On Effects of Perforated Domains on Parameter-Dependent Free Vibration

Free vibration characteristics of thin perforated shells of revolution vary depending not only on the dimensionless thickness of the shell but also on the perforation structure. All holes are assumed to be free, that is, without any kinematical constraints. For a given conguration there exists a critical value of the dimensionless thickness below which homogenisation fails, since the modes do not have corresponding counterparts in the non-perforated reference shell. For a regular g g-perforation pattern, the critical thickness is reached when the lowest mode has an angular wave number of g=2. This observation is supported both by geometric arguments and numerical experiments. The numerical experiments have been carried out have been computed in 2D with high-order nite element method supporting Pitkaranta's mathematical shell model

On moduli of rings and quadrilaterals: algorithms and experiments

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds