# Frontiers of Mechanical Engineering

 Front. Mech. Eng.    2019, Vol. 14 Issue (2) : 171-189     https://doi.org/10.1007/s11465-019-0530-5
 RESEARCH ARTICLE |
Concurrent optimization of structural topology and infill properties with a CBF-based level set method
Long JIANG1, Yang GUO2, Shikui CHEN1(), Peng WEI3, Na LEI4, Xianfeng David GU2
1. Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
2. Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
3. State Key Laboratory of Subtropical Building Science, School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, China
4. DUT-RU International School of Information Science & Engineering, Dalian University of Technology, Dalian 116620, China
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 Abstract In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distance-regularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design. Corresponding Authors: Shikui CHEN Just Accepted Date: 29 December 2018   Online First Date: 18 February 2019    Issue Date: 22 April 2019
 Cite this article: Long JIANG,Yang GUO,Shikui CHEN, et al. Concurrent optimization of structural topology and infill properties with a CBF-based level set method[J]. Front. Mech. Eng., 2019, 14(2): 171-189. URL: http://journal.hep.com.cn/fme/EN/10.1007/s11465-019-0530-5 http://journal.hep.com.cn/fme/EN/Y2019/V14/I2/171
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 Fig.1  The different loading cases for a unit cell. (a) Load case 1; (b) Load case 2; (c) Load case 3; (d) Load case 4 Fig.2  Level set model for implicit boundary representation. (a) The three-dimensional level set function; (b) the embedded boundary (highlight in red) as the zero level set Fig.3  Illustration of two kernel functions. (a) The Gaussian RBF; (b) the newly constructed CBF Fig.4  (a) The distance regularization energy P of Eq. (11); (b) the corresponding diffusive rate D Fig.5  (a) The new distance regularization energy Pn of Eq. (13); (b) the corresponding diffusive rate Dn Fig.6  The distance regularization effect. TO: Topology optimization. (a) The initial binary image; (b) the binary value level set function based on the binary image; (c) the distance regularized level set function; (d) the zoom-in view of the binary level set function; (e) the zoom-in view of the distance regularized level set function Fig.7  The material property interpolation. (a) The conventional level set implicit boundary representation by the level set function and the zero level-set; (b) the material region $Ω$, the design boundary $Γ$ and the design domain D represented by the conventional level set model; (c) the proposed shell-infill representation by the level set function and the level $Φ=0$ and $Φ=Δ$ ($Δ$ is the selected shell width); (d) within the material region $Ω$, the shell region $Ω s$ and the infill region $Ωi$ are bounded by the shell boundary $Γs$ and the infill boundary $Γ i$ in the design domain D, respectively Fig.8  The coordinates change of a unit cell Fig.9  The isotropy polar plot Fig.10  The flow chart for the concurrent topology optimization process Fig.11  The boundary condition for designing the Michell-type structure with fixed-fixed supports and multiple loads Fig.12  Convergence history and the design evolution for the macroscale Michell-type structure with fixed-fixed supports and multiple loads. Upper: The level set function evolution; lower: The corresponding design evolution Fig.13  Convergence history and the isotropy of the infill structure. (a) The convergence history and design evolution for the infill structure: The level set function evolution (upper) and the corresponding design evolution (lower); (b) the isotropic polar plot of the corresponding infill structure throughout the optimization process: Reference standard circle (red) and isotropic polar plot of the current infill structure (blue); (c) the elastic tensor of the corresponding infill structure Fig.14  Convergence history and the design evolution for the macroscale NPR structure. Upper: The level set function evolution; Lower: The corresponding design evolution Fig.15  Convergence history and the isotropy of the infill structure. (a) The convergence history and design evolution for the infill structure: The level set function evolution (Upper) and the corresponding design evolution (lower); (b) the isotropic polar plot of the infill structure throughout the optimization process: Reference standard circle (red) and isotropic polar plot of the final infill structure (blue); (c) the elastic tensor of the corresponding infill structure Fig.16  The NPR effect verification. (a) A 3×3 array of the macroscale structure; (b) one unit cell of the macroscale structure (the outer shell is shown in blue and the inner infill is shown in green); (c) one unit cell of the infill structure Fig.17  The local shape preserving effect of conformal mapping Fig.18  The conventional conformal mapping Ricci flow method with four control points Fig.19  The multi-control-point conformal mapping process. (a) The shell-infill structure optimization result; (b) half of the original design; (c) the infill region; (d) the meshed infill region with multiple control points (red) and one central point (green); (e) the zoom-in view of the triangular mesh; (f) the isotropic infill unit cell; (g) the mapped infill structure; (h) half of the shell-infill mapped structure; (i) the final shell-infill multiscale structure Fig.20  The conformal mapping result for the Mitchell-type structure Fig.21  The conformal mapping result for the NPR structure Fig.22  The 3D printed multiscale Mitchell-type structure. (a) The computational result; (b) the 3D printed result