Level set band method: A combination of density-based and level set methods for the topology optimization of continuums

Peng WEI, Wenwen WANG, Yang YANG, Michael Yu WANG

PDF(3416 KB)
PDF(3416 KB)
Front. Mech. Eng. ›› 2020, Vol. 15 ›› Issue (3) : 390-405. DOI: 10.1007/s11465-020-0588-0
RESEARCH ARTICLE

Level set band method: A combination of density-based and level set methods for the topology optimization of continuums

Author information +
History +

Abstract

The level set method (LSM), which is transplanted from the computer graphics field, has been successfully introduced into the structural topology optimization field for about two decades, but it still has not been widely applied to practical engineering problems as density-based methods do. One of the reasons is that it acts as a boundary evolution algorithm, which is not as flexible as density-based methods at controlling topology changes. In this study, a level set band method is proposed to overcome this drawback in handling topology changes in the level set framework. This scheme is proposed to improve the continuity of objective and constraint functions by incorporating one parameter, namely, level set band, to seamlessly combine LSM and density-based method to utilize their advantages. The proposed method demonstrates a flexible topology change by applying a certain size of the level set band and can converge to a clear boundary representation methodology. The method is easy to implement for improving existing LSMs and does not require the introduction of penalization or filtering factors that are prone to numerical issues. Several 2D and 3D numerical examples of compliance minimization problems are studied to illustrate the effects of the proposed method.

Keywords

level set method / topology optimization / density-based method / level set band

Cite this article

Download citation ▾
Peng WEI, Wenwen WANG, Yang YANG, Michael Yu WANG. Level set band method: A combination of density-based and level set methods for the topology optimization of continuums. Front. Mech. Eng., 2020, 15(3): 390‒405 https://doi.org/10.1007/s11465-020-0588-0

References

[1]
Cheng K T, Olhoff N. An investigation concerning optimal design of solid elastic plates. International Journal of Solids and Structures, 1981, 17(3): 305–323
CrossRef Google scholar
[2]
Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224
CrossRef Google scholar
[3]
Bendsøe M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202
CrossRef Google scholar
[4]
Rozvany G I, Zhou M, Birker T. Generalized shape optimization without homogenization. Structural Optimization, 1992, 4(3–4): 250–252
CrossRef Google scholar
[5]
Bendsøe M P, Sigmund O. Material interpolation schemes in topology optimization. Archive of Applied Mechanics, 1999, 69(9–10): 635–654
[6]
Stolpe M, Svanberg K. An alternative interpolation scheme for minimum compliance topology optimization. Structural and Multidisciplinary Optimization, 2001, 22(2): 116–124
CrossRef Google scholar
[7]
Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896
CrossRef Google scholar
[8]
Querin O M, Steven G P, Xie Y M. Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Engineering Computations, 1998, 15(8): 1031–1048
CrossRef Google scholar
[9]
Huang X, Xie Y M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in Analysis and Design, 2007, 43(14): 1039–1049
CrossRef Google scholar
[10]
Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 1988, 79(1): 12–49
CrossRef Google scholar
[11]
Sethian J A, Wiegmann A. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics, 2000, 163(2): 489–528
CrossRef Google scholar
[12]
Osher S J, Santosa F. Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum. Journal of Computational Physics, 2001, 171(1): 272–288
CrossRef Google scholar
[13]
Allaire G, Jouve F, Toader A M. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, 194(1): 363–393
CrossRef Google scholar
[14]
Wang M Y, Wang X M, Guo D M. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 227–246
CrossRef Google scholar
[15]
Sui Y K, Ye H L. Continuum Topology Optimization Methods ICM. Beijing: Science Press, 2013 (in Chinese)
[16]
Tong L Y, Lin J Z. Structural topology optimization with implicit design variable—Optimality and algorithm. Finite Elements in Analysis and Design, 2011, 47(8): 922–932
CrossRef Google scholar
[17]
Wei P, Ma H T, Wang M Y. The stiffness spreading method for layout optimization of truss structures. Structural and Multidisciplinary Optimization, 2014, 49(4): 667–682
CrossRef Google scholar
[18]
Cao M J, Ma H T, Wei P. A modified stiffness spreading method for layout optimization of truss structures. Acta Mechanica Sinica, 2018, 34(6): 1072–1083
CrossRef Google scholar
[19]
Zhang W S, Yuan J, Zhang J, A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Structural and Multidisciplinary Optimization, 2016, 53(6): 1243–1260
CrossRef Google scholar
[20]
Zhang W S, Li D D, Kang P, Explicit topology optimization using IGA-based moving morphable void (MMV) approach. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112685
CrossRef Google scholar
[21]
Sigmund O. A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127
CrossRef Google scholar
[22]
Huang X, Xie Y M. Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. New York: Wiley, 2010
[23]
Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer, 2002
[24]
Sethian J A. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. London: Cambridge University Press, 1999
[25]
Xia Q, Shi T, Wang M Y, Simultaneous optimization of cast part and parting direction using level set method. Structural and Multidisciplinary Optimization, 2011, 44(6): 751–759
CrossRef Google scholar
[26]
van Dijk N P, Maute K, Langelaar M, van Keulen F. Level-set methods for structural topology optimization: A review. Structural and Multidisciplinary Optimization, 2013, 48(3): 437–472
CrossRef Google scholar
[27]
Xia Q, Shi T. Optimization of structures with thin-layer functional device on its surface through a level set based multiple-type boundary method. Computer Methods in Applied Mechanics and Engineering, 2016, 311: 56–70
CrossRef Google scholar
[28]
Jiang L, Guo Y, Chen S K, Concurrent optimization of structural topology and infill properties with a CBF-based level set method. Frontiers of Mechanical Engineering, 2019, 14(2): 171–189
CrossRef Google scholar
[29]
Xia Q, Shi T, Xia L. Stable hole nucleation in level set based topology optimization by using the material removal scheme of BESO. Computer Methods in Applied Mechanics and Engineering, 2019, 343: 438–452
CrossRef Google scholar
[30]
Barrera J L, Geiss M J, Maute K. Hole seeding in level set topology optimization via density fields. Structural and Multidisciplinary Optimization, 2020, 61: 1319–1343
CrossRef Google scholar
[31]
Shu L, Wang M Y, Fang Z D, Level set based structural topology optimization for minimizing frequency response. Journal of Sound and Vibration, 2011, 330(24): 5820–5834
CrossRef Google scholar
[32]
Shu C W, Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Journal of Computational Physics, 1989, 83(1): 32–78
CrossRef Google scholar
[33]
Jiang G S, Peng D P. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM Journal on Scientific Computing, 2000, 21(6): 2126–2143
CrossRef Google scholar
[34]
Peng D P, Merriman B, Osher S, A PDE-based fast local level set method. Journal of Computational Physics, 1999, 155(2): 410–438
CrossRef Google scholar
[35]
Adalsteinsson D, Sethian J A. The fast construction of extension velocities in level set methods. Journal of Computational Physics, 1999, 148(1): 2–22
CrossRef Google scholar
[36]
Guo X, Zhao K, Wang M Y. A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function. Control and Cybernetics, 2005, 34(1): 255–282
[37]
Belytschko T, Xiao S P, Parimi C. Topology optimization with implicit functions and regularization. International Journal for Numerical Methods in Engineering, 2003, 57(8): 1177–1196
CrossRef Google scholar
[38]
Wei P, Wang M Y. Piecewise constant level set method for structural topology optimization. International Journal for Numerical Methods in Engineering, 2009, 78(4): 379–402
CrossRef Google scholar
[39]
Yamada T, Izui K, Nishiwaki S, A topology optimization method based on the level set method incorporating a fictitious interface energy. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45–48): 2876–2891
CrossRef Google scholar
[40]
de Ruiter M J, van Keulen F. Topology optimization using a topology description function. Structural and Multidisciplinary Optimization, 2004, 26(6): 406–416
CrossRef Google scholar
[41]
Wang S Y, Wang M Y. Radial basis functions and level set method for structural topology optimization. International Journal for Numerical Methods in Engineering, 2006, 65(12): 2060–2090
CrossRef Google scholar
[42]
Luo Z, Wang M Y, Wang S Y, A level set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering, 2008, 76(1): 1–26
CrossRef Google scholar
[43]
Wei P, Li Z Y, Li X P, An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Structural and Multidisciplinary Optimization, 2018, 58(2): 831–849
CrossRef Google scholar
[44]
Wang Y J, Benson D J. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers of Mechanical Engineering, 2016, 11(4): 328–343
CrossRef Google scholar
[45]
Wei P, Paulino G H. A parameterized level set method combined with polygonal finite elements in topology optimization. Structural and Multidisciplinary Optimization, 2020, 61: 1913–1928
CrossRef Google scholar
[46]
Burger M, Hackl B, Ring W. Incorporating topological derivatives into level set methods. Journal of Computational Physics, 2004, 194(1): 344–362
CrossRef Google scholar
[47]
Ye Q, Guo Y, Chen S K, Topology optimization of conformal structures on manifolds using extended level set methods (X-LSM) and conformal geometry theory. Computer Methods in Applied Mechanics and Engineering, 2019, 344: 164–185
CrossRef Google scholar
[48]
Wei P, Wang M Y, Xing X H. A study on X-FEM in continuum structural optimization using a level set model. Computer Aided Design, 2010, 42(8): 708–719
CrossRef Google scholar
[49]
Li L, Wang M Y, Wei P. XFEM schemes for level set based structural optimization. Frontiers of Mechanical Engineering, 2012, 7(4): 335–356
CrossRef Google scholar
[50]
Geiss M J, Barrera J L, Boddeti N, A regularization scheme for explicit level-set XFEM topology optimization. Frontiers of Mechanical Engineering, 2019, 14(2): 153–170
CrossRef Google scholar
[51]
Bruyneel M, Duysinx P. Note on topology optimization of continuum structures including self-weight. Structural and Multidisciplinary Optimization, 2005, 29(4): 245–256
CrossRef Google scholar
[52]
Zhu J H, He F, Liu T, Structural topology optimization under harmonic base acceleration excitations. Structural and Multidisciplinary Optimization, 2018, 57(3): 1061–1078
CrossRef Google scholar
[53]
Wang Y Q, Chen F F, Wang M Y. Concurrent design with connectable graded microstructures. Computer Methods in Applied Mechanics and Engineering, 2017, 317: 84–101
CrossRef Google scholar
[54]
Kang Z, Wang Y G, Wang Y Q. Structural topology optimization with minimum distance control of multiphase embedded components by level set method. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 299–318
CrossRef Google scholar
[55]
Guest J K, Prevost J H, Belytschko T. Achieving minimum length scale in topology optimization using nodal design variables and projection functions. International Journal for Numerical Methods in Engineering, 2004, 61(2): 238–254
CrossRef Google scholar
[56]
Wang F W, Lazarov B S, Sigmund O. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011, 43(6): 767–784
CrossRef Google scholar
[57]
Da D C, Xia L, Li G Y, Evolutionary topology optimization of continuum structures with smooth boundary representation. Structural and Multidisciplinary Optimization, 2018, 57(6): 2143–2159
CrossRef Google scholar
[58]
Rockafellar R T. The multiplier method of Hestenes and Powell applied to convex programming. Journal of Optimization Theory and Applications, 1973, 12(6): 555–562
CrossRef Google scholar
[59]
Liu K, Tovar A. An efficient 3D topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2014, 50(6): 1175–1196
CrossRef Google scholar

Acknowledgments

The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 11372004), the State Key Laboratory of Subtropical Building Science (Grant No. 2016 KB13), and the State Key Laboratory of Structural Analysis for Industrial Equipment (Grant No. GZ18109).

Open Access

This article is licensed under Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution, and reproduction in any medium or format, as long as appropriate credit is given to the original author(s) and the source, a link is provided to the Creative Commons license, and any changes made are indicated.ƒImages or other third-party materials in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.ƒTo view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

RIGHTS & PERMISSIONS

2020 The Author(s) 2020. This article is published with open access at link.springer.com and journal.hep.com.cn
AI Summary AI Mindmap
PDF(3416 KB)

Accesses

Citations

Detail

Sections
Recommended

/