Efficient multigrid version of the finite-element method for the solution of problems of the theory of elasticity

Abstract

We propose an economic version of the multigrid finite-element method based on the concept of substructures for computing large complex finite-element systems. The product of matrices derived by the incomplete decomposition of stiffness matrices of substructures is used as a relaxation operator of the multigrid process. The main advantages of the described method are its explicit decomposition, high rate of convergence, and considerable economy in memory and arithmetic operations. The efficiency of the proposed version of the finite-element method as compared with the well-known methods is illustrated by the solution of several problems, namely, the axially symmetric and three-dimensional Boussinesq problems and the three-dimensional problem of fracture mechanics about stretching a plate with a lateral crack. It follows from the numerical examples that the indicated approach provides a considerably higher rate of convergence and requires substantially less memory for the solution. The proposed method enables the solution of large problems with personal computers for which the calculation rate and memory capacity play the decisive role in the restriction of the order of problems.