FAST ALGORITHMS FOR MULTIGRID SOLVERS OF 3-DIMENSIONAL BOUNDARY-VALUE-PROBLEMS IN STRUCTURAL-ANALYSIS

A large number of analyses in structural mechanics may be treated using the three-dimensional boundary value formulation of the problem. The use of the standard finite element techniques, typically comprising local stiffness matrix assembling and a direct eliminating procedure for solving resulting systems of equations, is usually computationally very demanding. It is natural to employ here multi-grid methods giving a good convergence in solving boundary value elliptic problems. But in spite of the fact that these methods avoid a costly eliminating procedure, the difficulties associated with programming costs, grid generation, storage requirements and introducing a coarse problem concept, remain acute. There are some potential methods to overcome these difficulties by allowing some simplifications. We propose one of them associated with constructing a simplified data structure based on logically rectangular grids which leads to developing efficient vectorial algorithms for executing basic multi-grid operations. These algorithms appeared to be easy to program and good for pre- and post-processing, suitable for parallel processing and very efficient when implemented as a software package for heat condition and stress-strain state analyses of various engineering structures, such as reactor vessels, dam structures, building fragments and others. We propose the substructuring algorithm in the framework of a multi-grid method which is useful for treating problems with cyclic and complicated domains. It is based on dividing the domain into simple parts each admitting the simplified data structure. We also represent the multi-grid method previously developed to demonstrate the possibility of using the algorithmic approaches. In conclusion we consider the example of an actual problem illustrating the 3-D finite element analysis of a reactor vessel.