Theory and Applications of Multigrid and Domain Decomposition Methods in Computational Mechanics

小行星9600133 在有限元方法应用于计算力学问题时，会遇到诸如闭锁现象、不可压缩约束、裂纹区域和高阶方程等困难。 通过结合有限元、宏单元、奇异函数和混合公式等技术，可以开发出使用相对较少未知数的稳健有限元方法来克服这些困难。 研究人员开发多重网格和区域分解算法，这样的有限元方法。 多重网格算法计算方程组的解的成本与未知数的数量成正比，因此是求解大型系统的最有效的方法。 区域分解算法将计算分布在物理区域的不同部分上，因此非常适合于并行计算机。 它们是目前高性能计算中的两个主要思想。 从这个项目的结果可以导致弹性，塑性，板，梁，壳和流体力学问题的快速求解器。 特别是，一个新的多重网格方法的应力强度因子的计算，这是在材料的宏观现象，如疲劳和断裂的研究中的重要量。 这些问题出现在建筑物，桥梁，飞机和船舶等结构的设计中，以及结构如何执行的分析中。 因此，该项目提供了解决制造问题以及监测和更新民用基础设施的工具。

9600133 Brenner In the application of finite element methods to problems in computational mechanics, difficulties such as the phenomenon of locking, the incompressibility constraint, cracked domains and higher order equations are encountered. By combining the techniques of nonconforming elements, macro elements, singular functions and mixed formulations, robust finite element methods that employ relatively few numbers of unknowns can be developed to overcome these difficulties. The investigator develops multigrid and domain decomposition algorithms for such finite element methods. Multigrid algorithms compute the solutions of systems of equations at a cost proportional to the number of unknowns and therefore are the most efficient methods for solving large systems. Domain decomposition algorithms distribute the computation over different parts of the physical domain and hence are very suitable for parallel computers. They are currently two of the major ideas in high performance computing. The results from this project can lead to fast solvers for problems in elasticity, plasticity, plates, beams, shells, and fluid mechanics. In particular, a new multigrid method is developed for the computation of stress intensity factors, which are important quantities in the study of the macroscopic phenomena of materials such as fatigues and fractures. These problems arise in the design of structures such as buildings, bridges, planes, and ships, and in the analysis of how structures perform. Hence the project provides tools for solving problems of manufacturing and of monitoring and renewing civil infrastructure.