Fluid Mechanics and Flight Mechanics

Research progress of RBF dynamic mesh technology and its application in aeroelasticity

  • Chao YANG ,
  • Zhicheng ZOU ,
  • Changchuan XIE ,
  • Chao AN ,
  • Cunyi HU
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  • 1.School of Aeronautic Science and Engineering,Beihang University,Beijing 100191,China
    2.Hangzhou International Innovation Institute,Beihang University,Hangzhou 311115,China
    3.School of Automation Science and Electrical Engineering,Beihang University,Beijing 100191,China

Received date: 2024-07-15

  Revised date: 2024-09-11

  Accepted date: 2024-09-18

  Online published: 2024-09-23

Supported by

Young Elite Scientists Sponsorship Program by CAST(YESS20230417);Young Elite Scientists Sponsorship Program by BAST(BYESS2023345)

Abstract

In multidisciplinary coupled calculations represented by aeroelastic calculations, structural deformation will lead to deformation of the fluid solution domain. It is necessary to develop a dynamic mesh deformation technology with good versatility, high computational efficiency and strong applicability to meet the calculation needs of aerodynamic forces. The dynamic mesh technology based on the Radial Basis Function (RBF) interpolation method has strong deformation ability and is applicable to the deformation calculation of any type of mesh. It is considered to be a dynamic mesh technology with good application prospects. This article introduces the basic theory of dynamic mesh technology based on RBF, and analyzes the selection scheme of basis function and compact radius of RBF. The research progress of acceleration algorithm and accuracy improvement method for mesh deformation technology based on RBF is summarized, and the hybrid dynamic mesh technology based on RBF is introduced. A brief summary of the current research status of RBF-based dynamic mesh technology in aeroelastic calculations is also made.

Cite this article

Chao YANG , Zhicheng ZOU , Changchuan XIE , Chao AN , Cunyi HU . Research progress of RBF dynamic mesh technology and its application in aeroelasticity[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2025 , 46(5) : 530945 -530945 . DOI: 10.7527/S1000-6893.2024.30945

References

1 杨超, 吴志刚. 飞行器气动弹性原理[M]. 3版. 北京: 北京航空航天大学出版社, 2024: 1-2.
  YANG C, WU Z G. Aeroelastic principle of aircraft[M]. 3rd ed. Beijing: Beijing University of Aeronautics & Astronautics Press, 2024: 1-2 (in Chinese).
2 杨超, 吴志刚, 谢长川. 气动弹性设计基础[M]. 3版. 北京: 北京航空航天大学出版社, 2021: 11-14.
  YANG C, WU Z G, XIE C C. Aeroelastic design foundation[M]. 3rd ed. Beijing: Beijing University of Aeronautics & Astronautics Press, 2021: 11-14 (in Chinese).
3 林言中, 陈兵, 徐旭. 基于径向基函数插值的气动弹性计算方法[J]. 北京航空航天大学学报201440(7): 953-958.
  LIN Y Z, CHEN B, XU X. Numerical method of aeroelasticity based on radial basis function interpolation[J]. Journal of Beijing University of Aeronautics and Astronautics201440(7): 953-958 (in Chinese).
4 GUO T Q, LU D X, LU Z L, et al. CFD/CSD-based flutter prediction method for experimental models in a transonic wind tunnel with porous wall[J]. Chinese Journal of Aeronautics202033(12): 3100-3111.
5 YE K, CHEN S B, YE Z Y. Numerical investigation of nonlinear aeroelastic characteristics of a supersonic drag-reduction spike[J]. Computers & Fluids2024273: 106201.
6 马砾, 招启军, 赵蒙蒙, 等. 基于CFD/CSD耦合方法的旋翼气动弹性载荷计算分析[J]. 航空学报201738(6): 120762.
  MA L, ZHAO Q J, ZHAO M M, et al. Computation analyses of aeroelastic loads of rotor based on CFD/CSD coupling method[J]. Acta Aeronautica et Astronautica Sinica201738(6): 120762 (in Chinese).
7 许云涛, 檀大林, 杨超. 基于CFD/CSD耦合的高速射弹尾拍载荷特性研究[J]. 北京航空航天大学学报202349(9): 2539-2546.
  XU Y T, TAN D L, YANG C. Study on tail-slap load characteristics of high-speed projectile based on CFD/CSD coupling[J]. Journal of Beijing University of Aeronautics and Astronautics202349(9): 2539-2546 (in Chinese).
8 TAMURA S, YUMITORI T. Accurate aeroelastic analysis of the wing structure with control surface[C]∥ Proceedings of the AIAA Scitech 2024 Forum. Reston: AIAA, 2024.
9 张伟伟. 基于CFD技术的高效气动弹性分析方法研究[D]. 西安: 西北工业大学, 2006: 50-51.
  ZHANG W W. Research on efficient aeroelastic analysis method based on CFD technology?[D]. Xi’an: Northwestern Polytechnical University, 2006: 50-51 (in Chinese).
10 张兵, 韩景龙. 带旋转修正的弹簧-TFI混合动网格方法[J]. 航空学报201132(10): 1815-1823.
  ZHANG B, HAN J L. Spring-TFI hybrid dynamic mesh method with rotation correction[J]. Acta Aeronautica et Astronautica Sinica201132(10): 1815-1823 (in Chinese).
11 杨超, 杨澜, 谢长川. 大展弦比柔性机翼气动弹性分析中的气动力方法研究进展[J]. 空气动力学学报201836(6): 1009-1018.
  YANG C, YANG L, XIE C C. Development of aeodynamic methods in aeroelastic analysis for high aspect ratio flexible wings[J]. Acta Aerodynamica Sinica201836(6): 1009-1018 (in Chinese).
12 杨国伟. 计算气动弹性若干研究进展[J]. 力学进展200939(4): 406-420.
  YANG G W. Recent progress on computational aeroelasticity[J]. Advances in Mechanics200939(4): 406-420 (in Chinese).
13 DEGAND C, FARHAT C. A three-dimensional torsional spring analogy method for unstructured dynamic meshes[J]. Computers & Structures200280(3-4): 305-316.
14 STEIN K, TEZDUYAR T, BENNEY R. Mesh moving techniques for fluid-structure interactions with large displacements[J]. Journal of Applied Mechanics200370(1): 58-63.
15 陈炎, 曹树良, 梁开洪, 等. 基于温度体模型的动网格生成方法及在流固耦合振动中的应用[J]. 振动与冲击201029(4): 1-5, 227.
  CHEN Y, CAO S L, LIANG K H, et al. A new dynamic grids based on temperature analogy and its application in vibration engineering with fluid-solid interaction[J]. Journal of Vibration and Shock201029(4): 1-5, 227 (in Chinese).
16 ALLEN C. Automatic structured multiblock mesh generation using robust transfinite interpolation?[C]?∥18th AIAA Computational Fluid Dynamics Conference. Reston: AIAA, 2007.
17 WITTEVEEN J, BIJL H. Explicit mesh deformation using inverse distance weighting interpolation?[C]?∥19th AIAA Computational Fluid Dynamics. Reston: AIAA, 2009.
18 肖天航, 昂海松, 仝超. 大幅运动复杂构形扑翼动态网格生成的一种新方法[J]. 航空学报200829(1): 41-48.
  XIAO T H, ANG H S, TONG C. A new dynamic mesh generation method for large movements of flapping-wings with complex geometries[J]. Acta Aeronautica et Astronautica Sinica200829(1): 41-48 (in Chinese).
19 BECKERT A, WENDLAND H. Multivariate interpolation for fluid-structure-interaction problems using radial basis functions[J]. Aerospace Science and Technology20015(2): 125-134.
20 周璇, 李水乡, 陈斌. 非结构动网格生成的弹簧-插值联合方法[J]. 航空学报201031(7): 1389-1395.
  ZHOU X, LI S X, CHEN B. Spring-interpolation approach for generating unstructured dynamic meshes[J]. Acta Aeronautica et Astronautica Sinica201031(7): 1389-1395 (in Chinese).
21 郑冠男, 杨国伟. 基于背景网格的混合网格变形方法[J]. 振动工程学报201124(5): 473-481.
  ZHENG G N, YANG G W. Hybrid grid deformation method based on background grid[J]. Journal of Vibration Engineering201124(5): 473-481 (in Chinese).
22 DE BOER A, VAN DER SCHOOT M S, BIJL H. Mesh deformation based on radial basis function interpolation[J]. Computers & Structures200785(11-14): 784-795.
23 LI G J, KUMAR JAIMAN R, LIU H Z. Coupled dynamics of steady jet flow control for flexible membrane wings[J]. AIAA Journal202462(6): 2264-2281.
24 CHWALOWSKI P, STANFORD B, JACOBSON K, et al. Flutter prediction report in support of the high angle working group at the third aeroelastic prediction workshop[C]∥AIAA Scitech 2024 Forum. Reston: AIAA, 2024.
25 YE K, YANG M B, QIN L Z, et al. Effects of structural geometric nonlinearities on the transonic aeroelastic characteristics of wing[J]. Aerospace Science and Technology2024149: 109161.
26 SHI Y Y, HE S, CUI G W, et al. Oscillation quenching and physical explanation on freeplay-based aeroelastic airfoil in transonic viscous flow[J]. Chinese Journal of Aeronautics202336(10): 124-136.
27 任浩源, 王毅, 王亮, 等. 基于热/力试验的折叠舵连接刚度与颤振分析[J]. 航空学报202344(14): 227927.
  REN H Y, WANG Y, WANG L, et al. Connection stiffness and flutter analysis of folding fin based on thermal-mechanical test[J]. Acta Aeronautica et Astronautica Sinica202344(14): 227927 (in Chinese).
28 杨澜. 大柔性机翼气动建模方法及气动弹性分析研究[D]. 北京: 北京航空航天大学, 2021: 60-62.
  YANG L. Aerodynamic modeling methodology and aeroelastic analysis of very flexible wings[D]. Beijing: Beihang University, 2021: 60-62 (in Chinese).
29 张伟伟, 高传强, 叶正寅. 气动弹性计算中网格变形方法研究进展[J]. 航空学报201435(2): 303-319.
  ZHANG W W, GAO C Q, YE Z Y. Research progress on mesh deformation method in computational aeroelasticity?[J]. Acta Aeronautica et Astronautica Sinica201435(2): 303-319 (in Chinese).
30 RENDALL T C S, ALLEN C B. Unified fluid-structure interpolation and mesh motion using radial basis functions[J]. International Journal for Numerical Methods in Engineering200874(10): 1519-1559.
31 赵奥博, 郑冠男, 黄程德, 等. 地效翼的颤振特性研究[J]. 振动与冲击202342(17): 265-274.
  ZHAO A B, ZHENG G N, HUANG C D, et al. Flutter characteristics of ground effect wings[J]. Journal of Vibration and Shock202342(17): 265-274 (in Chinese).
32 ESTRUCH O, LEHMKUHL O, BORRELL R, et al. A parallel radial basis function interpolation method for unstructured dynamic meshes[J]. Computers & Fluids201380: 44-54.
33 FRIGOLETTO B E, FIDKOWSKI K, CESNIK C E. Output-based mesh adaptation for high-order fluid-structure interaction of flexible wings[C]?∥Proceedings of the AIAA Scitech 2024 Forum. Reston: AIAA, 2024.
34 MASTERS J S, DENNY A, COTHRAN W. Scalable deformation of unstructured computational meshes[C]∥ AIAA Scitech 2019 Forum. Reston: AIAA, 2019.
35 HUANG C D, LIU W, YANG G W. Numerical studies of static aeroelastic effects on grid fin aerodynamic performances[J]. Chinese Journal of Aeronautics201730(4): 1300-1314.
36 苏波, 石启印, 钱若军. 径向基函数应用于流固耦合分析初探[J]. 工程力学201330(1): 59-63.
  SU B, SHI Q Y, QIAN R J. Preliminary study on the use of radial basis funcion in fluid-structure interaction analysis[J]. Engineering Mechincs201330(1): 59-63 (in Chinese).
37 WENDLAND H. On the smoothness of positive definite and radial functions[J]. Journal of Computational and Applied Mathematics1999101(1-2): 177-188.
38 林言中, 陈兵, 徐旭. 径向基函数插值方法在动网格技术中的应用[J]. 计算物理201229(2): 191-197.
  LIN Y Z, CHEN B, XU X. Radial basis function interpolation in moving mesh technique[J]. Chinese Journal of Computational Physics201229(2): 191-197 (in Chinese).
39 HOLGER W. Konstruktion und untersuchung radialer basisfunktionen mit kompaktem trager[D]. Gottingen: University of Gottingen, 1996: 31.
40 VAN STEEN F. Mesh deformation using radial basis function interpolation for numerical optimisation of internal flow applications?[D]. Delft: Delft University of Technology, 2023: 8.
41 GAO X, XU C F, DONG Y D, et al. Efficient and robust parallel mesh motion solver using radial basis functions[J]. Journal of Aerospace Engineering201831(3): 04018019.
42 MORELLI M, BELLOSTA T, GUARDONE A. Efficient radial basis function mesh deformation methods for aircraft icing[J]. Journal of Computational and Applied Mathematics2021392: 113492.
43 LI K, KOU J Q, ZHANG W W. Aeroelastic reduced-order modeling for efficient static aeroelastic analysis considering geometric nonlinearity[J]. Journal of Fluids and Structures2024124: 104055.
44 HUANG C D, HUANG J, SONG X, et al. Three dimensional aeroelastic analyses considering free-play nonlinearity using computational fluid dynamics/computational structural dynamics coupling[J]. Journal of Sound and Vibration2021494: 115896.
45 ABERGO L, GUARDONE A. Unsteady adjoint optimization in SU2 based on the hybrid harmonic balance-URANS approach[C]?∥AIAA Aviation 2023 Forum. Reston: AIAA, 2023.
46 方洪, 禹彩辉, 张星, 等. 基于径向基函数法的网格变形技术研究进展[C]∥2017全国仿真技术学术会议. 2017: 22-27, 59.
  FANG H, YU C H, ZHANG X, et al. Review of mesh deformation technology based on radial basis function[C]∥2017 National Academic Conference on Simulation Technology. 2017: 22-27, 59 (in Chinese).
47 ZHAO Z, MA R, HE L, et al. An efficient large-scale mesh deformation method based on MPI/OpenMP hybrid parallel radial basis function interpolation[J]. Chinese Journal of Aeronautics202033(5): 1392-1404.
48 王刚, 雷博琪, 叶正寅. 一种基于径向基函数的非结构混合网格变形技术[J]. 西北工业大学学报201129(5): 783-788.
  WANG G, LEI B Q, YE Z Y. An efficient deformation technique for hybrid unstructured grid using radial basis functions[J]. Journal of Northwestern Polytechnical University201129(5): 783-788 (in Chinese).
49 RENDALL T, ALLEN C. Efficient mesh motion using radial basis functions with data reduction algorithms[C]∥ 46th AIAA Aerospace Sciences Meeting and Exhibit. Reston: AIAA, 2008.
50 RENDALL T C S, ALLEN C B. Reduced surface point selection options for efficient mesh deformation using radial basis functions[J]. Journal of Computational Physics2010229(8): 2810-2820.
51 魏闯, 杨龙, 李春鹏, 等. ARI_OPT气动优化软件研究进展及应用[J]. 航空学报202041(5): 623370.
  WEI C, YANG L, LI C P, et al. Reseach progress and application of ARI-OPT software for aerodynamic shape optimization[J]. Acta Aeronautica et Astronautica Sinica202041(5): 623370 (in Chinese).
52 BERETTA L, MORELLI M C, GUARDONE A, et al. Numerical investigation of three-dimensional ice formation on a wing with leading edge tubercles[C]?∥ AIAA Scitech 2022 Forum. Reston: AIAA, 2022.
53 LIU S Y, WANG Y B, QIN N, et al. Quantification of airfoil aerodynamic uncertainty due to pressure-sensitive paint thickness[J]. AIAA Journal201958(4): 1432-1440.
54 高翔, 廖海翔, 徐传福. 基于支持向量回归的动网格技术研究[J]. 空气动力学学报202240(5): 146-157.
  GAO X, LIAO H X, XU C F. Dynamic mesh technology based on support vector regression[J]. Acta Aerodynamica Sinica202240(5): 146-157 (in Chinese).
55 RENDALL T, ALLEN C. Parallel efficient mesh motion using radial basis functions with application to multi-bladed rotors[C]?∥26th AIAA Applied Aerodynamics Conference. Reston: AIAA, 2008.
56 SU X Y, SHENG C H, ALLEN C. An efficient mesh deformation approach based on radial basis functions in unstructured flow solver[C]?∥Proceedings of the 20th AIAA Computational Fluid Dynamics Conference. Reston: AIAA, 2011.
57 WANG G, MIAN H H, YE Z Y, et al. Improved point selection method for hybrid-unstructured mesh deformation using radial basis functions?[J]. AIAA Journal201453(4): 1016-1025.
58 LI H, YE Z Y. Effects of rotational motion on dynamic aeroelasticity of flexible spinning missile with large slenderness ratio[J]. Aerospace Science and Technology201994: 105384.
59 WANG G, CHEN X, LIU Z K. Mesh deformation on 3D complex configurations using multistep radial basis functions interpolation?[J]. Chinese Journal of Aeronautics201831(4): 660-671.
60 魏其, 李春娜, 谷良贤, 等. 一种基于径向基函数和峰值选择法的高效网格变形技术[J]. 航空学报201637(7): 2156-2169.
  WEI Q, LI C N, GU L X, et al. An efficient mesh deformation method based on radial basis functions and peak-selection method[J]. Acta Aeronautica et Astronautica Sinica201637(7): 2156-2169 (in Chinese).
61 STROFYLAS G A, LYGIDAKIS G N, NIKOLOS I K. An agglomeration strategy for accelerating RBF-based mesh deformation?[J]. Advances in Engineering Software2017107: 13-37.
62 FANG H, ZHANG H, SHAN F L, et al. Efficient mesh deformation using radial basis functions with a grouping-circular-based greedy algorithm?[J]. Journal of Computational Physics2021433: 110200.
63 WANG H D, WANG X D, LIU X Y, et al. Improved radial basis functions mesh deformation based on parallel point selection strategy and incremental LDLT decomposition?[J]. Aerospace Science and Technology2023141: 108522.
64 LI C, XU X H, WANG J Y, et al. A parallel multiselection greedy method for the radial basis function-based mesh deformation[J]. International Journal for Numerical Methods in Engineering2018113(10): 1561-1588.
65 郭中州, 何志强, 赵文文, 等. 高效非结构网格变形与流场插值方法[J]. 航空学报201839(12): 122411.
  GUO Z Z, HE Z Q, ZHAO W W, et al. Efficient mesh deformation and flowfield interpolation method for unstructured mesh[J]. Acta Aeronautica et Astronautica Sinica201839(12): 122411 (in Chinese).
66 BENTLEY J L. Multidimensional binary search trees used for associative searching[J]. Communications of the ACM197518(9): 509-517.
67 王海洋. 基于虚拟控制点和径向基函数的网格变形方法研究[D]. 杭州: 杭州电子科技大学, 2023: 18-46.
  WANG H Y. Mesh deformation method based on virtual control points and radial basis functions[D]. Hangzhou: Hangzhou Dianzi University, 2023: 18-46 (in Chinese).
68 XIAO Z F, WANG H Y, XU G. Improved mesh deformation based on radial basis functions[C]∥AIAA Scitech 2023 Forum. Reston: AIAA, 2023.
69 MICHLER A K. Aircraft control surface deflection using RBF-based mesh deformation[J]. International Journal for Numerical Methods in Engineering201188(10): 986-1007.
70 谢亮, 徐敏, 安效民, 等. 基于径向基函数的网格变形及非线性气动弹性时域仿真研究[J]. 航空学报201334(7): 1501-1511.
  XIE L, XU M, AN X M, et al. Research of mesh deforming method based on radial basis functions and nonlinear aeroelastic simulation[J]. Acta Aeronautica et Astronautica Sinica201334(7): 1501-1511 (in Chinese).
71 谢亮, 徐敏, 张斌, 等. 基于径向基函数的高效网格变形算法研究[J]. 振动与冲击201332(10): 141-145.
  XIE L, XU M, ZHANG B, et al. Space points reduction in grid deforming method based on radial basis functions[J]. Journal of Vibration and Shock201332(10): 141-145 (in Chinese).
72 XIE L, LIU H. Efficient mesh motion using radial basis functions with volume grid points reduction algorithm[J]. Journal of Computational Physics2017348: 401-415.
73 ABERGO L, MORELLI M, GUARDONE A. Aerodynamic optimization based on a discrete adjoint framework and radial basis function mesh deformation in SU2[C]∥ AIAA Aviation 2021 Forum. Reston: AIAA, 2021.
74 ABERGO L, MORELLI M, GUARDONE A. Aerodynamic shape optimization based on discrete adjoint and RBF[J]. Journal of Computational Physics2023477: 111951.
75 XIE L, KANG Z C, HONG H F, et al. Local mesh deformation using a dual-restricted radial basis functions method[J]. Aerospace Science and Technology2022130: 107940.
76 LIU W, HUANG C D, YANG G W. Time efficient aeroelastic simulations based on radial basis functions[J]. Journal of Computational Physics2017330: 810-827.
77 SELIM M M, KOOMULLIL R P, SHEHATA A S. Incremental approach for radial basis functions mesh deformation with greedy algorithm[J]. Journal of Computational Physics2017340: 556-574.
78 FANG H, HU Y K, YU C H, et al. An efficient radial basis functions mesh deformation with greedy algorithm based on recurrence Choleskey decomposition and parallel computing?[J]. Journal of Computational Physics2019377: 183-199.
79 LIU J, FANG H, SHAN F L, et al. An improved peak-selection algorithm using block-based recurrence Cholesky decomposition for mesh deformation[J]. AIP Advances202111(9): 095004.
80 COULIER P, DARVE E. Efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method[J]. Computer Methods in Applied Mechanics and Engineering2016308: 286-309.
81 HACKBUSCH W, KHOROMSKIJ B N, KRIEMANN R. Hierarchical matrices based on a weak admissibility criterion[J]. Computing200473(3): 207-243.
82 高翔. 非结构CFD并行网格变形算法及其应用[D]. 长沙: 国防科技大学, 2018: 12-13.
  GAO X. Parallel unstructured mesh deformation algorithms and their applications in CFD[D]. Changsha: National University of Defense Technology, 2018: 12-13 (in Chinese).
83 RENDALL T, ALLEN C. Parallel efficient mesh motion using radial basis functions with application to multi-bladed rotors[C]?∥26th AIAA Applied Aerodynamics Conference. Reston: AIAA, 2008.
84 GILLEBAART T, BLOM D S, VAN ZUIJLEN A H, et al. Adaptive radial basis function mesh deformation using data reduction[J]. Journal of Computational Physics2016321: 997-1025.
85 FLOATER M S, ISKE A. Multistep scattered data interpolation using compactly supported radial basis functions[J]. Journal of Computational and Applied Mathematics199673(1-2): 65-78.
86 KEDWARD L, ALLEN C B, RENDALL T C S. Efficient and exact mesh deformation using multiscale RBF interpolation?[J]. Journal of Computational Physics2017345: 732-751.
87 NIU J P, LEI J M, HE J D. Radial basis function mesh deformation based on dynamic control points[J]. Aerospace Science and Technology201764: 122-132.
88 曾铮, 王刚, 叶正寅. RBF整体网格变形技术与多体轨迹仿真[J]. 空气动力学学报201533(2): 170-177.
  ZENG Z, WANG G, YE Z Y. Enhanced RBF mesh deformaion method and multi-body trajectory simulation[J]. Acta Aerodynamica Sinica201533(2): 170-177 (in Chinese).
89 WANG G, CHEN X, XING Y, et al. Multi-body separation simulation with an improved general mesh deformation method?[J]. Aerospace Science and Technology201771: 763-771.
90 FIELD D A. Laplacian smoothing and Delaunay triangulations[J]. Communications in Applied Numerical Methods19884(6): 709-712.
91 路宽, 宋文萍, 郭恒博, 等. 基于空间嵌套径向基函数的高效并行网格变形方法[J]. 航空学报202345(16): 129433.
  LU K, SONG W P, GUO H B, et al. An efficient parallel mesh deformation technique based on spatially-nested radial basis functions[J]. Acta Aeronautica et Astronautica Sinica202345(16): 129433 (in Chinese).
92 GAITONDE A, FIDDES S. A moving mesh system for the calculation of unsteady flows[C]∥Proceedings of the 31st Aerospace Sciences Meeting. Reston: AIAA, 1993.
93 孙岩, 邓小刚, 王运涛, 等. RBF_TFI结构动网格技术在风洞静气动弹性修正中的应用[J]. 工程力学201431(10): 228-233.
  SUN Y, DENG XG, WANG Y T, et al. Application of structural dynamic grid method based on RBF_TFI on wind tunnel static aeroelastic modification[J]. Engineering Mechanics201431(10): 228-233 (in Chinese).
94 DING L, LU Z L, GUO T Q. An efficient dynamic mesh generation method for complex multi-block structured grid[J]. Advances in Applied Mathematics and Mechanics20146(1): 120-134.
95 王运涛, 孟德虹, 孙岩, 等. 超大规模气动弹性数值模拟软件研制(2017)[J]. 空气动力学学报201836(6): 1019-1026.
  WANG Y T, MENG D H, SUN Y, et al. Software development of ultra-scale numerical simulaiton for aero-elastic problem (2017)[J]. Acta Aerodynamics Sinica201836(6): 1019-1026 (in Chinese).
96 杨起, 刘伟, 杨小亮, 等. 三角翼机翼摇滚主动控制多学科耦合数值模拟[J]. 航空学报202142(12): 124685.
  YANG Q, LIU W, YANG X L, et al. Multidisplinary interactions numerical simulation for active control of delta wing rock[J]. Acta Aeronautica et Astronautica Sinica202142(12): 124685 (in Chinese).
97 周铸, 黄江涛, 高正红, 等. 民用飞机气动外形数值优化设计面临的挑战与展望[J]. 航空学报201940(1): 522370.
  ZHOU Z, HUANG J T, GAO Z H, et al. Challenges and prospects of numerical optimization design for large civil aircraft aerodynamic shape[J]. Acta Aeronautica et Astronautica Sinica201940(1): 522370 (in Chinese).
98 张淼, 刘铁军, 马涂亮, 等. 基于CFD方法的大型客机高速气动设计[J]. 航空学报201637(1): 244-254.
  ZHANG M, LIU T J, MA T L, et al. High speed aerodynamic design of large civil transporter based on CFD method?[J]. Acta Aeronautica et Astronautica Sinica201637(1): 244-254 (in Chinese).
99 刘学强, 李青, 柴建忠, 等. 一种新的动网格方法及其应用[J]. 航空学报200829(4): 817-822.
  LIU X Q, LI Q, CHAI J Z, et al. A new dynamic grid algrithm and its application[J]. Acta Aeronautica et Astronautica Sinica200829(4): 817-822 (in Chinese).
100 孙岩, 邓小刚, 王光学, 等. 基于径向基函数改进的Delaunay图映射动网格方法[J]. 航空学报201435(3): 727-735.
  SUN Y, DENG X G, WANG G X, et al. Improvement on delaunay graph mapping dynamic grid method based on radial basis functions[J]. Acta Aeronautica et Astronautica Sinica201435(3): 727-735 (in Chinese).
101 孙岩, 孟德虹, 王运涛, 等. 基于径向基函数与混合背景网格的动态网格变形方法[J]. 航空学报201637(5): 1462-1472.
  SUN Y, MENG D H, WANG Y T, et al. Dynamic grid deformation method based on radial basis function and hybrid background grid[J]. Acta Aeronautica et Astronautica Sinica201637(5): 1462-1472 (in Chinese).
102 孙岩, 江盟, 孟德虹, 等. 国家数值风洞工程结构网格变形程序SGDP V1.0开发与应用[J]. 空气动力学学报202038(4): 668-676.
  SUN Y, JIANG M, MENG D H, et al. Development and application of structured grid deformation program SGDP V1.0 in national numerical windtunnel project[J]. Acta Aerodynamica Sinica202038(4): 668-676 (in Chinese).
103 孙岩, 王昊, 江盟, 等. NNW-FSI软件静气动弹性耦合加速策略设计与实现[J]. 航空学报202142(9): 625738.
  SUN Y, WANG H, JIANG M, et al. Design and implementation of coupling acceleration strategy in static aeroelastic module of NNW-FSI software[J]. Acta Aeronautica et Astronautica Sinica202142(9): 625738 (in Chinese).
104 郭秋亭, 孙岩, 郭正, 等. 风洞试验雷诺数/静气动弹性效应分离方法[J]. 航空学报202243(11): 526312.
  GUO Q T, SUN Y, GUO Z, et al. Separation method for Reynolds number/static aeroelastic coupling effect in wind tunnel test[J]. Acta Aeronautica et Astronautica Sinica202243(11): 526312 (in Chinese).
105 黄江涛, 高正红, 白俊强, 等. RBF 径向基函数与Delaunay图映射技术在飞行器型架外形设计中应用研究[J]. 空气动力学学报201432(3): 328-333.
  HUANG J T, GAO Z H, BAI J Q, et al. Aircraft jig shape deisign based on radial basis functions and Delaunay graphic mapping?[J]. Acta Aerodynamica Sinica201432(3): 328-333 (in Chinese).
106 WANG Y B, QIN N, ZHAO N. Delaunay graph and radial basis function for fast quality mesh deformation[J]. Journal of Computational Physics2015294: 149-172.
107 王昊达, 刘南, 张颖, 等. 基于径向基函数和Delaunay图映射的高效高鲁棒性的非结构网格变形方法[J]. 气体物理20238(6): 41-54.
  WANG H D, LIU N, ZHANG Y, et al. Efficient and robust unstructured grid deformation method based on radial basis function and Delaunay graph mapping[J]. Physics of Gases20238(6): 41-54 (in Chinese).
108 LEFRAN?OIS E. A simple mesh deformation technique for fluid-structure interaction based on a submesh approach[J]. International Journal for Numerical Methods in Engineering200875(9): 1085-1101.
109 LIU Y, GUO Z, LIU J. RBFs-MSA hybrid method for mesh deformation[J]. Chinese Journal of Aeronautics201225(4): 500-507.
110 FANG H, GONG C Y, YU C H, et al. Efficient mesh deformation based on Cartesian background mesh[J]. Computers & Mathematics with Applications201773(1): 71-86.
111 TANG H, ZHENG G N, ZHANG Y C. Investigation of radial basis function dynamic mesh method with rotation correction based on adaptive background mesh[J]. Computers & Fluids2024276: 106264.
112 LYRIO J A A, RADE D A, AZEVEDO J A L F. High-fidelity fluid-structure interaction applied to static aeroelasticity in transonic flows[J]. Aerospace Science and Technology2024153: 109477.
113 AUBERT S, MASTRIPPOLITO F, RENDU Q, et al. Planar slip condition for mesh morphing using radial basis functions?[J]. Procedia Engineering2017203: 349-361.
114 LIU X Y, WANG H, ZHAO Z, et al. Gridder-HO: rapid and efficient parallel software for high-order curvilinear mesh generation[J]. Advances in Engineering Software2024197: 103739.
115 GAGLIARDI F, GIANNAKOGLOU K C. A two-step radial basis function-based CFD mesh displacement tool[J]. Advances in Engineering Software2019128: 86-97.
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