Accurate and Efficient Partial Inductance Evaluation for the PEEC Method on a Uniform Cubic Grid Using a 1-D Array of 2-D Interpolants

Abstract

Accurate and efficient evaluation of partial inductances is essential in large-scale Fast Fourier Transform (FFT)-accelerated Partial Element Equivalent Circuit (PEEC) simulations. This paper presents a novel method for partial inductance evaluation between uniform cubic cells on an integer Cartesian grid. The proposed approach exploits the structured domain of unique configurations arising from translation invariance and cubic lattice symmetry. By decomposing the resulting three-faced pyramidal configuration space into X-indexed Y–Z layers, the original three-dimensional (3D) problem is reformulated as a 1D (one-dimensional) Array of 2D (two-dimensional) Interpolants (n2D). The inherently triangular layer domains are extended to square support regions for compatibility with standard two-dimensional interpolation schemes on a uniform grid. The method is evaluated using nearest-neighbor, linear, cubic, and spline interpolations. Its accuracy, memory requirements, preparatory phase complexity, and usage-phase performance are analyzed and compared with analytical, decomposition-based, and interpolation-based approaches. The results demonstrate that the proposed n2D method, by exploiting the layered structure of the three-faced pyramidal domain of unique configurations, achieves a substantially improved accuracy–memory trade-off compared with direct 3D interpolation. For comparable memory, it attains accuracy improvements exceeding two orders of magnitude. The spline variant further extends the attainable accuracy range, achieving maximum relative errors below $10^{\mathrm {-9}}$ while remaining within practical memory limits and exceeding the accuracy of compared interpolation-based methods. Usage-phase evaluation times range from 59 ns to 77 ns per query depending on interpolation order. The proposed n2D method provides an efficient computational module for high-accuracy partial inductance evaluation applicable to PEEC simulations.