Multigrid Methods for Elasto-Plastic Problems with the Stress Concentration

Alexander V. Trofimov
Dnepropetrovsk State University, Ukraine

Abstract

It's well known that the fracture processes in the solid bodies is determined mostly by the stress distribution near concentrators: micro-holes, crack tips etc. The plastic deformations plays an important role in such processes. Usually in fracture mechanics we solve the problem in the original ``continuous'' domain (without the holes and cracks) in order to obtain the local stress field (near concentrators) and then we deal with the elastico-plastic problem for the concentrator in unbounded domain.

We propose a technique of numerical solving the elastico-plastic boundary-value problems for several types of stress concentrators based on multigrid methods. We investigated some varieties of iteration process (such as V- and W-cycles, Richardson's extrapolation etc). The choice of the most efficient relaxation scheme is also discussed. Comparisons of the computational work of the multigrid methods and the several popular one-grid methods were performed. We also propose a method of the correspondent software development based on the object-oriented approach. We consider C++ class hierarchy that provides a flexible and convenient way of program packages design; C++ class library for the fracture mechanics problems was developed.