Plane Wave Time Domain Algorithm Research Articles

Analysis of transient wave scattering from rigid bodies using a Burton–Miller approach

Transient scattering from closed rigid bodies can be analyzed using a variety of time domain integral equations, e.g., the Kirchhoff integral equation and its normal derivative. Unfortunately, when the spectrum of the incident field includes one or more of the resonance frequencies of the corresponding interior problem, the solutions to these time domain integral equations become corrupted with spurious interior modes. In this article, this phenomenon is demonstrated via numerical experiments, and a Burton–Miller-type time domain combined field integral equation is proposed as a remedy. To verify that the solutions to this Burton–Miller-type equation are not corrupted by interior modes, various numerical results are presented. It is anticipated that this equation, when used in conjunction with fast time domain integral equation solvers (e.g., plane wave time domain algorithms), will enable the accurate analysis of transient wave scattering from acoustically large bodies.

Acceleration of integral equation based transient analysis of acoustic scattering phenomena using the plane wave time domain algorithm

The numerical analysis of transient scattering from homogeneous bodies residing in an unbounded medium is often performed using integral equation (IE) based methods. The major advantages of these methods over differential equation based techniques are that (i) the radiation condition is implicitly imposed in the formulation, and that (ii) the number of spatial unknowns, Ns, scales as the surface area of the scatterer. However, IE techniques are notoriously costly as their computational complexity scales as O(N2s) per time step. This complexity can be reduced considerably by using the recently introduced plane wave time domain (PWTD) algorithm [A. A. Ergin et al., J. Comput. Phys. 146, 157–180 (1998)]. In this presentation, the PWTD algorithm will be briefly reviewed and its incorporation into an IE-based transient solver for analyzing scattering from rigid bodies will be elucidated. It will be shown that the computational cost of a transient analysis using two-level and multilevel PWTD enhanced solvers scales as O(Ns1.5<th>log<th>Ns) and O(Ns<th>log2<th>Ns) per time step, respectively. Numerical examples that validate these complexity estimates and demonstrate the efficacy of the proposed methods in analyzing scattering from realistic structures will be presented.

Exact global boundary condition with reduced computational complexity for grid truncation in acoustic finite difference time domain simulations

The numerical analysis of transient acoustic radiation by and scattering from inhomogeneous bodies is often carried out using the finite difference time domain (FDTD) method. However, FDTD grids must be truncated with absorbing boundary conditions when the structure of interest resides in an unbounded medium. Local boundary conditions have to be imposed on a convex outer boundary, thereby often inflating the problem size. This drawback is alleviated if the grid is truncated using exact boundary conditions that rely on a retarded-time boundary integral (RTBI) [L. Ting and M. J. Miksis, J. Acoust. Soc. Am. 80, 1825–1827 (1986)]. Unfortunately, for a 3-D computational domain comprising of O(N3) FDTD cells, where N denotes the number of cells per unit length, the evaluation of the RTBI using classical methods requires O(N4) operations per time step, which renders the RTBI approach impractical. In this presentation, it will be shown that this computational bottleneck can be alleviated by employing the plane wave time domain algorithm [A. A. Ergin et al., J. Comput. Phys. 146, 157–180 (1998)] using which the RTBI can be evaluated in O(N2<th>log2<th>N) operations. Numerical results that demonstrate the efficacy and reduced complexity of the method will be presented.

The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena

This article describes a plane-wave time-domain (PWTD) algorithm that facilitates the fast evaluation of transient wave fields produced by surface scattering. The algorithm presented relies on a Whittaker-type expansion of transient fields in terms of propagating plane waves. The incorporation of the PWTD scheme into existing matching-on-in-time- (MOT-) based integral-equation solvers is elucidated. It is shown that the computational cost of performing a surface-scattering analysis, using two-level and multilevel PWTD-enhanced MOT schemes, scales as O(N/sub t/N/sub s//sup 1.5/ log N/sub s/) and O(N/sub t/N/sub s/log/sup 2/N/sub s/), respectively, when the surface source density is represented by N/sub s/ spatial and N/sub t/ temporal samples. Hence, the computational cost of the proposed algorithms scales much more favorably than that of classical MOT schemes, which scale as O(N/sub t/N/sub s//sup 2/). Therefore, PWTD-enhanced MOT schemes make possible the analysis of broadband scattering from structures of unprecedented dimensions.

Fast Evaluation of Three-Dimensional Transient Wave Fields Using Diagonal Translation Operators

This paper presents novel plane wave time domain (PWTD) algorithms which accelerate the computational analysis of transient surface scattering phenomena. The proposed PWTD algorithms permit the fast evaluation of transient fields satisfying the wave equation. The cost associated with the computation of fields atNsobservers produced by a surface bound source density represented in terms ofNsspatial samples forNttime steps scales asO(NtN2s) if classical time domain integral-equation-based methods are used. It is shown that this cost can be reduced toO(NtN4/3slogNs) andO(NtNslogNs) using two-level and multilevel PWTD schemes, respectively. These algorithms are the time domain counterparts of frequency domain fast multipole methods and make feasible the practical broadband analysis of scattering from large and complex bodies.