A regularization scheme for explicit level-set XFEM topology optimization

Abstract

Regularization of the level-set (LS) field is a critical part of LS-based topology optimization (TO) approaches. Traditionally this is achieved by advancing the LS field through the solution of a Hamilton-Jacobi equation combined with a reinitialization scheme. This approach, however, may limit the maximum step size and introduces discontinuities in the design process. Alternatively, energy functionals and intermediate LS value penalizations have been proposed. This paper introduces a novel LS regularization approach based on a signed distance field (SDF) which is applicable to explicit LS-based TO. The SDF is obtained using the heat method (HM) and is reconstructed for every design in the optimization process. The governing equations of the HM, as well as the ones describing the physical response of the system of interest, are discretized by the extended finite element method (XFEM). Numerical examples for problems modeled by linear elasticity, nonlinear hyperelasticity and the incompressible Navier-Stokes equations in two and three dimensions are presented to show the applicability of the proposed scheme to a broad range of design optimization problems.

Keywords

level-set regularization / explicit level-sets / XFEM / CutFEM / topology optimization / heat method / signed distance field / nonlinear structural mechanics / fluid mechanics

References

[1] Bendsøe M P, Sigmund O. Topology Optimization. Berlin: Springer, 2004

[2] Sigmund O, Maute K. Topology optimization approaches: A comparative review. Structural and Multidisciplinary Optimization, 2013, 48(6): 1031–1055

[3] Deaton J D, Grandhi R V. A survey of structural and multidisciplinary continuum topology optimization: Post 2000. Structural and Multidisciplinary Optimization, 2014, 49(1): 1–38

[4] 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

[5] Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. Vol. 153. New York: Springer, 2003

[6] Osher S, Paragios N. Geometric Level Set Methods in Imaging, Vision, and Graphics. New York: Springer, 2003

[7] Gibou F, Fedkiw R, Osher S. A review of level-set methods and some recent applications. Journal of Computational Physics, 2018, 353: 82–109

[8] van Dijk N P, Maute K, Langelaar M, Level-set methods for structural topology optimization: A review. Structural and Multidisciplinary Optimization, 2013, 48(3): 437–472

[9] 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

[10] Allaire G, Jouve F, Toader A M. A level-set method for shape optimization. Comptes Rendus Mathématique, 2002, 334(12): 1125–1130 (in French)

[11] Wang M Y, Wang X, Guo D. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 227–246

[12] 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

[13] Wang S, 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

[14] Luo Z, Wang M Y, Wang S, A level set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering, 2008, 76(1): 1–26

[15] Kreissl S, Pingen G, Maute K. An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method. International Journal for Numerical Methods in Fluids, 2011, 65(5): 496–519

[16] van Dijk N P, Langelaar M, van Keulen F. Explicit level-set-based topology optimization using an exact Heaviside function and consistent sensitivity analysis. International Journal for Numerical Methods in Engineering, 2012, 91(1): 67–97

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

[18] Haber R B, Jog C S, Bendsøe M P. A new approach to variable-topology shape design using a constraint on perimeter. Structural Optimization, 1996, 11(1): 1–12

[19] 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

[20] Otomori M, Yamada T, Izui K, Level set-based topology optimisation of a compliant mechanism design using mathematical programming. Mechanical Science, 2011, 2(1): 91–98

[21] Gomes J, Faugeras O. Reconciling distance functions and level sets. Journal of Visual Communication and Image Representation, 2000, 11(2): 209–223

[22] Zhu B, Zhang X. A new level set method for topology optimization of distributed compliant mechanisms. International Journal for Numerical Methods in Engineering, 2012, 91(8): 843–871

[23] Li C, Xu C, Gui C, Level set evolution without re-initialization: A new variational formulation. In: Proceedings of 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (CVPR’05). San Diego: IEEE, 2005, Vol. I, 430–436

[24] Coffin P, Maute K. Level set topology optimization of cooling and heating devices using a simplified convection model. Structural and Multidisciplinary Optimization, 2016, 53(5): 985–1003

[25] Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics, 1994, 114(1): 146–159

[26] Osher S. Book review: Level set methods and fast marching methods: Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Mathematics of Computation, 2001, 70(233): 449–451

[27] Li C, Xu C, Gui C, Distance regularized level set evolution and its application to image segmentation. IEEE Transactions on Image Processing, 2010, 19(12): 3243–3254

[28] Zhu B, Zhang X, Fatikow S. Structural topology and shape optimization using a level set method with distance-suppression scheme. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 1214–1239

[29] Jiang L, Chen S. Parametric structural shape & topology optimization with a variational distance-regularized level set method. Computer Methods in Applied Mechanics and Engineering, 2017, 321: 316–336

[30] Burger M, Osher S J. A survey in mathematics for industry a survey on level set methods for inverse problems and optimal design. European Journal of Applied Mathematics, 2005, 16(2): 263–301

[31] Hartmann D, Meinke M, Schröder W. The constrained reinitialization equation for level set methods. Journal of Computational Physics, 2010, 229(5): 1514–1535

[32] Fu J, Li H, Xiao M, Topology optimization of shell-infill structures using a distance regularized parametric level-set method. Structural and Multidisciplinary Optimization, 2018, 1–14 (in press)

[33] Crane K, Weischedel C, Wardetzky M. Geodesics in heat. ACM Transactions on Graphics, 2013, 32(5): 1–11

[34] Crane B K, Weischedel C, Wardetzky M. The heat method for distance computation. Communications of the ACM, 2017, 60(11): 90–99

[35] Kreissl S, Maute K. Levelset based fluid topology optimization using the extended finite element method. Structural and Multidisciplinary Optimization, 2012, 46(3): 311–326

[36] Sethian J A. Fast marching methods. SIAM Review, 1999, 41(2): 199–235

[37] Wong T, Leung S. A fast sweeping method for eikonal equations on implicit surfaces. Journal of Scientific Computing, 2016, 67(3): 837–859

[38] Daux C, Moës N, Dolbow J, Arbitrary branched and intersecting cracks with the extended finite element method. International Journal for Numerical Methods in Engineering, 2000, 48(12): 1741–1760

[39] Fries T P, Belytschko T. The extended/generalized finite element method: An overview of the method and its applications. International Journal for Numerical Methods in Engineering, 2010, 84(3): 253–304

[40] Hansbo A, Hansbo P. A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 2004, 193(33–35): 3523–3540

[41] Makhija D, Maute K. Numerical instabilities in level set topology optimization with the extended finite element method. Structural and Multidisciplinary Optimization, 2014, 49(2): 185–197

[42] Tran A B, Yvonnet J, He Q C, A multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within XFEM. International Journal for Numerical Methods in Engineering, 2011, 85(11): 1436–1459

[43] Burman E, Hansbo P. Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. Mathematical Modelling and Numerical Analysis, 2014, 48(3): 859–874

[44] Schott B, Rasthofer U, Gravemeier V, A face-oriented stabilized Nitsche-type extended variational multiscale method for incompressible two-phase flow. International Journal for Numerical Methods in Engineering, 2015, 104(7): 721–748

[45] Villanueva C H, Maute K. CutFEM topology optimization of 3D laminar incompressible flow problems. Computer Methods in Applied Mechanics and Engineering, 2017, 320(Suppl C): 444–473

[46] Burman E, Claus S, Hansbo P, CutFEM: Discretizing geometry and partial differential equations. International Journal for Numerical Methods in Engineering, 2015, 104(7): 472–501

[47] Nitsche J A. On a variational principle for the solution of Dirichlet problems under the use of subspaces which are subject to no boundary conditions. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1971, 36(1): 9–15 (in German)

[48] Amestoy P R, Guermouche A, L’Excellent J Y, Hybrid scheduling for the parallel solution of linear systems. Parallel Computing, 2006, 32(2): 136–156

[49] Svanberg K. The method of moving asymptote—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373

[50] Sharma A, Villanueva H, Maute K. On shape sensitivities with Heaviside-enriched XFEM. Structural and Multidisciplinary Optimization, 2017, 55(2): 385–408

[51] Geiss M J, Maute K. Topology optimization of active structures using a higher-order level-set-XFEM-density approach. In: Proceedings of 2018 Multidisciplinary Analysis and Optimization Conference, AIAA AVIATION Forum, (AIAA 2018-4053). Atlanta, 2018

[52] Villanueva C H, Maute K. Density and level set-XFEM schemes for topology optimization of 3-D structures. Computational Mechanics, 2014, 54(1): 133–150

[53] Geiss M J, Bodeti N, Weeger O, Combined level-set-XFEM-density topology optimization of 4D printed structures undergoing large deformation. Journal of Mechanical Design, 2018 (in press)

[54] Sharma A, Maute K. Stress-based topology optimization using spatial gradient stabilized XFEM. Structural and Multidisciplinary Optimization, 2018, 57(1): 17–38

[55] Guo X, Zhang W, Zhong W. Explicit feature control in structural topology optimization via level set method. Computer Methods in Applied Mechanics and Engineering, 2014, 272: 354–378

[56] Kreissl S, Pingen G, Maute K. Topology optimization for unsteady flow. International Journal for Numerical Methods in Engineering, 2011, 87(13): 1229–1253

[57] Makhija D, Maute K. Level set topology optimization of scalar transport problems. Structural and Multidisciplinary Optimization, 2015, 51(2): 267–285

[58] Borrvall T, Petersson J. Topology optimization of fluids in Stokes flow. International Journal for Numerical Methods in Fluids, 2003, 41(1): 77–107

[59] Dunning P D, Alicia Kim H. A new hole insertion method for level set based structural topology optimization. International Journal for Numerical Methods in Engineering, 2013, 93(1): 118–134

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The Author(s) 2019. This article is published with open access at link.springer.com and journal.hep.com.cn