本文將採用有限差分法中的虛擬邊界法於非交錯性直角座標網格系統，模擬不規則形狀物體之熱傳導以及金屬材料透過模具成型之問題。使用此方法在進行求解時，不需要配合物體的複雜幾何形狀做網格產生以及進行不同的網格座標轉換，如此一來能使計算速度大幅增加，並且保持答案的準確性。
從不規則形狀物體之熱傳導的模擬結果中可以發現利用此方法所計算出的溫度場具有相當高的準確性。而雖然金屬成型的模擬結果有所錯誤，但並不是此方法出現問題，而是在應力分析時缺少應力波方程式才導致這樣的結果。

In this study, we apply virtual boundary method of finite difference method with non-staggered coordinate grid system to simulate heat conduction of the irregularly shaped object and the metal forming problem. This method doesn't need to match the complex geometry of the object to do the grid generation and the conversion in different grid coordinates, so that the calculating time can be reduced and keep the accuracy at the same time.
From the simulation results of the heat conduction of irregularly shaped object, the answer calculated by this method is very accurate. While the simulation results of metal forming are wrong, this is not a problem with this method, but rather the lack of stress wave equation in stress analysis causes this result.

摘要 II
ABSTRACT III
目錄 IV
圖目錄 VI
符號說明 VII
第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 1
1.3 研究目的 3
第二章 數值分析 4
2.1 網格系統 4
2.2 虛擬邊界法 4
2.3 虛擬邊界之計算 6
2.3.1 給定邊界速度 6
2.3.2 給定邊界牽引 6
2.3.3 混合邊界牽引 9
第三章 案例模擬 11
3.1.1 不規則形狀物體的熱傳導(一) 11
3.1.2 不規則形狀物體的熱傳導(二) 14
3.1.3 邊界區域驗證 15
3.2 金屬成型 16
第四章 結論 25
參考文獻 26

[1] E. G. Thompson and S. W. Yu, A flow formulation for rate equilibrium equations, International Journal for Numerical Methods in Engineering, Vol. 30, 1619-1632 (1990).
[2] Z. G. Voyiadjis and M. Foroozesh, A finite strain, total Lagrangian finite element solution for metal extrusion problems, Comp. Meth. Appl. Mech. Eng, Vol. 86, 337-370(1991).
[3] M. S. Galada, G.A.E. Oravas and M. A. Dokainish, A consistent Eulerian formulation of large deformation problems in static and dynamics, Int. J. Non-linear Mech, Vol. 18 21-35(1983).
[4] R. B. Haber, A mixed Eulerian-Lagrangian displacement model for large-deformation analysis in solid mechanics, Comput. Meth. Appl. Mech. Eng, Vol. 43, 277-292(1984).
[5] D. J. Benson, An efficient, accurate simple ALE method for nonlinear finite element programs, Comput. Meth. Appl. Eng, Vol. 72, 269-301(1989).
[6] P. J. G. Schreurs, F. E. Veldpaus and W. A. M. Brekelmans, Simulation of forming precess using the arbitrary Eulerian-Lagrangian formulation, Comput. Mech. Eng, Vol. 58,19-36(1986).
[7] M. S. Gadala and J. Wang, Elasto-plastic finite element simulation of rolling and compression between wedge-shaped dies, Journal of Materials Processing Technology, Vol. 97,132-147 (2000).
[8] H. N. Bayoumi and M. S. Gadala, A complete finite element treatment for the fully coupled implicit ALE formulation, Computational Mechanics, Vol. 33, 435-452 (2004).
[9] K. S. Al-Athel and M. S. Gadala, Eulerian volume of solids (VOS) approach in solid mechanics and metal forming, Computer Methods in Applied Mechanical and Engineering, Vol. 200, 2145-2159 (2011).
[10] P. K. Banerjee, The Boundary Element Methods in Engineering, Ch. 13, McGraw-Hill, London (1994).
[11] N. R. Chitkara and W. Johnson, Plane strain compression of pre-shaped material between wedge-shaped dies, Int. J. mech. Sci. Pergamon Press, Vol. 14, 151-184 (1972).
[12] S. L. Lee and D. W. Lin, Transient conjugate heat transfer on a naturally cooled body of arbitrary shape, International Journal of Heat and Mass Transfer, Vol. 40, 2133-2145 (1997).
[13] S. L. Lee and C. R. Ou, Integration scheme for elastic deformation and stresses, ASME J. Applied Mechanics, Vol. 66, 978-985 (1999).
[14] K. W. Chen, Implicit Virtual Boundary Method for Moving Boundary Problems on Non-Staggered Cartesian Grid System, Master’s thesis, Nation Tsing Hua University (2013).
[15] G. S. Cyue, Implicit Virtual Boundary Method for Moving Flat Plates of Zero Thickness, Master’s thesis, Nation Tsing Hua University (2014).
[16] S. L. Lee, G. S. Cyue and K. W. Chen, Implicit virtual boundary method for moving boundary problems on non-staggered Cartesian patch grids, Journal of Mechanics, (2017).