在這篇論文中，我們使用了沉浸邊界法模擬二維與三維非黏滯性流體與固體移動邊界交互作用之問題。在處理固定與移動邊界問題，我們接施以外力的方式讓流場模擬出沉浸的固體。數值的精準度使用了decaying vortex來做測試，從結果可以看出數值誤差有達到二接精準度。接著使用沉浸邊界法模擬了多個問題: 非對稱圓柱之二維流場、圓柱在靜止流體中來回震盪、三維球體因重力作用掉落之問題、三維空間中兩顆球體掉落之問題。所有的模擬結果皆跟實驗與理論值接近，顯現出以沉浸邊界法模擬沉浸邊界是可以模擬出現時的狀況。
所有的程式撰寫皆建立在PETSc這個具有平行運算與矩陣球解的工具上。使用平行處理可以更加快我們處理二維與三維的龐大問題上。結果與討論的最後對程式進行了二維與三維的效率測試，在16個CPU以下二維與三維具有類似的加速效果，而在32個CPU時二維的加速效率則比三維的高。然而，整體的加速效率仍然差理論值有一段距離，這是目前可以再加強的部分。

In the present study, we use the immersed-boundary technique to simulate two and three-dimensional viscous incompressible flows interacting with moving solid boundaries. For both stationary and moving boundary problems, we apply the solid
body forcing within the solid node and provide a procedure to implement the forcing nodes between the non-stationary fluid and solid body. The accuracy of numerical scheme is first examined by decaying vortex test, and the results show that these are second-order accurate with respect to the L2 norm and the L∞ norm. Further test problems are simulated to examine the validity of the present immersed-boundary technique such as 2-D flow over an asymmetrically-placed cylinder, in-line oscillating cylinder in fluid at rest, and 3-D simulation of a sphere settling under gravity. In addition to one moving object, two spheres sedimenting in a closed container filled with a viscous fluid are investigated. There must be a collision model to prevent
the spheres penetrating into each other. All the computed results are in generally good agreement with experimental measurements. This indicates the capability of the present implementation in solving flows with moving solid objects.
The above implementations and techniques are all constructed on the software of PETSc, which is associated with a parallel solver. By using the parallel solver, we can enhance the capability of computational power for two- and three-dimensional fluid-solid interaction simulations. The scalability results show that the speedup performance for two-dimensional and three-dimensional test cases are almost the same, except when the number of processors increases to 32 where two-dimensional case has better performance. However, the overall scalability for both cases deviates
from the theoretical values. There is still much room for improving the parallel efficiency.

1 Introduction
1.1 Introduction
1.2 Literature Survey
1.3 Objectives and Motivations
2 Numerical Methods
2.1 Methodology of the Immersed-Boundary Method
2.1.1 Mathematical Formulation
2.1.2 Numerical Scheme
2.1.3 Forcing Strategies
2.2 Determinations of lift and drag forces
2.3 Determinations of particle’s collision force
2.4 Complete solution procedure
3 Numerical Results
3.1 Code validation
3.2 Flow over an asymmetrically-placed cylinder
3.3 In-line oscillating cylinder in fluid at rest
3.4 Simulation of a sphere settling under gravity
3.4.1 Lubrication Force
3.5 Two spheres sedimenting in a closed container filled with a viscous fluid
3.6 Parallel Efficiency
4 Conclusions

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