Time-domain Boundary Condition Research Articles

Higher‐order impedance boundary conditions for finite difference solutions to the wave equation.

Higher‐order finite difference methods for the solution of the wave equation are well established. The interchangeability of temporal and spatial derivatives allows for exact representations of the differential equation in difference forms to the order of accuracy desired. Realistic impedance conditions, however, germane to problems of an architectural acoustics origin, are typically given as ratios of pressure and velocity in the frequency domain. This work develops fourth order‐accurate time domain boundary conditions for the types of materials typically encountered in the architectural acoustics field. Accuracy and agreement of numerical simulations to scale model measurements are compared and discussed, as is computational cost relative to lower‐order methods.

Analysis of Class-E Based RF Power Amplifiers Using Harmonic Modeling

An analysis of the Class-E power amplifier is given using harmonic modeling that facilitates its characterization for a wide load-space regime expected in applications such as RF plasma generation for semiconductor manufacturing. Using a hybrid matrix-framework made up of the dominant harmonics of the circuit variables aided by the time-domain boundary conditions of the power switch voltage waveform, a quasi-linear algorithm is developed to accurately determine the voltage and the current levels expected on all components of the power amplifier for arbitrary loading and operating conditions. The application of the algorithm for design optimization is discussed, and its extension to phase-controlled Class-E power amplifier pairs is described.

Thin-Wire Model Using Subcellular Extensions in the Finite-Difference Time-Domain Analysis of Thin and Lossy Insulated Cylindrical Structures in Lossy Media

Recently, a subcellular thin-wire model for the finite-difference time-domain (FDTD) simulation of resistively coated cylinders with lossless insulating and surrounding media was presented. In this paper, it is shown that this model can be extended to lossy cases. The material discontinuity between lossy insulating and surrounding media is corrected as the time-domain boundary condition. The convolution term of the boundary condition is solved by employing a recursive technique. Applying the contour-path integration to the FDTD unit cells around the wire, one may find the coarse-grid-based equation with the correction term and factors for the material discontinuity and the quasi-static field behavior around the wire. In the 2-D cylindrical coordinates with rotational symmetry, the validity of the proposed model is confirmed by an impedance analysis of insulated and resistive antennas according to the electrical properties of insulating and surrounding media, as well as the choice of cell size.

Time-domain simulations of sound propagation in a stratified atmosphere over an impedance ground

Finite-difference time-domain simulations of broadband sound propagation in a stratified atmosphere are presented. A method recently proposed to obtain an impedance time-domain boundary condition is implemented in a linearized Euler equations solver, which enables to study long range sound propagation over an impedance ground. Some features of the pressure pulse evolution with time are analyzed in both upward-and downward-refracting conditions, and the time-domain simulations are compared to parabolic equation calculations in the frequency domain to show the effectiveness of the proposed impedance boundary condition.

Time‐domain modeling of porous media acoustics

Sound waves propagating in porous media are subject to strong dissipation and dispersion. This paper elaborates upon several recent publications by the authors regarding the time‐domain description of these effects. The foundation is a relaxational description of the viscous and thermal dissipation in a rigid porous medium, which is shown to possess an exact, analytical conversion from the frequency to the time domain. The complex density and bulk modulus operators transform to temporal convolutions between a causal response function and the acoustic field variables. When the convolutions are neglected, the equations reduce to the well known Zwikker‐Kosten phenomenological model. The relaxation function can be inverted to provide an equivalent time‐domain formulation from the complex volume and compressibility operators. Although a direct time‐domain transformation of the specific impedance from the relaxation model has not been found, an accurate broadband approximation thereof can be transformed. The resulting time‐domain boundary condition (TDBC) describes the absorptive and reactive response of the material. The response of the boundary decays slowly, as the inverse square root of time, but efficient numerical procedures are formulated that allow the TDBC to be approximated with a small number of recursive filters.

Padé approximation in time-domain boundary conditions of porous surfaces

Formulation and implementation of time-domain boundary conditions (TDBCs) at the surface of a reactive porous material are made challenging by the slow decay, complexity, or noncausal nature of many commonly used models of porous materials. In this paper, approaches are described that improve computational efficiency and enforce causality. One approach involves approximating the known TDBC for the modified Zwikker-Kosten impedance model as a summation of decaying exponential functions. A second approach, which can be applied to any impedance model, involves replacing the characteristic admittance with its Padé approximation. Then, approximating fractional derivatives with decaying exponentials, a causal and recursive TDBC is formulated.

Time-domain boundary conditions in atmospheric acoustics

In atmospheric acoustics, formulation of a reactive, time-domain boundary condition (TDBC) at the ground surface is a challenging problem since many commonly used models of the ground are noncausal and have a slowly decaying response. A correctly formulated TDBC can significantly simplify finite-difference time-domain simulation of outdoor sound propagation, which is a rapidly developing field in computational acoustics. In the present paper, approaches are developed that enforce causality and improve computational efficiency of TDBCs. First, an approach is developed that allows one to derive a causal TDBC for any impedance model of the ground. A Pade approximation is used to represent the characteristic admittance. Then, using fractional derivatives, a causal TDBC is formulated. This method is illustrated by formulation of TDBCs for the Zwikker-Kosten (ZK) and Attenborough models of the ground. Furthermore, by approximating the fractional derivatives as a summation of decaying exponentials, an effective recursive algorithm for implementation of these TDBCs is outlined. Second, it is demonstrated that approximating the known TDBC for the ZK model as a summation of decaying exponential functions significantly improves the computational efficiency of the method.

Treatment of boundary conditions by finite difference time domain method

In this paper, we propose a simple method that considers boundary conditions in a finite difference time domain (FDTD) scheme by varying density, sound speed and flow resistance. A method based on a Rayleigh model is also proposed, and by these methods, we can design the frequency characteristics of normal incident absorption coefficient arbitrarily. These methods have three advantages: 1. easy coding, 2. easy designing of a frequency characteristic of normal incident absorption coefficient and 3. easy configuration of material thickness. For example, by our method, we can simulate the sound field in a reverberation chamber with a thick material such as glass wool. To confirm the accuracy of the model used, we compare the normal incident absorption coefficient with a one-dimensional exact solution. Results show that the model is sufficiently accurate. Although our method requires a high cost for calculation power and memory, a practical increase in elapsed time can be ignored. This method provides an easy way of analyzing the inner region of a material.

A smoothed particle interpolation scheme for transient electromagnetic simulation

In this paper, the fundamentals of a mesh-free particle numerical method for electromagnetic transient simulation are presented. The smoothed particle interpolation methodology is used by considering the particles as interpolation points in which the electromagnetic field components are computed. The particles can be arbitrarily placed in the problem domain: No regular grid, nor connectivity laws among the particles, have to be initially stated. Thus, the particles can be thickened only in distinct confined areas, where the electromagnetic field rapidly varies or in those regions in which objects of complex shape have to be simulated. Maxwell's equations with the assigned boundary and initial conditions in time domain are numerically solved by means of the proposed method. Validation of the model is carried out by comparing the results with those obtained by the FDTD method for a one-dimensional (1-D) case study in order to easily show the capability of the proposed scheme

Helmholtz-Like Resonator Self-Sustained Oscillations, Part 2: Detailed Flow Measurements and Numerical Simulations

A global description of the effect of the neck geometry on self-sustained oscillations of a grazing e ow along a Helmholtz-like resonator has been given in a companion paper. Detailed e ow measurements taken by means of hot-wire anemometry and numerical simulations based on the Euler equations for inviscid and two-dimensional compressible e ows are now given. Vortex shedding is obtained in an inviscid e ow simulation by considering a neck geometry with sharp edgesat which the code predictse ow separation. Although two-dimensional e owcalculations are attractive because of their computational efe ciency, they are not able to represent the three-dimensional acoustical radiation from the resonator into free space without special frequency-dependent boundary condition treatments. Frequency-independent time-domain boundary conditions are considered. In view of the crudeness of this approximation, the agreement between theory and experiments is quite fair. The effects of changes in the geometry of theneck are qualitatively predicted by the model. The detailed e owinformation provides someinsight into the ine uence of the shape of the upstream edge of the neck that could not be obtained from analytical models proposed in the companion paper.

FDTD Analysts of Dielectric-Loaded Longitudinally Slotted Rectangular Waveguides

A versatile electromagnetic (EM) computational algorithm, based on the Finite-Difference Time-Domain (FDTD) technique, is developed to analyze longitudinally oriented, square-ended, single slot fixtures and slot-pair configurations cut in the broad wall of a WR-975 guide operating at a frequency of 915 MHz. The finite conductivity of the wave guide walls is accountedfor by employing a time-domain Surface-Impedance Boundary Conditions (SIBC) formulation. The proposed FDTD algorithm has been validated against measurements performed on a probe-excited slot cut along the center line of the broad wall of a WR-284 guide and available experimental data for energy coupled from a longitudinal slot pair in the broad wail of a WR-340 guide. Numerical results are presented to exploit the influence of the constitutive parameters of the processed material as well as protective insulating window slabs mounted on the exterior surface of the slots. Particular attention is given to the resonant length, scattering parameters, and the electric field distribution within lossy objects placed in the near-field region over a range of slot offsets and workloads with extensive results being reportedfor the first time. It is shown that the FDTD technique can accurately predict the coupling and power absorption characteristics in loads located in the near field zone of the slotted wave guide structures and, therefore, should prove to be a powerful design tool applicable to a wide class of slotted waveguide applicators that may be difficult to analyze using other available techniques.

TIME-DOMAIN SIMULATION OF ACOUSTIC SOURCES OVER AN IMPEDANCE PLANE

An implicit high-order compact unconditionally stable finite-difference time-domain (FDTD) method is proposed here for numerical solution of point sources over an impedance plane. In this method, the linearized Euler equations are split into three directional sets and twelve simple wave components, or six when equivalent sources are adopted with no mean flow. Each component is solved using a fourth-order Padé approximant in space and second-order trapezoidal integration in time. The concept of reflection coefficient is used and algebraically modeled to develop time-domain impedance-equivalent boundary conditions. Comparisons with established methods for reflections of harmonic or impulsive sources demonstrate the applicability of this method for general impedance value, source type, or their arbitrary distributions. Examples of using typical wool felt and grass ground impedances are given to illustrate its practicality and effectiveness. This method provides a means through which time-domain theories and procedures for in-situ characterization of impedance surfaces can be developed.

A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution

We present a multidomain pseudospectral method for the accurate and efficient time-domain computation of scattering by body-of-revolution (BOR) perfectly electrically conducting objects. In the BOR formulation of the Maxwell equations, the azimuthal dependence of the fields is accounted for analytically through a Fourier series. The numerical scheme in the (r,z) plane is developed in general curvilinear coordinates and the method of characteristics is applied for patching field values in the individual subdomains to obtain the global solution. A modified matched-layer method is used for terminating the computational domain and special attention is given to proper treatment of the coordinate singularity in the scattered field formulation and correct time-domain boundary conditions along edges. Numerical results for monochromatic plane wave scattering by smooth and nonsmooth axis-symmetric objects, including spheres, cone-spheres, and finite cylinders, is compared with results from the literature, illustrating the accuracy and computational efficiency associated with the use of properly constructed spectral methods. To emphasize the versatility of the presented framework, we compute plane wave scattering by a missile and find satisfactory agreement with method-of-moment (MoM) computations.

Unified two-dimensional model for grating dynamics in photorefractive materials

A two-dimensional treatment suitable both for perpendicular (Pockels readout optical modulators, etc.) and for parallel (image amplifiers, etc.) photorefractive devices is presented. A partial differential equation of the third order, valid for both families of devices, is set up and solved analytically for the relevant initial and boundary conditions in the full time domain. The results are illustrated in a few examples showing the relative significance of bulk and surface charges, and some new features of the solution for the perpendicular configuration.

Time-domain impedance boundary conditions for computational aeroacoustics

It is an accepted practice in aeroacoustics to characterize the properties of an acoustically treated surface by a quantity known as impedance. Impedance is a complex quantity. As such, it is designed primarily for frequency-domain analysis. Time-domain boundary conditions that are the equivalent of the frequency-domain impedance boundary condition are proposed. Both single frequency and model broadband time-domain impedance boundary conditions are provided. It is shown that the proposed boundary conditions, together with the linearized Euler equations, form well-posed initial boundary value problems. Unlike ill-posed problems, they are free from spurious instabilities that would render time-marching computational solutions impossible.

A comparison of time domain boundary conditions for acoustic waves in wave guides

We consider several types of boundary conditions in the context of time domain models for acoustic waves. Experiments with four different duct terminations (hardwall, free radiation, foam, wedge) were carried out in a wave duct from which reflection coefficients over a wide frequency range were measured. These reflection coefficients are used to estimate parameters in the time domain boundary conditions, and a comparison of the relative merits of the models in describing the data is presented. Boundary conditions that yield a good fit of the model to the experimental data were found for all duct terminations except the wedge.

Time-domain inverse scattering using the local shape function (LSF) method

A non-linear inverse scattering algorithm is presented that uses a local shape function (LSF) approximation to parametrize very strong scatterers in the presence of a transient excitation source. The LSF approximation was presented recently in the context of continuous-wave (CW) excitation and was shown to give good reconstructions of strong scatterers such as metallic objects. It is shown that the local (binary) shape function may be implemented as a volumetric boundary condition in a finite-difference time domain (FDTD) forward scattering solver. The inverse scattering problem is then cast as a non-linear optimization problem where the N-dimensional Frechet derivative of the scattered field is computed as a single backpropagation and correlation using the FDTD forward solver. Connection between the new algorithm and a similar method employing the distorted Born approximation is shown. Computer simulations show that the LSF method employing a FDTD forward solver has superior convergence properties over the corresponding distorted-Born algorithm.