Free vibration analysis of elastic rods by the enhanced Ck-generalized finite element method

Resumo

Mesh-based smooth approximations endowed with the first degree of consistency can be built through the Ck-Generalized Finite Element Method (Ck-GFEM). This is accomplished by incorporating the Moving Least Squares (MLS) method in its procedure when computing the Partition of Unity (PoU) functions. Despite the widened PoU’s supports demanded, the enhanced polynomial reproducibility yields solutions with a cost, in terms of degrees of freedom, similar to those approximations provided by the original version of GFEM, with C0 continuity, for a certain polynomial degree of the ansatz. Additionally, in the case of static loading, such modified PoU functions allow some advantages when using special enrichment functions for localized features, leading to an increase in the conditioning of the stiffness matrices comparable to the original GFEM (Torres, 2024). The present work discusses some results for the free vibration analysis of linear elastic straight rods, considering different levels of k-continuity and polynomial degrees, with both attributes uniformly applied along the domain. It is emphasized that the k-order of continuity can be adjusted separately from the polynomial degree. The findings indicate that a certain level of smoothness allows better use of the degrees of freedom in the sense that a greater number of discrete natural frequencies have physical meaning. This is in contrast to conventional finite element approximations with C0 regularity, for which higher-order discrete modes deviate significantly from continuous ones.