependent loads. Struct Multidiscip Optim 37(2):121–129. doi:10.1007/s00158-007-0221-4

  28. Zheng B, Chang Cj, Gea HC (2008) Topology optimization with design-dependent pressure loading. Struct Multidiscip Optim 38(6):535–543. doi:10.1007/s00158-008-0317-5

  29. Zhou M, Rozvany G (1991) The coc algorithm, part ii: Topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336

Download references


The authors would like to thank Dr. Li Chunming from University of Pennsylvania School of Medicine for providing the source code of DRLSE. This work was supported by National Natural Science Foundation of China (Grant No. 51109132) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110073120015). Critical comments from reviewers are also greatly appreciated.

Author information

Correspondence to Min Zhao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Zhao, M. & Ge, T. Structural topology optimization with design-dependent pressure loads. Struct Multidisc Optim 53, 1005–1018 (2016). https://doi.org/10.1007/s00158-015-1376-z

Download citation


  • Topology optimization
  • Design-dependent loads
  • Boundary identification
  • Image segmentation
  • Level set