Dual‐energy extension of the uniform‐approximation approach

Abstract

In the talk an extension of the uniform-approximation approach, that relies on the truncation of the elastic energy after a certain power of a geometric scaling factor, towards an analogous truncation of the dual energy is proposed. On the one hand, this allows for a derivation of fully defined boundary value problems including compatible displacement boundary conditions. On the other hand, the extension enables us to provide an a priori error estimate for the systematic error of the arising approximative models with respect to the exact three-dimensional solution.The approach is compared to the approach of a fixed kinematic assumption for the displacement field which is widely used in engineering. We show that the later approach leads in general to a more complex model for a comparable approximation accuracy so that the consistent approximation approach is to prefer. The general idea of energetic approximationAnalytical models for thin structural members, like plate and beam theories are defined on two-or one-dimensional domains, like the mid-plane or the neutral axis. Therefore, in general they possess a systematic error compared to the three-dimensional continuum theory of elasticity -even if the model can be solved exactly. So called structured approaches aim to derive hierarchies of models with increasing approximation accuracy from the three-dimensional model of elasticity by dimension reduction techniques. One principal branch of structured approaches relies on the truncation of the elastic energy. The uniform approximation technique is one of the dimension reduction techniques belonging to this principal branch.The basic idea of energetic approximation can be motivated from a deeper lying theorem in elasticity that follows from Korn's inequality. It basically states that the quadratic error in the weak solution norm (component-wise H 1 ) of any admissible displacement field v to the exact solution u can be estimated via the difference in the elastic energy