In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two-level additive Schwarz method with a mixed-dimensional coarse preconditioner. The key components of the preconditioner include (1) a three-dimensional coarse preconditioner covering the aneurysm; (2) a one-dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three-dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier–Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second-order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton-Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient-specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

中文翻译：

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用于加速牛顿-克雷洛夫法的动脉瘤特异性预处理技术及其在血流模拟中的应用


在本文中，我们开发了一种模拟动脉瘤动脉中的血流的算法，并重点关注构建稳健且高效的多级预处理器，以加速线性和非线性求解器的收敛。这项工作的动机是观察到在局部动脉瘤区域，血流通常相当复杂，有一个或多个涡流，但在动脉的健康部分，血液的主要成分沿着动脉的中心线流动。基于这一观察，我们引入了一种带有混合维粗预处理器的新型两级加性 Schwarz 方法。预处理器的关键组成部分包括（1）覆盖动脉瘤的三维粗预处理器； (2)覆盖动脉健康断面中心线的一维粗预处理器； （3）覆盖整个动脉细网格的三维重叠子域预处理器的集合； (4)由径向基函数构造的扩展/限制算子。血流通过非稳态不可压缩纳维-斯托克斯方程进行建模，阻力流出边界条件通过完全非结构化网格上的稳定有限元方法和时间上的二阶向后微分公式进行离散化。由此产生的大型非线性代数系统通过新的预处理器加速的牛顿-克雷洛夫算法以两种方式求解：（1）通过求解粗略预处理器定义的线性系统来获得牛顿的初始猜测； (2) 雅可比系统的 Krylov 求解器由新的预处理器进行预处理。 数值实验表明，所提出的预处理器对于患有动脉瘤的患者特定动脉中的复杂流动是非常有效和稳健的，并且它显着减少了线性和非线性迭代的数量。