The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

Abstract

SUMMARYThis paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four-noded element for the ReissnerMindlin plate is derived in this paper, which is free of shear locking. Such a locking-free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency-dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost.

Characterization Of Cracks And Delaminations Using Pwas Ad Lamb Wave Based Time-Frequency Methods

“…However, it is possible to construct an enriched spectral element for a given h-p refinement using incomplete set of dispersion curves ( ) , k ω [28]. In the next section we discuss the SFEM for modeling Lamb wave interaction with delamination.…”

“…where ( ) For those kinematic descriptions of material points in the plane of wave propagation (e.g., higher order beam, plate and layered system) where the complete wave vector is to be considered instead of a scalar wavenumber as in case of simple beam, it is not always possible to construct an exact spectral element. However, it is possible to construct an enriched spectral element for a given h-p refinement using incomplete set of dispersion curves ( ) , k ω [28]. In the next section we discuss the SFEM for modeling Lamb wave interaction with delamination.…”

Doubly Spectral Finite Element Method for Stochastic Field Problems in Structural Dynamics

“…where ∆K e (ω) is the random part of the matrix. Following equation (24), this matrix can be conveniently expressed as ∆K e (ω) = Γ(ω) ∆K e (ω)Γ T (ω)…”

“…In the context of structural dynamics, spectral methods have been used in random vibration problems [15][16][17][18] and for the discretisation of displacement fields in the frequency domain. [19][20][21][22][23][24] Several application of such method has been discussed in details in ref. 25 In spite of the fact that both approaches use spectral decomposition (one for the random fields and while the other for the dynamic displacement fields), there has been very little overlap between them in literature.…”

Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework

“…The enrichment can also be intrinsic, based on the recent work by Fries and Belytschko [8]. In this paper, we focus on the extrinsic partition of unity enrichment and in general, the field variables are approximated by [1,4,9,10,6,11,7,12]: u h (x) = I∈N fem N I (x)q I + enrichment functions (1) where N I (x) are standard finite element shape functions, q I are nodal variables associated with node I. XFEM, one of the aforementioned partition of unity methods, was successfully applied for crack propagation and other fields in computational physics [13,14,15,16,17,18,19,20,21,22] and recently open source XFEM codes were released to help the development of the method [23] and numerical implementation and efficiency aspects were studied [24]. XFEM is quite a robust and popular method which is now used for industrial problems [25] and under implementation by leading computational software companies.…”