论文标题
浸入边界双层（IBDL）方法
The Immersed Boundary Double Layer (IBDL) Method
论文作者
论文摘要
Peskin的浸入边界（IB）方法（J.Comput。Phys。，1977）对于涉及流体结构相互作用或复杂几何形状的问题很有用。通过使用独立于几何形状的常规网格，IB框架产生了一种可靠的数值方案，该方案可以有效处理浸泡的可变形结构。 IB方法还适用于规定运动和其他PDE的问题，并具有给定的边界数据。这些问题的IB方法传统上涉及仅满足边界条件的惩罚力，否则它们被称为约束问题。在后一种方法中，必须通过求解与条件差的第一类积分方程相对应的方程来找到未知力。因此，此操作可能需要大量的Krylov方法的迭代，并且由于时间依赖性问题需要在每个时间步骤求解，因此此方法可以无效而无需预处理。该论文引入了一种新的，条件良好的IB公式，用于边界价值问题，称为浸入边界双层（IBDL）方法。我们为Poisson，Helmholtz，Brinkman，Stokes和Navier-Stokes方程式制定它，并在其他约束方法上演示其效率。在这个新的公式中，未知边界分布的方程对应于条件良好的第二种积分方程，可以通过Krylov方法的少量迭代有效地求解。此外，迭代计数与网格大小和边界点间距无关。该方法会从边界收敛，并与局部插值结合使用，它会在整个PDE域中收敛。此外，虽然原始约束方法仅适用于Dirichlet问题，但IBDL公式也可以用于Neumann边界条件。
The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. The IB method has also been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces that only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. This dissertation introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. We formulate it for Poisson, Helmholtz, Brinkman, Stokes, and Navier-Stokes equations and demonstrate its efficiency over the other constraint method. In this new formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and the boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions.
