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

对于许多类型的单物理场问题，已有成熟的技术可以解决，但对于耦合的多物理场问题，特别是大规模并行计算机，则迫切需要鲁棒的、可扩展的技术。该建议的重点是一些新的区域分解为基础的非线性预处理技术的数值解的一些高度非线性，耦合系统的偏微分方程（PDE）所产生的多物理应用。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。目前的方法通常涉及一个小心的分裂领域和使用的领域的迭代，以获得一个解决方案的耦合问题。这种方法具有许多优点，例如易于实现，因为只需要单个场求解器，但也存在缺点。例如，场之间的某些非线性相互作用可能无法完全捕获，并且对于非定常问题，难以设计稳定的时间积分方案。此外，当在大规模并行计算机上实现时，逐场迭代的顺序性质大大降低了并行效率。为了克服这些缺点，全耦合的方法进行了研究，以获得完整的物理模拟。这种完全耦合的方法的成功几乎完全取决于一个非线性代数系统求解器，是强大的和可扩展的。然而，传统的非线性迭代方法并不能很好地解决问题，例如Newton类方法由于解中存在局部非光滑成分和缺乏良好的初始估计而收敛速度非常慢。新算法的动机是最近推出的PI和他的同事为解决代数非线性方程组，具有不平衡的非线性非线性的非线性预处理方法。可扩展性是通过将多重网格方法的算法。几个重要的应用将进行研究，包括使用耦合的Navier-Stokes方程和弹性方程的顺应性动脉中的血液流动的模拟。

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.