以完全非線性無網格方法數值模擬三維邊界移動造波

Abstract

本文主旨係以無網格方法探討邊界移動之完全非線性造波數值模擬。本無網格數值方法之建立是以三維線性基本解（Fundamental solution）求解拉普拉斯方程式（Laplace equation）以得到速度勢（Velocity potential），並以二維高斯幅狀基底函數計算自由液面梯度，再配合邊界條件便可求出自由液面高程；而時間域之處理，則是以二階中央插分法（亦稱蛙跳法）對時間離散所得到之顯式前進差分計算。 數值模擬方面，先以水平方向不透水邊界進行簡諧運動以模擬自由振盪問題（Sloshing problem），檢驗其質量守恆與波形，並與前人研究互相比較，探討線性與非線性模式之水面波。模擬結果顯示，本模式在模擬自由振盪問題與文獻上之資料相當地吻合，而且更可表現出其非線性之效應。接著，再以底床運動進行垂向造波，藉以探討其水面波之運動，包含其生成、傳遞及溯昇等於斜坡上的過程，嘗試瞭解海嘯運動之變化情形。

The objective of this paper focuses on the application of meshless method to simulate fully nonlinear water wave generation exerted by moving boundaries. In order to construct this meshless numerical model, three dimensional fundamental solution of Laplace equation is chosen to solve velocity potential, and two dimensional Gaussian radial basis function is used to calculate gradient of free surface. The elevation of free surface can be solved by using the fully nonlinear free-surface boundary conditions. In addition, an explicit time marching technique is developed by utilizing the leap-frog second-order central difference scheme. For numerical simulation, firstly, sloshing problem is proceeded with horizontal simple-harmonic motion boundary. Mass conservation and shape of wave are checked with other study to compare the differences between linear and nonlinear waves. It shows that, present numerical results agree very well with other research results. Furthermore, present results reveal more nonlinear characteristics. In the second case, nonlinear wave generation is simulated with a vertical moving bed boundary for purpose of figuring out tsunami wave motion, including processes of its generation, propagation, and transformation on a sloping bed.