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

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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.