Time-domain Equation Research Articles

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

Motion limiting nonlinear dynamics and frequency characteristics of pipes reinforced with carbon nanotube agglomerations coupled with piezoelectric actuator

In the current work, for the first time, the motion limiting nonlinear dynamics and frequency analysis of a nanopipe reinforced with carbon nanotube agglomerations (CNTAs) is presented. The current composite structure is coupled with a piezoelectric actuator for electrical purposes. The first-order shear deformation theory (FSDT) is presented for displacement fields of two layers (composite and piezoelectric). Nonlocal strain gradient theory (NSGT) is used to consider the size effects. For mathematical modeling of the nanocomposite layer, the effective Poisson’s ratio, Young’s modulus, thermal expansion in addition to carbon nanotubes (CNTs) clusters/polymer density are presented. The deflection of the pipe is described through Cartesian and cylindrical coordinates. The nonlinear problem equations are extracted as two different 1-D models which are perpendicular to each other. Generalized differential quadrature method (GDQM) and Multiple scales solution procedure (MSSP) were used for solving the nonlinear equations in displacement and time domains, respectively. Finally, the outcomes of the current report demonstrate that the area ratio of piezoelectric layer, CNTs’ volume fraction, boundary conditions, and size-dependent parameter have an important role in the nonlinear dynamics of the composite nanopipes coupled with the piezoelectric layer.

Numerical Simulation of Elastic Wave Field in Viscoelastic Two-Phasic Porous Materials Based on Constant Q Fractional-Order BISQ Model.

The fractional-order differential operator describes history dependence and global correlation. In this paper, we use this trait to describe the viscoelastic characteristics of the solid skeleton of a viscoelastic two-phasic porous material. Combining Kjartansson constant Q fractional order theory with the BISQ theory, a new BISQ model is proposed to simulate elastic wave propagation in a viscoelastic two-phasic porous material. The corresponding time-domain wave propagation equations are derived, and then the elastic waves are numerically simulated in different cases. The integer-order derivatives are discretised using higher-order staggered-grid finite differences, and the fractional-order time derivatives are discretised using short-time memory central differences. Numerical simulations and analysis of the wave field characterisation in different phase boundaries, different quality factor groups, and multilayered materials containing buried bodies are carried out. The simulation results show that it is feasible to combine the constant Q fractional-order derivative theory with the BISQ theory to simulate elastic waves in viscoelastic two-phasic porous materials. The combination can better describe the viscoelastic characteristics of the viscoelastic two-phasic porous materials, which is of great significance for further understanding the propagation mechanism of elastic waves in viscoelastic two-phasic porous materials and viscoelastic two-phasic porous materials containing buried bodies. This paper provides a theoretical forward simulation for fine inversion and reconstruction of layer information and buried body structure in viscoelastic two-phasic porous materials.

P3 approximation equation of light transport in a slab medium: steady-state and time domains

P3 equations in a slab medium in the steady state and the time domain are proposed in this study. The spatially resolved diffuse reflectance and transmittance of the steady-state P3 equation are presented. Two time-domain equations in a slab medium, the optical parameter method and the double diffusion coefficient method, are established. The Monte Carlo simulation acted as an experimental result for comparison, and the results from the diffusion equation and P3 were compared against the Monte Carlo results. The results showed that the detection distance was less than twice the thickness of the medium, and the P3 equation results were in good agreement with the Monte Carlo simulation results. For the steady state, the P3 equation was more accurate than the diffusion equation when the distance was greater than 0.6 mm and less than 2 mm. However, when the absorption coefficient increased, the P3 steady-state equation was more accurate than the diffusion equation. In the time domain, the optical parameter method was more accurate than the double diffusion coefficient method and the diffusion equation. Therefore, the P3 approximation model can replace the diffusion model, and it can be applied to study tissue optics.

Moving force identification based on learning dictionary with double sparsity

Moving force identification (MFI) is essential for the bridge safety as it is one of the major loads acting on the bridge deck. MFI techniques based on force dictionary are promising owing to their prominent performance in solving ill-posed problems and calculation efficiency. Since the specific forms of authentic moving forces are complex and unknown, a fixed dictionary normally adopted tends to fail in expressing moving forces sparsely enough. In this study, dictionary learning (DL) is introduced into the field of MFI to design a better fit force dictionary based on the measured response data. A novel MFI method based on learning dictionary with double sparsity is proposed. Firstly, the MFI equation in time domain which describes the relationship between moving force and measured structural responses is established. Then a sparse dictionary model is designed in which the force dictionary is assumed to be expressed sparsely over the base dictionary, and the moving force is sparse over the force dictionary simultaneously. Moreover, the sparse K-singular-value-decomposition (K-SVD) algorithm is employed to realize the learning process through alternatively updating between double sparse codes. Finally, the learned force dictionary and moving forces are estimated through base force dictionary and double sparse codes. Numerical simulations and experimental studies are carried out to investigate the performance of the proposed method, and the results clearly certify its effectiveness and robustness.

Motion Evaluation of Floating Point Absorber-type Wave Energy Converter by Kane's Equation

In this paper, we derive the equations of longitudinal motion for a floating point absorber type wave energy converter (FPAWEC) by Kane's equation. Since FPAWECs have a nonholonomic constraint, we selected Kane's equation which is based on the principle of virtual power. The time-domain motion equations of the FPAWEC with several control strategies were developed and its motion evaluations were conducted in regular and irregular waves. The motion evaluation results were reasonably consistent with the measured ones, both in regular and irregular waves. The dominant pitch motion in irregular waves was observed around pitch natural frequency. Using the derived equations, we investigated the pitch moment components and found that the pitch moment on the float had the largest contributions to pitching motion with the pitch natural frequency.

Passively Q-Switched Dual-Wavelength Laser Operation With Coaxially End-Pumped Composite Laser Materials

A passively Q-switched dual-wavelength laser with tunable power ratio and pulse interval is demonstrated based on the coaxial diode end-pumping configuration by varying the pump wavelength. A theoretical model was built by a set of time-domain rate equations illustrating the dynamic process of both the pulse generation of two wavelengths from a shared pump source and a saturable absorber. The simulation showed that both the power ratio and the time interval between the pulses at two resonant wavelengths in the Q-switched mode could be tuned by balancing the gains in two laser crystals by varying the pump wavelength, realized by controlling the working temperature of the pump laser diode. The experiment was performed with Nd:YAG/Nd:YAP composite laser crystals and a Cr4+:YAG slice working as the saturable absorber for pulse generation. The continuous-wave and Q-switched total output power reached 5.62 W and 2.12 W, corresponding to overall optical-optical conversion efficiencies of 37.5% and 14.1%, respectively. The manipulation of power ratio and pulse interval between 1064 nm and 1080 nm agreed well with the simulation results.

Total Harmonic Distortion of a Piezoelectric MEMS Loudspeaker in an IEC 60318-4 Coupler Estimation Using Static Measurements and a Nonlinear State Space Model.

We propose a method to evaluate the Total Harmonic Distortion generated by a cantilever-based PZT loudspeaker inside an IEC 60318-4 coupler. The model is validated using experimental data of a commercial loudspeaker. Using the time domain equations of the equivalent electrical circuit of the loudspeaker inside the coupler and a state space formulation, the acoustic pressure response is calculated and compared to the measurement of the manufacturer. Next, the stiffness, transduction and capacitance nonlinear functions are evaluated with a Double-Beam Laser Interferometer (DBLI) and a nanoindenter on test devices and on the commercial loudspeaker. By introducing the nonlinear functions into the model as amplitude-dependent parameters, the THD generated by the loudspeaker is calculated and compared to the value provided by the manufacturer. The good agreement between the measurement and the simulation could allow for a rather quick simulation of the performance of similarly designed loudspeakers at the early stage of the design, by only estimating the static linearity of the main nonlinearity sources.

Evolution equation for nonlinear Lucassen waves, with application to a threshold phenomenon.

A nonlinear, fractional, surface wave equation with a spatial derivative of second order was developed by Kappler, Shrivastava, Schneider, and Netz [Phys. Rev. Fluids 2, 114804 (2017)] for propagation along an elastic interface coupled to a viscous incompressible liquid. Linear theory for the attenuation and dispersion was developed originally by Lucassen [Trans. Faraday Soc. 64, 2221 (1968)]. Kappler et al. introduced a fractional time derivative to account for the Lucassen wave attenuation and dispersion, and they included quadratic and cubic nonlinearity associated with compression of the elastic interface. Presented here is an integrated form of their time domain equation for progressive waves that is first order in the spatial derivative. Solutions of this evolution equation capture the main features of waveforms predicted by the full model equation of Kappler et al., especially the formation and propagation of shocks, while the evolution equation can be solved numerically with substantially less computational cost. Approximate analytical expressions obtained from the evolution equation for the nonlinear propagation speed and attenuation of a compression pulse reveal that a threshold phenomenon discussed by Kappler et al. is due to competition between quadratic and cubic nonlinearity associated with a lipid monolayer interface.

Discrete time domain analysis of radiation detector noise

In this paper, the analysis of radiation detector noise is carried out based on digital signal processing in the discrete time domain. To generate accurate noise models, detector noise sources are analyzed and modeled in the discrete region. Poisson noise generator was used for shot noise, and White Gaussian Noise Generator was used for Johnson Noise and Channel Thermal Noise. For flicker noise, White Gaussian Noise Generator with 1/f filter was applied. The output noise was calculated by convolution sum of the system impulse function and noise input. In order to improve the accuracy noise analysis, the time domain equation was derived based on the advanced model compared to the existing noise model. CSA model is based on a realistic limited-bandwidth OP-AMP. For the realistic detector model, Laplace transformed function of CSA and shaper out signal for parallel and series noise were derived and inversed for time domain signal. The results of the existing frequency domain analysis match well with the discrete time domain analysis. The discrete time domain analysis can increase the accuracy of noise investigation because it can visualize results of existing frequency domain models and provide different representation methods for the same results.

Analytical solution for thermoelastic oscillations of nonlocal strain gradient nanobeams with dual-phase-lag heat conduction

In order to examine the impact of structural and thermal scale parameters on thermoelastic vibrations of Euler-Bernoulli nanobeams, this article intends to provide a size-dependent generalized thermoelasticity model with the help of nonlocal strain gradient theory (NSGT) in conjunction with dual-phase-lag (DPL) heat conduction model. To highlight the role of each scale parameter in size-dependent motion and heat conduction equations, normalized forms of these nonclassical coupled thermoelastic equations are extracted by introducing and exploiting some dimensionless parameters. By exploiting power series expansion as a general solution for arbitrary boundary conditions, system of partial differential equations reduces to system of ordinary differential equations in time domain. The Laplace transform is then utilized to attain the analytical solution of this set of differential equations. By conducting a case study on a hinged-hinged nanobeam, with the aim of illustrating the impact of nonclassical structural and thermal parameters on thermoelastic vibrations, size-dependent numerical results are compared with those estimated by classical elasticity theory and heat conduction model. Meaningful disparity between classical and nonclassical results reveals the pivotal role of scale parameters in realistic assessment of thermoelastic behavior of nanobeams. Outcomes also indicate that corresponding to the relationship between the magnitudes of two scale parameters of NSGT, the nonclassical model of nanobeam can exhibit either softening or hardening behavior compared to the classical one. This affirms the ability of NSGT to justify both stiffness softening and stiffness hardening phenomenon in nanostructures.

Three-dimensional thermodynamic behaviors for a FML structure subjected to 3-D moving dual-ellipse distribution laser heat source

In present work, the three-dimensional transient heat transfer equation for a fiber metal laminated (FML) plate is established, and temperature field under dual-ellipse distribution laser processing is obtained through the Newton-Cotes method. Then the three-dimensional constitutive equations are derived, and the transient thermodynamic governing equations are obtained based on the Hamilton variational principle subsequently. Finally, the finite difference method and Newmark method are utilized to solve the thermodynamic equations in both space and time domains, respectively. The research aims at studying three-dimensional thermodynamic behaviors for a FML structure subjected to dual-ellipse distribution laser heat source and giving the effects of Fourier series term, finite difference point, aspect ratio, fiber arrangement angle and laser moving speed on the thermodynamic behaviors for the FML structure.

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

ABSTRACTTime-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

Modified accelerate iteration for optical property reconstruction based on time-domain radiative transfer equation

A modified diffusion synthetic acceleration method is proposed to speed up the convergence process of TD-RTE calculation by employing the finite element method, combined with OpenMP parallel technology to further enhance the computational efficiency. In addition, the sequential quadratic programming algorithm is applied with Gaussian Markov random field (GGMRF) regularization to reconstruct the absorption and scattering parameter fields simultaneously. The accuracy and efficiency of the proposed model are investigated. It was observed that the modified DSA model is more efficient with the increase of the optical thickness, where the acceleration ratio is improved to 6 for the optical thicker media. In addition, similar acceleration effect was witnessed for both scattering dominated and non-scattering dominated media. Furthermore, by employing the modified DSA model into the reconstruction, it is interesting to remark that not only the reconstruction time is significantly reduced with the acceleration ratio of 8–11, but also the reconstructed image is further improved. All the results indicate that the developed scheme is efficient and robust based on TD-RTE, showing great potential in optical imaging.

The model of a permanent magnet DC motor in time domain and frequency domain based on Bond Graph modelling and design of position control using PID controller

During the research work, the modelling of a permanent magnetic DC (PMDC) motor was realised in time domain and frequency domain via 4 different types of model construction. To achieve this, the Bond Graph model of the motor was determined, and based on them the system equations were defined in time domain firstly and then in frequency domain by using the Laplace transform. According to the system equations in time domain, the state-space model of the motor was also realised and implemented. Based on the system equations in frequency domain, the transfer function of the position control loop was realised. As the further part of the research work, PID control-based position control was designed in both time domain and frequency domain. The controller was tuned according to the Ziegler-Nichols method. Finally, the models were compared from the point of view of behaviour during the tuning process. The models were implemented in LabVIEW environment produced by the National Instruments.

Spectral causality and the scattering of waves

Causality—the principle stating that the output of a system cannot temporally precede the input—is a universal property of nature. Here, we show that analogous input-output relations can also be realized in the spectral domain by leveraging the peculiar properties of time-modulated non-Hermitian photonic systems. Specifically, we uncover the existence of a broad class of complex time-modulated metamaterials that obey the time-domain equivalent of the well-established frequency-domain Kramers–Kronig relations (a direct consequence of causality). We find that, in the scattering response of such time-modulated systems, the output frequencies are inherently prohibited from spectrally preceding the input frequencies, and hence we refer to these systems as “spectrally causal.” We explore the consequences of this newly introduced concept for several relevant applications, including broadband perfect absorption, temporal cloaking of an “event,” and truly unidirectional propagation along a synthetic dimension. By emulating the concept of causality in the spectral domain and providing new tools to extend the field of temporally modulated metamaterials (“chrono-metamaterials”) into the complex realm, our findings may open unexplored opportunities and enable relevant technological advances in various areas of photonics and, more broadly, of wave physics and engineering.

Automated microwave monitoring of hidden objects for strategic and security applications

An automated microwave technique for the monitoring of hidden objects for strategic applications has been presented. The proposed approach is quite novel as it implements and integrates the time-domain method in free-space with the indigenously developed automated scanning setup to obtain the dielectric profile of hidden objects in standard sand or behind the concrete structure. The existing imaging methods for concealed media cannot reveal detailed information of objects with restrictions to access the other side of the test media. The overall approach is implemented by measuring the S-parameters of the test region containing the hidden objects in the wide frequency range of 2-10 GHz using the horn-lens antenna and the network analyzer. The S-parameters are transformed into the equivalent time-domain using the standard IFFT algorithm to retrieve the test region's 2D microwave image and the approximate size of concealed objects, which helps in situations such as through-the-wall imaging, and landmine detection.

Acoustic Propagation in a Medium with Spatially Distributed Relaxation Processes and a Possible Explanation of a Frequency Power Law Attenuation

A possible medium is considered with a large number of diverse but possibly similar relaxation processes. The equations for single spatially located relaxation processes of general type are developed and extended to a large spatially extended system. Each process is characterized by a relaxation time and a relaxation strength and is excited by the local pressure in an incident acoustic wave. A constant frequency model is initially assumed with complex amplitudes associated with acoustic pressure and relaxation responses. A smearing process results in the collective relaxation responses being regarded as a continuous function of relaxation times. An expression for entropy perturbation is developed in terms of the relaxation responses. The equations of fluid mechanics then lead to an expression for the pressure perturbation in terms of the dilatation and the relaxation processes. The extended result yields a time-domain wave equation for propagation through an inhomogeneous medium with distributed relaxation processes. Expressions for frequency-dependent attenuation and phase velocity are derived in which relaxation is characterized by a single function giving strength as a continuous function of relaxation time. Possible choices for this function are discussed, and it is shown that some choices lead to an attenuation varying nearly as a power of the frequency over any fixed and possibly large range of frequencies. A model set of parameters leads to good agreement with attenuation and phase velocity measurements for a suspension of human red blood cells in saline solution, as communicated to the authors by Treeby and analogous to those reported by Treeby, Zhang, Thomas, and Cox.

Modified first harmonic approximation-based modeling of SN-compensated inductive power transfer links operating at load-independent-voltage-output frequency

Some popular applications of wireless power transfer (such as electric vehicle charging) include a transmitter residing on a static platform (charging station) and a dynamic receiver (vehicle). In such applications, compensation may be excluded from the receiver in order to minimize, simplify and lower the cost. If transmitter compensation is also maximally simplified series to none (SN) compensation topology is created. It was recently shown that classical first harmonic approximation (FHA) based analysis fails to accurately predict RMS values of system currents, thus overestimating the achievable coil-to-coil efficiency. Consequently, this paper establishes modified FHA (MFHA) equivalent circuit of SN-compensated inductive power transfer link (IPTL), utilizing analytical expressions of primary and secondary currents based on time-domain differential equations (DE) based solution. It is shown that the outcomes of proposed MFHA and DE methods are very close, justifying the proposed methodology. Simulations and experiments demonstrate excellent matching, validating the presented analysis.

Transient and steady-state faults location in intelligent distribution networks compensated with D-STATCOM using time-domain equations and distributed line model

Intelligent electric distribution networks compensated with D-STATCOM are active networks in which data of terminals is important for better distribution of energy. Similar to all electricity network devices, energy distribution lines in such networks are also subject to transient and steady-state AC faults, e.g., short circuit. Sections of the network in which the fault occurs are separated by relays. In such a condition, fault location and restoring the normal status of the network are important. Fault location in these networks is performed based on synchronous data of two terminals because data of all buses are transmitted to the main bus In this paper, a time-domain equation method (TDEM) is proposed for locating faults on smart distribution grid (SDG) using distributed line model based on time-domain equations. The proposed method locates the point of transient and steady-state fault in SDG considering DGs and compensators. In this paper, the D-STATCOM and wind power plant are considered in SDG. The suggested method does not need the D-STATCOM and DGs model for applying the algorithm. It only uses data of less than half of the cycle for executing the algorithm. The numerical simulation confirms that the proposed fault location method is accurate and it provides a reliable solution for automatic fault location problem in distribution networks.