# Eigenfrequency Analysis FE-Adaptivity and a Nonlinear Eigenproblem Algorithm

[Doktorsavhandling]

Engineering often pose the problem of computing the frequencies with which a system oscillates and the corresponding displacement patterns the system forms, and in case damping is present then the rates of which the amplitudes of the displacement patterns decay are also required. It is necessary to compute these quantities within a prescribed accuracy and the effort required for the computations should be minimised. To this end the present study deals with adaptive Finite Element (FE) eigenfrequency analysis of undamped vibrating systems, and with the development of an efficient solution algorithm for the nonlinear equations resulting from damped vibrations using nonproportional damping. A viscoelastic constitutive relation is used to model the nonproportional damping in this study.

An adaptive FE strategy of the h -version is proposed in order to minimise the effort for the accurate computation of the eigenfrequencies and the eigenmodes of free undamped vibrations. The discretisation error for an eigenvalue is estimated by the difference between the original FE solution and its improved solution and this quantity is scaled by an estimate of the effectivity index to produce an accurate estimation of the true error. Improved eigenmodes are obtained by Superconvergent Patch Recovery of the Displacements (SPRD) and improved eigenvalues are obtained by forming Rayleigh quotients for the improved eigenmodes. The distribution of the discretisation error is estimated for each element in the FE triangulation by means of error indicators. Several possible error indicators are tested. The triangulation is refined according to the error distribution using an advancing front technique. The process is repeated until the discretisation error meets a prescribed tolerance.

The thesis also shows that the SPRD solution can be used as a good starting vector in an iterative global updating technique. The global updating technique is based on minimising the Rayleigh quotient with the Preconditioned Conjugate Gradient (PCG) method and produces a displacement field close to a higher order FE solution.

Finally, an algorithm is proposed for the solution of nonlinear eigenproblems. The algorithm is tested on the nonlinear equations obtained by taking nonproportional damping into account. The algorithm requires a small number of matrix factorisations to locate several latent pairs and it makes the algorithm suitable for large systems where it is expensive to factorise the matrix. The algorithm is based on rational Krylov, the approach can be interpreted as a linearisation of the nonlinear eigenproblem by Regula falsi and solution of the resulting linear eigenproblem by Arnoldi where the Regula falsi iteration and the Arnoldi recursion are knit together. Moreover, a deflation technique is incorporated in the algorithm. The solution algorithm for the nonlinear eigenproblem can be applied to locate latent pairs to any matrix equation nonlinear in one parameter.

**Nyckelord: **finite element, adaptivity, nonlinear eigenvalue, superconvergent patch recovery, rational krylov

Denna post skapades 2006-08-28. Senast ändrad 2013-09-25.

CPL Pubid: 702