Efficient simulation of elastic guided waves interacting with notches, adhesive joints, delaminations and inclined edges in plate structures

Highlights

• An approach to model guided wave propagation in plate structures is presented.

• The interaction of guided waves with notches, edges, delaminations and joints is simulated.

• Only line elements are reqired to discretize the structure’s thickness direction.

• Boundary conditions can be applied on the plate surface without requiring discretization.

• CPU times are typically reduced by 2–3 orders of magnitude compared to a full finite element discretization.

Abstract

This paper presents an approach to model transmission and reflection phenomena of elastic guided waves in plates. The formulation is applied to plate structures containing notches, inclined edges, delaminations or (adhesive) joints. For these cases, only the thickness direction of the structure needs to be discretized at several locations, while the direction of propagation is described analytically. Consequently, the number of degrees of freedom is very small. Semi-infinite domains can be modeled, in which case the radiation condition is fulfilled exactly. Traction boundary conditions are introduced on the plate surface without requiring a mesh along the surface. Results are validated against conventional finite element implementations, showing the accuracy of the proposed approach and a reduction of the computational costs by typically 2–3 orders of magnitude.

Introduction

The mathematical description and simulation of elastic guided waves, particularly in the ultrasonic frequency range, has concerned researchers and engineers for decades (see e.g. [1], [2], [3], [4], [5], [6]) and is still an active research field [7], [8], [9]. Especially the practical applications of guided waves in the context of non-destructive testing [10], [11], [12], structural health monitoring [13], [14] and material characterization [15], [16] have triggered research in this field and created a demand for efficient simulation tools. Even today, with modern computers and very general and easy-to-use finite element implementations at hand, the simulation of guided waves is still considered a challenging and time-consuming task. This is due to the high frequencies and small wavelengths (relative to the dimensions of the structure), requiring fine spatial and temporal discretizations in numerical schemes.

Depending on the application, numerical methods are applied to analyze dispersion properties of guided waves in a given structure or to perform a full dynamic simulation in the time or frequency domain. Often, the structure is long enough to be considered as (semi-) infinite in the numerical model, which is challenging to achieve in mesh-based approaches like finite elements or finite differences [17].

The problem of computing dispersion curves for elastic waveguides of uniform cross-section has been solved for many cases, and highly efficient algorithms are available. Homogeneous or layered plates or cylinders can be modeled using analytical approaches [18] or, e.g. spectral decomposition techniques [19]. For more complex cases, such as strongly inhomogeneous media, materials involving anisotropy and damping or general three-dimensional cross-sections, a finite element based discretization of the cross-section is typically applied. This approach is common between the thin layer method (TLM) [20], [21], the semi-analytical finite element (SAFE) method [22], [23] and the scaled boundary finite element method (SBFEM) [24], [25]. The TLM was originally formulated for two-dimensional soil layers, by discretizing the thickness direction and describing the direction of wave propagation analytically. The SAFE method can be considered as the extension of this approach to three dimensions by discretizing a two-dimensional cross-section using classical finite elements. The SBFEM is a more general concept for solving partial differential equations on arbitrary star-convex domains by discretizing their boundary only [26], [27]. However, to compute the modes in structures of uniform cross-section, the SBFEM reduces to a simplified formulation that is similar to the SAFE or TLM method, while different solution procedures are employed [28]. To improve efficiency, discretizations based on spectral elements of very high order [29] or NURBS [30] have been applied as well as advanced solution procedures based on mode-tracing [31], [32]. For waveguides that are periodic (but not necessarily uniform), the waveguide finite element (WFE) method offers a suitable alternative by modeling a representative substructure and applying periodic boundary conditions [33], [34].

When a simulation of a signal propagating through a waveguide and/or interacting with geometric features is desired (rather than only analyzing the general dispersive properties) the situation becomes more complicated. Numerous methods have been applied to this problem, such as finite differences [35], [36], finite volumes [37], [38], the local interaction simulation approach [39], [40] and of course finite and spectral elements [41], [42]. Since these methods require a full discretization of the geometry, the total number of degrees of freedom tends to be large. In an earlier work using the SBFEM, a transient simulation of a waveguide problem was addressed by dividing the structure into subdomains and discretizing the boundary of each subdomain [43]. This concept significantly reduces computational costs compared to a full finite element discretization but still requires large numbers of degrees of freedom in extended homogeneous sections of the waveguide.

More recently, it has been demonstrated that for ultrasonic waves in plates of constant cross-section, dynamic stiffness matrices can be applied very effectively [44], similar to the case of layered soils as discussed in detail e.g. in [45], [46]. The stiffness matrices describe the relationship between displacements and forces on the cross-section. They can be computed for finite as well as semi-infinite plate sections (Fig. 1). In the case of a section of finite length L, the stiffness matrix relates the displacements and forces on the cross-section at the beginning (${\mathbf{u}}_{\mathbf{i}}\text{,}{\mathbf{f}}_{\mathbf{i}}$) and end (${\mathbf{u}}_{\mathbf{e}}\text{,}{\mathbf{f}}_{\mathbf{e}}$) of the section. For a semi-infinite plate, the stiffness matrix involves only degrees of freedom on the cross-section at the beginning of the plate. Once the stiffness matrix has been computed, boundary conditions can be applied at the nodes and the displacement response is solved for based on the discretized cross-section only.

In this paper, it will be demonstrated how to couple several domains in order to efficiently model geometries of practical relevance. Fig. 2 shows simplified models of a notch, an inclined plate, a delamination, and an adhesive joint. What these cases have in common is the fact that the SBFEM requires only the thickness direction to be discretized at a few locations. Hence, the resulting number of degrees of freedom is very small. The domains of uniform cross-section can be of arbitrary length without affecting the computational costs. Note that to model the inclined edge, the general SBFEM formulation for arbitrary star-convex domains is employed rather than the simplified formulation for plates. Still, only the thickness direction of the plate is discretized.

Furthermore, in this paper an approach is presented to account for traction boundary conditions on the plate surface without adding degrees of freedom. Since the plate surface is not discretized, the boundary conditions need to be considered in the analytical solution procedure. The tractions are described as a polynomial of arbitrary order in the spatial coordinate. The numerical examples in this paper show that the approach can be applied to plate structures consisting of several materials or containing arbitrary steps or notches.