以局部化徑向基底函數微分積分法求解二維自由液面問題

Localized Radial Basis Function Differential Quadrature Method for Two-Dimensional Free Surface Problems

黃翊恆(Yi-Heng Huang)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/碩士(2011年)

https://doi.org/10.6342/NTU.2011.02128

摘要

本研究應用一個改良的區域化之徑向基底函數微分積分法(local radial basis function differential quadrature, LRBF-DQ method)，以求解二維非線性自由液面的問題。此處提出的區域化之徑向基底函數微分積分法是一個具有高階精確度，而此數值方法不需要建立網格，因此得以得到較佳之計算效率。此數值方法是利用所求點的附近點之線性加權函數值之總和來離散化，徑而去得到所求點之微分值。以往傳統的微分積分法較易受限於計算矩陣非常條件不良(ill-condition)以及對於計算網格型式過於限制等缺點，本方法在模擬求解控制方程式之近似解時可以比傳統的數值方法更加的穩定。在自由液面問題的數值模擬中，非線性項影響很大的情況下，往往隨著計算時間的變長，誤差就會跟著越來越大，因而最後容易導致結果的錯誤。本研究會先以一個非線性影響很小的自由震盪問題當作驗證，在自由液面的問題中若非線性的影響很小，那就可以把問題當作是一個類似線性的自由液面問題，因此就有解析解可以去比對其結果的對錯，得以確保數值模式的正確與否，再來就會利用此數值模式去模擬一些非線性項影響較大的自由震盪問題。藉由這些數值的試驗，證實本數值模式有能力並能夠精確的解決非線性項影響較大的自由液面問題。

關鍵字

區域化之徑向基底函數微分積分法 ； 非線性 ； 自由液面 ； 數值模式 ； 無網格 ； 自由震盪問題

並列摘要

In this study, a modified local radial basis function differential quadrature (LRBF-DQ) method is applied to solve the two-dimensional non-linear free surface problems. The LRBF-DQ method presented here has high order accuracy. This numerical scheme is a meshless approach, so that the better efficiency of calculation is obtained. It is discretized by a weighted linear sum of functional values at the points neighboring its desired knot so as to obtain the differential value of the desired point. The conventional DQ method is easier to subject to the ill-condition of the computed matrix and has a higher limit to the computing mesh. This method is more stable than the conventional numerical scheme when the approximate solution of the governing equation is solved by numerical simulation. In simulating the numerical value of the free surface, the error will become higher as the calculation time is longer in the case that the influence of the non-linear item is very high. Therefore, it is easy to result in wrong conclusions. A sloshing problem with a little influence on the non-linearity will be used for verification at first in this study. If the non-linearity has a little influence on the problem of the free surface, this problem can be considered as a pseudo-linear problem of the free surface. Therefore, the analytical solution can be used to compare its results so as to ensure the numerical model is correct. In addition, this numerical mode will be used to simulate the sloshing problems which are highly affected by some non-linear items. With the experiment of these numerical values, it proves that the numerical model is capable of accurately solving the problems of the free surface which is highly affected by the non-linear items.

並列關鍵字

local radial basis function differential quadrature (LRBF-DQ) method ； non-linear ； free surface ； numerical model ； meshless ； sloshing problems

參考文獻

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