一种保持二阶精度的反距离加权空间插值算法

计算物理 ›› 2020, Vol. 37 ›› Issue (4): 393-402.

一种保持二阶精度的反距离加权空间插值算法

柴国亮¹, 苏军伟¹, 王乐²

1. 西安交通大学人居环境与建筑工程学院, 陕西西安 710049;
2. 西安石油大学机械工程学院, 陕西西安 710065

收稿日期:2019-04-26 修回日期:2019-08-16 出版日期:2020-07-25 发布日期:2020-07-25
通讯作者: 苏军伟,E-mail:sujunwei@mail.xjtu.edu.cn
作者简介:柴国亮(1994-),硕士研究生,研究方向为固液两相流动数值模拟,E-mail:peter79634@stu.xjtu.edu.cn
基金资助:
国家重大科技专项（2016ZX05011001）、国家自然科学基金（21306145）资助项目

An Inverse Distance Weighting Spatial Interpolation Algorithm with Second Order Accuracy

CHAI Guoliang¹, SU Junwei¹, WANG Le²

1. School of Human Settlements and Civil Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China;
2. School of Mechanical Engineering, Xi'an Shiyou University, Xi'an, Shaanxi 710056, China

Received:2019-04-26 Revised:2019-08-16 Online:2020-07-25 Published:2020-07-25

摘要/Abstract

摘要： 针对传统反距离加权（IDW）插值精度较低的缺陷，发展一种高精度插值算法.该插值算法采用迭代亏量校正技术（IDeC）对一次反距离加权插值结果进行修正，通过有限次迭代，理论上将计算精度提高至二阶.在基于结构化网格、非结构化网格的数值验证中，该插值算法的计算精度均保持在二阶左右.应用该算法针对二维圆形和三维球形界面重构时，算法提高了重构界面的光滑度，且计算精度保持为二阶.双层网格插值实验中，算法将速度和压力的绝对误差降低45%以上，得到的压力等值线更接近于初始场.

关键词: 反距离加权插值, IDeC, 二阶精度, 界面重构, 重叠网格

Abstract: For low precision of traditional inverse distance weighting (IDW) interpolation, a high-precision interpolation algorithm is developed. Iterative defect correction (IDeC) is applied to correct IDW result, with which a second order accuracy is achieved theoretically within finite iterations. Numerical validations based on structured and unstructured grid are performed. It shows that, with this algorithm the second order accuracy is kept. Furthermore, the algorithm is adopted to surface reconstruction of a 2D circle and a 3D sphere, in which smoothness of reconstructed surface is improved and second order accuracy is maintained. In a two-layer mesh interpolation experiment, absolute error of velocity and pressure is reduced by more than 45%, and pressure contours are closer to the initial field.

Key words: inverse distance weighting, IDeC, second order accuracy, surface reconstruction, overlapping grids

中图分类号:

O302

柴国亮, 苏军伟, 王乐. 一种保持二阶精度的反距离加权空间插值算法[J]. 计算物理, 2020, 37(4): 393-402.

CHAI Guoliang, SU Junwei, WANG Le. An Inverse Distance Weighting Spatial Interpolation Algorithm with Second Order Accuracy[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(4): 393-402.

参考文献

[1] SHEPARD D. A two-dimensional interpolation function for irregularly-spaced data[C]//Proceedings of 1968 ACM National Conference. 1968:517-524.
[2] WITTEVEEN J, BIJL H. Explicit mesh deformation using inverse distance weighting interpolation[M]//19th AIAA Computational Fluid Dynamics. 2009:3996.
[3] 董志南, 郑拴宁, 赵会兵, 等. 基于空间插值的风场模拟方法比较分析[J]. 地球信息科学学报, 2015, 17(1):37-44.
[4] VARATHARAJAN R, MANOGARAN G, PRIYAN M, et al. Visual analysis of geospatial habitat suitability model based on inverse distance weighting with paired comparison analysis[J]. Multimedia Tools and Applications, 2018, 77(14):17573.
[5] 于洋, 卫伟, 陈利顶, 等. 黄土高原年均降水量空间插值及其方法比较[J]. 应用生态学报, 2015, 26(4):999-1006.
[6] 高歌, 龚乐冰, 赵珊珊, 等. 日降水量空间插值方法研究[J]. 应用气象学报, 2007, 18(5):732-736.
[7] CHEN F W, LIU C W. Estimation of the spatial rainfall distribution using inverse distance weighting (IDW) in the middle of Taiwan[J]. Paddy and Water Environment, 2012, 10(3):209-222.
[8] 丁卉, 余志, 徐伟嘉, 等. 3种区域空气质量空间插值方法对比研究[J]. 安全与环境学报, 2016, 16(3):309-315.
[9] de MESNARD L. Pollution models and inverse distance weighting:Some critical remarks[J]. Computers & Geosciences, 2013, 52:459-469.
[10] LU G Y, WONG D W. An adaptive inverse-distance weighting spatial interpolation technique[J]. Computers & Geosciences, 2008, 34(9):1044-1055.
[11] 马晨旭, 许才军, 张琼, 等. 引入变差函数的反距离加权法及应用研究[J]. 大地测量与地球动力学, 2010, 30(1):83-87.
[12] 余小东, 武莹, 何腊梅. 反距离加权网格化插值算法的改进及比较[J]. 工程地球物理学报, 2013, 10(6):900-904.
[13] 李超, 陈钱, 顾国华, 等. 改进的峰值检测反距离加权插值算法[J]. 红外与激光工程, 2013, 42(2):533-536.
[14] 李正泉, 吴尧祥. 顾及方向遮蔽性的反距离权重插值法[J]. 测绘学报, 2015, 44(1):91-98.
[15] 孔龙星, 汤晓安, 张俊达, 等. 顾及局部特性的自适应3D矢量场反距离权重插值法[J]. 系统工程与电子技术, 2016, 38(7):1697-1702.
[16] 樊子德, 李佳霖, 邓敏. 顾及多因素影响的自适应反距离加权插值方法[J]. 武汉大学学报(信息科学版), 2016, 41(6):842-847.
[17] BÖHMER K, HEMKER P, STETTER H. The defect correction approach[M]//Defect Correction Methods. Springer, 1984:1-32.
[18] NATARAJAN G, SOTIROPOULOS F. IDeC(k):A new velocity reconstruction algorithm on arbitrarily polygonal staggered meshes[J]. Journal of Computational Physics, 2011, 230(17):6583-6604.
[19] ARORA K, DESHPANDE S M. Accuracy and robustness of kinetic meshfree method[M]//Meshfree Methods for Partial Differential Equations V. Springer, 2011:173-188.
[20] DÉSIDÉRI J. Convergence behaviour of defect correction for hyperbolic equations[J]. Journal of Computational and Applied Mathematics, 1993, 45(3):357-365.
[21] 覃燕梅, 冯民富, 尹蕾. Navier-Stokes方程的一种等阶稳定化亏量校正有限元法[J]. 计算数学, 2010, 32(1):1-14.
[22] BARTH T, JESPERSEN D. The design and application of upwind schemes on unstructured meshes[C]. Proceedings of the 27th Aerospace Sciences Meeting, F, 1989.
[23] STETTER H J. The defect correction principle and discretization methods[J]. Numerische Mathematik, 1978, 29(4):425-443.
[24] 李鹏, 高振勋, 蒋崇文. 重叠网格方法的研究进展[J]. 力学与实践, 2014, 36(5):551-565.
[25] ZHANG Lin, GE Yongbin. High-oder fully implicit scheme and multigrid method for two-dimensional semilinear diffusion reaction equations[J]. Chinese Journal of Computational Physics, 2020, 37(3):307-319.
[26] YIN Liang, YANG Chao, MA Shizhuang. Parallel numerical simulations of thermal convection in the earth's outer core based on domain decomposition and multigrid[J]. Chinese Journal of Computational Physics, 2019, 36(1):1-14.
[27] ZHOU Zhiyang, XU Xiaowen, SHU Shi, et al. An adaptive two-level preconditioner for 2-D 3-T radiation diffusion equations[J]. Chinese Journal of Computational Physics, 2012, 29(4):475-483.
[28] SHU Shi,HUANG Yunqing, YANG Ying, et al. A class of algebraic multigrid algorithms with three-dimensional equal algebraic structures[J]. Chinese Journal of Computational Physics, 2005, 22(6):488-492.
[29] LIU Zhaoxia, CHANG Qianshun. Implicit numerical simulations of an anisotropic diffusion model driven by diffusion tensors[J]. Chinese Journal of Computational Physics, 2005, 22(4):365-370.
[30] TAYLOR G I, GREEN A E. Mechanism of the production of small eddies from large ones[J]. Proc R Soc Lond A, 1937, 158(895):499-521.