Adaptive Cross Approximation Algorithm for Accelerating BEM in Eddy Current Nondestructive Evaluation

This paper presents the adaptive cross approximation (ACA) algorithm to accelerate boundary element method (BEM) for eddy current nondestructive evaluation (NDE) problem. The eddy current problem is formulated by boundary integral equation and discretized into matrix equations by BEM. Stratton-Chu formulation is selected and implemented for the conductive medium which does not has low frequency breakdown issue. The ACA algorithm has the advantage of purely algebraic and kernel independent. It starts with hierarchically partitioning the object to get diagonal blocks, near blocks and far blocks. The far-block interactions which are rank deficient can be compressed by ACA algorithm meanwhile the elements for diagonal-block interactions and near-block interactions are stored and computed by BEM. We apply modified ACA (MACA) for more memory saving while keeping almost same accuracy compared with original ACA. For numerical testing, several practical NDE examples such as coil above a half space conductor, tube in a fast reactor and Testing Electromagnetic Analysis Methods ( TEAM) workshop benchmark problem are presented to show the robust and efficiency of our method. With the aid of ACA, for electrically small problems, the complexity of both the memory requirement and CPU time for BEM are reduced to O(N log N).

本文提出自适应交叉近似（ACA）算法，用于加速边界元法（BEM）以解决涡流无损检测（NDE）问题。涡流问题由边界积分方程构建，并通过边界元法离散为矩阵方程。针对不存在低频失效问题的导电介质，选择并应用了斯特拉顿 - 朱（Stratton - Chu）公式。ACA算法具有纯代数且与核无关的优点。它首先对物体进行分层划分，以得到对角块、近块和远块。ACA算法可压缩秩亏的远块相互作用，同时对角块相互作用和近块相互作用的元素由边界元法存储和计算。我们应用改进的ACA（MACA）以节省更多内存，同时与原始ACA相比保持几乎相同的精度。在数值测试方面，给出了几个实际的无损检测示例，如半空间导体上方的线圈、快堆中的管道以及测试电磁分析方法（TEAM）研讨会基准问题，以展示我们方法的稳健性和高效性。借助ACA，对于电小问题，边界元法的内存需求和CPU时间复杂度都降低到O(N log N)。