3-D held solvers have become popular tools for parasitic capacitance extraction of full custom circuits, IPs, and packages. Traditional held solvers based on hnite difference method/hnite element method/floating random walk truncate the held on the outer boundary of the numerical region by applying Dirichlet or Neumann boundary conditions. However, to ensure high accuracy in the vicinity of circuit elements, a substantial gridded region between the circuit and the outer boundary is necessary in order to reduce distortion of the held caused by these truncation conditions. In this paper, we make a fundamental contribution to the application of numerical held solvers by proposing a class of absorbing boundary conditions, which when implemented, signihcantly reduce the distortion of the held at the numerical boundary, and consequently, throughout the numerical region. The absorbing boundary condition we propose will allow the held throughout the numerical region to behave as though there is no numerical boundary, accurately mimicking the helds in an actual circuit. As a result, the size of the numerical region can be signihcantly reduced, which in turn reduces the run time without sacrihcing accuracy. A mathematical development of the proposed absorbing boundary condition is presented. It is shown that the error of the proposed nth order absorbing boundary is O(1/rn+2), while the error of the traditional Neumann boundary is O(1/r2), where r is the size of the numerical region. Experimental results for capacitance extraction with interconnects in multilayer dielectrics and silicon on insulator show the proposed methods improve the run time and accuracy of numerical solutions of Laplace's equation over previous boundary conditions for uniform or nonuniform meshes.

The electromagnetic field due to a line source radiating in the presence of a two-dimensional composite wedge composed of a number of conducting and dielectric materials is obtained. The Fourier transform path integral method (FTPI) is described and used to perform the numerical analysis. An important feature of the FTPI method is that it is based on a global solution to the Helmholtz scalar wave equation. As such the method avoids numerical enforcement of boundary conditions and the necessity of reformulating the analytical/numerical equations for each geometric configuration. The total scattered field is presented for several cases where one of the dielectric wedge sections is lossy, including examples of microwave scattering from a crested ocean surface and an air-ocean-sea ice interface.>