本文將採用隱式虛擬邊界法於非交錯性直角坐標網格系統解決零厚度平板之沉浸邊界或移動邊界問題，並搭配補綴網格的使用，使計算速度可大幅提升且可保持答案之精確性。此方法可準確計算零厚度平板之沉浸邊界問題，不論平板為垂直或是傾斜任意的角度，本文更將此方法應用於蜻蜓飛行之流場計算，完全符合蜻蜓翅膀厚度極小亦可視為無厚度之情形，無須將翅膀設定為有厚度的類橢圓形，且成功模擬出蜻蜓飛行之流場與壓力場之答案，也獲得蜻蜓飛行之垂直力與水平力，由模擬結果可獲得許多不同以往之想法。而此方法經由本文成功解決零厚度之沉浸邊界問題後，便可適用於各類沉浸邊界或移動邊界之問題，無論有無厚度或不規則形狀，皆可獲得良好之答案。

We apply an implicit virtual boundary method with non-staggered coordinate grid system to zero thickness flat plate. Also we use a patch grid to reduce the calculating time and keep the accuracy at the same time. This method can solve the immersed boundary problem for a zero thickness flat plate accurately, no matter it is vertical or with arbitrary angle. We also apply a method to calculate a flying dragonfly, which we won’t need to assume the wings as elliptical geometries with thin thickness, but directly solve them as a zero thickness problem and successfully get the answer of flow and pressure field instead. Also, we may get the drag and lift coefficients. Due to the simulation in this case, we can gain many ideas different from the past. Consequently, the implicit virtual boundary method can successfully applied to many different problems.

[1]M. R. Castelli, P. Cioppa and E. Benini, Numerical Simulation of the Flow Field around a Vertical Flat Plate of Infinite Extent, World Academy of Science, Engineering and Technology, Vol. 6, 235-240 (2012).
[2]M. R. Castelli, P. Cioppa and E. Benini, Numerical Simulation of the Flow Field around a 30° Inclined Flat Plate, World Academy of Science, Engineering and Technology, Vol. 6, 730-735 (2012).
[3]C. S. Peskin, Flow patterns around heart valves: A numerical method, Journal of Computational Physics, Vol. 10, 252-271 (1972).
[4]C. S. Peskin and D. M. McQueen, A three-dimensional computational method for blood flow in the heart: (I) immersed elastic fibers in a viscous incompressible fluid, Journal of computational physics, Vol. 81, 372-405 (1989).
[5]D. M. McQueen and C. S. Peskin, A three-dimensional computational method for blood flow in the heart: (II) contractile fibers, Journal of computational physics, Vol. 82, 289-297 (1989).
[6]C. S. Peskin and D. M. McQueen, Cardiac fluid dynamics, Critical Reviews in Biomedical Engineering, Vol. 82, 451-459 (1992).
[7]J. Mohd-Yusof, Combined Immersed Boundaries/B-Splines Methods for Simulations of Flows in Complex Geometries, Center for turbulence research annual research briefs (1997).
[8]E.A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof, Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, Journal of computational physics, Vol. 161, 35-60 (2000).
[9]Y. H. Tseng and J. H. Ferziger, A ghost-cell immersed boundary method for flow in complex geometry, Journal of computational physics, Vol. 192, 593-623 (2003).
[10]J. Yang and E. Balaras, An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, Journal of computational physics, Vol. 215, 12-40 (2006).
[11]S. L. Lee and R. Y. Tzong, Artificial pressure for pressure-linked equation, International Journal of Heat and Mass Transfer, Vol. 35, 2705-2716 (1992).
[12]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).
[13]S. L. Lee, Weighting function scheme its application on multidimensional conservation equations, International Journal of Heat and Mass Transfer, Vol. 32, 2065–2073 (1989).
[14]S. L. Lee, A strongly implicit solver for two-dimensional elliptic differential equations, Numerical Heat Transfer, Vol. 16, 161-178 (1989).
[15]S. Taneda and H. Honji, Unsteady flow past a flat plate normal to the direction of motion, Journal of the Physical Society of Japan, Vol. 30, 262 (1971).
[16]Z. J. Wang and D. Russell, Effect of Forewing and Hindwing Interactions on Aerodynamic Forces and Power in Hovering Dragonfly Flight, Physical review letters 99, 148101 (2007).
[17]Z. J. Wang, Dissecting insect flight, Annual Review of Fluid Mechanics, Vol. 37, 183-210 (2005).