Fast integral equation solvers for layered media and chip packaging
Broadband, low-frequency-stable field solvers for packaging: (a) a discrete exterior calculus volume solver coupled to a boundary integral equation, which imposes the radiation condition exactly; (b) hierarchical off-diagonal low-rank compression of the sparse mass matrix inverse; (c) impedance extraction from DC to 100 GHz compared with a three-dimensional finite element reference. Figure by Sizhe Zhu.
Background. Layered-media structures, interconnects, and chip packages now approach a thousand wavelengths in size at their operating frequencies. Low-frequency breakdown, high material contrast, and the cost of evaluating layered-medium Green’s functions make classical formulations and general-purpose commercial solvers impractical at this scale.
Objective. Develop scalable integral equation solvers for electrically large quasi-planar and layered systems, with the long-term target of full-wave simulation of circuits approaching a thousand wavelengths on a single compute node.
Approach. Kernel-independent fast multipole methods that accelerate layered Green’s functions and other three-dimensional kernels without analytical expansions; hierarchical off-diagonal low-rank inversion of the sparse mass matrices that arise in these formulations; potential-based formulations that avoid low-frequency breakdown (see the hybrid discrete exterior calculus and surface integral equation framework and our high-contrast-stable volume integral equations); and interconnect solvers for packaging.
Main results. Our hierarchical off-diagonal low-rank inversion of sparse mass matrices received an Honorable Mention in the student paper competition of the 2026 IEEE International Symposium on Antennas and Propagation, and a hybrid kernel-independent fast multipole algorithm for three-dimensional Green’s functions was presented at the same meeting.
Simulation of an ultra-large geometry (163 wavelengths) with three million unknowns. Figure by Sina Vaezi.
Significance. This work is conducted through the Consortium for Electromagnetic Technologies, which Dr. Gomez co-directs with Professors Weng Cho Chew and Dan Jiao; current members are ASML, ASUS, Cadence, and Siemens EDA. The same fast multipole machinery is planned to accelerate full-wave-guided optical proximity correction for lithography.
Publications.
S. Zhu (G), B. Zhang, W. C. Chew, and L. J. Gomez, "Fast Inversion of Sparse Mass Matrices Using HODLR Compression," IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, July 2026 (Honorable Mention, student paper competition).
S. S. Vaezi (G) and L. J. Gomez, "A Hybrid Kernel-Independent Fast Multipole Algorithm for Three-Dimensional Green’s Functions," IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, July 2026.