局部化無網格數值法求函數及其衍生函數內插值技術之研究

本研究提出一個以局部化無網格(meshless)微分積分法(Local differential quadrature method)為基礎的內插(interpolation)方法，並選用徑向基底函數(radial basis functions)中的多元二次(Multiquadrics)方程式做為形狀函數(shape function)。此內插方法係利用斜率與控制方程式作為考量，以利滿足幾何與物理上的特性。為了驗證其精確性與穩定性，文中應用此內插方法於數個二維和三維問題，並將其數值結果與解析解(或數值解)、線性多項式內插法(Linear polynomial fitting method)、二次多項式內插法(Quadratic polynomial fitting method)的結果加以比較，且分析其誤差。研究結果顯示，本文提出的內插方法可以在測試的案例中提供較精確且穩定的內插成果。此外，為了在對流控制的問題之下亦能獲得良好的模擬結果，本研究將上風法(upwind)應用至局部化微分積分法，進而得到精確的數值結果並提高計算效率。此上風法亦被應用至內插的技術，並且有良好的表現。

關鍵字

無網格；局部化微分積分法；徑向基底函數；內插；上風法

並列摘要

An interpolation technique based on the local radial basis functions differential quadrature (LRBF-DQ) method is developed to interpolate the unknown data by a set of irregularly scattered data. By employing the multiquadric function (MQ) as the test functions, the LRBF-DQ method is a meshless numerical scheme with high accuracy. In this study, the unknown data are interpolated by utilizing the field gradients or the governing equations of the problem, thus the interpolated data satisfies the geometric properties or the physical principles. Several 2D and 3D examples are presented to validate the current interpolation method. The results interpolated by the presented method are compared with those interpolated by the linear polynomial fitting (LPF) method and the quadratic polynomial fitting (QPF) method. The interpolation results show that the presented methods are more accurate and robust than the conventional interpolation methods. Moreover, for the sake of tackling the strongly convection-dominated problems, the upwind scheme is applied to the LRBF-DQ method. The convection phenomena can be described more accurately and efficiently by the upwind technique based LRBF-DQ method than the conventional LRBF-DQ method. The proposed upwind technique based LRBF-DQ method is further applied to the data interpolation, and the results also have good performances.

並列關鍵字

meshless ； local differential quadrature ； radial basis functions ； interpolation ； upwind

參考文獻

[1] Franke, R. (1982), “Scattered data interpolation: tests of some methods”, Mathematics of Computation., 38, pp.181-200

[2] Hardy, R.L. (1971), “Multiquadric equations of topography and other irregular surfaces”, J. Geophys. Res., 176, pp.1905-1915

[3] Duchon, J. (1976), “Interpolation des donctions de deux variables suivant le principe de flexion des plaques minces”, RAIRO Analyse Numeriques, 10, pp.5-12

[4] Madych, W.R. and Nelson, S.A. (1990), “Multivariate interpolation and conditionally positive definite functions, Π”, Mathematics of Computation, 54, pp.211-230

[5] Madych, W.R. and Nelson, S.A. (1992), “Miscellaneous error bounds for multiquadric and related interpolators”, Comput. Math. Applic., 24, pp.121-138

國際替代計量

局部化無網格數值法求函數及其衍生函數內插值技術之研究

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