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It is important to model the substrate coupling for mixed-signal circuit designs today. This paper presents an improved direct boundary element method (DBEM) for substrate resistance calculation, where only the boundary of substrate volumes is discretized and only the free-space Green function is used. At first, we discard some inessential unknowns to compress the linear system without accuracy loss. Then we make the coefficient matrix sparser. In this way, solving the linear system is greatly accelerated. Experiments on various substrates validate that DBEM is several to tens of times faster than DCT-accelerated Green's function methods and the eigendecomposition method, while preserving high accuracy. Besides, another experiment shows that this method is versatile for irregular substrates.

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It is important to model substrate couplings for SoC/mixed-signal circuit designs. After introducing the continuation equation of full current in lossy substrates, we present a new direct boundary element method (DBEM), which can handle the substrates with arbitrary doping profiles. Three techniques can speed up the DBEM remarkably, which include reusing coefficient matrices for multiple-frequency calculation, condensing the linear system, and sparsifying coefficient matrix. Numerical experiments illustrate that DBEM has high accuracy and high efficiency, and is versatile for arbitrary doping profiles.