Abstract
The regularization of the level set function is important to ensure numerical stability in the level set topology optimization. For the parametric level set method, introducing the distance potential functional is a popular way to regularize the distance of level set field and bypass the common re-initialization procedure in the conventional level set method. However, the conventional distance potential functional does not have sufficient control on the gradient of level set function and therefore it may lead to the lack of smoothness and regularity for the optimization results. This paper presents a modified potential functional which incorporates the information of the interfaces into the domain of functions so that the signed distance property can be enforced directly along the interfaces. In addition, the diffusion term is employed to further smooth the structural boundaries. The Compactly Supported Radial Basis Functions (CSRBF) are used to parametrize the level set function and the cut element method is adopted to discretize the governing equations. The linear elasticity problems including compliance minimization problems and compliant mechanism problem are tested to validate the proposed method.
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Replication of results
The numerical examples of the compliance minimum problem can be implemented by using the MATLAB codes provided as the supplementary material.
References
Aage N, Giele R, Andreasen CS (2021) Length scale control for high-resolution three-dimensional level set–based topology optimization. Struct Multidisc Optim. https://doi.org/10.1007/s00158-021-02904-4
Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method
Allaire G, Jouve F, Michailidis G (2016) Thickness control in structural optimization via a level set method. Struct Multidisc Optim 53:1349–1382. https://doi.org/10.1007/s00158-016-1453-y
Amstutz S, Andrä H (2006) A new algorithm for topology optimization using a level-set method. J Comput Phys 216:573–588. https://doi.org/10.1016/j.jcp.2005.12.015
Andreasen CS, Elingaard MO, Aage N (2020) Level set topology and shape optimization by density methods using cut elements with length scale control. Struct Multidisc Optim 62:685–707. https://doi.org/10.1007/s00158-020-02527-1
Azegami H, Fukumoto S, Aoyama T (2013) Shape optimization of continua using NURBS as basis functions. Struct Multidisc Optim. https://doi.org/10.1007/s00158-012-0822-4
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsoe MP, Sigmund O (2004) Topology optimization: theory, methods and applications. Springer, New York
Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84:253–304. https://doi.org/10.1002/nme.2914
Fu J, Li H, Xiao M, Gao L, Chu S (2019) Topology optimization of shell-infill structures using a distance regularized parametric level-set method. Struct Multidisc Optim. https://doi.org/10.1007/s00158-018-2064-6
Geiss MJ, Barrera JL, Boddeti N, Maute K (2019) A regularization scheme for explicit level-set XFEM topology optimization. Front Mech Eng 14:153–170. https://doi.org/10.1007/s11465-019-0533-2
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically–-a new moving morphable components based framework. J Appl Mech 81:197. https://doi.org/10.1115/1.4027609
Jansen M (2019) Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints. Struct Multidisc Optim 59:1775–1788. https://doi.org/10.1007/s00158-018-2162-5
Jiang L, Chen S (2017) Parametric structural shape & topology optimization with a variational distance-regularized level set method. Comput Methods Appl Mech Eng 321:316–336. https://doi.org/10.1016/j.cma.2017.03.044
Kitago M, Gopi M (2006) Efficient and prioritized point subsampling for CSRBF compression. In Eurographics symp point-based graph, vol 9
Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidisc Optim 46:311–326. https://doi.org/10.1007/s00158-012-0782-8
Kumar AV, Parthasarathy A (2011) Topology optimization using B-spline finite elements. Struct Multidisc Optim. https://doi.org/10.1007/s00158-011-0650-y
Li C, Xu C, Gui C, Fox MD (2010) Distance regularized level set evolution and its application to image segmentation. IEEE Trans Image Process 19:3243–3254. https://doi.org/10.1109/TIP.2010.2069690
Lin Y, Zhu W, Li J, Ke Y (2021) Structural topology optimization using a level set method with finite difference updating scheme. Struct Multidisc Optim. https://doi.org/10.1007/s00158-020-02779-x
Luo Z, Tong L, Kang Z (2009) A level set method for structural shape and topology optimization using radial basis functions. Comput Struct. https://doi.org/10.1016/j.compstruc.2009.01.008
Makhija D, Maute K (2014) Numerical instabilities in level set topology optimization with the extended finite element method. Struct Multidisc Optim 49:185–197. https://doi.org/10.1007/s00158-013-0982-x
Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49. https://doi.org/10.1016/0021-9991(88)90002-2
Qian X (2013) Topology optimization in B-spline space. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2013.06.001
Sharma A, Villanueva H, Maute K (2017) On shape sensitivities with heaviside-enriched XFEM. Struct Multidisc Optim. https://doi.org/10.1007/s00158-016-1640-x
Shen Y, Feng J (2018) Parametric level set-based multimaterial topology optimization of heat conduction structures. Math Probl Eng. https://doi.org/10.1155/2018/9804123
Song Y, Ma Q, He Y, Zhou M, Wang MY (2020) Stress-based shape and topology optimization with cellular level set in B-splines. Struct Multidisc Optim. https://doi.org/10.1007/s00158-020-02610-7
Suresh K, Takalloozadeh M (2013) Stress-constrained topology optimization: a topological level-set approach. Struct Multidisc Optim 48:295–309. https://doi.org/10.1007/s00158-013-0899-4
Svanberg K (1987a) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373. https://doi.org/10.1002/nme.1620240207
Svanberg K (1987b) The method of moving asymptotes–-a new method for structural optimization. Int J Numer Methods Eng 24:359–373. https://doi.org/10.1002/nme.1620240207
Tikhonov AN, Goncharsky AV, Stepanov VV, Yagola AG (1995) Numerical methods for the solution of ill-posed problems
van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidisc Optim 48:437–472. https://doi.org/10.1007/s00158-013-0912-y
Wang Y, Benson DJ (2016) Isogeometric analysis for parameterized LSM-based structural topology optimization. Comput Mech. https://doi.org/10.1007/s00466-015-1219-1
Wang S, Wang MY (2006) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65:2060–2090. https://doi.org/10.1002/nme.1536
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246. https://doi.org/10.1016/s0045-7825(02)00559-5
Wei P, Li Z, Li X, Wang MY (2018) An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Struct Multidisc Optim. https://doi.org/10.1007/s00158-018-1904-8
Wendland H (2004) Scattered data approximation. Cambridge University Press, Cambridge
Wu J, Aage N, Westermann R, Sigmund O (2018) Infill optimization for additive manufacturing-approaching bone-like porous structures. IEEE Trans vis Comput Graph. https://doi.org/10.1109/TVCG.2017.2655523
Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199:2876–2891. https://doi.org/10.1016/j.cma.2010.05.013
Zhu B, Zhang X, Fatikow S (2015) Structural topology and shape optimization using a level set method with distance-suppression scheme. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2014.08.017
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This research was supported by the Science and Technology Program of Zhejiang Province (Grant No. 2020C01039). The authors wish to thank the anonymous reviewers for their valuable comments on the early drafts of the manuscript.
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Lin, Y., Zhu, W., Li, J. et al. A distance regularization scheme for topology optimization with parametric level sets using cut elements. Struct Multidisc Optim 65, 88 (2022). https://doi.org/10.1007/s00158-021-03098-5
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DOI: https://doi.org/10.1007/s00158-021-03098-5