A coupled finite and boundary spectral element method for linear water-wave propagation problems

TL;DR

This work addresses efficient, accurate simulation of linear water-wave propagation over variable bathymetries in unbounded domains by coupling a spectral boundary element method (BSEM) for the exterior with a spectral element method (SEM) for the interior. The BSEM employs a complete fundamental solution for variable depth and automatically enforces the Sommerfeld radiation condition, while the SEM provides flexible modeling of arbitrary bathymetries inside. The coupled BSEM-SEM achieves spectral convergence with relatively few degrees of freedom, as demonstrated on three benchmarks, including scattering over circular and elliptical shoals, with strong agreement to experimental and reference solutions. The approach offers a robust framework for coastal wave problems and can be extended to MMSE, wave breaking, currents, and wave-structure interactions, enabling efficient simulations of realistic engineering scenarios.

Abstract

A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22-34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld's radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre-Gauss-Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p-convergence of spectral methods.

Figures (15)

• Figure 1: Example of a spectral boundary element. An element of order $p=7$ is represented in the global $x-y$-axis (left) and in the local coordinate $\xi$ (rigth). Also the nodal function corresponding to the fifth node is shown
• Figure 2: Singular integral around $S_{\epsilon}$ when the collocation point is located inside the element. The distance to the singularity $\rho=|\xi-\xi'|$ is defined to extract the singular kernel
• Figure 3: A finite spectral element of order $p=4$ is represented in the physical ($x-y$)-coordinate system and in the normalized $(\xi,\zeta)\in[-1,1]^2$ reference coordinate system (top). Two representative nodal basis functions of the element are also showed (bottom)
• Figure 4: Illustration of the physical model with dashed lines representing iso-depth contours. A region with an arbitrary bathymetry is considered on $\Omega_{F}$ closed by $\Gamma_C$. On $\Omega_{B}$, the domain has a bathymetry with by straight and parallel contour lines
• Figure 5: Wave train with an incident angle $\theta=\pi/6$ propagating through a rectangular domain with black solid lines representing Dirichlet boundary conditions and red dashed lines showing Neumann boundary conditions