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

9600133 Brenner在有限元方法应用于计算力学问题时，会遇到诸如锁定现象、不可压缩约束、裂纹域和高阶方程等困难。通过结合非协调元、宏元、奇异函数和混合格式的技术，可以开发出使用相对较少的未知数的稳健有限元方法来克服这些困难。研究人员开发了这种有限元方法的多重网格和区域分解算法。多重网格算法以与未知数成正比的代价计算方程组的解，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域的不同部分，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想法。这个项目的结果可以导致快速解决弹性、塑性、板、梁、壳和流体力学中的问题。特别是，发展了一种新的多重网格法来计算应力强度因子，它是研究材料疲劳和断裂等宏观现象的重要量值。这些问题出现在建筑、桥梁、飞机和船舶等结构的设计中，以及结构如何发挥作用的分析中。因此，该项目为解决制造问题以及监测和更新民用基础设施提供了工具。

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.