研究流體與結構體的交互作用在許多科學或工程應用問題上扮演著重要的角色，而這類的計 算流體問題通常會涉及複雜的結構體幾何形狀。傳統上結構體配合法是經常被使用於模擬具 有複雜邊界流場問題的動力行為，在該方法中，不可壓縮的那維爾-史托克方程的空間變數 是被離散在符合沉浸結構體邊界的曲線或非結構化網格上，因此可以很容易地賦予內部邊界 條件。然而，當結構體在流場中會產生型變或會隨流體而運動時，此結構體配合法的空間變 數離散必須在每個時間步驟重新網格化空間定義域，但是網格生成可能變成沈重的計算成本 負擔，應該避免在每個時間步驟重新進行空間定義域的網格化，因此尋求立基於笛卡爾網格 的非邊界配合方法的幫助是顯而易見的期望，例如所謂的沉浸邊界法，應可較有效率地解決 複雜的流體與結構體耦合的動力問題。 在這三年期的研究計畫中，我們主要的目的是發展模擬流體與結構體耦合的動力行為的 高效率數值計算方法，特別是關注在下列三個子題： 1. 直接施力沉浸邊界投影法求解流體與固體交互作用的耦合方程，其中沉浸在流體裡的 固體依循統御方程而運動。 2. 直接施力沉浸邊界投影法求解流體與彈性體交互作用的耦合方程，其中彈性結構是準 靜態、等向均質及僅具小形變。 在上述的這兩種直接施力沉浸邊界投影法中，固體或彈性體所在區域將被視為是流體 的一部份，但是在此區域引入一個額外的虛擬力，使得該區域行為真的就像是一個固 體或彈性體。實際上，該虛擬力會被加進那維爾-史托克方程的動量方程式之中，以便 協調流體與固體或彈性體之間的交互作用，同時使得流體符合正確的沉浸邊界條件。 我們亦將研究沉浸邊界法結合人工壓縮法求解流構耦合問題，其中流體的不可壓縮條 件“r u = 0”將被人工壓縮條件“#¶tp + r u = 0”所取代，其中# > 0是一個與時間離 散步長相關的微小參數，這個子題的研究重點將放在這種懲罰式方法在完全離散後的 線性代數方程組高效率的求解方式。 ;The study of fluid-structure interaction problems is of great importance in many applications of sciences and engineering, and it usually involves complex structure geometries. The body-fitted approach is a conventional method that is frequently used to simulate flow with a complex boundary. In that approach, the incompressible Navier-Stokes equations are spatially discretized on a curvilinear or unstructured grid that conforms to the immersed structure boundaries. Therefore, the internal boundary conditions can be imposed easily. However, the body-fitted discretizations have to re-mesh the spatial domain at every time step correspondingly when the body deforms or moves in the fluid. Since the grid generation can become a large computational overhead, it would be desirable to avoid the need of re-meshing at each time step. Thus, one should seek the help of the Cartesian grid based non-boundary conforming methods, such as the so-called immersed boundary (IB) method, to address the complex fluid-structure interaction problems. The main purpose of this three-year project is to develop efficient numerical methods for simulating the dynamics of fluid-structure interaction problems. We will focus on the following three topics: 1. The direct-forcing IB projection method for the fluid-solid interaction, where the immersed solid object is moving in the fluid governed by the equations of motion. 2. The direct-forcing IB projection method for the fluid-elastic body interaction, where the elastic structure is quasi-static, isotropic and homogeneous which undergoes small deformations. In the above two direct-forcing IB projection methods, the solid/elastic body domain is treated like a fluid with an additional virtual force field applied to it so that it would act like a solid/elastic body. Actually, this virtual force is added to the momentum equations to accommodate the interaction between the solid/elastic body and fluid such that the boundary condition at the immersed boundary is exactly satisfied. 3. We will also study the IB method combined with the artificial compressibility method for solving fluid-structure interaction problems, in which the incompressibility constraint r u = 0 is replaced by the artificial compressibility equation, #¶tp +r u = 0, and # > 0 is a small parameter depending on the time step length. The development of efficient solvers for the linear systems associated with the fully discrete vector-valued problems to be solved will be the focus of this penalty approach.
