Electromigration (EM) induced stress evolution is a major reliability challenge in nanometer-scale VLSI interconnects. Accurate EM analysis requires solving stress-governing partial differential equations over large interconnect trees, which is computationally expensive using conventional finite-difference methods. This work proposes two fast EM stress analysis techniques based on rational Krylov subspace reduction. Unlike traditional Krylov methods that expand around zero frequency, rational Krylov methods enable expansion at selected time constants, aligning directly with metrics such as nucleation and steady-state times and producing compact reduced models with minimal accuracy loss. Two complementary frameworks are developed: a frequency-domain extended rational Krylov method, ExtRaKrylovEM, and a time-domain rational Krylov exponential integration method, EiRaKrylovEM. We show that the accuracy of both methods depends strongly on the choice of expansion point, or shift time, and demonstrate that effective shift times are typically close to times of interest such as nucleation or post-void steady state. Based on this observation, a coordinate descent optimization strategy is introduced to automatically determine optimal reduction orders and shift times for both nucleation and post-void phases. Experimental results on synthesized structures and industry-scale power grids show that the proposed methods achieve orders-of-magnitude improvements in efficiency and accuracy over finite-difference solutions. Using only 4 to 6 Krylov orders, the methods achieve sub-0.1 percent error in nucleation time and resistance change predictions while delivering 20 to 500 times speedup. In contrast, standard extended Krylov methods require more than 50 orders and still incur 10 to 20 percent nucleation time error, limiting their practicality for EM-aware optimization and stochastic EM analysis.

翻译：电迁移（EM）引发的应力演化是纳米级超大规模集成电路互连中的主要可靠性挑战。精确的电迁移分析需要在大型互连树上求解应力控制偏微分方程，使用传统有限差分方法计算成本高昂。本文提出了两种基于有理Krylov子空间降阶的快速电迁移应力分析技术。与在零频率处展开的传统Krylov方法不同，有理Krylov方法能够在选定的时间常数处展开，直接与成核时间和稳态时间等指标对齐，从而以最小精度损失生成紧凑的降阶模型。我们开发了两个互补框架：频域扩展有理Krylov方法ExtRaKrylovEM，以及时域有理Krylov指数积分方法EiRaKrylovEM。研究表明，两种方法的精度均强烈依赖于展开点（即移位时间）的选择，并证明有效的移位时间通常接近目标时间（如成核时间或空洞形成后稳态时间）。基于此观察，我们引入坐标下降优化策略，自动确定成核阶段和空洞后阶段的最优降阶阶数与移位时间。在合成结构和工业级电源网格上的实验结果表明，所提方法在效率和精度上较有限差分解实现了数量级提升。仅使用4至6阶Krylov基，该方法在成核时间和电阻变化预测中即可实现低于0.1%的误差，同时获得20至500倍的加速。相比之下，标准扩展Krylov方法需要超过50阶且仍会产生10%至20%的成核时间误差，这限制了其在电迁移感知优化和随机电迁移分析中的实用性。