On the hyper-singular boundary integral equation methods for dynamic poroelasticity: three dimensional case

TL;DR

This paper extends hyper-singular boundary integral equation methods for dynamic poroelasticity from 2D to 3D, introducing new formulations and spectral analysis to improve accuracy and computational efficiency.

Contribution

It develops a new 3D hyper-singular boundary integral equation approach for dynamic poroelasticity, with reformulated operators, spectral analysis, and regularized equations for better numerical performance.

Findings

The proposed method achieves high accuracy in 3D poroelastic problems.

Spectral properties reveal eigenvalues depend only on two Lamé constants.

Numerical results demonstrate the efficiency of the Chebyshev-based solver.

Abstract

In our previous work [SIAM J. Sci. Comput. 43(3) (2021) B784-B810], an accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions has been developed. This work is devoted to studying the more complex and difficult three-dimensional problems with Neumann boundary condition and both the direct and indirect methods are adopted to construct combined boundary integral equations. The strongly-singular and hyper-singular integral operators are reformulated into compositions of weakly-singular integral operators and tangential-derivative operators, which allow us to prove the jump relations associated with the poroelastic layer potentials and boundary integral operators in a simple manner. Relying on both the investigated spectral properties of the strongly-singular operators, which indicate that the corresponding eigenvalues accumulate at three points whose values are only dependent on two Lam\'e constants, and the spectral properties of the Calder\'on relations of the poroelasticity, we propose low-GMRES-iteration regularized integral equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed methodology by means of a Chebyshev-based rectangular-polar solver.