Dispersion analysis of composite acousto-elastic waveguides

doi:10.1016/j.compositesb.2017.07.040

Composites Part B: Engineering

Volume 130, 1 December 2017, Pages 200-216

https://doi.org/10.1016/j.compositesb.2017.07.040 Get rights and content

Abstract

Propagation characteristics of guided waves are widely utilized for nondestructive characterization of laminated composites which are often immersed in fluid. These propagation characteristics are quantified through dispersion and attenuation curves, and need to be computed for a large number of estimated structure and material property combinations, over a wide range of frequencies. To solve this central problem, an efficient approach is proposed for dispersion analysis of three types of immersed waveguides: laminated plates; laminated rods and pipes; composite waveguides with generic cross-section. The approach is based on Semi-Analytical Finite Element (SAFE) method enhanced with two novel discretization methods: Complex-length Finite Element Method (CFEM) for the solid domain and Perfectly Matched Discrete Layers (PMDL) for the surrounding fluid. The resulting approach is computationally more efficient than the existing methods in that it keeps the underlying eigenvalue problem linear and substantially smaller. The paper presents the basic ideas of the proposed approach, specific algorithms for determining the discretization parameters, and open-source implementation of the resulting waveguide models. Several numerical examples are presented to illustrate the method's efficiency. Finally, the theoretical predictions from the method are validated using experimental observations for several structural members.

Introduction

Guided waves are widely used for non-destructive evaluation [1], [2], [3], [4], [5], [6], [7], [8] and structural health monitoring [9], [10], [11], [12], with specific applications that include corrosion screening of oil and gas pipelines [13], [14], [15], [16], [17], [18], characterization of moduli, damage and delamination of composites [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], characterization of fluid loaded on structures [33], [34], [35], [36], near surface geophysics and earthquake engineering [37], [38], [39], [40] as well as ultrasound imaging of biological structures such as arteries [41], [42], [43]. The attractiveness of guided waves stems from their long-distance propagation without substantial attenuation. However, in many circumstances, waveguides are immersed in fluid [44], [45], [46], [47], [48] or embedded in solid [49], [50], [51], [52], [53], [54], [55], [56], resulting in energy loss into the surrounding medium, leading to significant reduction of inspection range. Thus, special attention is given to open waveguides in obtaining accurate estimates of attenuation, phase and group velocities, which is termed as dispersion analysis and is the focus of this paper.

In recent years, several methods have been developed for dispersion analysis of acousto-elastic (immersed elastic) waveguides. Examples include analytical methods that have been extensively applied to simple geometries such as plates [57], [58], [59], [60], [61], [62], [63], [64] and cylindrical structures [65], [66], [67], [68], [69], [70], [71]. Analytical methods are typically based on transfer matrix method [72], [73], [74] or global matrix method [75], [76], [77]. This family of matrix methods is not computationally demanding; however, its application is limited to simple geometries and solution becomes more complicated by considering multiple layers. Besides, the required root-finding procedure (especially for leaky or damped modes) may fail at high frequencies (see e.g. Fig. 6.3 in Ref. [78]). Improved root-finding algorithms have been suggested (e.g. in Ref. [79]), but require increased computational effort.

Unlike analytical methods, numerical approaches based on finite element semi-discretization are versatile in modeling waveguides with arbitrary geometry, boundary conditions and material property distribution. This family of methods appears to have been first proposed in geotechnical earthquake engineering and geophysics, i.e. the Thin Layer Method (TLM) [80], [81], and extended to ultrasonic applications under the name of Semi-Analytical Finite Element (SAFE) method [82], [83], [84], [85], [86], [87]. The idea is to discretize the cross-section of the waveguide (in one or two dimensions) using finite element method (FEM), and use analytical solution along the axis of the waveguide.

Focusing on immersed waveguides, to model the surrounding fluid, SAFE method has been coupled with different approaches. One scheme couples the boundary element method (BEM) with FEM which is referred to as 2.5D FEM-BEM [88]. This method leads to a nonlinear eigenvalue problem that requires a more expensive solution strategy compared to linear eigenvalue problems. Similarly, using exact boundary condition employed in e.g. Ref. [89] requires an iterative solution due to nonlinearity. For the special case of immersed plates, the nonlinear eigenvalue problem can be simplified into a cubic eigenvalue problem using a change of variables [90], which too is computationally expensive.

Other hybrid SAFE methods consist of using non-reflecting boundaries [91] or absorbing layers [33]. These methods require minimal modifications to the existing SAFE codes and have the advantage of keeping the eigenvalue problems linear, which can be solved by standard packages. The disadvantage is that a large region of the surrounding fluid has to be discretized, which significantly increases the cost of the eigenvalue problem.

Motivated by eliminating the drawbacks of the existing approaches, we propose an efficient approach for modeling different types of acousto-elastic waveguides, namely (a) immersed laminated plates; (b) immersed laminated rods and fluid-filled pipes; and (c) composite waveguides with generic cross-sections. Specifically, we focus on the discretization of the surrounding infinite fluid and incorporate the method of Perfectly Matched Discrete Layers (PMDL) [92], [93] into the SAFE formulation. PMDL can be considered as an efficient discretization of the highly successful perfectly matched layers (PML, [94]). For each class of waveguides, we examine the underlying wave propagation characteristics and provide simple guidelines for choosing the PMDL parameters. Through several numerical examples, we show that the proposed approach results in accurate dispersion curves with just a handful of PMDL elements, leading to reduced computational cost.

In a second, complementary contribution, we adapt the recently developed Complex-length Finite Element Method (CFEM) [95], [96] for efficient discretization of the interior solid. CFEM is applicable to cross-sections that can be divided into homogeneous intervals (in 1D) or parallelograms (in 2D), and is shown to have convergence properties identical to expensive spectral finite elements, but achieved with minimal modification to inexpensive linear finite elements. In cases where CFEM is not applicable, we utilize higher order finite elements to discretize the interior (the details are discussed in the appropriate sections).

Finally, the methods resulting from combining the ideas of PMDL, CFEM and high-order FEM are implemented into an open-source software named WaveDisp [97] which computes the dispersion and attenuation curves for various types of waveguides. As illustrated later in the paper, the resulting dispersion curves are verified to be highly accurate, and validated with experimental observations.

Section snippets

Overview

In this paper we address dispersion analysis of acousto-elastic waveguides. Specifically, we consider three different cases shown in Fig. 1, i.e. immersed plates, immersed rods and fluid-filled pipes and immersed waveguides with generic cross-section. In all cases, solid domain and the infinite fluid medium are semi-discretized using the framework of semi-analytical finite element (SAFE) method.

Discretization of solid domain is performed by the recently developed complex-length finite element

Plates immersed in fluid

As mentioned in the introduction, numerical methods based on SAFE and FEM have been extensively used for investigation of composite plates (see e.g. Refs. [5], [98], [99], [100], [101], [102], [103], [104], [105], [106]). In this section, we present a brief summary of the SAFE formulation for immersed plates shown in Fig. 1 (a). We then classify the solid-born wavemodes into different types and present the appropriate scheme to treat each type of these modes.

Governing equations

The geometry of the immersed cylindrical waveguide is shown in Fig. 5 (a). In the solid domain, elastodynamic equation for the harmonic waves of the form $\tilde{u} (x, t) = u (x, ω) e^{- i ω t}$ with $x = (r, θ, z)$ is given in (1). In this problem, $u = {u_{r} u_{θ} u_{z}}^{T}$ is the displacement vector and the stress vector $σ = D ε = {σ_{r r} σ_{θ θ} σ_{z z} σ_{θ z} σ_{r z} σ_{r θ}}^{T}$ is related to the strain vector $ε = L_{ε} u = {\{ε_{r r} ε_{θ θ} ε_{z z} γ_{θ z} γ_{r z} γ_{r θ}\}}^{T}$ through $D_{6 \times 6}$ where the nonzero entries for an isotropic medium are $D_{11} = D_{22} = D_{33} = ρ_{S} c_{p}^{2}$ , $D_{44} = D_{55} = D_{66} = ρ_{S} c_{s}^{2}$ and $D_{12} = D_{21} = D_{13} = D_{31} = D_{23} = D_{32} = ρ$

Governing equations

Geometry of the immersed waveguides with general cross-section is shown in Fig. 8 (a). In the solid domain, elastodynamic equation for the harmonic waves of the form $\tilde{u} (x, t) = u (x, ω) e^{- i ω t}$ with $x = (x, y, z)$ is given in (1) where, in this problem, $u = {u_{x} u_{y} u_{z}}^{T}$ is the displacement vector and the stress vector $σ = D ε = {σ_{x x} σ_{y y} σ_{z z} σ_{y z} σ_{x z} σ_{x y}}^{T}$ is related to the strain vector $ε = L_{ε} u = {\{ε_{x x} ε_{y y} ε_{z z} γ_{y z} γ_{x z} γ_{x y}\}}^{T}$ through $D_{6 \times 6}$ where the nonzero entries for an isotropic medium are $D_{11} = D_{22} = D_{33} = ρ_{S} c_{p}^{2}$ , $D_{44} = D_{55} = D_{66} = ρ_{S} c_{s}^{2}$ and $D_{12} = D_{21}$

Validation

In order to validate the proposed computational methods, we compare predictions from the methods with experimental results from the literature. The examples include immersed hard/soft plates, a fluid-filled and immersed pipe and an immersed rectangular rod. As described below, all examples confirm the validity of the proposed waveguide models.

Conclusions

We have proposed an efficient approach for dispersion analysis of different types of composite waveguides immersed in acoustic fluid: plates; circular rods; pipes and fluid-filled pipes; and prismatic waveguides with arbitrary cross-section. The underlying formulation is based on the semi-analytical finite element (SAFE) method, enhanced by recently developed discretization methods to increase the computational efficiency. For discretization of solid domain, we used complex-length finite

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1016514 and Grant No. CMMI-1635291.

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