本文以無網格法所建立之二維數值模式，模擬受外力作用下之運動水槽，其
內部流體擺盪的情形。數值水槽的數學模式主要是求解拉普拉斯方程和非線性的
自由液面邊界條件，並在移動坐標系統下進行計算。本模式所使用的無網格法
── 基本解法是在計算域邊界，以及計算域外之虛擬邊界上分佈計算點。相較於
傳統基於網格結構之計算方法，此方法在處理移動邊界問題時，具有易於建構且
變化彈性大之優勢。本研究以駐波於水槽自由擺盪之案例進行初步數值實驗，並
確定了模式中如虛擬邊界形狀、佈點數量、時間步長等相關參數之設置。
電腦程式以 Python 語言撰寫。本模式並發展簡易的完全非線性自由液面追蹤
法，持續更新計算域邊界，成功模擬完全非線性流體擺盪行為。本模式應用於解
析水平橫向運動水槽，模擬結果分別和 Chen 與 Chiang (1999) 及 Frandsen (2004)
之研究結果進行比對。在非共振與擬共振外力作用下，本模式的模擬結果皆與前
人之模擬結果吻合良好。本模式更進一步應用於模擬雙方向 (水平與垂直) 運動
水槽內液體擺盪。並針對垂直運動時液體擺盪之穩定與非穩定分析 (Benjamin and
Ursell, 1954) 進行模擬。藉由設定 25 組水槽運動參數，分別討論不同運動參數下
液體擺盪變化之差異，進一步分析此特性對液體擺盪之影響。本模式成功模擬水
平單向與水平垂直雙向水槽運動，證明無網格基本解法可以應用於解析 2D 水槽
完全非線性流體擺盪行為。

A two dimension numerical model for sloshing response of the partially filled con-
tainer is developed by a meshless method in this study. Laplace equation is considered
as the governing equation for potential flow, and fully non-linear free surface boundary
conditions are implemented on the free surface. The numerical model is established in
the moving coordinate system attached to the container. The present meshless method,
method of fundamental solutions, only need to distribute nodes on the physical bound-
aries and the virtual boundaries outside the computational domain. Comparing with the
traditional mesh-based numerical method, this method has the advantage of being easy
to construct and flexible in dealing with the moving boundary problem. A preliminary
numerical validation was carried out by the case of a free oscillating standing wave. The
parameters such as the position of the virtual boundary, the number of nodes, and the time
step size are also determined.
The program is written in Python. A simple method for tracking fully non-linear free
surface is implemented, and the physical boundaries are updated with time. The numerical
results under horizontal motions were compared with the solutions of Chen and Chiang
(1999) and Frandsen (2004), respectively. The comparison shows that the presented re-
sults coincide with their solutions very well in both resonance and off-resonance cases.
The present numerical model is further applied to the sloshing response under coupled
motions (horizontal and vertical). Based on the study of Benjamin and Ursell (1954) who
discussed the stability of sloshing response under vertical motion, 25 cases are conducted
in various sloshing parameters under coupled motions. The present numerical model has
been successfully applied to investigating sloshing response under coupled motions.

[摘要+i]
[ABSTRACT+ii]
[ACKNOWLEDGEMENTS+iv],
[TABLE OF CONTENTS+vi],
[LIST OF FIGURES+vii],
[LIST OF TABLES+x],
[NOMENCLATURE+xii],
[CHAPTER 1. INTRODUCTION+1],
[1.1. Backgrounds+1],
[1.2. Literature reviews+2],
[1.2.1. Works in analysis of Sloshing effect+2],
[1.2.2. Overview of Meshless method+4],
[1.2.3. Applications of Method of fundamental solutions+6],
[1.3. Research Objectives+7],
[CHAPTER 2. MATHEMATICAL FORMULATION+9],
[2.1. Governing equations of fluid in a moving tank+10],
[2.2. Boundary conditions of fluid in a moving tank+12],
[2.2.1. Free surface boundary conditions+12],
[2.2.2. Solid wall boundary conditions+13],
[2.2.3. Initial conditions+14],
[2.3. Summary of chapter+14],
[CHAPTER 3. MODEL FOR NUMERICAL ANALYSIS+16],
[3.1. The implementation of MFS+16],
[3.2. The implementation of RBF method+19],
[3.3. Conjugate gradient method for inverting ill-conditioned matrices+22],
[3.4. Computational process+24],
[3.5. Summary of chapter+27],
[CHAPTER 4. STABILITY ANALYSIS AND THE BENCHMARK TESTS+28],
[4.1. Distribution of the virtual boundary+28],
[4.1.1. Geometry of the virtual boundary+30],
[4.1.2. Distance between physical boundary and virtual boundary+32],
[4.2. Stretching distribution of the nodes near free surface+33],
[4.3. Convergence Study of number of nodes and time step size+36],
[4.4. Summary of chapter+40],
[CHAPTER 5. SLOSHING RESPONSE UNDER SINGLE MOTION+41],
[5.1. Fully non-linear free surface+42],
[5.2. Sloshing response under horizontal motions+43],
[5.2.1. Preliminary validation+44],
[5.2.2. Sloshing response under different excitation frequency+49],
[5.3. Sloshing response under vertical motions+54],
[5.3.1. Preliminary validation+54],
[5.3.2. Sloshing response under stable and unstable zones of vertical motion+55],
[5.4. Summary of chapter+58],
[CHAPTER 6. COUPLED HORIZONTAL AND VERTICAL MOTIONS+59],
[6.1. Preliminary study of stable zone and the unstable zone under coupled motion+59],
[6.2. Investigation along the path on stability map+61],
[6.3. Multi-directional sloshing with different excitation frequency in horizontal motion+65],
[6.4. Summary of chapter+78],
[CHAPTER 7. CONCLUDING REMARKS+79],
[7.1. Conclusions+79],
[7.2. Recommendations for the future research+80],
[REFERENCES+82],
[APPENDIX . PYTHON PROGRAM+84]

[1] G. W. Housner, “Dynamic pressures on accelerated fluid containers,” Bulletin of the
Seismological Society of America, vol. 47, no. 1, pp. 15–35, 1957.
[2] H. N. Abramson, W. H. Chu, and D. D. Kana, “Some studies of nonlinear lateral
sloshing in rigid containers,” Journal of Applied Mechanics, vol. 33, no. 4, pp. 777–
784, 1966.
[3] O. M. Faltinsen, “A Nonlinear Theory of Sloshing in Rectangular Tanks,” Journal
of Ship Research, vol. 18, no. 4, pp. 224–241, 1974.
[4] R. Ibrahim, Liquid sloshing dynamics. 2005.
[5] O. M. Faltinsen, “A Numerical Nonlinear Method of Sloshing in Tanks With Two-
Dimensional Flow,” Journal of Ship Research, vol. 22, no. 3, pp. 193–202, 1978.
[6] T. Nakayama and K. Washizu, “Nonlinear analysis of liquid motion in a container
subjected to forced pitching oscillation,” International Journal for Numerical Meth-
ods in Engineering, vol. 15, no. 8, pp. 1207–1220, 1980.
[7] T. Nakayamaf and K. Washizus, “The Boundary Element Method Applied To Non-
linear Sloshing Problems The Analysis Of Two-Dimensional,” Internasioanl Jour-
nal For Numerical Methods In Engineering, vol. 17, no. August 1980, pp. 1631–
1646, 1981.
[8] C. W. Hirt and B. D. Nichols, “Volume of fluid (VOF) method for the dynamics of
free boundaries,” Journal of Computational Physics, vol. 39, no. 1, pp. 201–225,
1981.
[9] B.-F. Chen and H.-W. Chiang, “Complete 2D and Fully Nonlinear Analysis of ideal
fluid in tanks,” Journal of Engineering Mechanics, vol. 125, no. 1, pp. 70–78, 1999.
[10] J. B. Frandsen, Sloshing motions in excited tanks, vol. 196. 2004.
[11] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless meth-
ods: An overview and recent developments,” Computer Methods in Applied Me-
chanics and Engineering, vol. 139, no. 1, pp. 3–47, 1996.
[12] C.-C. Tsai, Z.-H. Lin, and T.-W. Hsu, “Using a local radial basis function collo-
cation method to approximate radiation boundary conditions,” Ocean Engineering,
vol. 105, no. May 2016, pp. 231–241, 2015.
[13] M. A. Golberg, “The method of fundamental solutions for Poisson’s equation,” En-
gineering Analysis with Boundary Elements, vol. 16, no. 3, pp. 205–213, 1995.
[14] A. Karageorghis, “The method of fundamental solutions for the calculation of the
eigenvalues of the Helmholtz equation,” Applied Mathematics Letters, vol. 14, no. 7,
pp. 837–842, 2001.
[15] C. C. Tsai, “Meshless BEM for three-dimensional stokes flows,” CMES - Computer
Modeling in Engineering and Sciences, vol. 3, no. 1, pp. 117–128, 2001.
[16]
fundamental solutions for unsteady Stokes equations,” Engineering Analysis with
Boundary Elements, vol. 30, no. 10, pp. 897–908, 2006.
[17] N. J. Wu, T. K. Tsay, and D. L. Young, “Meshless numerical simulation for fully
nonlinear water waves,” International Journal for Numerical Methods in Fluids,
vol. 50, no. 2, pp. 219–234, 2006.
[18] N.-J. Wu, T.-K. Tsay, and D. Young, “Computation of nonlinear free-surface flows
by a meshless numerical method,” Journal of Waterway, Port, Coastal and Ocean
Engineering, vol. 134, no. 2, pp. 97–103, 2008.
[19] G. R. Liu and Y. T. Gu, An introduction to meshfree methods and their programming.
2005.
[20] A. Poullikkas, A. Karageorghis, and G. Georgiou, “Methods of fundamental solu-
tions for harmonic and biharmonic boundary value problems,” Computational Me-
chanics, vol. 21, no. 4-5, pp. 416–423, 1998.
[21] W. Hui and Q. Qinghua, “Some problems with the method of fundamental solution
using radial basis functions,” Acta Mechanica Solida Sinica, vol. 20, no. 1, pp. 21–
29, 2007.
[22] F. Benjamin, T.B., Ursell, “The Stability of the Plane Free Surface of a Liquid in
Vertical Perioidic Motion,” 1954.