{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:52:32Z","timestamp":1760403152375,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,24]],"date-time":"2020-04-24T00:00:00Z","timestamp":1587686400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51875220"],"award-info":[{"award-number":["51875220"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work presents a topology optimization method for symmetric hierarchical lattice structures with substructuring. In this method, we define two types of symmetric lattice substructures, each of which contains many finite elements. By controlling the materials distribution of these elements, the configuration of substructure can be changed. And then each substructure is condensed into a super-element. A surrogate model based on a series of super-elements can be built using the cubic B-spline interpolation. Here, the relative density of substructure is set as the design variable. The optimality criteria method is used for the updating of design variables on two scales. In the process of topology optimization, the symmetry of microstructure is determined by self-defined microstructure configuration, while the symmetry of macro structure is determined by boundary conditions. In this proposed method, because of the educing number of degree of freedoms on macrostructure, the proposed method has high efficiency in optimization. Numerical examples show that both the size and the number of substructures have essential influences on macro structure, indicating the effectiveness of the presented method.<\/jats:p>","DOI":"10.3390\/sym12040678","type":"journal-article","created":{"date-parts":[[2020,4,29]],"date-time":"2020-04-29T01:29:15Z","timestamp":1588123755000},"page":"678","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Substructure-Based Topology Optimization for Symmetric Hierarchical Lattice Structures"],"prefix":"10.3390","volume":"12","author":[{"given":"Zijun","family":"Wu","sequence":"first","affiliation":[{"name":"Hubei Key Laboratory of Digital Textile Equipment, Wuhan Textile University, Wuhan 430073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Renbin","family":"Xiao","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1016\/j.cma.2014.05.022","article-title":"Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework","volume":"278","author":"Xia","year":"2014","journal-title":"Comput. 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