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Coupled model- and solution-adaptivity in the finite element method. (English) Zbl 0926.74127

Summary: Modeling of elastic thin-walled beams, plates and shells as one- and two-dimensional 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 one- and two-dimensional boundary value problems. 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. Here we apply the expansion strategy 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 of the dimensional error is possible and reasonable, which is demonstrated by examples.
We also present an error estimator for the dual problem, namely a posteriori 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 is 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.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K99 Thin bodies, structures
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