This paper introduces a material-aware strength-of-connection measure for smoothed aggregation algebraic multigrid methods, aimed at improving robustness for scalar partial differential equations with heterogeneous and anisotropic material properties. Classical strength-of-connection measures typically rely only on matrix entries or geometric distances, which often fail to capture weak couplings across material interfaces or align with anisotropy directions, ultimately leading to poor convergence. The proposed approach directly incorporates material tensor information into the coarsening process, enabling a reliable detection of weak connections and ensuring that coarse levels preserve the true structure of the underlying problem. As a result, smooth error components are represented properly and sharp coefficient jumps or directional anisotropies are handled consistently. A wide range of academic tests and real-world applications, including thermally activated batteries and solar cells, demonstrate that the proposed method maintains robustness across material contrasts, anisotropies, and mesh variations. Scalability and parallel performance of the algebraic multigrid method highlight the suitability for large-scale, high-performance computing environments.

翻译：本文针对具有异质与各向异性材料属性的标量偏微分方程，提出一种面向材料的连接强度度量方法，用于平滑聚合代数多重网格方法，旨在提升其鲁棒性。传统的连接强度度量通常仅依赖于矩阵元素或几何距离，往往无法捕捉材料界面处的弱耦合或对齐各向异性方向，最终导致收敛性能不佳。所提出的方法将材料张量信息直接纳入粗化过程，从而能够可靠地检测弱连接，并确保粗化层级保留原问题的真实结构。因此，光滑误差分量得以恰当表示，剧烈的系数跳跃或方向性各向异性也能得到一致处理。大量学术测试与实际应用（包括热激活电池与太阳能电池）表明，所提方法在材料对比度、各向异性及网格变化下均能保持鲁棒性。代数多重网格方法的可扩展性与并行性能凸显了其适用于大规模高性能计算环境的优势。