本文以無網格方法建立的二維數值模式，求解卜桑方程式(Poisson Equation)，模擬非線性造波之自由液面變形及孤立波造波。本數值模式主要架構為Wu and Tsay (2013)和Wu et al.(2015)發展之二維無網格模式，此模式的特色為以局部近似(Local approximation)求解，這個方式增加了計算上的效率。Wu et al.(2015)於文章中求解描述速度勢分佈的拉普拉斯方程式(Laplace equation)得到速度勢分佈和速度分佈與速度梯度分佈，再解一次描述壓力分佈的卜桑方程式求出壓力分佈。本研究提出的模式將其改為直接解算卜桑方程式求出壓力分佈，並對此模式進行問題驗證。
在給定邊界與造波板運動方程式的條件下，水槽內配置點位的位置、速度、加速度與壓力值均可以得知，並藉由稱為跳蛙法(leap-frog approach)的時間離散法做點位資訊更新，本研究以兩種問題對該模式進行驗證，分別以Liu and Lin (2008)水槽晃動試驗和Chambarel et al.(2009)之驗證孤立波撞擊直立壁試驗，前者採取Liu and Lin (2008)之實驗數據及文獻所述之流體體積法模式做對照；後者則參照Chambarel et al.(2009)孤立波互撞之試驗數據與文獻中Cooker et al.(1997)孤立波撞擊直立壁之數據進行比對。
前者水槽晃動的模擬結果與文獻波高計數據有不錯的吻合度，但因共振效應造成液面振盪幅度劇增時，本模式模擬出的液面高度會發生略為低估的狀況。後者孤立波在擴散的過程中可發現隨著非線性效應之提高，波後的尾波效應和波高衰減的量會隨之增加；而兩孤立波對撞的模擬結果則非常接近文獻中Chambarel et al.(2009)使用之邊界積分法的結果，並隨著孤立波的非線性程度提高，模式結果亦會出現較為低估的趨勢。
論文最後對模式的模擬成果進行討論，本研究所使用之模式以局部多項式近似法計算速度梯度值並進行壓力的解算，比起上一版本之模式需先計算流場速度勢來而後才能進一步計算壓力，本模式在獲取壓力分佈的流程上更為直接，文末並對該解算壓力場之數值模式的未來發展提出些許建議。

The two-dimensional mesh-free numerical model for the simulation of pressure distribution beneath nonlinear free surface will be established in this study. The main framework of the present model is based on Wu and Tsay (2013). In this study, robust local polynomial collocation method is used, and it is developed in the way that the boundary point in the collocations have to satisfy governing equation and boundary conditions. This method is more efficient and accurate than the RBF-collocation method.
According to the text book of turbulent flows (Pope, 2000), no matter the viscous effect is considered or not, the momentum equation of incompressible flow can be reduced to a Poisson equation governing the pressure distribution. Following this, Wu et al. (2015) used their numerical results of the free surface potential flow simulation to obtain the pressure distribution in a rectangular sloshing tank. Based on this previous model, a new algorithm is proposed in this study. After solving the pressure equation, the pressure gradient in the flow field and then the accelerations of fluid particles will be obtained. Therefore, the fluid particle trajectories are accurately estimated and the movement of the free surface is precisely predicted. Because the pressure equation is directly solved in this model, the results include the pressure distribution on solid surfaces.
To test the performance of the present model in solving non-linear wave, two simulations are carried out for the study. First, a 2-D rectangular tank with both non-resonance and resonance shaking frequencies are respectively simulated. The results are compared with the experimental data by Liu and Lin (2008), moreover, the pressure variation on the solid boundary and the pressure distribution in the domain are also observed in this study.
The second one is the head-on collision of two equal solitary waves. The results are equivalent to the reflection of one solitary wave by a vertical wall when the viscosity is neglected (Chambarel et al. ,2009). The results show that this model can capture the flow dynamics pretty well when the non-linear value is not large, indicating that the present model can well simulate important flow phenomenon.

1. Chambarel, J., Kharif, C., and Touboul, J. (2009). Head-on collision of two solitary waves and residual falling jet formation, Nonlin. Processes Geophys., 16, 111-122.
2. Eisuke Kita, Jun'ichi Katsuragawa, Norio Kamiya, (2004). Application of Trefftz-type boundary element method to simulation of two-dimensional sloshing phenomenon. Engineering Analysis with Boundary Elements, Volume 28, Issue 6,Pages 677-683
3. J. Fenton (1972). A ninth-order solution for the solitary wave. J. Fluid Mech. 52 257–271.
4. Liu D, Lin P. (2008). A numerical study of three-dimensional liquid sloshing in tanks. Journal of Computational Physics 227:3921–3939, 2008
5. Wu, N. J., T.-K. Tsay and D. Young (2008). Computation of nonlinear free-surface flows by a meshless numerical method. Journal of waterway, port, coastal, and ocean engineering 134(2): 97-103.
6. Wu, N. J. and K. A. Chang (2011). Simulation of free‐surface waves in liquid sloshing using a domain‐type meshless method. International Journal for Numerical Methods in Fluids 67(3): 269-288.
7. Wu, N. J. and T. K. Tsay (2013). A robust local polynomial collocation method. International Journal for Numerical Methods in Engineering 93(4): 355-375.
8. Wu, N.-J. Tsay, T.-K.,Chen Y.-Y. (2014).Generation of stable solitary waves by a piston-type wave maker. Wave Motion, 51,pp.240-255.
9. Wu NJ, Hsiao SC, Wu HL (2015). Application of Local Polynomial Collocation Method to the Simulation of 3D Liquid Sloshing. Journal of Coastal and Ocean Engineering Vol.15(2): pp. 67-83.
10. Wu NJ, Hsiao SC, Wu HL (2016). Mesh-free simulation of liquid sloshing subjected to harmonic excitations. Engineering Analysis with Boundary Elements, Volume 64,2016,Pages 90-100.
11. Nan-Jing Wu, Shih-Chun Hsiao, Hsin-Hung Chen, Ray-Yeng Yang (2016). The study on solitary waves generated by a piston-type wave maker. Ocean Engineering, Volume 117, Pages 114-129.
12. 黃大祐(2012), 以無網格方法模擬海底崩移所造成之波浪傳遞與溯升. 國立成功大學水利及海洋工程研究所碩士論文
13. 闕立愷(2014), 以無網格法模擬波浪與結構物作用分析. 國立成功大學水利及海洋工程研究所碩士論文