Building Spectral Element Dynamic Matrices Using Finite Element Models of Waveguide Slices and Elastodynamic Equations

Abstract

Structural spectral elements are formulated using the analytical solution of the applicable elastodynamic equations and, therefore, mesh refinement is not needed to analyze high frequency behavior provided the elastodynamic equations used remain valid. However, for modeling complex structures, standard spectral elements require long and cumbersome analytical formulation. In this work, a method to build spectral finite elements from a finite element model of a slice of a structural waveguide (a structure with one dimension much larger than the other two) is proposed. First, the transfer matrix of the structural waveguide is obtained from the finite element model of a thin slice. Then, the wavenumbers and wave propagation modes are obtained from the transfer matrix and used to build the spectral element matrix. These spectral elements can be used to model homogeneous waveguides with constant cross section over long spans without the need of refining the finite element mesh along the waveguide. As an illustrating example, spectral elements are derived for straight uniform rods and beams and used to calculate the forced response in the longitudinal and transverse directions. Results obtained with the spectral element formulation are shown to agree well with results obtained with a finite element model of the whole beam. The proposed approach can be used to generate spectral elements of waveguides of arbitrary cross section and, potentially, of arbitrary order.

Wave finite element‐based superelements for forced response analysis of coupled systems via dynamic substructuring

“…The motivation behind the use of the matrix (20) is to provide a matrix, on the right-hand side of Equation (21), which is likely to be well-conditioned [8]. This may be explained for damped structures, because the 2-norms of the products of the off-diagonal block termse.g.,ˆ 1 is much smaller than one, because the number of substructures N is usually large (i.e., k k 2N 2 is small).…”

“…One strategy that solves this issue consists in expressing the condensed dynamic stiffness matrices of straight periodic structures by means of WFE wave modes . This appears to be an efficient alternative to the spectral element methods that make use of analytical waves for expressing the condensed dynamic stiffness matrices of waveguides. This is understood because the WFE method works well in the MF range, as opposed to the analytical methods that are constraint by low‐frequency assumptions.…”

Vibroacoustic modeling of periodic cylindrical shells with internal fluid via wave finite element method

“…More recently, Silva et al (2011) studied the longitudinal wave propagation in a periodic rod constituted by a sequence of unit cells, each cell consisting of two layers of alternating different materials by the same reasoning of Hussein et al (2006). They confirmed the existence of longitudinal wave band gaps both numerically and experimentally.…”

Dynamic analysis of periodic structures via wave-based numerical approaches and substructuring techniques = Análise dinâmica de estruturas periódicas utilizando uma abordagem de propagação de ondas e técnicas de sub-estruturação

“…For the sake of clarity, in this thesis, WFE is always used as the acronym for wave finite element. Some important milestones on the development of the WFE method are presented in Figure 1 The WFE method has been applied to address the dynamic analysis of various engineering systems, such as beams (Mace et al, 2005;Waki et al, 2009b;Nascimento, 2009;Silva et al, 2013b), truss beams (Signorelli and von Flotow, 1988), simply-supported plates (Mace et al, 2005;Silva and Arruda, 2012), multi-layered systems (Mencik and Ichchou, 2008), fluidfilled pipes (Manconi et al, 2009), curved structures (Zhou and Ichchou, 2010;Silva et al, 2013a), composite panels (Chronopoulos et al, 2013, flat shells (Mencik, 2013), cylinders (Renno and Mace, 2014), stiffened and non-stiffened cylindrical shells (Renno and Mace, 2014;Silva et al, 2014b). Also, using the WFE method, the problem of multiple periodic waveguides coupled through a common elastic coupling element has been addressed (Mencik and Ichchou, 2005).…”

“…On the other hand, expressions for the condensed dynamic stiffness matrices of straight periodic structures by means of the WFE method were also proposed (Duhamel et al, 2006;Mead, 2009). This approach appears to be an efficient alternative to the spectral element method (Doyle, 1997;Lee, 2009;Silva et al, 2013b) that make use of analytical waves for expressing the condensed dynamic stiffness matrices of waveguides. This is so because the WFE method works well in the MF range, as opposed to the analytical methods, which are limited by LF assumptions.…”