0
Research Papers: Design Automation

A Level Set Method With a Bounded Diffusion for Structural Topology Optimization

[+] Author and Article Information
Benliang Zhu

Guangdong Key Laboratory of
Precision Equipment and Manufacturing
Technology,
South China University of Technology,
Guangzhou 510640, Guangdong, China
e-mail: meblzhu@scut.edu.cn

Rixin Wang, Hai Li

Guangdong Key Laboratory of
Precision Equipment and Manufacturing
Technology,
South China University of Technology,
Guangzhou 510640, Guangdong, China

Xianmin Zhang

Guangdong Key Laboratory of Precision
Equipment and Manufacturing Technology,
South China University of Technology,
Guangzhou 510640, Guangdong, China
e-mail: zhangxm@scut.edu.cn

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 28, 2017; final manuscript received April 9, 2018; published online May 11, 2018. Assoc. Editor: James K. Guest.

J. Mech. Des 140(7), 071402 (May 11, 2018) (11 pages) Paper No: MD-17-1662; doi: 10.1115/1.4039975 History: Received September 28, 2017; Revised April 09, 2018

In level-set-based topology optimization methods, the spatial gradients of the level set field need to be controlled to avoid excessive flatness or steepness at the structural interfaces. One of the most commonly utilized methods is to generalize the traditional Hamilton−Jacobi equation by adding a diffusion term to control the level set function to remain close to a signed distance function near the structural boundaries. This study proposed a new diffusion term and built it into the Hamilton-Jacobi equation. This diffusion term serves two main purposes: (I) maintaining the level set function close to a signed distance function near the structural boundaries, thus avoiding periodic re-initialization, and (II) making the diffusive rate function to be a bounded function so that a relatively large time-step can be used to speed up the evolution of the level set function. A two-phase optimization algorithm is proposed to ensure the stability of the optimization process. The validity of the proposed method is numerically examined on several benchmark design problems in structural topology optimization.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Bendsøe, M. P. , and Sigmund, O. , 2003, Topology Optimization: Theory, Methods and Applications, Springer, Berlin.
Bendsøe, M. P. , and Kikuchi, N. , 1988, “ Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 197–224. [CrossRef]
Rozvany, G. I. N. , Zhou, M. , and Birker, T. , 1992, “ Generalized Shape Optimization Without Homogenization,” Struct. Optim., 4(3–4), pp. 250–252. [CrossRef]
Xie, Y. M. , and Steven, G. P. , 1997, Evolutionary Structural Optimization, Springer, Berlin. [CrossRef]
Allaire, G. , Jouve, F. , and Toader, A. M. , 2004, “ Structural Optimization Using Sensitivity Analysis and a Level Set Method,” J. Comput. Phys., 194(1), pp. 363–393. [CrossRef]
Deng, X. , Wang, Y. , Yan, J. , Tao, L. , and Wang, S. , 2016, “ Topology Optimization of Total Femur Structure: Application of Parameterized Level Set Method Under Geometric Constraints,” ASME J. Mech. Des., 138(1), p. 011402.
Yamada, T. , Izui, K. , and Nishiwaki, S. , 2011, “ A Level Set-Based Topology Optimization Method for Maximizing Thermal Diffusivity in Problems Including Design-Dependent Effects,” ASME J. Mech. Des., 133(3), p. 031011. [CrossRef]
Maute, K. , Tkachuk, A. , Wu, J. , Qi, H. J. , Ding, Z. , and Dunn, M. L. , 2015, “ Level Set Topology Optimization of Printed Active Composites,” ASME J. Mech. Des., 137(11), p. 111402.
van Dijk, N. P. , Maute, K. , Langelaar, M. , and van Keulen, F. , 2013, “ Level Set Methods for Structural Topology Optimization: A Review,” Struct. Multidiscip. Optim., 48(3), pp. 437–472. [CrossRef]
Wu, J. , Luo, Z. , Li, H. , and Zhang, N. , 2017, “ Level-Set Topology Optimization for Mechanical Metamaterials Under Hybrid Uncertainties,” Comput. Methods Appl. Mech. Eng., 319(1), pp. 414–441. [CrossRef]
Ghasemi, H. , Park, H. S. , and Rabczuk, T. , 2017, “ A Level-Set Based Iga Formulation for Topology Optimization of Flexoelectric Materials,” Comput. Methods Appl. Mech. Eng., 313, pp. 239–258. [CrossRef]
Lawry, M. , and Maute, K. , 2018, “ Level Set Shape and Topology Optimization of Finite Strain Bilateral Contact Problems,” Int. J. Numer. Methods Eng., 113(8), pp. 1340–1369.
Wang, Y. , Luo, Z. , Zhang, N. , and Qin, Q. , 2016, “ Topological Shape Optimization of Multifunctional Tissue Engineering Scaffolds With Level Set Method,” Struct. Multidiscip. Optim., 54(2), pp. 333–347.
Behrou, R. , Lawry, M. , and Maute, K. , 2017, “ Level Set Topology Optimization of Structural Problems With Interface Cohesion,” Int. J. Numer. Methods Eng., 112(8), pp. 990–1016.
Zhu, B. , Zhang, X. , Wang, N. , and Fatikow, S. , 2016, “ Optimize Heat Conduction Problem Using Level Set Method With a Weighting Based Velocity Constructing Scheme,” Int. J. Heat Mass Transfer, 99, pp. 441–451. [CrossRef]
Zhu, B. , Zhang, X. , and Fatikow, S. , 2014, “ A Velocity Predictor–Corrector Scheme in Level Set-Based Topology Optimization to Improve Computational Efficiency,” ASME J. Mech. Des., 136(9), p. 091001. [CrossRef]
Jiang, L. , and Chen, S. , 2017, “ Parametric Structural Shape & Topology Optimization With a Variational Distance-Regularized Level Set Method,” Comput. Methods Appl. Mech. Eng., 321, pp. 316–336. [CrossRef]
Hartmann, D. , Meinke, M. , and Schröder, W. , 2010, “ The Constrained Reinitialization Equation for Level Set Methods,” J. Comput. Phys., 229(5), pp. 1514–1535. [CrossRef]
Yamasaki, S. , Nishiwaki, S. , Yamada, T. , Izui, K. , and Yoshimura, M. , 2010, “ A Structural Optimization Method Based on the Level Set Method Using a New Geometry-Based Re-Initialization Scheme,” Int. J. Numer. Methods Eng., 83(12), pp. 1580–1624. [CrossRef]
Wang, S. , and Wang, M. Y. , 2006, “ Radial Basis Functions and Level Set Method for Structural Topology Optimization,” Int. J. Numer. Methods Eng., 65(12), pp. 2060–2090. [CrossRef]
Luo, Z. , Wang, M. Y. , Wang, S. , and Wei, P. , 2008, “ A Level Set-Based Parameterization Method for Structural Shape and Topology Optimization,” Int. J. Numer. Methods Eng., 76(1), pp. 1–26. [CrossRef]
Wei, P. , and Wang, M. Y. , 2009, “ Piecewise Constant Level Set Method for Structural Topology Optimization,” Int. J. Numer. Methods Eng., 78(4), pp. 379–402. [CrossRef]
Luo, Z. , Tong, L. , Wang, M. Y. , and Wang, S. , 2007, “ Shape and Topology Optimization of Compliant Mechanisms Using a Parameterization Level Set Method,” J. Comput. Phys., 227(1), pp. 680–705. [CrossRef]
Guo, X. , 2014, “ Doing Topology Optimization Explicitly and Geometrically: A New Moving Morphable Components Based Framework,” ASME J. Appl. Mech., 81(8), p. 081009. [CrossRef]
Zhang, W. , Zhou, J. , Zhu, Y. , and Guo, X. , 2017, “ Structural Complexity Control in Topology Optimization Via Moving Morphable Component (Mmc) Approach,” Struct. Multidiscip. Optim., 56(3), pp. 535–552.
Guo, X. , Zhang, W. , Zhang, J. , and Yuan, J. , 2016, “ Explicit Structural Topology Optimization Based on Moving Morphable Components (MMC) With Curved Skeletons,” Comput. Methods Appl. Mech. Eng., 310, pp. 711–748. [CrossRef]
Yamada, T. , Izui, K. , Nishiwaki, S. , and Takezawa, A. , 2010, “ A Topology Optimization Method Based on the Level Set Method Incorporating a Fictitious Interface Energy,” Comput. Methods Appl. Mech. Eng., 199(45–48), pp. 2876–2891. [CrossRef]
Li, C. , Xu, C. , Gui, C. , and Fox, M. D. , 2010, “ Distance Regularized Level Set Evolution and Its Application to Image Segmentation,” IEEE Trans. Image Process., 19(12), pp. 3243–3254. [CrossRef] [PubMed]
Zhu, B. , Zhang, X. , and Fatikow, S. , 2015, “ Structural Topology and Shape Optimization Using a Level Set Method With Distance-Suppression Scheme,” Comput. Methods Appl. Mech. Eng., 283, pp. 1214–1239. [CrossRef]
Wang, M. , Wang, X. M. , and Guo, D. M. , 2003, “ A Level Set Method for Structural Topology Optimization,” Comput. Methods Appl. Mech. Eng., 192(1–2), pp. 227–246. [CrossRef]
Zhou, M. , and Wang, M. Y. , 2012, “ A Semi-Lagrangian Level Set Method for Structural Optimization,” Struct. Multidiscip. Optim., 46(4), pp. 487–501. [CrossRef]
Allaire, G. , Gournay, F. D. , Jouve, F. , and Toader, A. M. , 2005, “ Structural Optimization Using Topological and Shape Sensitivity Via a Level Set Method,” Control Cybern., 34(1), pp. 59–80. https://eudml.org/doc/209353
Zhu, B. , and Zhang, X. , 2012, “ A New Level Set Method for Topology Optimization of Distributed Compliant Mechanisms,” Int. J. Numer. Methods Eng., 91(8), pp. 843–871. [CrossRef]
Gilboa, G. , Sochen, N. , and Zeevi, Y. Y. , 2002, “ Forward-and-Backward Diffusion Processes for Adaptive Image Enhancement and Denoising,” IEEE Trans. Image Process., 11(7), pp. 689–703. [CrossRef] [PubMed]
Sethian, J. A. , 1999, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Version, and Material Science, Cambridge University Press, Cambridge, UK.
Kim, N. H. , and Chang, Y. , 2005, “ Eulerian Shape Design Sensitivity Analysis and Optimization With a Fixed Grid,” Comput. Methods Appl. Mech. Eng., 194(30–33), pp. 3291–3314. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

A diagram of a possible diffusive rate function that is bounded

Grahic Jump Location
Fig. 2

A diagram of the diffusion rate of Eq. (19)

Grahic Jump Location
Fig. 3

A diagram of the energy density function r of Eq. (21)

Grahic Jump Location
Fig. 4

Different categories of elements when conducting finite element analysis

Grahic Jump Location
Fig. 5

The Michell-type structure design problem: (a) the design domain and (b) the initial configuration

Grahic Jump Location
Fig. 6

The intermediate results and the optimized configuration of the bridge design problem: (a) step 1, (b) step 5, (c) step 10, (d) step 20, (e) step 30, and (f) step 37 (optimized)

Grahic Jump Location
Fig. 7

Corresponding level set functions in Fig. 6: (a) step 1, (b) step 5, (c) step 10, (d) step 20, (e) step 30, and (f) step 37 (optimized)

Grahic Jump Location
Fig. 8

Convergence curves of the bridge problem

Grahic Jump Location
Fig. 9

Convergence curve of the ℜ(|∇ϕ|) when using the IDS method to solve the bridge problem

Grahic Jump Location
Fig. 10

Optimization process in using the DS method proposed in Ref. [29] and a different initial configuration: (a) initial, (b) step 5, (c) step 10, (d) step 20, (e) step 30, and (f) step 44 (optimized)

Grahic Jump Location
Fig. 11

Optimization process in using the proposed IDS method and a different initial configuration: (a) initial, (b) step 5, (c) step 10, (d) step 20, (e) step 30, and (f) step 34 (optimized)

Grahic Jump Location
Fig. 12

The convergence histories of the mean compliance and volume fraction of the optimization processes shown in Figs. 10 and 11

Grahic Jump Location
Fig. 13

The convergence histories of the mean compliance and volume fraction. The ϖ for determining the convergence criteria is enhanced by setting it to 0.1%.

Grahic Jump Location
Fig. 16

The final designs of the short cantilever obtained using different mesh refinements: (a) 40 × 40, (b) 80 × 80, (c) 120 × 120, and (d) 160 × 160

Grahic Jump Location
Fig. 17

The final designs of the short cantilever obtained using different mesh refinements: (a) 40 × 40, (b) 80 × 80, (c) 120 × 120, and (d) 160 × 160

Grahic Jump Location
Fig. 14

Optimization process in using the proposed method and another different initial configuration: (a) initial, (b) step 5, (c) step 10, (d) step 20, (e) step 30, and (f) step 34 (optimized)

Grahic Jump Location
Fig. 15

The design domain of the short cantilever (left) and the utilized initial configuration (right)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In