# Adaptive Finite Element Procedures in Structural Dynamics

[Doktorsavhandling]

This thesis deals with *a posteriori* error estimation and adaptivity in finite element procedures for the analysis of structural dynamic problems.

First, error estimation and adaptivity for the Newmark time integration method are studied. Based on a postprocessing technique, *a posteriori* local error estimates for displacements, velocities and, thus, also the total energy norm error estimate are derived. An adaptive time-stepping procedure is described. It adjusts the time step size automatically so that the estimated local errors are controlled within a specified tolerance.

Secondly, a postprocessed type of *a posteriori* error estimate and an *h*-adaptive procedure for the semidiscrete finite element method for two-dimensional continuous problems are presented. In space, the superconvergent patch recovery technique (SPR) is used for determining higher order accurate solutions and, thus, a spatial error estimate. In time, the local error estimate developed above is adopted. The presented *h*-adaptive finite element procedure is able to update the spatial mesh and the time step so that the discretization errors both in space and time are under control.

Then, a time-discontinuous Galerkin finite element method (or DG method) is studied, by which both displacements and velocities are approximated as piecewise linear functions in time and may be discontinuous at the discrete time levels. An iterative solution algorithm is proposed for solving the resulted system of coupled equations. By using the jumps of the displacements and the velocities in the total energy norm as a local error indicator, an adaptive time-stepping procedure similar to the one for the Newmark method is implemented. It is shown that, while the Newmark method is only of second-order accuracy and is unable to avoid the effect of the spurious high modes, the DG method considered here is of third-order accuracy and is capable of filtering out the effect of spurious high modes.

Finally, the DG method is applied to two-dimensional continuous problems. The method approximates both displacements and velocities as piecewise linear functions in space and time simultaneously and permits them being discontinuous at the discrete time levels. The adaptive procedure is based on using the Zienkiewicz-Zhu error estimate in space and the jumps of displacements and velocities in the total energy norm as an error indicator in time. It is demonstrated that this space-time finite element method is of second-order accuracy in space (in *L _{2}*) and third-order accuracy in time, and the adaptive procedure is capable of updating the spatial mesh and time step automatically so as to control the discretization errors within specified tolerances, thus, making the solutions reliable and the computation efficient.

**Nyckelord: **structural dynamics, finite element, error estimation, a posteriori, h-adaptivity, postprocessing, superconvergent patch recovery, Newmark method, automatic time-stepping, semidiscretization, space-time, discontinuous Galerkin

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

CPL Pubid: 1278