Numerical approach of free and forced elastic vibrations using high-regularity Hermitian partition of unities

Abstract

In this paper, shape functions with regularity up to $$C^{2}$$ were developed for the four-node quadrilateral finite element considering the partition of unity property. This high-regularity approximation space was applied to approach the natural frequencies of free vibration of some in-plane elastic problems as well as the elastic wave response of forced vibration caused by the application of impulsive loading. The obtained results show that it was possible to numerically approximate a greater number of accurate natural frequencies with the devised procedure when a comparison is established with the $$C^{0}$$ Lagrangian and serendipity elements of 4, 8 (serendipity), 16, and 25 nodes. Also, the numerical predictions for the elastic wave propagation problem using the derived approximation space presented small oscillations improving the representation of the displacement field.