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A fully automatic computational framework for beam structure design from continuum structural topology optimization

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Abstract

This paper proposes a new fully automatic computational framework from continuum structural topology optimization to beam structure design. Firstly, the continuum structural topology optimization is performed to find the optimal material distribution. The centers of the elements (i.e., vertices) in the final topology are considered as the original model of the skeleton extraction. Secondly, the Floyd-Warshall algorithm is used to calculate the geodesic distances between vertices. By combining the geodesic distance-based mapping function and a coarse-to-fine partition scheme, the original model is partitioned into regular components. The skeleton can be extracted by using edges to link the barycenter of the components and decomposed into branches by identified joint vertices. Each branch is normalized into a straight line. After mesh generation, a beam finite element model is established. Compared to other methods in the literature, the beam structures reconstructed by the proposed method have a desirable centeredness and keep the homotopy properties of the original models. Finally, the cross-sectional areas of members in the beam structure are considered as the design variables, and the sizing optimization is performed. Four numerical examples, both 2D and 3D, are employed to demonstrate the validity of the automatic computational framework. The proposed method extracts a parameterized beam finite element model from the topology optimization result that bridges the gap between the topology optimization of continuum structures and the subsequent optimization or design that enables a fully automatic design of beam-like structures.

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Acknowledgements

This work was supported by the Basic General Scientific Research Program of Higher Education of Jiangsu Province [22KJD460003], the Science Foundation of Jiangsu Vocational Institute of Architectural Technology [JYJBZX22-05], the Enterprise Practice Training Program of Young Teachers in Vocational Colleges of Jiangsu Province [2023QYSJ031], and the National Natural Science Foundation of China [52805123, 52165029, 52305244]. The author would like to acknowledge the valuable comments and suggestions by Professors Jinzhou Yang and Yunkai Gao for this research.

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Authors and Affiliations

  1. School of Intelligent Manufacturing, Jiangsu Vocational Institute of Architectural Technology, Xuzhou, 221116, Jiangsu, China

    Chao Ma

  2. School of Automotive Studies, Tongji University, Shanghai, 201804, China

    Chao Ma

  3. College of Mechanical and Electrical Engineering, Hainan University, Haikou, 570100, Hainan, China

    Na Qiu

  4. School of Engineering, Anhui Agricultural University, Hefei, 230036, China

    Xiang Xu

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Correspondence to Chao Ma.

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The details of the methodology are included in the paper. Thus, we believe that the results can be reproduced with limited effort. The main program in this study can be found at https://github.com/Angry-kun/skel.

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Appendix: overview of skeleton extraction using the Reeb graph

Appendix: overview of skeleton extraction using the Reeb graph

The Reeb graph, a structure composed of 1D elements, is topologically equivalent to the original model. It is established by connecting nodes which are derived by contracting the level sets of a mapping function defined on the original model. In Morse theory, given a model M and a mapping function f defined on M, (vi, f(vi)) is equivalent to (vj, f(vj)) if and only if f(vi) = f(vj) and vertices vi and vj are in the same connected component of f −1(f(vi)). Each component is represented by a node. Nodes are linked by edges if corresponding components connect with each other, resulting a Reeb graph.

Take for instance the model shown in Fig. 

Fig. 27
figure 27

The schematic of constructing the Reeb graph: a The first partition, b the second partition, and c the third partition

Full size image

27 (Hilaga et al. 2001), the height function is employed as the mapping function, i.e., the mapping function f returns to the value of the y-coordinate of vertex in the model. In Fig. 27a, there is only one region r0 and one connected component s0. Thus, the Reeb graph consists of one node n0. In Fig. 27b, r0 is partitioned into two regions r1 and r2, while s0 is partitioned into three connected components s1, s2, and s3. Thus, there are three nodes n1, n2, and n3. Based on the connectivity of the components, the Reeb graph consists of three nodes and two edges. Following this coarse-to-fine partition scheme, the original model can be partitioned into finer level. Finally, a Reeb graph that is topologically equivalent to the original model is obtained as is shown in Fig. 27c.

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Ma, C., Qiu, N. & Xu, X. A fully automatic computational framework for beam structure design from continuum structural topology optimization. Struct Multidisc Optim 66, 250 (2023). https://doi.org/10.1007/s00158-023-03704-8

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