An accelerated boundary element method via cross approximation of integral kernels for large-scale cathodic protection problems

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.

Automated summary

In this work, a new, more efficient HM-ACA/BEM method is proposed, which applies the boundary element method to large-scale cathodic protection problems through hierarchical matrices and adaptive cross approximation techniques.