Graph Products Applied to the Regular and Locally Modified Regular Structures Using Iterative Methods

Abstract

In this chapter, graph products are employed to study various kinds of regular structural patterns. The emphasis is on the eigensolution of the finite element models associated with such structures. However, the methods developed here can also be used for static and dynamic analysis as well. In Sect. 10.2, various symmetric and regular structural patterns and their corresponding canonical matrix forms are investigated. It is demonstrated that using the idea of matrix decomposition, one can simplify the eigenproblem associated with the regular model under consideration. In Sect. 10.3, we extend our investigation to structural models with a dominant regular pattern, which need to be slightly perturbed or modified in order to be considered as purely regular. There are plenty of such examples in structural mechanics applications; we can refer to the local refinement of a regularly meshed finite element model, small cut-outs extracted from a structural model and non-regular constraints imposed on a regular model as a few examples. The idea of matrix decomposition is further extended in order to develop numerical methods to deal with such cases. The concept of modification seems also to be attractive in dealing with nonconforming matrix forms, such as those associated with translational regular patterns. Using this concept in conjunction with substructuring techniques, an approximate method is presented in Sect. 10.4 for efficient solution of the corresponding eigenproblem.