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

Abstract

SUMMARYNode-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 -FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by ∈[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, 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 , 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) -FEM can even be tuned for obtaining nearly exact natural frequencies.

Node-based Smoothed Point Interpolation Method (NS-PIM)

“…Some special stabilization techniques have been developed for NS-PIM models to solve dynamic problems [24,25]. We will not discuss this matter in detail in this book, and interested reader may refer to [24,25].…”

“…It has been a dream of many to find a general systematical way to obtain an upper bound of the exact solution for practical problems, so that we can then bound the solution from both sides and our numerical solution can be properly "certified". It has been discovered [4,15,25,28,30] recently that the NS-PIM and the NS-RPIM with sufficient softening effects can provide an upper bound solution in energy norm for such problems. The theory on the softening effects and upper bounds has been presented in Chapter 5.…”

“…25 Comparison of convergence rates and accuracy of the numerical results in displacement norm obtained using linear FEM and two NS-RPIM models for the rectangular cantilever problem.…”

Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems

“…Even when a unconditionally stable time-integration schemed is used to solve transient dynamic problems [17], unphysical numerical responses can appear. An α-FEM combining NS-FEM and standard FEM proposed in [28] can be used to stabilize NS-FEM by introducing the stiffening effects from the standard FEM stiffness matrix with a small α [26]. In this work, a stabilization procedure for NS-FEM is proposed based on the scheme in [29], by means of adding to the smoothed potential energy functional of NS-FEM a stabilization term which contains the square of the residual of the equilibrium equation.…”

“…Those tests were conducted regular size models where the overhead computational time is trivial. LC-PIM, SFEM, ES-FEM and NS-FEM have been successfully used to analyze linear and nonlinear solids [4][5][6][12][13][14][15][16][17][18]21,22], linear and nonlinear plates and shell structures [7,9,10,[23][24][25], free and forced vibration problems [17,26], piezoelectric structures [24], and so on.…”