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

Abstract

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.

Transient Modeling of Corrosion Protection Systems With BEM-Electrical Circuit Hybrid Model

“…𝐀, 𝐄 ∈ ℝ (2N+1)×(2N+1) , 𝐁 ∈ ℝ (2N+1)×N , 𝑥 ∈ ℝ 2N+1 , 𝑢 ∈ ℝ N . Let us notice that in (7), 𝑈 𝑃𝑂𝐿 and 𝐼 𝑃𝑂𝐿 are nonlinearly coupled by (5).…”

“…The Boundary Element Method (BEM) is the most used because only the interface between water/hull has to be discretized [3]. Combined with the method of images to represent air/water and sea floor/water interfaces, it has shown excellent performances for complex geometries [4], [5].…”

Sacrificial Anode Cathodic Protection System Modelling Using Direct Boundary Element Method

“…To solve the nonlinear system of matrix equations, the Newton-Raphson method was applied. In accordance with the Newton-Raphson method, the current density vector can be developed into a Taylor series [18]. Therefore, current density vector can be determined from following equation [6]:…”

Vision‐based real‐time marine and offshore structural health monitoring system using underwater robots

“…Using CV techniques and geometric analyses, a method was proposed to remesh finite element models for updating the geometric changes caused by structural damages (Youqi Zhang & Lin., 2022). However, most existing damage recognition technologies have focused on land (Oh et al., 2017), which leads to the fact that damage identification of underwater structures is still in its infancy (Gortsas et al., 2022; Meng et al., 2022), especially for the imperfect underwater environment (e.g., blurred, turbid, dim, etc. ).…”