內嵌界面問題之矩陣分解法

Abstract

嵌入邊界法(immersed boundary method)是一種模擬不可壓縮流體的數學模型，他的特色在於解決有無質量的嵌入邊界的情況。而解決嵌入邊界法的問題，矩陣分解法(matrix factorization method)是一種利用類似分部步驟法(fractional step method)的方法，將嵌入邊界法分解成三個步驟，在前兩個步驟我們會面對解一個對稱正定的線性系統，在此我們可以利用共軛梯度法(conjugate gradient method)來解決這個問題。在這份論文之中，我們利用矩陣分解法來模擬流場通過各種嵌入之物體，包括流過靜止與可動的圓柱、兩個靜止的圓柱、以及機翼形狀的物體。

The immersed boundary method is a model to simulate a viscous incompressible fluid with immersed massless boundary. It comes from the Navier-Stokes equation of viscous incompressible fluid with the interaction term between the immersed boundary and the fluid. The matrix factorization method is a formulation of immersed boundary method, and the idea is the fractional step method for Navier-Stokes equation. The immersed boundary problem could be factorized to three steps, and the conjugate gradient method can be applied to solve the first and second step. In this paper, we use the matrix factorization method simulate the flow past stationary or movable immersed object, including the flow past a stationary and a moving cylinder, the flow past two stationary cylinders, and the flow past a winglike object.