Finite difference method

Abstract: A numerical procedure is described that will simplify the analysis of the EMP response of structures with dielectric or poorly conducting segments.

Abstract: A flexible and simple way of introducing stress-free boundary conditions for including three-dimensional (3D) topography in the finite-difference method is presented. The 3D topography is discretized in a staircase by stacking unit material cells in a staggered-grid scheme. The shear stresses are distributed on the 12 edges of the unit material cell so that only shear stresses appear on the free surface and normal stresses always remain embedded within the solid region. This configuration makes it possible to implement stress-free boundary conditions at the free surface by setting the Lame coefficients λ and μ to zero without generating any physically unjustified condition. Arbitrary 3D topographies are realized by changing the distribution of λ and μ in the computational domain. Our method uses a parsimonious staggered-grid scheme that requires only 3/4 of the memory used in the conventional staggered-grid scheme in which six stress components and three velocity components need to be stored. Numerical tests indicate that 25 grids per wavelength are required for stable calculation. The finite-difference results are compared with those of the boundary-element method for the two-dimensional (2D) semi-circular canyon model. We also present the responses of a segment of semi-circular canyon and hemispherical cavity to vertically incident plane P, SV , and SH waves and discuss the response of a Gaussian hill to an isotropic point source embedded in the hill. In the segment of semi-circular canyon, the later portions of the synthetics are characterized by phases scattered from the two vertical side walls. The hemispherical cavity and 2D semi-circular canyon both show focusing of energy at the bottom of the cavity, although the focusing effect is stronger in the former geometry. Focusing and defocusing effects due to the strong topography of the Gaussian hill produce a strong amplification of displacements at a spot located on the flank opposite to the source. Backscattering from the top of the hill is also clearly seen.

Abstract: Direct numerical simulations and computational aeroacoustics require an accurate finite difference scheme that has a high order of truncation and high-resolution characteristics in the evaluation of spatial derivatives. Compact finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial truncation orders. An analytic method with a systematic procedure to achieve maximum resolution characteristics is devised for multidiagonal schemes, based on the idea of the minimization of dispersive (phase) errors in the wave number domain, and these are applied to the analytic optimization of multidiagonal compact schemes. Actual performances of the optimized compact schemes with a variety of truncation orders are compared by means of numerical simulations of simple wave convections, and in this way the most effective compact schemes are found for tridiagonal and pentadiagonal cases, respectively. From these comparisons, the usefulness of an optimized high-order tridiagonal compact scheme that is more efficient than a pentadiagonal scheme is discussed. For the optimized high-order spatial schemes, the feasibility of using classical high-order Runge-Kutta time advancing methods is investigated.