Wave propagation in slowly varying waveguides using a finite element approach
Introduction
Wave-based methods commonly assume that waveguide properties are homogeneous in the direction of wave propagation, limiting the application of such approaches. This assumption arises mainly because analytical solutions for non-homogeneous waveguides are only possible for very particular cases, for example acoustic horns, ducts, rods and beams, e.g. Refs. [[1], [2], [3], [4]]. Moreover, randomly varying material and geometric properties along the axis of propagation play a significant role in the so-called mid-frequency region.
Wave solutions for non-homogeneous waveguides can be found by applying the classical WKB approximation [[5], [6], [7], [8]]. Named after Wentzel, Kramers and Brillouin, it was initially developed for solving the Schrödinger equation in quantum mechanics. The formulation assumes that the waveguide properties vary slowly enough spatially such that there are no or negligible reflections due to these local changes, even if the net change is large. It maintains the wave-like interpretation of non-uniform waveguides, but it is restricted to available analytical solutions. Pierce [9] presented a physical interpretation for inhomogeneous beams and plates, deriving the leading terms of the approximation in terms of conservation of power. Recently, Nielsen and Sorokin [10] generalised the WKB method with applications to Rayleigh and Timoshenko beam theories in curved waveguides, using a FE model as a benchmark solution. Additionally, they have also shown that the energy flux conservation property leads to the exact amplitude function to the leading order of the approximation. Moreover, Foucaud et al. [11] have experimentally shown the validity of the WKB model for a plate of varying thickness immersed in a fluid connected to an acoustic black hole, modelling a passive cochlea. They have also shown experimentally the variation of the wavenumber as a function of position, a working assumption of the WKB method.
The fact that the wavenumber is a function of the position in a slowly varying waveguide can lead to the situation in which, for a given frequency, there could exist a part of the waveguide in which a particular wave mode is propagating in one region while it becomes non-propagating in another region. The transition between these two regions, a cut-on or cut-off transition, is known as a critical section or turning point [12] and leads to wave reflection due to the sudden change of the characteristics of wave propagation, even though the mechanical properties are slowly varying. In this region, the WKB approximation is no longer valid and a uniformly valid solution is required (e.g. Refs. [8,13,14]).
In engineering applications, manufacturing variability introduces variability in homogeneous or periodic structures affecting the dynamic performance of the designed structure [15]. Typically, this variability is such that it can be spatially varying with some degree of spatial correlation. Random field theory provides the means for a probabilistic representation of this variability and typically involves expressions for the probability density function together with a model for the spatial variability of the properties, given by a correlation function and correlation length, for a second order homogeneous random field [16]. The most commonly used methods of representing a random field in a mechanical model include the use of series expansions, such as the Karhunen-Loève (KL) decomposition or the Polynomial Chaos expansion [17]. Fabro et al. [18] have applied the WKB method to investigate wave propagation in rods and beams with spatially correlated randomness, using random field theory. An experimental validation was also presented showing the validity of the approximation for low order beam theory using a set of randomly distributed added masses [18] and for chopped fibre composite beams [19].
Waveguides with complicated cross-sections, such as laminates or sandwich-structured composites, typically have no analytical solution available, so the WKB formalism cannot be directly applied. For such cases involving homogeneous or periodic waveguides, the wave and finite element (WFE) approach can be applied. The WFE is a wave-based method that is used to predict the wavenumbers and wave modes of a waveguide from a finite element (FE) model of a small segment of the waveguide, by postprocessing the mass and stiffness matrices typically obtained from a commercial or in-house FE package. This method has been applied to a number of cases in structural dynamics including free and forced vibration [[20], [21], [22], [23], [24], [25], [26], [27], [28]]. Because of its computational efficiency, the WFE approach offers a suitable framework for the analysis of randomness in periodic structures, but its applications have been limited to spatially homogeneous randomness [[29], [30], [31]], even though it can be used to model non-uniform cross-sections.
In this work, a WFE approach is proposed to model waveguides with slowly varying properties using conservation of the time-averaged energy flow, which is in principle equivalent to the WKB approximation to its leading order, as described by Pierce [9]. Therefore, the wave properties are calculated using the WFE approach and they are expressed as a function of the position along the waveguide. The phase change of a propagating wave over a distance is calculated using a Gauss-Legendre quadrature scheme for numerical integration of the local wavenumber. The WFE method is used to evaluate the wavenumbers at each integration point, and these are kept to a minimum to reduce computation cost while still being able to capture the non-homogeneity to a given specified level of accuracy. The wave amplitude change is calculated using conservation of power.
Various numerical examples comprising a rod, beam and plate strip with non-uniform material and geometrical properties are considered. Results obtained from deterministic and stochastic examples are compared to standard FE and analytical solutions, the latter when available. Spatially correlated randomness is given by a KL expansion. Results show good agreement and require only a few WFE evaluations, providing a suitable framework to account for spatially correlated randomness in waveguides.
In the next section, the WKB approximation is briefly reviewed and described in terms of propagation and reflection matrices for the forced response of a finite length waveguide. In section 3, the WFE approach is reviewed and, in Section 4, a framework for introducing slowly varying properties along the lines of the WKB assumption to a WFE formulation is proposed. Section 5 presents numerical results for waveguides with properties that vary along the waveguide. Rods, beams and plate strips with varying and known (i.e. deterministic) material and geometric properties are considered. Section 6 introduces random variability using the KL expansion for Gaussian and non-Gaussian random fields. Even though results in this work are given for an analytical expression of the KL expansion for an exponentially decaying correlation function, it is straightforward to extend them for other numerical solutions. Section 6 then presents numerical examples of rods, beams and plate strips with stochastic properties. Finally, concluding remarks are given in section 7.
Section snippets
The WKB approximation
The WKB formulation has been applied in many fields of engineering, including acoustics [12,32] and structural dynamics [9,10,18,33,34]. However, the WKB approximation breaks down if the properties change rapidly or when the travelling wave reaches a local cut-off section where the wave mode ceases to propagate. This transition, also known as a turning point or critical section, leads to an internal reflection, breaking down the main assumption in the theory and requiring a different
The wave and finite element approach
In this section, a brief review of the WFE approach is presented for one-dimensional periodic or homogeneous waveguide. A section of the waveguide of axial length is cut from the structure and, assuming harmonic motion, its dynamic stiffness matrix can be obtained from a conventional FE analysis, such that , where and are, respectively, the stiffness, damping and mass matrices, is the vector of nodal degrees of freedom and is the vector of nodal forces. The dynamic
Wave and finite element approach with slowly varying properties
Under the assumptions of the WKB approximation, it is necessary to calculate the phase change considering the locally defined wavenumber as well as the amplitude change caused by the slowly varying waveguide properties. In this section, the WFE approach is used to estimate at a number of points to calculate the phase change from position to . A numerical integration using a Gauss-Legendre (GL) quadrature scheme is applied, i.e. [36]
Numerical simulations for one-dimensional waveguides
In this section, some numerical examples are given for rods, beams and plate strips, with given variability in material and geometrical properties. The input mobility, i.e. velocity per unit force at the excitation position, is calculated for the rod and beam examples. In the rod example, the case of spatially varying Young's modulus is considered and the results obtained from the proposed approach and a standard model FE are presented. For the beam, two variable properties are considered.
Random variability of waveguide properties
Spatially distributed randomness can be modelled by the well-established random field theory using a probability measure. There are a number of methods available in the literature for generating random fields [16,17,38,39], including formulations using series expansions that are able to represent the field using deterministic spatial functions and random uncorrelated variables. The KL expansion is a special case where these deterministic spatial functions are orthogonal and derived from the
Concluding remarks
A method was proposed to extend the applicability of the WKB approach to cases where no analytical solution to the equations of motion exists by using a Finite Element (FE) approach. The phase change and the attenuation constant require the numerical evaluation of the locally defined wavenumber at various points, which are kept to a minimum, being evaluated at locations defined by a Gauss-Legendre (GL) quadrature scheme. In addition, the WKB solution implies conservation of power (in the
Acknowledgements
The authors gratefully acknowledge the financial support of the Brazilian National Council of Research (CNPq) Process number 445773/2014-6, the Federal District Research Foundation (FAPDF) Process number 0193001040/2015 and the Royal Society for the Newton International Exchanges Fund.
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