Aggregation Multilevel Iterative Solver for Analysis of Large-Scale Finite Element Problems of Structural Mechanics: Linear Statics and Natural Vibrations

Abstract

The preconditioned conjugate gradient method with aggregation multilevel preconditioning is employed for solution of large-scale finite element linear static and natural vibration problems. The aggregation approach, proposed by Prof. V.E.Bulgakov, is applied to create the aggregation multilevel preconditioning. Formulation of the restriction-prolongation operators and preparation of the coarse level matrix is based on clear mechanical interpretation. Application of the element-by-element (EBE) procedure makes evaluation of the coarse level matrix almost as fast as the assembly of the global stiffness matrix. Use of the shift improves properties of the aggregation multilevel preconditioning for solution of natural vibration problems. The PCG algorithm with acceleration by shifts is developed further on. Next, the implicit approach to solution of the additional equation set with the preconditioning ”shifted” respectively is applied. There is a correlational relationship between the well-known inverse iteration procedure with the shift and the PCG method accelerated by shifts. Efficiency of the approach proposed is illustrated by large-scale examples. These methods are incorporated in the Robot Millennium software (RoboBAT Software Company, Krakow, Poland, http://robot-structures.com/us/).