In continuous aperture arrays (CAPAs), careful consideration of the underlying physics is essential, among which electromagnetic (EM) mutual coupling plays a critical role in beamforming performance. Building on a physically consistent mutual coupling model, the beamforming design is formulated as a functional optimization whose optimality condition leads to a Fredholm integral equation. The incorporation of the coupling model, however, substantially increases computational complexity, necessitating efficient and accurate integral equation solvers. In this letter, we propose two efficient solvers: 1) a coordinate-transformation-based kernel approximation that preserves the operator structure while alleviating discretization demands, and 2) a direct lower-upper (LU)-based solver that stably handles the Nyström-discretized system. Numerical results demonstrate improved accuracy and reduced computational overhead compared to conventional methods, with the LU-based solver emerging as an efficient and scalable solution for large-scale CAPA optimization via offline factorization.

翻译：在连续孔径阵列中，充分考虑其底层物理特性至关重要，其中电磁互耦效应对波束赋形性能起着关键作用。基于物理一致的互耦模型，波束赋形设计被构建为一个泛函优化问题，其最优性条件导出一个Fredholm积分方程。然而，耦合模型的引入显著增加了计算复杂度，因此需要高效且精确的积分方程求解器。本文提出两种高效求解器：1）基于坐标变换的核函数近似方法，在保持算子结构的同时降低离散化需求；2）基于直接LU分解的求解器，可稳定处理Nyström离散化系统。数值结果表明，与传统方法相比，所提方法在提升精度的同时降低了计算开销，其中基于LU分解的求解器通过离线分解为大规模连续孔径阵列优化提供了高效且可扩展的解决方案。