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

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

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

[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

[3] Bendsøe M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202

[4] Rozvany G I, Zhou M, Birker T. Generalized shape optimization without homogenization. Structural Optimization, 1992, 4(3–4): 250–252

[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

[7] Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[21] Sigmund O. A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127

[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

[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

[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

[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

[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

[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

[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

[32] Shu C W, Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Journal of Computational Physics, 1989, 83(1): 32–78

[33] Jiang G S, Peng D P. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM Journal on Scientific Computing, 2000, 21(6): 2126–2143

[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

[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

[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

[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

[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

[40] de Ruiter M J, van Keulen F. Topology optimization using a topology description function. Structural and Multidisciplinary Optimization, 2004, 26(6): 406–416

[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

[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

[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

[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

[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

[46] Burger M, Hackl B, Ring W. Incorporating topological derivatives into level set methods. Journal of Computational Physics, 2004, 194(1): 344–362

[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

[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

[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

[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

[51] Bruyneel M, Duysinx P. Note on topology optimization of continuum structures including self-weight. Structural and Multidisciplinary Optimization, 2005, 29(4): 245–256

[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

[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

[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

[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

[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

[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

[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

[59] Liu K, Tovar A. An efficient 3D topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2014, 50(6): 1175–1196