Time-domain Formulation Research Articles

An accurate and efficient computational method for time-domain aeroacoustic scattering

This paper presents a convective time-domain equivalent-source method for determining the scattered acoustic pressure field in a uniform moving medium. The proposed method is based on the solution of the time-domain convective Ffowcs Williams–Hawkings (FW–H) equation while the strengths of equivalent sources are determined by the required pressure gradient boundary condition on the scattering surface. The scattered acoustic pressure can be calculated once the strengths of equivalent sources have been determined. The current paper adopts the recently published analytical time-domain formulation for the acoustic pressure gradient in a moving medium to evaluate the incident pressure gradient on the scattering surface. This makes the proposed method considerably more efficient and accurate than a direct method. The total acoustic pressure consists of the scattered and the incident components. The latter can be obtained by the time-domain acoustic pressure formulation of the convective FW–H equation. Causes of possible instability in the proposed method are analyzed and an effective stabilizing method is proposed. Three test cases are considered to demonstrate the validity of the proposed method: a point monopole source field scattered by: (1) a rigid sphere in a stationary medium, (2) an infinite flat plate in uniform flow parallel to its surface, and (3) a cylinder of infinite length in axial uniform flow. To demonstrate the usefulness of the proposed method in practical engineering applications, the scattering of a point monopole source field by a slender wing in uniform flow is considered.

Time-Domain Evaluation of Full-Wave Scattering Center Models

This letter describes how to evaluate a previously derived full-wave scattering center model directly in time domain. The direct time-domain evaluation is particularly important in the case of moving targets, where the time dependency of the radar cross section (RCS) prevents the efficient usage of the scattering center model in frequency domain. The time-domain formulation generates the scattered signal directly in baseband.

On the Design of Two-Dimensional Quincunx Filterbanks with Directional Vanishing Moment Based on Eigenfilter Approach

This paper presents a two-dimensional (2-D) eigenfilter technique to design 2-D perfect reconstruction (PR) filterbanks with directional vanishing moment (DVM). In this paper, we first introduce the DVM constraint for 2-D directional filterbank design. The proposed DVM constraint is imposed in 2-D eigenfilter formulation to obtain maximally flat 2-D low-pass analysis filter. Then, we use this designed analysis filter in 2-D time-domain formulation of PR condition to obtain complementary low-pass synthesis filters. The PR and DVM constraints are imposed to obtain 2-D low-pass synthesis filter. It is shown that the proposed 2-D filters satisfy the PR criteria with prescribed DVM. The performance of 2-D PR filters is evaluated in image denoising application. The performance of designed filterbank is compared with well-known existing methods in terms of peak signal-to-noise ratio to validate the results.

The Frisch scheme for EIV system identification: time and frequency domain formulations

Several estimation methods have been proposed for identifying errors-in-variables systems, where both input and output measurements are corrupted by noise. One of the more interesting approaches is the Frisch scheme. The method can be applied using either time or frequency domain representations. This paper investigates the general mathematical and geometrical aspects of the Frisch scheme, illustrating the analogies and the differences between the time and frequency domain formulations.

Personal Sound Zones by Subband Filtering and Time Domain Optimization

Personal Sound Zones (PSZ) systems aim to render independent sound signals to multiple listeners within a room by using arrays of loudspeakers. One of the algorithms used to provide PSZ is Weighted Pressure Matching (wPM), which computes the filters required to render a desired response in the listening zones while reducing the acoustic energy arriving to the quiet zones. This algorithm can be formulated in time and frequency domains. In general, the time-domain formulation (wPM-TD) can obtain good performance with shorter filters and delays than the frequency-domain formulation (wPM-FD). However, wPM-TD requires higher computation for obtaining the optimal filters. In this article, we propose a novel approach to the wPM algorithm named Weighted Pressure Matching with Subband Decomposition (wPM-SD), which formulates an independent time-domain optimization problem for each of the subbands of a Generalized Discrete Fourier Transform (GDFT) filter bank. Solving the optimization independently for each subband has two main advantages: 1) lower computational complexity than wPM-TD to compute the optimal filters; 2) higher versatility than the classic wPM algorithms, as it allows different configurations (sets of loudspeakers, filter lengths, etc.) in each subband. Moreover, filtering the input signals with a GDFT filter bank (as in wPM-SD) requires lower computational effort than broadband filtering (as in wPM-TD and wPM-FD), which is beneficial for practical PSZ systems. We present experimental evaluations showing that wPM-SD offers very similar performance to wPM-TD. In addition, two cases where the versatility of wPM-SD is beneficial for a PSZ system are presented and experimentally validated.

Angular Distribution and Polarization of Superradiant Emission from Atomic Ensembles

A density-matrix approach is developed to provide a theoretical description of the intensity, angular distribution, and polarization of superradiative emission from an ensemble of many-electron atomic systems. The many-electron atomic systems are described as cooperatively interacting by means of forces that can be long range. Particular emphasis is given to the coherent excitation of the collective atomic-ensemble states, which may be produced by incident laser radiation. The initial excitation and spontaneous emission processes may be described as independent. Both frequency-domain and time-domain formulations of the density-matrix approach are developed. The collective atomic-ensemble states are specified in a detailed hyperfine representation, corresponding to successively coupling the individual hyperfine angular momenta F pertaining to the many-electron atoms. A less detailed fine-structure angular-momentum representation may also be used. In the density-operator approach, account can be taken of the coherent excitation of a particular subspace of the initial atomic-ensemble states. For a comprehensive and unified development of time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations, a reduced-density-matrix (quantum-open-systems) approach is introduced. The non-equilibrium atomic-ensemble-state kinetics and the homogeneous spectral-line shapes can thereby be systematically and self-consistently determined. The collective atomic-ensemble states may be obtained using a variety of different methods. These states can be determined using a dressed-state approach, in which the required states are calculated in the presence of an electromagnetic field.

Two-dimensional finite-difference time-domain formulation for sound propagation in a temperature-dependent elastomer-fluid medium.

This study focuses on the two-dimensional (2-D) finite-difference time-domain (FDTD) formulations to investigate the acoustic wave propagation in elastomers contained in a fluid region under different thermal conditions. The developed FDTD formulation is based on a direct solution of the time-domain wave equation and the Havriliak-Negami (H-N) dynamic mechanical response of the elastomers. The H-N representation, including double fractional derivative operators, can be accurately transferred from the frequency-domain to the time-domain by using Riemann-Liouville theory and the Grunwald-Letnikov operator for fractional derivative approximations. Since the Williams-Landel-Ferry shift function is related to the relaxation time for different thermal conditions, the proposed scheme represents a simple and accurate prediction of acoustic wave propagation for varying thermal conditions. The pulse-wave propagation in a viscous fluid field is simulated by investigating the Navier-Stokes equations. The acoustic properties of different elastomers in a variety of temperatures are obtained by means of the proposed FDTD formulation and validated by a good agreement with the experimental data over a wide frequency range. Additionally, the 2-D examples relevant to wave propagation in different elastomers contained in a fluid field are implemented. The proposed FDTD formulation can be used to predict 2-D acoustic wave propagation in different thermal conditions accurately.

A Precise-Integration Time-Domain Formulation Based on Auxiliary Differential Equation for Transient Propagation in Plasma

The electromagnetic wave propagation through plasma medium is one of the most important research fields in computational electromagnetics. A numerical formulation based on both the auxiliary differential equation (ADE) and the precise-integration time-domain (PITD) method for solving the plasma problems is proposed to break through the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the current density J is introduced as the auxiliary variable to deal with the complex permittivity of the plasma which is dependent on the frequency, and the precise integration (PI) technique makes the selectable maximum time-step size become much larger and removes the impact of the time-step size to the numerical dispersion error. Numerical experimentations of the typical plasma problems verify and validate the reliability of the proposed formulation. Through the numerical results, it can be found that the maximum allowable time-step size of the new method is much larger than that of the CFL limit of the FDTD method, and the calculation error of the new method is nearly independent of the time-step size. As a consequence, the execution time is significantly reduced by using a larger time-step size.

FDTD Formulation Based on High-Order Surface Impedance Boundary Conditions for Frequency-Dependent Lossy Multi-Conductor Transmission Lines

A finite-difference time-domain (FDTD) formulation for a uniform frequency-dependent lossy multi-conductor transmission line (MTL) is presented. It is based on a distributed parameter MTL model using partial resistances and partial inductances. In order to consider both skin and proximity effects, the partial internal impedances are computed with a boundary element method (BEM) formulation, enforcing a high-order surface impedance boundary condition (SIBC). The proposed FDTD formulation is numerically validated with two test cases.

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

Time domain imaging of extended transient noise sources using phase coherence.

An acoustic imaging algorithm is proposed herein for transient noise source time reconstruction. Time domain formulations are not well suited for acoustic imaging because of the size of the resulting system to be inversed. Based on the phase coherence principle widely used in ultrasound imaging and image processing, the first step of the algorithm consists in proposing the phase coherence metric used to reject pixels that are unlikely to contribute to the radiated sound field. This translates in a reduction of the domain size and ill-posedness of the problem. In the second step, the inverse problem is solved using the Tikhonov regularization and the generalized cross-validation to extract the vibration field on the imaging domain. Two test cases are considered: a simulated baffled piston and a panel submitted to a mechanical impact in anechoic conditions. The actual vibration field of the panel is measured with an optical technique for reference. In both numerical and experimental cases, the reconstructed vibration field using the proposed approach compares well with their respective reference. The results confirm that transient excitations can be localized and quantified with the proposed approach, in contrast with the classical time-domain beamforming that dramatically overestimates its magnitude.

Smart Graphene-Based Frequency Selective Surface Designed to Act as Single or Dual Band Device with Reconfigurable Operation Bands

In this paper, a finite-difference time-domain (FDTD) formulation, based on the exponential matrix and on thin material sheet methods is developed for modeling subcellular thin graphene sheets. This formulation is validated by reproducing graphene frequency selective surfaces (FSS) known from literature. Then, we propose in this work a smart graphene FSS device. Smartness is obtained by means of a unity cell formed by a graphene ring with a graphene sheet placed in its aperture. By properly regulating the chemical potentials of the graphene elements, two frequency-tunable modus operandi are obtained: single- or dual-band rejection modes. When the device operates in its dual-band rejection mode, either of the rejection bands can be shifted individually in the frequency spectrum. Additionally, both rejection bands can also be reconfigured simultaneously.

A general time-domain formulation of equation-of-motion coupled-cluster theory for linear spectroscopy.

A time-dependent (TD) formulation of equation-of-motion (EOM) coupled-cluster (CC) theory is developed, which, unlike other similar TD-EOM-CC approaches [D. R. Nascimento and A. E. DePrince III, J. Chem. Theory Comput. 12, 5834-5840 (2016)], can be applied to any type of linear electronic spectroscopy. The TD-EOM-CC method is formally equivalent to the standard frequency-domain formulation of EOM-CC theory, with a potential computational advantage of a comparatively low memory footprint. This general TD-EOM-CC framework is applied to the linear absorption and electric circular dichroism spectra of several small oxirane derivatives.

Long-Time Instability of Pseudospectral Time-Domain Method in Curvilinear Coordinates

We found that the pseudospectral time-domain (PSTD) formulation discussed in the literature exhibits long-time instability when applied to problems with curvilinear geometry. To overcome this problem, we have developed a provably stable PSTD formulation in curvilinear coordinates. In this article, we discuss the new PSTD formulation and present a rigorous proof of stability. Numerical results are given to validate the new formulation. The new formulation requires only a simple change in the form of the spatial-derivative operators in curvilinear coordinates and so can easily be implemented in the existing PSTD codes.

Analytical formulation to estimate the dynamic energy loss in electrical steels: Effectiveness and limitations

This article aims to investigate the accuracy in modeling and characterization of the dynamic energy loss of modern laminated electrical steels in use for power electronics applications. Measurements by an Epstein Frame are performed in case of sinusoidal and non-sinusoidal waveform of the magnetic induction, at different frequencies and peak values. The Steinmetz and the time domain statistical loss theory, either in frequency domain are used, and the related parameters are fitted experimentally. The dynamic energy loss is also modeled and estimated by means of a finite element method (FEM) formulation in time domain, coupled with a vector Preisach model to represent the constitutive law of the magnetic material. The series of the results reported indicates effectiveness and limitations of the modeling approaches considered. Some considerations about the excess loss are also given.

Time-Domain Bloch Boundary Conditions for Efficient Simulation of Thermoacoustic Limit Cycles in (Can-)Annular Combustors

Abstract Thermo-acoustic eigenmodes of annular or can-annular combustion chambers, which typically feature a discrete rotational symmetry, may be computed in an efficient manner by utilizing the Bloch-wave theory. Unfortunately, the application of the Bloch-wave theory to combustion dynamics has hitherto been limited to the frequency domain. In this study, we present a time-domain formulation of Bloch boundary conditions (BBC), which allows to employ them in time domain simulations, e.g., computational fluid dynamics (CFD) simulations. The BBCs are expressed as acoustic scattering matrices and translated to complex-valued state-space systems. In a hybrid approach an unsteady, compressible CFD simulation of the burner-flame zone is coupled via characteristic-based state-space boundary conditions to a reduced order model of the combustor acoustics that includes BBCs. The acoustic model with BBC accounts for cross-can acoustic coupling and the discrete rotational symmetry of the configuration, while the CFD simulation accounts for the nonlinear flow–flame acoustic interactions. This approach makes it possible to model limit cycle oscillations of (can-)annular combustors at drastically reduced computational cost compared to CFD simulations of the full configuration and without the limitations of weakly nonlinear approaches that utilize a flame describing function. In this study, the suggested approach is applied to a generic multican combustor. Results agree well with a fully compressible CFD simulation of the complete configuration.

Efficient, wide-band coupled structural-acoustic computations combining time and frequency domain finite elements/equivalent sources

Time and frequency domain computations are complementary, and here, the overall goal is to combine them to generate wide-band solutions of coupled structural-acoustic problems. Modal frequency response computations are efficient for low frequencies but become inefficient as the number of basis functions required to represent the response grows large. Modal formulations are required for large problems because the coupled equation system becomes densely populated when acoustic pressure forces are included. In contrast, the equation system remains sparsely populated in time domain formulations when the time step size is chosen appropriately, allowing large problems to be solved efficiently in terms of nodal degrees-of-freedom. However, the analysis times depend linearly on the number of time steps and a large number are generally required to represent the ringing response of low frequency modes. Here, the number of time steps is limited by applying an exponential taper to the time domain solution and truncating the computations. This reduces leakage effects, but does not affect components of response that have damped out before it is applied. Examples are given to illustrate the computations and the tradeoff between number of basis functions in the modal frequency response computations and number of time steps in the transient analyses.Time and frequency domain computations are complementary, and here, the overall goal is to combine them to generate wide-band solutions of coupled structural-acoustic problems. Modal frequency response computations are efficient for low frequencies but become inefficient as the number of basis functions required to represent the response grows large. Modal formulations are required for large problems because the coupled equation system becomes densely populated when acoustic pressure forces are included. In contrast, the equation system remains sparsely populated in time domain formulations when the time step size is chosen appropriately, allowing large problems to be solved efficiently in terms of nodal degrees-of-freedom. However, the analysis times depend linearly on the number of time steps and a large number are generally required to represent the ringing response of low frequency modes. Here, the number of time steps is limited by applying an exponential taper to the time domain solution and truncat...

Correction to: A Time-Domain Formulation of Elastic Waves in Heterogeneous Unbounded Domains

There are a few errors in the published version of the article. Three equations should be corrected, and one equation should be added to the original article.

Research on an Optimization Method for a Partially Responsive Continuous Phase Modulated (CPM) Signal Based on an Optimal Generic Function

This paper establishes an optimal generic function model in order to obtain a continuous phase modulated (CPM) signal with a smoother phase modulation function. This is achieved by finding the solution to the symbol signals at different lengths of the CPM function. In the solution process, the unknown amount that needs to be solved is reduced by using the even function symmetry characteristic of the signal to be solved. For each different form of the signal, the time domain form of the CPM function and the corresponding normalized energy spectral density are compared under the influence of the phase modulation signal length and the generic function parameter n. The data transmission rate is improved by introducing inter-symbol interference, and the modulation process is realized using six-way parallel transmission when the CPM function is 6T. The simulation results show that the CPM function obtained by establishing an optimal generic function model has high-quality time-frequency characteristics. The real-time phase trajectory and the high-order derivative are both continuous, and the modulated signal has constant envelope characteristics. The CPM function has a fast rolling-off in the frequency domain and small out-of-band radiation, which greatly improves the characteristics of the frequency band utilization.

Encoding the electrodynamics in spatiotemporal boundaries

Maxwell’s equations represented by differential operators describe the local dependence between both electric and magnetic fields in every location of space and time. This description responds to the field concept proposed by Michael Faraday and formalized by James Clerk Maxwell. The discretized version in the form of the finite-difference time-domain (FDTD) technique relies on a proposal of local computation of the respective differential operators making use of the central difference approximation of the second derivative of functions through the Taylor series expansion. This article introduces a novel time-symmetric “non-local” technique from the mathematical formalism of electromagnetic potentials in the wave equation and its physical interpretation in the Minkowski spacetime. In the proposed case study, the “non-local” proposal is 1643 faster than FDTD, i.e., more than three orders of magnitude, and it uses a time-step that is 4096 times greater than the Courant–Friedrichs–Lewy limit without encountering stability problems. New electromagnetic potentials are calculated from the previous and distant ones located at vertices of spatiotemporal regions called causal diamonds that tessellates spacetime without the need for computations inside them. We show that the performance gain is proportional to the size of the spacetime tessellations because the ratio “domain to boundary” increases as the domain extension does. Consequently, our novel “non-local” approach provides a reduction in computational complexity and a more comprehensible explanation of their fundamental physical aspects, without contradicting the principles of the successful classical field theory. Program summaryProgram Title: SOME_modelsProgram Files doi:http://dx.doi.org/10.17632/7sgh6grm7g.1Licensing provisions: GPLv3Programming language: CNature of problem: The Courant–Friedrichs–Lewy (CFL) condition imposes a numerical bound to the size of the Yee cell in the FDTD approach. This restriction causes excessive oversampling, degrading the computation time of this technique.Solution method: We applied the parallelogram rule to solve the wave equation for the case of (1+1) time–space dimensions. With this rule, we proposed a novel algorithm that allows encoding potentials along the characteristic lines that tessellate the spacetime. The new technique produces higher performance gains than those achieved through traditional FDTD.