Robust boundary integral equations for the solution of elastic scattering problems via Helmholtz decompositions

TL;DR

This work develops robust second-kind boundary integral equations for 2D elastic scattering by decomposing elastic fields into Helmholtz compressional and shear components with wave-numbers $k_p$ and $k_s$. Building on the regularization approach of Boubendir et al., the authors construct regularized CFIER formulations that map to Helmholtz Cauchy data on the boundary, achieving invertible, order-zero operators for smooth scatterers and providing strong spectral properties for iterative solvers. They derive detailed gradient and Hessian trace formulas for Helmholtz layer potentials, connect Dirichlet and Neumann data to these quantities, and validate the approach numerically via Nyström discretizations, showing substantial improvements over classical CFIE formulations, especially at high frequencies. The results offer a viable alternative to Navier-based BIEs and set the stage for extensions to penetrable scatterers and to three dimensions, where coupling with Maxwell BIOs may be required.

Abstract

Helmholtz decompositions of the elastic fields open up new avenues for the solution of linear elastic scattering problems via boundary integral equations (BIE) [Dong, Lai, Li, Mathematics of Computation,2021]. The main appeal of this approach is that the ensuing systems of BIE feature only integral operators associated with the Helmholtz equation. However, these BIE involve non standard boundary integral operators that do not result after the application of either the Dirichlet or the Neumann trace to Helmholtz single and double layer potentials. Rather, the Helmholtz decomposition approach leads to BIE formulations of elastic scattering problems with Neumann boundary conditions that involve boundary traces of the Hessians of Helmholtz layer potential. As a consequence, the classical combined field approach applied in the framework of the Helmholtz decompositions leads to BIE formulations which, although robust, are not of the second kind. Following the regularizing methodology introduced in [Boubendir, Dominguez, Levadoux, Turc, SIAM Journal on Applied Mathematics 2015] we design and analyze novel robust Helmholtz decomposition BIE for the solution of elastic scattering that are of the second kind in the case of smooth scatterers in two dimensions. We present a variety of numerical results based on Nystrom discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers.

Figures (5)

• Figure 1: Geometries for the experiments considered in this section. Smooth curves: the unit circle, the kite and the cavity curve; Lipschitz domains: a square and the $L-$shaped domain; Notice that all the curves are of length $2\pi$.
• Figure 2: Numbers of GMRES iterations required to reach residuals of $10^{-5}$ for the CFIE and CFIER formulations for the circle (left), kite (middle) and the smooth cavity (right) in the case of Dirichlet boundary conditions and frequencies $\omega=10,20,40,80,160$ with Lamé parameters $\lambda=2$ and $\mu=1$ under plane wave incidence. We used Nyström discretizations corresponding to 8 points per the shorter wavelength. The numbers of iterations are independent of the direction and polarization of the plane wave.
• Figure 3: Numbers of GMRES iterations required to reach residuals of $10^{-5}$ for the CFIE and CFIER formulations for the circle (left), kite (middle) and the smooth cavity (right) in the case of Neumann boundary conditions and frequencies $\omega=10,20,40,80,160$ with Lamé parameters $\lambda=2$ and $\mu=1$ under plane wave incidence. We used Nyström discretizations corresponding to 8 points per the shorter wavelength. In order to illustrate the effect of discretization size on the iterative behavior of the CFIE formulations, we report iteration counts "CFIE refined" corresponding to discretizations refined by a factor of two. The numbers of iterations are independent of the direction and polarization of the plane wave.
• Figure 4: Eigenvalue distribution in the [id=catB]complex plane of the CFIER operators in the case of the kite geometry and $\omega=40$ for the Dirichlet (left) and Neumann (right) cases.
• Figure 5: Numbers of GMRES iterations required to reach residuals of $10^{-4}$ for the CFIE and CFIER formulations for the square and the L-shaped scatterers in the high frequency regime for the Dirichlet (left) and Neumann (right) boundary conditions and the same material parameters as in the previous cases, In the case of Neumann boundary conditions, the solvers based on CFIE formulations did not converge.

Theorems & Definitions (28)

• Remark 2.1
• Proposition 3.1
• Lemma 3.2
• Proposition 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Proposition 3.7
• Proposition 3.8
• Theorem 4.1