Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends

The main purpose of this paper is two-fold: first, to present a critical review of available errorcontrolled adaptive finite element methods—with absolute global and goal-oriented error estimates— for approximate solutions of a given mathematical model and the model error of the mathematical model considered and its dimensions, enhanced with our own recent results in fracture mechanics. The second related important matter is which physical/mathematical models—including various boundary layers and other disturbances—and which accuracy of the finite element solutions in which norm are prescribed and necessary for the safety and reliability of a certain engineering problem, e.g. in structural engineering, of course depending on materials, systems and the type as well as the magnitude of prescribed loadings (i.e. actions in a factorized concept) acting on a specific structure. It is hard to understand why the current codes for structural engineering design in Europe—especially the European Standards (Eurocodes) and the relevant German DIN-Codes (DIN: Deutsches Institut für Normung e. V.)—do not lay out any rules or bounds for the prescribed accuracy of the computation of material and structural resistance, i.e. the validity of mechanical modelling and the accuracy of associated numerical methods, usually today the finite element method. It can be observed that behind the existing codes there still lies the old thinking of, e.g., classical elastic beam theory of so-called second order for stability problems and the corresponding elastic potential for plates and shells in buckling. Modern flexible finite element modelling by deriving theories in variational form for thin-walled beams, plates and shells from rather general 3D theories, for both linear and non-linear problems, capturing inelastic deformations and layer effects by material and dimensional expansions are not addressed in the Eurocodes for structural engineering. The necessary consequence is that joint work of computational mechanics specialists and structural engineers, working in national and international code committees, is absolutely necessary, in order to avoid a lot of costly troubles and dangers of today by improving and updating the existing codes