Structural vibration analysis with random fields using the hierarchical finite element method

Abstract

Element based techniques, like the Finite Element method, are the standard approach in industry for low frequency applications in structural dynamics. However, mesh requirements can significantly increase the computational cost for increasing frequencies. In addition, randomness in system properties starts to play a significant role and its inclusion in the model further increases the computational cost. In this paper, a hierarchical finite element formulation is presented which incorporates spatially random properties. Polynomial and trigonometric hierarchical functions are used in the element formulation. Material and geometrical spatially correlated randomness are represented by the Karhunen-Loève expansion, a series representation for random fields. It allows the element integration to be performed only once for each term of the series which has benefits for a sampling scheme and can be used for non-Gaussian distributions. Free vibration and forced response statistics are calculated using the proposed approach. Compared to the standard h-version, the hierarchical finite element approach produces smaller mass and stiffness matrices, without changing the number of nodes of the element, and tends to be computationally more efficient. These are key factors not only when considering solutions for higher frequencies but also in the calculation of response statistics using a sampling method such as Monte Carlo simulation.

Stochastic aeroelastic analysis of laminated composite plate with variable fiber spacing

“…The material properties are assumed to follow normal distribution. 29,30,44 For large stochastic dimensions, sampling techniques such as the MCS are the most computationally efficient approaches available due to the curse of dimensionality. However, when a small number of random variables are considered, the use of Monte Carlo simulation (MCS) is computationally expensive due to the requirement of large sample size.…”

“…Fabro et al. 44 presented highly efficient stochastic hierarchical finite element method to investigate the free and forced vibration responses considering non-Gaussian distribution for the modeling of uncertain system properties.…”

Wave and vibration analysis of elastic metamaterial and phononic crystal beams with slowly varying properties

“…When considering spatially correlated random variability (see Appendix A), the spatial correlation length should be taken into account. It can be shown that the spatial frequency of the random field increases with the increasing order the KL expansion and that each term of the series have a equivalent spatial frequency representation [78,79]. The proposed approximation is equivalent to a mid-point type of random field discretization.…”

Stress assessment in reinforcement for columns with concrete creep and shrinkage through Brazilian technical normative

“…Equation 1shows the chosen trigonometric function for this study, which can be used for structures with varying geometries, as by Wahrhaftig [21] in comparison with computational analyses using the finite element method (FEM). The FEM, as an analysis technique in a similar context, was used by Alkloub et al [22] in order to analyze the buckling of reinforced concrete columns with a truncated cone shape and square cross section; by Rodrigues et al [23] to evaluate the buckling of reinforced concrete columns, eccentrically loaded, considering nonlinearities inherent to the system, including creep; and by Fabro et al [24], who employed polynomial and trigonometric interpolation functions to optimize the application in real structures with low frequency.…”