Coupled model- and solution-adaptivity in the finite-element method

Modeling of elastic thin-walled beams, plates and shells as 1D- and 2D-boundary value problems is valid in undisturbed subdomains. Disturbances near supports and free edges, in the vicinity of concentrated loads and at thickness jumps cannot be described in a sufficient way by 1D- and 2D-BVPs. In these disturbed subdomains dimensional (d)-adaptivity and model (m)-adaptivity have to be performed coupled with h- and/or p-adaptivity using hierarchically expanded test spaces in order to guarantee reliable and efficient overall results. The expansion strategy is applied for enhancing the spatial dimension and the model which is more efficient and evident for engineers than the reduction method. Using local residual error estimators of the primal problem in the energy norm by solving Dirichlet-problems on element patches, an efficient integrated adaptive calculation of the discretization-and the dimensional error is possible and reasonable, demonstrated by examples. We also present an error estimator of the dual problem, namely a posterior equilibrium method (PEM) for calculation of the interface tractions on local patches with Neumann boundary conditions, using orthogonality conditions. These tractions are equilibrated with respect to the global equilibrium condition of the stress resultants. An upper bound error estimator based on differences between the new tractions and the discontinuous tractions calculated from the stresses of the current finite element solution. The introduction of new element boundary tractions yields a method which can be regarded as a stepwise hybrid displacement method or as Trefftz method for local Neumann problems of element patches. An important advantage of PEM is the coupled computation of local discretization, dimensional-and model errors by an additive split.