Extended Level Set Methods (X-LSM) for Shape and Topology Optimization on Manifolds

A new method is proposed to approach the problem of structural shape and topology optimization on manifold (or free-form surfaces). A manifold is conformally mapped onto a 2D rectangle domain, where the level set functions are defined. With conformal mapping, the corresponding covariant derivatives on a manifold can be represented by the Euclidean differential operators multiplied by a scalar. Therefore, the topology optimization problem on a free-form surface can be formulated as a 2D problem in the Euclidean space. To evolve the boundaries on a free-form surface, we propose a modified Hamilton-Jacobi equation and solve it on a 2D plane following the conformal geometry theory. In this way, we can fully utilize the conventional level-set-based computational framework. Compared with other established approaches which need to project the Euclidean differential operators to the manifold, the computational difficulty of our method is highly reduced while all the advantages of conventional level set methods are well preserved. We hope the proposed computational framework can provide a timely solution to increasing applications involving innovative structural designs on free-form surfaces in different engineering fields.

Structure topology optimization of a vase surface

     Boundary Conditions                                                     Conformal Mapping

3D printed vase design

Structure topology optimization of a chair surface

Structure topology optimization on a torus surface (Genus number =1)

      Boundary Conditions                               Design evolution on torus surface

Representative publications:

Journal:

1. Qian Ye, Yang Guo, Shikui Chen, Na Lei, Xianfeng Gu, "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, 344 (2019): 164-185. (https://doi.org/10.1016/j.cma.2018.08.045)

Conference:

1. Qian Ye, Long Jiang, Shikui Chen, Xianfeng Gu,"Generative Design of Multifunctional Conformal Structures using Extended Level Set Methods (X-LSM) and Conformal Geometry Theory."  IUTAM Symposium on when topology optimization meets additive manufacturing–theory and methods, Dalian, China, October 8-12,2018.
2. Qian Ye, Yang Guo, Shikui Chen, Xianfeng David Gu and Na Lei, "Topology Optimization of Conformal Structures Using Extended Level Set Methods and Conformal Geometry Theory",ASME Proceedings of IDETC/CIE, August 26-29, 2018, Quebec City, Canada.