Accuracy Estimates in Free Vibration Analysis/Paklaidų Nustatymas Laisvųjų Svyravimų Uždaviniuose

Abstract

The construction of a posteriori error estimates to control numerical simulation procedure is very attractive subject for the researchers in the field of the finite element method. By now a considerable success has been achieved mainly on problems of linear elliptic type, such as linear elastostatics and stationary heat conduction problems, see e.g. Babuska et a!. (l] and Oden [2].Recently, an objective methodology for assessing the reliability of a posteriori error estimators has been developed by Babuska eta!. [3]. For free vibration problems, however, theory and computer implementation for error estimation are far from completed and need to be further exploited.When standard Galerkin finite element approximation is used, a priori error estimation is available for the generalised eigenvalue problem [4,5]. However, from the computational viewpoint applications of a priori error estimates, based upon knowledge of the general properties of solutions for the model equations and the approximation properties of the discretization methods, are in practice very limited as they provide only a qualitative assessment of the error and the asymptotic rate of convergence when the number of degrees of freedom in the approximation tends to infinity. A priori estimates provide indications of the error based upon upper bounds for Sobolev norms of the solution. However, they usually do not provide much information about the actual error in the discrete approximation. Instead, more precise information to evaluate the actual discretization error of the eigcnfrequencies can be gained only by a posteriori error estimates which utilise the finite clement solution itself.New methods to improve accuracy of the eigenfrequencies of the discretized engineering stmcture and to give error bounds have recently appeared. Friberg ct a!. [(>] propose an error estimate and an adaptive procedure for eigenpairs computation within a framework of the hierarchical finite clement method. This approach represents an iterative procedure for the selection of hierarchical refinements based on the activation of positive maximum indicators . 5 .