Integrated solution-and dimensional adaptivity of FEM by hierarchical projections

Modeling of elastic thin-walled beams, plates and shells as ID- and 2D-boundary value problems due to kinematical hypotheses, for example the normal hypothesis, is valid in undisturbed subdomains. Disturbances near supports and free edges, in the vicinity of concentrated loads and at thickness jumps cannot be described by ID- and 2D-BVP's. In these disturbed subdomains dimensional adaptivity has to be performed with respect to the 3D-theory. Dimensional adaptivity ($d$-adaptivity) coupled with a mixed $h$- or $p$-adaptivity becomes necessary in order to guarantee a reliable overall solution. The error analysis for a mixed $h$-(mesh refinement), $p$-(polynomial expansion) and $d$-(dhnensional expansion) adaptive process is treated. Error-estimators including the discretization and dimensional errors and additionally the model errors ($m$-adaptivity) are derived form an analytical point of view. Furthermore, the derivation of additional hierarchical test-spaces ($h$- and $p$-test-spaces) is given for effective and robust algorithms and good overall convergence. By these new combinations of error-estimations a quality jump of FEM is intended.