以無網格區域微分積分法求解多階微分項及內插值

Abstract

本研究提出一個改良徑向基底函數型態式區域微分積分法的計算流程並應用於求解兩項問題：微分值及內插值。首先，多階微分值在數值計算裡一直都是一項不易處理的問題。傳統上，在結構性網格做空間上的離散就可以求得多階微分值；但應用於非結構性網格時便非常難於求解。而本研究裡所提出的無網格微分值求解技術則不受限於網格的型態。再者，傳統方法通常不易求解多階微分值，例如FEM (using linear element)則受限於形狀函數(shape function)的影響；但本文採用的方法係以多階可微的徑向基底函數為權重函數，故可以有效地處理在計算多階微分值所面臨的難題。此外，本模式所採用的逼近方程式裡，第一種方法係利用調整形狀參數以符合所求問題的控制方程式，第二種方法則是以各方向、不同階數分別求解最適權重係數的概念，因此二者皆可精確地推算出各階微分值。另一方面，於任意撒點的計算域裡要求解內插值一直是項困難的問題。大多數的內插方法僅受限於正交網格上的應用；此外，這些方法皆未考慮控制方程式或斜率，但本研究所提之內插工具將分別考慮這兩項因素的方法做討論。本研究提出了一項易於操作且具高精確度的內插工具，而此項工具將滿足控制方程式或考慮斜率兩項方法結合最小平方法來穩定所求解的內插值。 為驗證本研究所提出的兩種方法於求解技術的精確度與穩定性，文中應用該二法於結構性及非結構性網格之上並與解析解和其它數值方法驗證比較。上述測試的結果顯示，本研究所發展的模式提供了使用者有效且精確的多階微分值及內插值之求解工具。

This thesis modified a local differential quadrature (LDQ) method with radial basis functions (RBFs) to deal with two kinds of problems: evaluating derivatives and interpolating data. First of all, it is difficult to obtain the differential values from numerical procedures in general. Most of the traditional derivative calculations can be only adopted to evaluate the differential values with the regular meshes. Moreover, the traditional numerical schemes are very restricted by the order of the shape function. The present technique is able to be applied to both of the structured and unstructured meshes due to a meshless numerical algorithm - RBF and LDQ method. In addition, the proposed model can be applied to estimate multi-order or mixed partial differential values because its test function (RBFs) is a multi-order differentiable function. Furthermore, the derivatives can be obtained quite accurately because the present scheme agrees with the governing equation for the first method and characteristic of weighting coefficients for the second method. Secondly, it is tough to interpolate the unknown data from the known scattered data. To be more accurately, most of the interpolation methods can be only applied to structured grid. Besides, most of them do not consider the governing equation and gradients. This investigation proposed a convenient and accurate tool to construct the unknown data from nearby relative knots. Present interpolation methods are operated under two conceptions: one is to consider the governing equations, and the other is to take account of the gradients. All of the results were tested with the structured and unstructured meshes and compared with exact solutions and other numerical techniques. Consequently, this study provides an effective algorithm to calculate the multi-order differential values and interpolate the new data accurately.