Local and global error estimations in linear structural dynamics

The study discusses the concept of error estimation in linear elastodynamics. Two different types of error estimators are presented. First ‘classical’ methods based on post-processing techniques are discussed starting from a semidiscrete formulation. The temporal error due to the finite difference discretization is measured independently of the spatial error of the finite element discretization. The temporal error estimators are applied within one time step and the spatial error estimators at a time point. The error is measured in the global energy norm. The temporal evolution of the error cannot be reflected. Furthermore the estimators can only evaluate the mean error of the whole spatial domain. As the second scheme local error estimators are presented. These estimators are designed to evaluate the error of local variables in a certain region by applying duality techniques. Local estimators are known from linear elastostatics and have later on been extended to nonlinear problems. The corresponding dual problem represents the influence of the local variable on the initial problem and may be related to the reciprocal theorem of Betti–Maxwell. In the present study this concept is transferred to linear structural dynamics. Because the dual problem is established over the total space–time domain, the spatial and temporal error of all time steps can be accumulated within one procedure. In this study the space–time finite element method is introduced as a single field formulation.