本文的主要目的是實現文獻[22]所提出的一種人造壓縮性直接施力沉
浸邊界法模擬二維非穩態流體與固體交互作用的行為，此方法結合了
人造壓縮性法和直接施力沉浸邊界法，並採用一種預測-修正策略。
我們依循[22]的主要想法，首先引入懲罰技術去弱化不可壓縮納維爾-史托克方程中的不可壓縮條件，同時引入一個只分布在固體上的虛擬力並將其附加到流體動量方程式上，用以處理沉浸固體上的無滑移邊界
條件。接著我們採用一階隱式尤拉法離散方程組中的時間變數，並使
用顯式一階的方法線性化其中的非線性對流項，然後運用直接施力沉
浸邊界法結合一種預測-修正策略求解時間離散後的方程組。關於方法中的空間離散方式，我們採用交錯網格的二階中央差分格式。本文執行了數個二維非穩態流體與固體交互作用的實驗來說明此演算法的效能，數值模擬結果顯示這個簡單的人造壓縮性直接施力沉浸邊界法對於流
固耦合問題可以取得相當合理的數值結果。

The main purpose of this thesis is to implement an artificial compressibility-immersed boundary method with
direct forcing proposed in [22] for simulating 2-D unsteady flows interacting with rigid solid objects.
This approach is based on the artificial compressibility method and the direct-forcing immersed boundary method combined with a prediction-correction strategy.
Following the ideas in [22], we employ the penalty technique to weaken the incompressibility condition in the incompressible Navier-Stokes equations and introduce a virtual force distributed on the whole solid object and imposed to the fluid momentum equations to accommodate the no-slip boundary condition at the immersed solid boundary. We then use the first-order implicit Euler scheme to discretize the temporal variable in the resulting system of equations and apply the explicit first-order approximation to linearize the nonlinear convection term. After that, we employ a direct forcing immersed boundary method with a prediction-correction strategy to solve the system of time-discretized equations. For the spatial discretization in this approach, we take the second-order central differences on the staggered grids. We illustrate the performance of the algorithm by performing several 2-D numerical experiments of unsteady flow interacting with solid object. From the numerical results, we find that this simple artificial compressibility immersed boundary method with direct forcing can achieve reasonable results for 2-D fluid-solid interaction problems.

[1] J. Blasco, M. C. Calzada, and M. Mar´ın, A fictitious domain, parallel numericalmethod for rigid particulate flows, Journal of Computational Physics, 228 (2009),
pp. 7596-7613.
[2] D. L. Brown, R. Cortez, and M. L. Minion, Accurate projection methods for the incompressible Naver-Stokes equatons, Journal of Computational Physics, 168 (2001),
pp. 464-499.
[3] M.-J. Chern, Y.-H. Kuan, G. Nugroho, G.-T. Lu, and T.-L. Horng, Direct-forcing immersed boundary modeling of vortex-induced vibration of a circular cylinder,
Journal of Wind Engineering and Industrial Aerodynamics, 134 (2014), pp. 109-121.
[4] M.-J. Chern, D. Z. Noor, C.-B. Liao, and T.-L. Horng, Direct-forcing immersed boundary method for mixed heat transfer, Communications in Computational Physics, 18 (2015), pp.1072-1094.
[5] M.-J. Chern, W.-C. Shiu, and T.-L. Horng, Immersed boundary modeling for interaction of oscillatory flow with cylinder array under effects of flow direction
and cylinder arrangement, Journal of Fluids and structures, 43 (2013), pp. 325-346.
[6] H. Choi and P. Moin, Effects of the computational time step on numerical solutions of turbulent flow, Journal of Computational Physics, 113 (1994), pp. 1-4.
[7] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Mathematics of Computation, 22 (1968), pp. 745-762.
[8] A. J. Chorin, On the convergence of discrete approximations to Navier-Stokes equations, Mathematics of Computation, 23 (1969), pp. 341-353.
[9] G. J. Dong and X. Y. Lu, Characteristics of flow over traveling wavy foils in a side-by-side arrangement, Physics of Fluid, 19, 057107 (2007).
[10] W. E and J.-G. Liu, Projection method I: convergence and numerical boundary layers, SIAM Journal on Numerical Analysis, 32 (1995), pp. 1017-1057.
[11] W. E and J.-G. Liu, Projection method III: spatial discretization on the staggered grid, Mathematics of Computation, 71 (2002), pp. 27-47.
[12] W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Communications in Mathematical Sciences, 1 (2003), pp. 317-332.
[13] E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof, Combined immersedboundary finite-difference methods for three-dimensional complex flow simulations,
Journal of Computational Physics, 161 (2000), pp. 35-60.
[14] R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph and J. P´eriaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, Journal of Computational Physics, 169 (2001), pp. 363-426.
[15] B. E. Griffith, An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner, Journal of Computational Physics, 228 (2009), pp. 7565-7595.
[16] A. Gronskis and G. Artana, A simple and efficient direct forcing immersed boundary method combined with a high order compact scheme for simulating flows with moving rigid boundaries, Computers & Fluids, 124 (2016) pp. 86-104.
[17] J.-L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, 195 (2006), pp. 6011-6045.
[18] J.-L. Guermond and P. Minev, High-order time stepping for the incompressible Navier-Stokes equations, SIAM Journal on Scientific Computing, 37 (2015), pp.
A2656-A2681.
[19] J.-L. Guermond and P. D. Minev, High-order time stepping for the Navier-Stokes equations with minimal computational complexity, Journal of Computational and
Applied Mathematics, 310 (2017), pp. 92-103.
[20] T.-L. Horng, P.-W. Hsieh, S.-Y. Yang, and C.-S. You, A simple direct-forcing immersed boundary projection method with prediction-correction for fluid-solid
interaction problems, Computers & Fluids, in revision, 2017.
[21] P.-W. Hsieh, M.-C. Lai, S.-Y. Yang, and C.-S. You, An unconditionally energy stable penalty immersed boundary method for simulating the dynamics of an inextensible interface interacting with a solid particle, Journal of Scientific Computing,64 (2015), pp. 289-316.
[22] P.-W. Hsieh, S.-Y. Yang, and C.-S. You, An artificial compressibility-immersed boundary method with direct forcing for simulating flow in complex geometries,
Preprint, 2017.
[23] T. J. R. Hughes,W. K. Liu, and A. Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, Journal of Computational
Physics, 30 (1979), pp. 1-60.
[24] T. Kajishima, S. Takiguchi, H. Hamasaki, and Y. Miyake, Turbulence structure of particle-laden flow in a vertical plane channel due to vortex shedding, JSME
International Journal, Series B, 44 (2001), pp. 526-535.
[25] T. Kajishima and S. Takiguchi, Interaction between particle clusters and particleinduced turbulence, International Journal of Heat and Fluid Flow, 23 (2002), pp.639-646.
[26] Y. Kim and M.-C. Lai, Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, Journal of Computational Physics, 229 (2010),
pp. 4840-4853.
[27] Y. Kim and C. S. Peskin, Penalty immersed boundary method for an elastic boundary with mass, Physics of Fluids, 19 (2007), 053103.
[28] J. Kim, D. Kim, and H. Choi, An immersed-boundary finite-volume method for simulations of flow in complex geometries, Journal of Computational Physics, 171 (2001), pp. 132-150.
[29] J.-G. Liu, J. Liu, and R. L. Pego, Stable and accurate pressure approximation for unsteady incompressible viscous flow, Journal of Computational Physics, 229(2010), pp. 3428-3453.
[30] T.-R. Lee, Y.-S. Chang, J.-B. Choi, D. W. Kim, W. K. Liu, and Y.-J. Kim, Immersed finite element method for rigid body motions in the incompressible NavierStokes
flow, Computer Methods in Applied Mechanics and Engineering, 197 (2008) pp. 2095-2316.
[31] D. Z. Noor, M.-J. Chern, and T.-L. Horng, An immersed boundary method to solve fluid-solid interaction problems, Computational Mechanics, 44 (2009), pp.
447-453.
[32] C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), pp. 479-517.
[33] C. S. Peskin, Flow patterns around heart valved: a numerical method, Journal of Computational Physics, 10 (1972), pp. 252-271.
[34] A. Quarteroni, F. Saleri, and A. Veneziani, Factorization methods for the numerical approximation of Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 188 (2000), pp. 505-526.
[35] J. Shen, A remark on the projection-3 method, International Journal for Numerical Methods in Fluids, 16 (1993), pp. 249-253.
[36] J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM Journal on Numerical Analysis, 32 (1995), pp. 386-403.
[37] J. M. Stockie and B. R. Wetton, Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes, Journal of Computational Physics, 154 (1999), pp. 41-64.
[38] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Revised Edition, Elsevier Science Publishers B.V., Amsterdam, 1984.
[39] M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, Journal of Computational Physics, 209 (2005), pp. 448-476.
[40] C.-S. You, On Two Immersed Boundary Methods for Simulating the Dynamics of Fluid-Structure Interaction Problems, PhD Dissertation, National Central University, June 2016.