This work proposes an accurate hyper-singular boundary integral equation method for dynamic poroelastic problems with Neumann boundary condition in three dimensions 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 Lamé constants, and the spectral properties of the Calderón 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.

本文提出了一种精确的超奇异边界积分方程方法，用于求解具有三维Neumann边界条件的动态多孔弹性问题，并采用直接和间接两种方法构造组合边界积分方程。将强奇异积分算子和超奇异积分算子重新构造为弱奇异积分算子和切向导数算子的组合，这使我们能够以简单的方式证明与多孔弹性层势和边界积分算子相关的跳跃关系。依赖于研究的强奇异算子的光谱特性，这表明相应的特征值在三个点处累积，其值仅取决于两个 Lamé 常数，和孔隙弹性的 Calderón 关系的光谱性质，我们提出了低 GMRES 迭代正则化积分方程。通过基于切比雪夫的矩形极坐标求解器，给出了数值示例来证明所提出方法的准确性和效率。