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

已有成熟的技术可用于解决许多类型的单一物理问题，但对于耦合的多物理问题，迫切需要健壮性和可扩展性的技术，特别是对于大型并行计算机。该方案的重点是一些新的基于区域分解的非线性预处理技术，用于多物理应用中出现的一些高度非线性的、耦合的偏微分方程组的数值解。这些偏微分方程组通常代表多个相互作用的场(例如，流体和固体)，每个场都由特定类型的方程来建模。目前的方法通常涉及对场的仔细拆分和使用逐场迭代来获得耦合问题的解。这种方法有许多优点，如易于实施，因为只需要单场解算器，但也有缺点。例如，场之间的某些非线性相互作用可能无法完全捕捉，对于非定常问题，很难设计稳定的时间积分格式。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，研究了全耦合方法，以获得完整的物理模拟。这种完全耦合方法的成功几乎完全依赖于一个健壮和可伸缩的非线性代数系统求解器。不幸的是，传统的非线性迭代方法并不能很好地发挥作用，例如，类牛顿迭代方法由于解中局部非光滑分量的存在以及缺乏良好的初值估计，往往收敛速度很慢。新的算法是由PI和他的同事最近引入的非线性预条件方法来求解具有不平衡非线性的代数非线性方程的。通过将多重网格方法结合到算法中来获得可伸缩性。将研究几个重要的应用，包括使用耦合的Navier-Stokes方程和弹性方程模拟顺应性动脉中的血液流动。

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.