Nonlinear Preconditioning Techniques for Coupled Multi-physics Problems on Massively Parallel Computers

成熟的技术可以解决许多类型的单物理问题，但对于耦合的多物理问题，迫切需要鲁棒和可扩展的技术，特别是对于大型并行计算机。本文重点研究了基于区域分解的非线性预处理技术，用于求解多物理场应用中出现的高度非线性耦合偏微分方程组的数值解。这些偏微分方程通常表示多个相互作用的场（例如，流体和固体），每个场都由某种类型的方程建模。当前的方法通常包括仔细地分割字段，并使用逐字段迭代来获得耦合问题的解。这种方法有很多优点，比如易于实现，因为只需要单个字段求解器，但也有缺点。例如，某些场之间的非线性相互作用可能无法完全捕获，对于非定常问题，难以设计稳定的时间积分方案。此外，当在大型并行计算机上实现时，逐字段迭代的顺序本质大大降低了并行效率。为了克服这些缺点，研究了全耦合方法以获得全物理模拟。这种完全耦合方法的成功几乎完全依赖于一个鲁棒和可伸缩的非线性代数系统求解器。然而，传统的非线性迭代方法并不能很好地发挥作用，如类牛顿方法由于解中存在局部非光滑分量和缺乏良好的初始猜测，往往收敛速度很慢。新算法的动机是由PI和他的同事最近引入的非线性预处理方法，用于解决具有不平衡非线性的代数非线性方程。通过将多网格方法引入到算法中，获得了可扩展性。将研究几个重要的应用，包括使用耦合Navier-Stokes和弹性方程模拟柔顺动脉中的血液流动。

Mature technologies are available for solving many types of single physics problems, but for coupled multi-physics problems, robust and scalable techniques are badly needed, especially for large scale parallel computers. The focus of the proposal is on some new domain decomposition based nonlinear preconditioning techniques for the numerical solution of some highly nonlinear, coupled systems of partial differential equations (PDEs) arising from multi-physics applications. These PDEs often represent multiple interacting fields (for example, fluid and solid), each is modeled by a certain type of equations. Current approaches usually involve a careful splitting of the fields and the use of field-by-field iterations to obtain a solution of the coupled problem. Such approaches have many advantages such as ease of implementation since only single field solvers are needed, but also exhibit disadvantages. For example, certain nonlinear interactions between the fields may not be fully captured, and for unsteady problems, stable time integration schemes are difficult to design. In addition, when implemented on large scale parallel computers, the sequential nature of the field-by-field iterations substantially reduces the parallel efficiency. To overcome the disadvantages, fully coupled approaches are investigated in order to obtain full physics simulations. The success of such a fully coupled approach depends almost entirely on a nonlinear algebraic system solver that is robust and scalable. Unfortunately, traditional nonlinear iterative methods do not work well, for example, Newton-like methods often converge very slowly because of the existence of local non-smooth components in the solution and the lack of good initial guess. The new algorithms are motivated by the nonlinear preconditioning methods recently introduced by the PI and his co-workers for solving algebraic nonlinear equations that have unbalanced nonlinearities. The scalability is obtained by incorporating the multigrid methods into the algorithms. Several important applications will be studied including the simulation of blood flows in compliant arteries using a coupled Navier-Stokes and elasticity equations.