Patrik Hager

Adaptive h-version eigenfrequency analysis

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.

Improved eigenfrequencies and eigenmodes in free vibration analysis

Abstract This paper presents an application of local and global updating methods to improve natural frequencies and mode shapes of the finite element solution in free vibration analysis. The local updating is based upon superconvergent patch recovery technique for displacements (SPRD). This approach provides reliable results for a sufficiently fine finite element mesh for the reason that it strongly depends on the original finite element solution. In order to overcome aforementioned shortcoming a preconditioned conjugate gradient method is applied for the global updating using p +1 finite element formulation. To increase the rate of convergence of the iterative procedure, a diagonal of the sum of stiffness and mass matrices is implemented as a preconditioning matrix in the general eigenvalue problem and initial trial eigenvectors are obtained by the SPRD technique. Numerical examples show for the eigenfrequencies an improved accuracy and an improved convergence rate of the error of the improved eigenfrequencies compared to the FE solution.

Error Estimation and h-adaptivity for Eigenfrequency Analysis of Plates in Bending: Numerical Results

Abstract This paper deals with error estimation and h-adaptive eigenfrequency finite-element analysis of Reissner–Mindlin plates. A termination criterion, two possible error indicators and a mesh refinement strategy for an h-adaptive eigenfrequency analysis without user tuned coefficients are presented. Superconvergent patch recovery of the displacement field is used for the error estimator in the termination criterion and to construct the error indicators. Numerical examples demonstrate the performance of the error estimator and the h-adaptive strategy. We also show the complexity of mesh refinement with respect to the appropriate mode by a numerical example.

Adaptive Eigenfrequency analysis by Superconvergent Patch Recovery

An adaptive technique of the h-version for vibration problems, utilizing a new error indicator and mesh refinement strategy, is proposed. It is based on improvement of the eigenmodes by Superconvergent Patch Recovery. A modification of the termination criterion of the adaptive strategy is proposed. The error indicator is tested by a numerical example and discussed. The adaptive technique is tested by some engineering examples.

Error estimation and adaptivity for h-version eigenfrequency analysis

Abstract The Superconvergent Patch Recovery technique is used to improve the displacement field of vibrating eigenmodes. The improved eigenmodes are used in the Rayleigh quotient to obtain improved eigenfrequencies, which replace the unknown exact solution in the error estimates. The improved eigenmodes are used to determine the error locally as error indicator and is the fundamental ingredient in the adaptive strategy.

Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

Abstract The paper presents postprocessing techniques based on locally improved finite element (FE) solutions of the basic field variables. This opens up the possibility to control both “strain energy” terms and “kinetic energy” terms in the governing equations. The proposed postprocessing technique on field variables is essentially a least square fit of the prime variables (displacements) at superconvergent points. Its performance is compared with other well-known techniques, showing a good performance. A h-adaptive FE strategy for acoustic problems is presented where, for adaptive mesh generation and remeshing the commercial software package i-deas has been applied and for the FE analysis the commercial software package sysnoise . The paper also presents an adaptive h-version FE approach to control the discretisation error in free vibration analysis. The postprocessing technique used here is a mix of local and global updating methods. Rapid convergence of the preconditioned conjugate gradient method is enhanced by choosing the initial trial eigenmodes as the superconvergent patch recovery technique for displacements improved FE eigenmodes. Numerical examples show nice properties of the final local and global updated solution as a basis for an error estimator and the error indicator in an adaptive process.