沉浸邊界法在囊泡問題之數值模擬

Abstract

過去數十年來，囊泡問題的動態數值問題一直是個熱門的議題。在本論文中，我們透過沉浸邊界法 (immersed boundary method)來描述囊泡的流體數學模型，公式，包含Eulerian座標系下的流體方程式以及建立在Lagrangian座標系中有關界面的變數，而這兩個座標系之間各個變數的轉換，則是藉由 Dirac delta function 來連結。本論文致力於發展簡易且精確的數值方法來模擬此流體界面問題。

首先，我們提出了一個fractional step immersed boundary method用於模擬不可延展界面之問題(不考慮bending effect)。我們證明了作用在表面張力spreading operator是surface divergence operator的斜自伴算子 (skew-adjoint)。利用這個特性，對於流體變數之離散我們可獲得一個對稱矩陣，並可使用fractional step方法解決此線性系統。我們比較了此數值方法的精確度，以及利用本方法研讀不可延展界面在shear flow底下之數值模擬。

再來我們研發出一種unconditionally stable immersed boundary method作為二維的囊泡在Navier-Stokes流體之模擬。我們用一種半隱示 (semi-implicit)的方法來表示囊泡之界面力，而與介面力相關的stretching factor則可透過其他的方程式獲得。我們證明出在本方法中，流體系統的總能量將隨時間而遞減。另外，利用projection method，對於流場我們推導出一個對稱正定的線性系統，此矩陣可利用多重網格法 (multi-grid method)有效率地計算其解。在數值實驗中，我們驗證本方法之精準度，並在效率上遠勝於傳統之顯示邊界力處裡方法。同樣我們也利用本方法研讀囊泡在二維座標底下之型變動態系統。

最後，我們延伸至三維軸對稱的囊泡問題。與其將表面張力作為一Lagrange's multiplier來迫使囊泡之不可延展性，我們定義一種spring-like表面張力作為本模型之逼近。囊泡的邊界可利用Fourier spectral來表示，並且我們可精準地計算出界面上的平均曲率、高斯曲率等等。透過一系列之數值模擬，我們展示出本方法之應用性與可靠性。我們使用本方法研讀囊泡在靜止流、重力場影響及Poiseuille flow下之動態型變問題。

Numerical simulation of vesicle dynamics has been a popular issue for many decades. In this dissertation, a mathematical formulation for suspension of vesicle in fluid is modeled by immersed boundary method, where a mixture of Eulerian fluid variables and curve-linear Lagrangian interfacial variables are used, and the linkage between these two variables is a smoothed Dirac delta function. The purpose of this dissertation is to develop accurate and efficient numerical schemes for simulating vesicle dynamics through immersed boundary method.

Firstly, we propose a fractional step immersed boundary method to mimic dynamical system of an inextensible interface (vesicle without bending effect). In addition to solving for the fluid variables such as the velocity and pressure, the present problem involves finding an extra unknown elastic tension such that the surface divergence of the velocity is zero along the interface. By taking advantage of skew-adjoint property between force spreading operator and surface divergence operator, the resultant linear system of equations is symmetric and can be solved by fractional steps so that only fast Poisson solvers are involved. The convergent tests for present fluid solver is performed and confirm the desired accuracy. The tank-treading motion for an inextensible interface under a simple shear flow has been studied extensively, and the results are in good agreement with those obtained in literature. This part of work has been published in SIAM Journal of Scientific Computing as in [37].

Secondly, we develop an unconditionally stable immersed boundary method to simulate 2D vesicle under a Navier-Stokes flow. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy of fluid system is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased. The numerical result shows the severe time step restriction in an explicit boundary forcing scheme is avoided by present method. The part of work has been published in East Asian Journal of Applied Mathematics as in [22].

Lastly, we extend to simulate three-dimensional axisymmetric vesicle suspended in a Navier-Stokes flow. Instead of introducing a Lagrange's multiplier to enforce the vesicle inextensibility constraint, we modify the model by adopting a spring-like tension to make the vesicle boundary nearly inextensible so that solving for the unknown tension can be avoided. We also derive a new elastic force from the modified vesicle energy and obtain exactly the same form as the originally unmodified one. In order to represent the vesicle boundary, we use Fourier spectral approximation so we can compute the geometrical quantities on the interface more accurately. A series of numerical tests on the present schemes have been conducted to illustrate the applicability and reliability of the method. We perform the convergence check for fluid variables for present schemes. Then we study the vesicle dynamics in quiescent flow, Poiseuille and under influence of gravity in detail. The numerical results are shown to be in good agreement with those obtained in literature. The part of work has been published in Journal of Computational Physics as in [23].