# On Adaptive Finite Element Analysis in Structural Dynamics of Shell-Like Structures - A Specific View on Practical Engineering Applications and Engineering Modelling

Adaptive Finite element analysis in structural analysis has reached a fairly mature status; however, in practice only fairly little is visible in engineering applications. This contribution gives an overview over some major aspects with a specific focus on structural dynamics of shell-like structures. Adaptive analysis in structural mechanics is mainly focusing on static problems where a large number of linear and nonlinear tasks in 3D continuum problems as well as in shell problems have been tackled. The developed procedures show really considerable improvements for the executed numerical problems. In statics on one side the competition among the methods is between low and high order approximation – the h- and p- or hp-enhancement and among error estimation between global and local estimation. For a good overview over the subject for a large number of problems it is referred to [11], for some mathematical background see [2]. For some simple benchmark problems the differences between the various approaches as presented in [5]. More recent developments are concerned with time dependent problems thus we are focusing here on structural dynamics [10]. In statics - linear or nonlinear – the spatial error distribution is the dominating quantity whereas in dynamics the consideration of the spatial error distribution over the complete considered time range and as well the consideration of the error due to time integration is needed. Here the consideration of dual problems – known in statics from the Betti-Maxwell principle and extended here with the according reciprocity idea, the Graffi-theorem [1] - allows checking the error in specific quantities at certain points in time, the so-called goal-oriented error computation or local error computation [3] [4]. On the basis of such error estimations, in principle, the adaptive modification of the finite element mesh as well as the time step is possible. However, while in a semi-discretization approach the time step could be fairly easily adjusted – which is frequently done in the so-called explicit FE programs using the central difference scheme - the modification of the finite element mesh introduces major problems. First the data have to be properly mapped between meshes avoiding non-physical artifacts and second the dual error estimation scheme has to take into account different time steps and meshes. Both actions introduce further errors into the analysis which can hardly be judged. In addition the effort for the numerical analysis concerning the computation as well as the required storage becomes overly large [6] leading to the conclusion that adaptive analysis of real world problems based on dual error estimation cannot be handled - at least with the current computer environment. Thus the focus of this contribution is on a discussion first on the importance of different parts of the error estimation and on the adaptive procedure and second how the major ingredients of the adaptive duality based analysis for practical engineering problems - restricting to shell problems - can still be used, regaining efficiency [7]. For some classes of shell type problems some simplifications can be suggested while still improving the quality of the analysis considerably by adaptive procedures [8]. In structural dynamics also eigenmodes and eigenvalues are important, thus improvements concerning these are also briefly discussed [9] [8]. Obviously the dominating quantity for achieving good results applying finite element methods in structural mechanics is a consistently refined mesh; not unexpected for high frequency excitations and interest of the engineers in these almost uniformly refined meshes with high mesh densities are required. It is shown, how the developed schemes can be applied to homogeneous problems and the limits concerning real world engineering models which include a large number of violations concerning standard continuum mechanics are presented. Also the procedures implemented in LS-DYNA [12], [13] for adaptive analysis are discussed with the background set above. Further some model adaptivity for large structural computations where some parts are – at least in some early states of the analysis – hardly deforming. This effect can be used to introduce rigid bodies in the analysis; the question then arises, how this can be handled. References

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