Finite difference method

Abstract: MIC and MMIC packages capable of good performance at frequencies as high as 60 GHz need to have small volume, low weight, microstrip and/or coplanar waveguide (CPW) compatibility and exhibit negligible electrical interference with the rest of the circuit. In order to acquire some of these characteristics, special provisions need to be made during circuit layout and design, resulting in high-density packages. The designed circuits have a large number of interconnects which are printed on electrically small surface areas and communicate through the substrate in a direct through-via fashion or electromagnetically through appropriately etched apertures. In a circuit environment of this complexity, parasitic effects such as radiation and cross talk are intensified, thus, making the vertical interconnection problem very critical. In this paper, transitions using through-substrate vias are considered and analyzed both in the time and frequency domains using the Finite Difference Time Domain (FDTD) technique and the Finite Element Method (FEM), respectively. The merits of each method in conjunction with accuracy, computational efficiency and versatility are discussed and results are compared showing excellent agreement. Specifically, a microstrip short-circuit, a microstrip ground pad, a CPW-to-microstrip through-via transition and a channelized CPW-to-microstrip transition are analyzed and their electrical performance is studied. >

Abstract: We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.

Abstract: In order to interpret the logging data from triaxial induction tools, inversion technology has been adopted. To make the inversion process reasonably fast, a fast forward method must be developed. Numerical methods, such as the finite element method and finite difference method, are flexible but slow for inversion purposes. In this paper, an analytic method is developed to simulate induction tools in deviated wells drilled in anisotropic formations. This method can be applied to the triaxial induction tools with transmitting and receiving coils oriented in three mutually perpendicular directions. The axis of the tool may intercept a formation with dip, azimuthal, and orientation angles. Formulations of the electromagnetic fields generated by these three transmitting coils are derived. The derivation uses the coefficient propagator method with the assumption of negligible borehole effect and invasion zones. This method overcomes the problem of numerical overflow without compromising the accuracy of the solution. Because the new induction tool has three transmitting and three receiving coils, a total of nine logs are obtained at each logging depth compared with one or two logs in regular induction or logging while drilling tools.