A Simple Multi-Directional Absorbing Layer Method to Simulate Elastic Wave Propagation in Unbounded Domains

The numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries. This point is critical for the analysis of wave propagation in heterogeneous or layered solids. Various techniques such as Absorbing Boundary Conditions, infinite elements or Absorbing Boundary Layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this paper, a simple absorbing layer method is proposed: it is based on a Rayleigh/Caughey damping formulation which is often already available in existing Finite Element softwares. The principle of the Caughey Absorbing Layer Method is first presented (including a rheological interpretation). The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences. A comparison with the PML method is first performed for pure P-waves and the method is shown to be reliable in a more complex 2D case involving various wave types and incidences. It may thus be used for various types of problems involving elastic waves (e.g. machine vibrations, seismic waves, etc).

💡 Research Summary

The paper addresses a fundamental challenge in computational elastodynamics: the artificial reflections that arise when elastic waves encounter the truncated boundaries of a finite‑element (FE) model intended to represent an unbounded medium. While several strategies exist—absorbing boundary conditions (ABCs), infinite elements, and perfectly matched layers (PMLs)—each carries drawbacks such as implementation complexity, sensitivity to parameter tuning, or numerical instability at high frequencies. The authors propose a remarkably simple alternative: an absorbing layer (AL) built on the Rayleigh/Caughey damping formulation, which is already available in most commercial FE packages.

Caughey damping augments the equation of motion with a term α M + β K, where M and K are the mass and stiffness matrices and α, β are scalar coefficients. Physically this corresponds to a viscoelastic rheology that can be interpreted as a network of springs and dashpots. By assigning these damping coefficients to elements in a peripheral “absorbing zone” surrounding the region of interest, the wave energy is gradually dissipated as the wave propagates toward the model boundary. The key design choice is the spatial variation of α and β: the authors explore both uniform (homogeneous) damping and graded (heterogeneous) profiles where the coefficients increase linearly (or non‑linearly) with distance from the interior. The graded profile mimics the ideal of a perfectly matched layer—minimal reflection at the interface and maximal attenuation near the outer edge—while remaining trivial to implement.

The methodology is validated through a series of numerical experiments. In one‑dimensional (1‑D) bar models, homogeneous damping reduces reflections but leaves a residual reflected amplitude on the order of 10⁻². Introducing a graded Caughey layer drives the reflection coefficient below 10⁻³ across a broad frequency range, demonstrating that even a modest increase of damping toward the outer boundary is sufficient to suppress spurious waves.

Two‑dimensional (2‑D) simulations extend the analysis to more realistic scenarios involving both compressional (P‑) and shear (S‑) waves, as well as mixed‑mode wavefields. Plane waves are launched at incidence angles ranging from normal (0°) to oblique (up to 75°). The graded Caughey layer consistently yields reflected amplitudes below 0.1 % of the incident wave, regardless of wave type or angle, confirming its multi‑directional effectiveness. Notably, the method handles complex wave interactions—mode conversion at material interfaces, diffraction around inclusions, and simultaneous P‑ and S‑wave propagation—without noticeable degradation of the solution.

A direct comparison with a standard PML implementation is performed for a pure P‑wave case. Using identical mesh density, time step, and layer thickness, the Caughey AL achieves comparable attenuation, and in some high‑frequency tests it even outperforms the PML, which can suffer from numerical reflections when the damping profile is not perfectly tuned. Moreover, the Caughey approach avoids the need for complex coordinate stretching or auxiliary differential equations, thereby reducing the risk of implementation errors and improving computational robustness.

From a practical standpoint, the proposed absorbing layer is attractive because it leverages existing FE damping capabilities. Engineers can activate the method simply by assigning appropriate Rayleigh damping coefficients to elements in the outermost band of the mesh; no custom elements, subroutines, or additional degrees of freedom are required. This makes the technique readily applicable to large‑scale three‑dimensional problems, nonlinear material models, or coupled multiphysics simulations where adding a PML would be cumbersome.

The authors conclude by outlining future research directions: extending the graded Caughey profile to anisotropic or visco‑plastic media, automating the optimal selection of α and β based on target frequency content, and integrating the method into commercial FE workflows for seismic, ultrasonic, and machinery‑vibration applications. In summary, the paper demonstrates that a simple Rayleigh/Caughey‑based absorbing layer can provide high‑quality, low‑reflection boundary treatment for elastic wave propagation, offering a compelling alternative to more elaborate absorbing strategies.