The boundary element method (BEM), combined with hierarchical matrices (HM) and adaptive cross approximation (ACA) techniques, is a powerful tool for treating large-scale cathodic protection (CP) problems in offshore engineering. Although HM-ACA/BEM achieves significant memory and time reduction compared to conventional BEM, its application to large-scale CP problems remains time consuming. In this work, a new, more efficient HM-ACA/BEM is proposed. The linear system coefficient matrix, obtained by BEM, is partitioned hierarchically into admissible and nonadmissible submatrices. The low-rank approximation of the admissible submatrices is accomplished indirectly, via a hybrid approach, utilizing approximations of the fundamental solutions and performing only a small number of integrations. The approximation of the fundamental solutions is obtained by a novel, two-step procedure, which uses a small number of Lebedev points. The proposed methodology is general and integral kernel independent, and thus can be applied to all differential equations solved by BEM. The efficiency of the proposed method is demonstrated by solving a large-scale CP problem dealing with the protection of an oil–steel offshore platform by sacrificial anodes.

中文翻译：

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一种基于积分核交叉逼近的大尺度阴极保护问题加速边界元法

边界元法 (BEM) 结合层次矩阵 (HM) 和自适应交叉逼近 (ACA) 技术，是处理海洋工程中大规模阴极保护 (CP) 问题的有力工具。尽管与传统 BEM 相比，HM-ACA/BEM 显着减少了内存和时间，但其在大规模 CP 问题中的应用仍然很耗时。在这项工作中，提出了一种新的、更有效的 HM-ACA/BEM。由边界元法获得的线性系统系数矩阵按层次划分为允许和不允许的子矩阵。允许子矩阵的低秩逼近是通过混合方法间接完成的，利用基本解的逼近并仅执行少量积分。基本解的近似值是通过一种新颖的两步程序获得的，该程序使用少量列别杰夫点。所提出的方法是通用的和积分核无关的，因此可以应用于由 BEM 求解的所有微分方程。通过解决处理牺牲阳极保护石油钢海上平台的大规模 CP 问题，证明了所提出方法的效率。