Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

Abstract

Node-based smoothed finite element method (NS-FEM) using triangular type of elements has been found capable to produce upper bound solutions (to the exact solutions) for force driving static solid mechanics problems due to its monotonic 'soft' behavior. This paper aims to formulate an NS-FEM for lower bounds of the natural frequencies for free vibration problems. To make the NS-FEM temporally stable, an α-FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by α∈[0,1], so that both the properties of stiff FEM and the monotonically soft NS-FEM models can be properly combined for a desired purpose. For temporally stabilizing NS-FEM, α is chosen small so that it acts like a 'regularization parameter' making the NS-FEM stable, but still with sufficient softness ensuring lower bounds for natural frequency solution. Our numerical studies demonstrate that (1) using a proper α, the spurious non-zero energy modes can be removed and the NS-FEM becomes temporally stable; (2) the stabilized NS-FEM becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum; (3) α-FEM can even be tuned for obtaining nearly exact natural frequencies. © 2010 John Wiley & Sons, Ltd.