非线性区域分解预条件子研究及其在流固耦合控制中的应用

In the field of high performance computing, the traditional research on domain decomposition methods mainly focuses on the linear solver level, and use the domain decomposition preconditioners to reduce the condition number of linear systems and then accelerate to obtain the solution of linear equations. However, in the application area of many problems, due to the high local nonlinearity of algebraic nonlinear system or the discontinuity of the solution, the nonlinear solver is difficult to converge or even not converge, when Newton-type methods are used to solve these large nonlinear systems of equations arising from the discretization of partial differential equations. As a result, the traditional parallel algorithms for scientific computing petascale computing is difficult to achieve high performance. Hence, the convergence issue in the nonlinear level is becoming a bottleneck in high performance computing. In the proposed project, we plan to investigate a class of important applications in the fluid-structure interaction areas, and pay attention to high scalable nonlinear solver and its application software by combining domain decomposition methods with the nonlinear preconditioning technique, such that it can not only significantly reduce the number of nonlinear iterations, but also efficiently implement on supercomputing platform. By the research, we will develop several critical algorithms and software for the target application problems and study their performance on machines including the current Tianhe and Xingyun with several billion unknowns and 10,000-100,000 processor-cores, and thereby provide substantial support for large-scale applications with wide prospects.

在高性能计算领域，传统的区域分解预条件子的研究主要关注于线性解法器层面，通过减少其线性系统的条件数来加速线性方程组的求解。然而，应用领域中许多问题，在求解由PDE离散形成的非线性方程组过程中，由于其代数方程组的局部高非线性或其解的不连续性，导致非线性解法器层面的难收敛或不收敛，从而传统的并行算法在千万亿次科学计算中难以实现较高的计算性能。鉴此，基于非线性预处理技术的算法研究成为高性能计算领域急需突破的瓶颈。本项目针对流固耦合领域中的典型问题，重点关注非线性迭代层面，结合区域分解和非线性预处理技术，研究高可扩展的非线性解法器和应用软件实现技术，使其既能显著降低非线性解法器的迭代次数，又能在超级计算平台上高效实现。通过本项目的研究，力争在以天河、星云等为代表的国产千万亿次平台上实现数十亿未知数规模以上的数值模拟，达到数万、甚至数十万核的高可扩展性，为具有广泛前景的大规模应用提供高效支撑。

在面向大型超级计算机的科学计算应用中，许多并行算法和并行软件计算效率低、难于扩展，无法有效应对动辄拥有数百万以上处理器核的众核环境、并高效利用超级计算资源。本项目针对计算流体力学中的两类典型应用——流体控制问题和多相流模拟问题，结合全隐式的并行离散格式、区域分解技术和全耦合的Newton-Krylov类方法，深入开展了开展高可靠性、低复杂度、大规模可扩展科学计算基础算法和实现技术的研究，为面向国产顶级超级计算机的可扩展算法和应用软件实现技术提供思路，努力推动国产超级计算机的应用。基于以上研究，项目执行期间以第一作者或通讯作者共在相关领域的SCI源刊上发表高水平科研论文11篇，其中包括SIAM Journal on Scientific Computing、Journal of Computational Physics、Computer Methods in Applied Mechanics and Engineering、Computer Physics Communications、Journal of Scientific Computing等顶级国际权威SCI期刊。