Frank D. Hastings

Bio: Frank D. Hastings is an academic researcher from Washington State University. The author has contributed to research in topics: Finite-difference time-domain method & Scattering. The author has an hindex of 5, co-authored 6 publications receiving 507 citations.

Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation

Abstract: A method is presented for application of the perfectly matched layer (PML) absorbing boundary condition (ABC) to the P‐SV velocity–stress finite‐difference method The PML consists of a nonphysical material, containing both passive loss and dependent sources, that provides ‘‘active’’ absorption of fields It has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies In this work, numerical simulations are used to compare the PML and an ‘‘optimal’’ second‐order elastic ABC [Peng and Toksoz, J Acoust Soc Am 95, 733–745 (1994)] Reflection factors are used to compare angular performance for continuous wave illumination; snapshots of potentials are used to compare performance for broadband illumination These comparisons clearly demonstrate the superiority of the PML formulation Within the PML there is a 60% increase in the number of unknowns per grid cell relative to the velocity–stress formulation However, the high quality of the PML ABC allows the use of a smaller grid, which can result in a lower overall computational cost

A Monte-Carlo FDTD technique for rough surface scattering

Abstract: A Monte-Carlo finite-difference time-domain (FDTD) technique is developed for wave scattering from randomly rough, one-dimensional surfaces satisfying the Dirichlet boundary condition. Both single-scale Gaussian and multiscale Pierson-Moskowitz surface roughness spectra are considered. Bistatic radar cross sections are calculated as a function of scattering angle for incident angles of 0, 45, 70, and 80 degrees measured from the vertical. The contour path FDTD method is shown to improve accuracy for incident angles greater than 45 degrees. Results compare well with those obtained using a Monte-Carlo integral equation technique. >

A finite-difference time-domain solution to scattering from a rough pressure-release surface

Abstract: The finite-difference time-domain (FDTD) method is a numerical technique that makes no explicit physical approximations to the underlying problem. The quality of a FDTD-based solution typically is determined by the discretization of the computational domain—the smaller the spacing, the more accurate the solution. Unfortunately, for large computational domains, i.e., ones spanning many wavelengths, the small spatial step size needed to obtain a high-fidelity solution may lead to a prohibitively large number of unknowns. Here it is shown how the FDTD method can be used to model accurately scattering from pressure-release surfaces above a homogeneous water column. To keep the computational cost manageable, a number of enhancements to the standard FDTD algorithm are employed. These enhancements include correcting for numerical dispersion along the specular direction of the incident insonification, using locally conformal cells at the pressure-release boundary, and propagating the field through the homogeneous water column via an analytic method. The accuracy of the FDTD approach is demonstrated by comparison with an integral equation-based reference solution to the same rough surface scattering problem [Thorsos, Proceedings of the Reverberation and Scattering Workshop, pp. 3.2–3.20 (1994) Naval Research Laboratory Book Contribution NRL/BE/7181-96-001].

Interactive software for undergraduate electromagnetics

Abstract: To demonstrate the relationship between physical reality and the equations used in electromagnetics, the authors have created interactive software using Mathematica with its notebook capability. The software is composed of different notebooks, each covering a specific topic, which are collectively called EM Notebooks. The notebooks are used in a workstation laboratory of 12 NeXT computers in conjunction with two required junior-level courses in electromagnetics. Each notebook consists of text, equations, and graphics. The equations are Mathematica commands that are used to evaluate electromagnetic formulas found in a typical undergraduate electromagnetics textbook. Equation parameters can be changed by a student permitting examination of an unlimited number of examples. In addition, much of the graphics can be animated. The animations provide a pedagogic tool unavailable in traditional textbooks. EM Notebooks must be used on a computer that runs Mathematica with the notebook facility. >

An FDTD method for analysis of scattering from rough fluid-fluid interfaces

Abstract: A finite-difference time-domain (FDTD) method for scattering by one-dimensional, rough fluid-fluid interfaces is presented, modifications to the traditional FDTD algorithm are implemented which yield greater accuracy at lower computational cost. These modifications include use of a conformal technique, in which the grid conforms locally to the interface, and a correction for the numerical dispersion inherent to the FDTD algorithm, Numerical results are presented for fluid-fluid cases modeling water-sediment interfaces. Two different roughness spectra, the single-scale Gaussian roughness spectrum and a multiscale modified power-law spectrum, are used. The Gaussian results are calculated as a function of the dimensionless parameters kh and kl, where k is the wavenumber in water, h is the rms surface height, and l is the surface correlation length. For the modified power-law spectrum, statistical parameters consistent with an insonification frequency of 7.5 kHz are used. Results are compared with those obtained using an integral equation technique both for scattering from single-surface realizations and for Monte Carlo averages of scattering from an ensemble of surface realizations. Scattering strengths are calculated as a function of scattering angle for an incident angle of 70/spl deg/ (20/spl deg/ grazing). The results agree well over all scattering angles for the cases examined.

Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media

Abstract: We present and analyze a perfectly matched, absorbing layer model for the velocity-stress formulation of elastodynamics. The principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components: a component orthogonal to the boundary and a component parallel to it. A system of equations governing these new unknowns then is constructed. A damping term finally is introduced for the component orthogonal to the boundary. This layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium. In practice, both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers. As we will show with several experiments, this model gives very satisfactory results; namely, the reflection coefficient, even in the case of heterogeneous, anisotropic media, is about 1% for a layer thickness of five space discretization steps.

An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation

Abstract: The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.

A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation

Abstract: SUMMARY The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first-order system in velocity and stress. We demonstrate how to use this condition for the same equation written as a second-order system in displacement. This facilitates use in the context of numerical schemes based upon such a system, e.g. the finite-element method, the spectral-element method and some finite-difference methods. We illustrate the efficiency of this second-order perfectly matched layer based upon 2-D benchmarks with body and surface waves.