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）引起的应力演化是纳米级VLSI互连中的主要可靠性挑战。精确的EM分析需要在大型互连树上求解应力控制偏微分方程，而使用传统有限差分方法进行求解计算成本极高。本文提出了两种基于有理Krylov子空间约简的快速EM应力分析技术。不同于在零频率附近展开的传统Krylov方法，有理Krylov方法能够在选定时间常数处展开，直接对准成核时间和稳态时间等指标，从而生成紧凑且精度损失极小的约简模型。我们开发了两种互补框架：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，两种方法的精度强烈依赖于展开点（即移位时间）的选择，并证明有效移位时间通常接近成核时间或空洞后稳态时间等关键时间点。基于这一发现，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约简阶数和移位时间。在合成结构及工业级电源网格上的实验结果表明，所提方法相比有限差分求解实现了数量级的效率与精度提升。仅使用4至6阶Krylov阶数，方法即可在成核时间和电阻变化预测中实现低于0.1%的误差，同时获得20至500倍的加速比。相比之下，标准扩展Krylov方法需要50阶以上仍会产生10%至20%的成核时间误差，限制了其在EM感知优化和随机EM分析中的实用性。