# Frontiers of Mechanical Engineering

 Front. Mech. Eng.    2020, Vol. 15 Issue (3) : 390-405     https://doi.org/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
Peng WEI1,2(), Wenwen WANG1, Yang YANG1, Michael Yu WANG3
1. School of Civil Engineering and Transportation, State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510641, China
2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
3. Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Hong Kong, China
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 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. Corresponding Author(s): Peng WEI Just Accepted Date: 27 May 2020   Online First Date: 19 June 2020    Issue Date: 03 September 2020
 Cite this article: Peng WEI,Wenwen WANG,Yang YANG, et al. Level set band method: A combination of density-based and level set methods for the topology optimization of continuums[J]. Front. Mech. Eng., 2020, 15(3): 390-405. URL: http://journal.hep.com.cn/fme/EN/10.1007/s11465-020-0588-0 http://journal.hep.com.cn/fme/EN/Y2020/V15/I3/390
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 Fig.1  Basic concept representing a boundary $∂Ω$ with a zero level set. Fig.2  Updating scheme for the level set function by solving Hamilton–Jacobi equation. (a) Two components for updating Φ; (b) all points move horizontally. Fig.3  Possible and impossible updated level set functions by solving Hamilton–Jacobi equation. Fig.4  Updating scheme of a zero level set method. (a) Updating scheme of Φ for Eq. (3); (b) all points move vertically. Fig.5  Density interpolation scheme of the level set band method by gradually reducing $Φb$. (a) Case with a large level set band between the upper and lower bounds in the beginning stage; (b) case with a small level set band after convergence. Fig.6  Illustration of the level set band method with an intermediate optimal example ($Δ=Φb /2= 3.5$). Fig.7  Level set band method can be considered a variation of the density-based method and the LSM by replacing the fixed bands $Φ b$ with an alterable one. Fig.8  Two cases of variation in the level set function without and with changing the objective function. (a) Original level set function; (b) the objective function is unchanged (Case 1); (c) the objective function is changed (Case 2). Fig.9  Design domain and boundary conditions of a cantilever beam. Fig.10  Optimization iteration process of the 2D cantilever beam with decreasing Δ from 5 to 0.5 by 0.1 at each step (from top to bottom: Steps 1, 10, 20, 30, 50, and 109). (a) Zero level set; (b) density distribution; (c) level sets on the lower and upper bound planes; (d) level set function and the lower and upper bound planes. Fig.11  Optimization iteration process of the 2D cantilever beam with fixed $Δ=0.5$ (from top to bottom: Steps 1, 10, 20, 30, 50, and 131). (a) Zero level set; (b) density distribution; (c) level sets on the lower and upper bound planes; (d) level set function and the lower and upper bound planes. Fig.12  Optimization process of the cantilever beam problem with uniformly distributed initial density. (a) Zero level set: Δ decreases from 5 to 0.5 by 0.1 at each step, and the iteration steps are 1, 10, 20, 30, 50, and 149 (from top to bottom); (b) density distribution: Δ decreases from 5 to 0.5 by 0.1 at each step, and the iteration steps are 1, 10, 20, 30, 50, and 149 (from top to bottom); (c) zero level set: fixed $Δ=0.5$, and the iteration numbers are 1, 10, 20, 30, 50, and 126 (from top to bottom); (d) density distribution: fixed $Δ=0.5$, and the iteration numbers are 1, 10, 20, 30, 50, and 126 (from top to bottom). Fig.13  Design domain and boundary conditions of a simply supported beam. Fig.14  Optimal results of the simply supported beam with different initial designs and three schemes. (a) $Δ=Δ0=minΔ=0.01$; (b) $Δ0=5$, $dΔ=−0.1$, $minΔ=0.01$; (c) $Δ0=5$, $dΔ=−0.05$, $minΔ=0.01$. Fig.15  Comparison of the objective functions and the total iteration steps of the simply supported beam problem with different initial designs and Δ values. Fig.16  Implementation of the proposed method in the optimization of a 3D cantilever beam. (a) Zero level set of the 3D cantilever beam problem (left to right: Steps 1, 15, 30, and 130); (b) density distribution of the 3D cantilever beam problem (left to right: Steps 1, 15, 30, and 130).
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