Adaptive h-version eigenfrequency analysis

Abstract

In dynamics it is important to compute natural frequencies and modes with a demanded accuracy. This paper presents an adaptive h-version finite element scheme to control the discretization error in free vibration analysis. Crucial parts of an adaptive analysis are the error estimator and the error indicator of the discretization error. The error indicator is based on the energy norm of the error. The estimate that constitutes both the error estimator and the error indicator are founded on post-processed eigenmodes in the absence of exact eigenmodes. The post-processing technique used is a mix of local and global updating methods. The local updating is based on the superconvergent patch recovery technique for displacements (SPRD). This approach provides reliable results for a sufficiently fine finite element mesh due to its strong dependence on the original finite element solution. In to overcome the necessity for meshes fine enough, a preconditioned conjugate gradient method is applied for the global updating using p+1 finite element formulation. The preconditioning matrix is implemented as a diagonal of the sum of stiffness and mass matrices in order to increase computational efficiency and the rate of convergence of the iterative procedure. Rapid convergence is enhanced by choosing the initial trial eigenmodes as the SPRD improved finite element eigenmodes. Finally we improve the global updated solution by applying the SPRD technique one more time. Numerical examples show the nice properties of the final local and global updated solution as a basis for an error estimator and an error indicator in an adaptive process.