Finite difference method

Abstract: In this paper we present a reformulation of the FDTD algorithm on nonorthogonal grids, which was originally proposed by Holland in 1983. Based on the matrix-vector notation of the finite integration technique (FIT), the new formulation allows to study a special type of instability, which is due to the spatial discretization and independent of the choice of the timestep. It is shown, that this type of instability can be avoided by a symmetric evaluation of the metric coefficients of the nonorthogonal grid. Two numerical examples demonstrate the stability properties and the high accuracy of the new method.

Abstract: This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is basedonacoupledscalar-vectorpotentialformulationforarbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. Thediscretizationofthepartialdifferentialequationsandthe enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. Theaccuracyandefficiencyofthealgorithmisassessedwith examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic,2Disotropic,and3Danisotropicrockformations.The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potentialformulationfavorablycompareinaccuracywithresults obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electricalconductivity.Ournumericalexercisesindicatethat the coupled scalar-vector potential equations provide a generalandconsistentalgorithmicformulationtosimulateborehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex modelsofelectricalconductivity.

Abstract: In this work, three-dimensional numerical simulations of acoustically excited flow through a millimeter-size circular orifice are conducted to assess its noise damping performance, with particular emphasis on applying the lattice Boltzmann method (LBM) as an alternative computational aeroacoustics tool. The model is intended to solve the discrete lattice Boltzmann equation (LBE) by using the pseudo-particle based technique. The LBE controls the particles associated with collision and propagation over a discrete lattice mesh. Flow variables such as pressure, density, momentum, and internal energy are determined by performing a local integration of the particle distribution at each time step. This is different from the conventional numerical investigation attempting to solve Navier-Stokes (NS) equations by using high order finite-difference or finite-volume methods. Compared with the conventional NS solvers, one of the main advantages of LBM may be a reduced computational cost. Unlike frequency domain simulations, the present investigation is conducted in time domain, and the orifice damping behavior is quantified over a broad frequency range at a time by forcing an oscillating flow with multiple tones. Comparing the numerical results with those obtained from the theoretical models, large eddy simulation, and experimental measurements, good agreement is observed.