{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:33:55Z","timestamp":1760229235702,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T00:00:00Z","timestamp":1654214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Key Laboratory of General Aviation Operation Technology (General Aviation Institute of Zhejiang JianDe)","award":["JDGA2020-4"],"award-info":[{"award-number":["JDGA2020-4"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The increasing structural flexibility of large aircraft leads to significant aeroelastic effects. More efficient topology optimization techniques are required for the design to further take advantage of aeroelasticity and obtain lightweight structures. This paper proposes a moving boundary meshfree topology optimization that combines the Galerkin method of weighted residuals and non-uniform rational B-splines (NURBS). The solution domain is described by the control points of NURBS and its property is calculated adaptively with an integration subtraction technique. The minimal compliance is searched for using the globally convergent method of moving asymptotes (GCMMA) by designing the locations of control points as subject to volume and flux constraints. The method is first applied to a typical two-dimensional design example with symmetric boundary conditions. The results show that the shape constraints can be conveniently applied, and smoother boundaries are obtained with fewer parameters. Then, a three-dimensional wing structure with asymmetric boundary conditions is optimized. A three-dimensional flight load that combines the high-order-panel and meshfree methods is employed to calculate the elastic loads and update asymmetric external loads during the optimization process. The designed wing satisfies engineering requirements and the presented method can solve the practical topology optimization problems of three-dimensional structures.<\/jats:p>","DOI":"10.3390\/sym14061154","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T08:01:18Z","timestamp":1654243278000},"page":"1154","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Aeroelastic Topology Optimization of Wing Structure Based on Moving Boundary Meshfree Method"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4824-4496","authenticated-orcid":false,"given":"Xiaozhe","family":"Wang","sequence":"first","affiliation":[{"name":"Institute of Unmanned System, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shanshan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiqiang","family":"Wan","sequence":"additional","affiliation":[{"name":"School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi","family":"Wang","sequence":"additional","affiliation":[{"name":"Zhejiang Key Laboratory of General Aviation Operation Technology, General Aviation Research Institute of Zhejiang Jiande, Jiande 311612, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/s11831-015-9151-2","article-title":"Topology Optimization in Aircraft and Aerospace Structures Design","volume":"23","author":"Zhu","year":"2016","journal-title":"Arch. Computat. Methods Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1038\/nature23911","article-title":"Topology optimization in aircraft and aerospace structures design computational morphogenesis for structural design","volume":"550","author":"Aage","year":"2017","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1007\/s00158-018-1948-9","article-title":"Optimal truss and frame design from projected homogenization-based topology optimization","volume":"57","author":"Larsen","year":"2018","journal-title":"Struct. Multidisc. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1007\/s00158-022-03208-x","article-title":"Efficient multi-stage aerodynamic topology optimization using an operator-based analytical differentiation","volume":"65","author":"Ghasemi","year":"2022","journal-title":"Struct. Multidisc. Optim."},{"key":"ref_5","first-page":"445","article-title":"Topology Optimization of Aircraft Components for Increased Sustainability","volume":"60","author":"Munk","year":"2022","journal-title":"AIAA J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s00158-022-03234-9","article-title":"Aerostructural topology optimization using high fidelity modeling","volume":"65","author":"Gomes","year":"2022","journal-title":"Struct. Multidisc. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1007\/s00158-020-02739-5","article-title":"An efficient multiscale optimization method for conformal lattice materials","volume":"63","author":"Wu","year":"2021","journal-title":"Struct. Multidisc. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s00158-017-1740-2","article-title":"The relevance of reliability-based topology optimization in early design stages of aircraft structures","volume":"57","author":"Baldomir","year":"2018","journal-title":"Struct. Multidisc. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1214","DOI":"10.1016\/j.cma.2014.08.017","article-title":"Structural topology and shape optimization using a level set method with distance-suppression scheme","volume":"283","author":"Zhu","year":"2015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"91134","DOI":"10.1109\/ACCESS.2019.2927820","article-title":"Isogeometric Bi-Directional Evolutionary Structural Optimization","volume":"7","author":"Yin","year":"2019","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s00161-018-0706-y","article-title":"Comparison of thermodynamic topology optimization with SIMP","volume":"31","author":"Jantos","year":"2019","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"115128","DOI":"10.1016\/j.compstruct.2021.115128","article-title":"Nonlinear supersonic post-flutter response of two-bay composite laminate curved panels","volume":"286","author":"Cabral","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1016\/j.cma.2017.03.034","article-title":"Reproducing kernel formulation of B-spline and NURBS basis functions: A meshfree local refinement strategy for isogeometric analysis","volume":"320","author":"Zhang","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0045-7825(96)01078-X","article-title":"Meshless methods: An overview and recent developments","volume":"139","author":"Belytschko","year":"1996","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.advengsoft.2014.06.006","article-title":"ESFM: An Essential Software Framework for Meshfree Methods","volume":"76","author":"Hsieh","year":"2014","journal-title":"Adv. Eng. Softw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"04017001","DOI":"10.1061\/(ASCE)EM.1943-7889.0001176","article-title":"Meshfree Methods: Progress Made after 20 Years","volume":"143","author":"Chen","year":"2017","journal-title":"J. Eng. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1830001","DOI":"10.1142\/S0219876218300015","article-title":"Meshfree Methods: A Comprehensive Review of Applications","volume":"15","author":"Garg","year":"2018","journal-title":"Int. J. Comput. Methods"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.enganabound.2021.06.019","article-title":"Numerical solution of the quasistatic contact problem with the Tresca friction in elastic-viscoplastic materials by the element-free Galerkin method","volume":"132","author":"Shen","year":"2021","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.camwa.2018.09.047","article-title":"An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems","volume":"77","author":"Li","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.1007\/s00366-019-00779-0","article-title":"The element-free Galerkin method based on moving least squares and moving Kriging approximations for solving two-dimensional tumor-induced angiogenesis model","volume":"36","author":"Dehghan","year":"2020","journal-title":"Eng. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"107529","DOI":"10.1016\/j.aml.2021.107529","article-title":"A fast element-free Galerkin method for the fractional diffusion-wave equation","volume":"122","author":"Li","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_22","unstructured":"Fung, Y.C., Tong, P., and Chen, X.H. (2017). Classical and Computational Solid Mechanics, World Scientific. [2nd ed.]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1109\/LMWC.2020.2992372","article-title":"Inverse Multiquadric Radial Basis Functions in Eigenvalue Analysis of a Circular Waveguide Using Radial Point Interpolation Method","volume":"30","author":"Sivaram","year":"2020","journal-title":"IEEE Microw. Wirel. Compon. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"115083","DOI":"10.1016\/j.compstruct.2021.115083","article-title":"Static and dynamic NURBS-based isogeometric analysis of composite plates under hygrothermal environment","volume":"284","author":"Gupta","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Montemurro, M., and Refai, K. (2021). A Topology Optimization Method Based on Non-Uniform Rational Basis Spline Hyper-Surfaces for Heat Conduction Problems. Symmetry, 13.","DOI":"10.3390\/sym13050888"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Liu, Y.Z., Wan, Z.Q., Yang, C., and Wang, X.Z. (2020). NURBS-Enhanced Meshfree Method with an Integration Subtraction Technique for Complex Topology. Appl. Sci., 10.","DOI":"10.3390\/app10072587"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Li, X.Y., and He, L.L. (2020). Shape Optimization Design for a Centrifuge Structure with Multi Topological Configurations Based on the B-Spline FCM and GCMMA. Appl. Sci., 10.","DOI":"10.3390\/app10020620"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Liu, Y.Z., Zhu, S.Y., Wan, Z.Q., and Yang, C. (2017, January 9\u201313). A High Efficiency Aeroelastic Analysis Method based on Rigid External Aerodynamic Force and Elastic Correction by High-Order Panel Method. Proceedings of the 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, USA.","DOI":"10.2514\/6.2017-1650"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1258","DOI":"10.1016\/j.cja.2018.01.024","article-title":"Integrated optimization on aerodynamics-structure coupling and flight stability of a large airplane in preliminary design","volume":"31","author":"Wang","year":"2018","journal-title":"Chin. J. Aeronaut."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1154\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:24:08Z","timestamp":1760138648000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1154"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,3]]},"references-count":29,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061154"],"URL":"https:\/\/doi.org\/10.3390\/sym14061154","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,3]]}}}