Adaptive Isogeometric Topology Optimization of Shell Structures Based on RMB-Splines

Li Jiawei; Li Hangchuan; Wang Xuan; Pan Maodong

doi:10.21656/1000-0887.460154

Volume 47 Issue 9

Sep. 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(9): 1222-1234

Li Jiawei, Li Hangchuan, Wang Xuan, Pan Maodong. Adaptive Isogeometric Topology Optimization of Shell Structures Based on RMB-Splines[J]. Applied Mathematics and Mechanics, 2026, 47(9): 1222-1234. doi: 10.21656/1000-0887.460154

Citation:

PDF( 6277 KB)

Adaptive Isogeometric Topology Optimization of Shell Structures Based on RMB-Splines

doi: 10.21656/1000-0887.460154

1.
School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, P.R. China
2.
College of Civil Engineering, Hefei University of Technology, Hefei 230009, P.R. China

Received Date: 2025-08-27
Rev Recd Date: 2025-12-09
Publish Date: 2026-09-01

Abstract

Abstract

An adaptive isogeometric topology optimization algorithm based on RMB-splines (reachable minimally supported b-splines) was proposed to optimize the material distribution problem of shell structures. This algorithm combines the SIMP model to interpolate material density, and uses a new type of local refinement spline RMB-splines to represent the displacement field and density field. By taking advantages of its local linear independence, minimum support, and efficient local refinement ability, it achieves the unification of the analysis domain and the design domain. An adaptive mesh refinement strategy was constructed to identify refinement regions from the density distribution and to refine the mesh locally along material boundaries. This effectively reduces the overall degree of freedom while ensuring the optimization accuracy. In the numerical implementation, the RMB-splines provides stable and efficient numerical support for the optimization process. The projection function was introduced for density filtering to smooth the material distribution, and the MMA algorithm was used to solve the optimization problem. The numerical examples of multiple shell structures show that, compared with the B-spline method, this method can significantly reduce the number of basis functions, improve computational efficiency, and obtain a clearer and more engineering-feasible topological layout, promising great application potential in the topological optimization of complex structures.
- isogeometric analysis,
- topology optimization,
- shell structure,
- RMB-spline

FullText(HTML)

References(25)

References

[1]	Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39/40/41): 4135-4195.
[2]	Seo Y D, Kim H J, Youn S K. Isogeometric topology optimization using trimmed spline surfaces[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(49/50/51/52): 3270-3296.
[3]	Hassani B, Khanzadi M, Tavakkoli S M. An isogeometrical approach to structural topology optimization by optimality criteria[J]. Structural and Multidisciplinary Optimization, 2012, 45(2): 223-233. doi: 10.1007/s00158-011-0680-5
[4]	Wang Y J, Benson D J. Isogeometric analysis for parameterized LSM-based structural topology optimization[J]. Computational Mechanics, 2016, 57: 19-35. doi: 10.1007/s00466-015-1219-1
[5]	Hou W, Gai Y, Zhu X, et al. Explicit isogeometric topology optimization using moving morphable components[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 694-712. doi: 10.1016/j.cma.2017.08.021
[6]	Hassani B, Tavakkoli S M, Ghasemnejad H. Simultaneous shape and topology optimization of shell structures[J]. Structural and Multidisciplinary Optimization, 2013, 48(1): 221-233. doi: 10.1007/s00158-013-0894-9
[7]	郝鹏, 张越一, 金灵智, 等. 基于等几何分析的板壳结构形状-拓扑协同优化[J]. 宇航总体技术, 2023, 7(5): 1-11. Hao Peng, Zhang Yueyi, Jin Lingzhi, et al. Simultaneous shape-topology optimization of shell structures based on isogeometric analysis[J]. Astronautical Systems Engineering Technology, 2023, 7(5): 1-11. (in Chinese)
[8]	Kang P, Youn S K. Isogeometric topology optimization of shell structures using trimmed NURBS surfaces[J]. Finite Elements in Analysis and Design, 2016, 120: 18-40. doi: 10.1016/j.finel.2016.06.003
[9]	Zhang W, Li D, Kang P, et al. Explicit topology optimization using IGA-based moving morphable void (MMV) approach[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112685. doi: 10.1016/j.cma.2019.112685
[10]	Zhang W, Jiang S, Liu C, et al. Stress-related topology optimization of shell structures using IGA/TSA-based moving morphable void (MMV) approach[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 366: 113036. doi: 10.1016/j.cma.2020.113036
[11]	吴涛, 王禹, 张正阳, 等. 基于等几何的板壳结构几何非线性拓扑优化[J]. 应用力学学报, 2026, 43(1): 110-122. Wu Tao, Wang Yu, Zhang Zhengyang, et al. Geometric nonlinear topology optimization of plate and shell structures based on isogeometric analysis[J]. Chinese Journal of Applied Mechanics, 2026, 43(1): 110-122. (in Chinese)
[12]	Xie X D, Wang S T, Xu M M, et al. A hierarchical spline based isogeometric topology optimization using moving morphable components[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112696. doi: 10.1016/j.cma.2019.112696
[13]	Xie X D, Wang S T, Wang Y J, et al. Truncated hierarchical B-spline-based topology optimization[J]. Structural and Multidisciplinary Optimization, 2020, 62: 83-105. doi: 10.1007/s00158-019-02476-4
[14]	Xie X D, Yang A D, Jiang N, et al. Adaptive topology optimization under suitably graded THB-spline refinement and coarsening[J]. International Journal for Numerical Methods in Engineering, 2021, 122(20): 5971-5998. doi: 10.1002/nme.6780
[15]	Xie X D, Yang A D, Jiang N, et al. Topology optimization using fully adaptive truncated hierarchical B-splines[J]. Applied Mathematical Modelling, 2021, 96: 131-151. doi: 10.1016/j.apm.2021.02.005
[16]	Xie X D, Yang A D, Wang Y J, et al. Fully adaptive isogeometric topology optimization using MMC based on truncated hierarchical B-splines[J]. Structural and Multidisciplinary Optimization, 2021, 63: 2869-2887. doi: 10.1007/s00158-021-02850-1
[17]	Deng J S, Chen F L, Li X, et al. Polynomial splines over hierarchical T-meshes[J]. Graphical Models, 2008, 70(4): 76-86. doi: 10.1016/j.gmod.2008.03.001
[18]	Gupta A, Mamindlapelly B, Karuthedath P L, et al. Adaptive isogeometric topology optimization using PHT splines[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 395: 114993. doi: 10.1016/j.cma.2022.114993
[19]	Wen Z P, Pan Q, Zhai X Y, et al. Adaptive isogeometric topology optimization of shell structures based on PHT-splines[J]. Computers & Structures, 2024, 305: 107565.
[20]	Groiss L, Jüttler B, Pan M D. Local linear independence of bilinear (and higher degree) B-splines on hierarchical T-meshes[J]. Computer Aided Geometric Design, 2023, 103: 102190. doi: 10.1016/j.cagd.2023.102190
[21]	Pan M D, Zou R J, Jüttler B. Algorithms and data structures for C^S-smooth RMB-splines of degree 2s+1[J]. Computer Aided Geometric Design, 2024, 114: 102389. doi: 10.1016/j.cagd.2024.102389
[22]	Timoshenko S P, Woinowsky-Krieger S. Theory of Plates and Shells[M]. New York: McGraw-Hill, 1959.
[23]	Wang F W, Lazarov B S, Sigmund O. On projection methods, convergence and robust formulations in topology optimization[J]. Structural and Multidisciplinary Optimization, 2011, 43(6): 767784.
[24]	Svanberg K. The method of moving asymptotes: a new method for structural optimization[J]. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359-373. doi: 10.1002/nme.1620240207
[25]	Pan Q, Zhai X Y, Kang H, et al. Isogeometric topology optimization of multi-patch shell structures[J]. Computer-Aided Design, 2024, 174: 103733. doi: 10.1016/j.cad.2024.103733