Traditional Time Domain Research Articles

Estimating the State-of-Charge of Lithium-Ion Battery Using an H-Infinity Observer Based on Electrochemical Impedance Model

The lithium-ion batteries in the electric vehicles are nonlinear systems with complex electrochemical dynamics, and estimation of battery state-of-charge (SOC) is affected by factors such as environmental temperature and battery current. Considering the above problems, the accurate estimation of battery SOC has always been a difficult and the critical issue of battery management system (BMS). In this paper, the constant phase element (CPE) is introduced to the traditional time domain circuit model by analyzing the electrochemical impedance spectra of lithium-ion batteries. Accordingly, an equivalent circuit model based on electrochemical impedance is constructed by using fractional order theory, which has specific physical significant, leading to the improved estimation accuracy to represent battery voltage. Moreover, the polarization resistance in the model is replaced by Butler-Volmer (BV) equation, which can solve the problem caused by large current and temperature variation during the actual operation of electric vehicles. Next, based on the model, an $\text{H}\infty $ observer is designed for battery SOC estimation, and the proposed SOC observer is tested by real-time experimental data of battery. The efficiency of the proposed model and observer are validated by some simulations and experiment tests.

Development of a Blended Time-Domain Program for Predicting the Motions of a Wave Energy Structure

Traditional linear time-domain analysis is used widely for predicting the motions of floating structures. When it comes to a wave energy structure, which usually is subjected to larger relative (to their geometric dimensions) wave and motion amplitudes, the nonlinear effects become significant. This paper presents the development of an in-house blended time-domain program (SIMDYN). SIMDYN’s “blend” option improves the linear option by accounting for the nonlinearity of important external forces (e.g., Froude-Krylov). In addition, nonlinearity due to large body rotations (i.e., inertia forces) is addressed in motion predictions of wave energy structures. Forced motion analysis reveals the significance of these nonlinear effects. Finally, the model test correlations examine the simulation results from SIMDYN under the blended option, which has seldom been done for a wave energy structure. It turns out that the blended time-domain method has significant potential to improve the accuracy of motion predictions for a wave energy structure.

Testing mean-reversion in agricultural commodity prices: Evidence from wavelet analysis

This study examines the validity of the random walk hypothesis for some selected soft agricultural commodity prices within the context of heterogeneous market hypothesis and mean reversion hypothesis. The study employs a battery of traditional unit root tests, GARCH-based models and a novel frequency-based wavelet analysis to analyze daily data sourced from 6th of Jan 1986 to 29th Dec 2018. Contrary to other existing studies that employed only traditional time domain unit root tests, our results reveal that soft commodity prices are mean reverting, suggesting the existence of potential excess returns for investors. Overall, our results show that the selected soft commodity series are inefficient when we factored in heteroscedascity and frequency domain into our model. Our study is an improvement on the existing studies as we analyze our data using both time and frequency domain estimates. Besides, unlike other studies that did not offer structural breaks, the current study provides structural break dates with major events in the global socioeconomic space, which are key to identifying the date of bubbles and potential signs of commodity price bubbles. Our findings have some critical implications for investors, policy makers

A novel symplectic MRTD scheme studies on numerical characteristics of plasma photonic crystals

A novel symplectic multi-resolution time-domain (S-MRTD) scheme is put forward for simulating the one-dimensional unmagnetized plasma photonic crystals (PPCs). The lossless plasma model is taken into account in S-MRTD scheme, and the detailed calculation formulas of the proposed scheme are provided. An one-dimensional perfectly matched layers (PML) technique is used to truncate the computational domain, and the corresponding formulas are also derived. From the perspective of numerical stability and dispersion characteristics, it is shown that the S-MRTD scheme is more effective than the traditional time-domain methods. The propagation process of electromagnetic (EM) waves in one-dimensional PPCs structure are simulated by S-MRTD scheme, and from the perspective of frequency domain, the reflection and transmission cofficients of EM pulse waves through the PPCs are analyzed. Meanwhile, The characteristics of photonic band gaps (PBGs) are studied based on different parameters. Numerical simulations show that plasma frequency, dielectric constant, periodic thickness, number of periods, the thickness ratio of plasma and medium are important factors affecting the characteristics of PBGs.

Distributed Formal Analysis for Power Networks With Deep Integration of Distributed Energy Resources

A scalable distributed formal analysis (DFA) via reachable set computation is presented to efficiently evaluate the stability of large-scale interconnected power networks under heterogeneous disturbances induced by high penetration of distributed energy resources (DERs). Based on rigorous mathematical derivation, DFA is able to directly compute the boundaries of all possible dynamics and provide stability information, which is unattainable by traditional time-domain simulations or direct methods. An $N+M$ decomposition approach is established to decouple a large-scale networked system and enable distributed reachable set calculations while also preserving the privacy of each subsystem. Numerical examples on a networked microgrid system show that DFA facilitates the efficient calculation and analysis of the impact DER disturbances can have on power network dynamics, which provides a potent means of optimizing the system's operation. Therefore, DFA provides an invaluable tool for designing and operating the interconnected power networks of the future, which will feature the deep integration of DERs.

Enhanced Distributed Fiber Optic Vibration Sensing and Simultaneous Temperature Gradient Sensing Using Traditional C-OTDR and Structured Fiber with Scattering Dots.

We present results demonstrating several beneficial effects on distributed fiber optic vibration sensing (DVS) functionality and performance resulting from utilizing standard single mode optical fiber (SMF) with femtosecond laser-inscribed equally-spaced simple scattering dots. This modification is particularly useful when using traditional single-wavelength amplitude-based coherent optical time domain reflectometry (C-OTDR) as sensing method. Local sensitivity is increased in quasi-distributed interferometric sensing zones which are formed by the fiber segments between subsequent pairs of the scattering dots. The otherwise nonlinear transfer function is overwritten with that of an ordinary two-beam interferometer. This linearizes the phase response to monotonous temperature variations. Furthermore, sensitivity fading is mitigated and the demodulation of low-frequency signals is enabled. The modification also allows for the quantitative determination of local temperature gradients directly from the C-OTDR intensity traces. The dots’ reflectivities and thus the induced attenuation can be tuned via the inscription process parameters. Our approach is a simple, robust and cost-effective way to gain these sensing improvements without the need for more sophisticated interrogator technology or more complex fiber structuring, e.g., based on ultra-weak FBG arrays. Our claims are substantiated by experimental evidence.

Optimization Algorithms of Neural Networks for Traditional Time-Domain Equalizer in Optical Communications

Neural networks (NNs) have been successfully applied to channel equalization for optical communications. In optical fiber communications, the linear equalizer and the nonlinear equalizer with traditional structures might be more appropriate than NNs for performing real-time digital signal processing, owing to its much lower computational complexity. However, the optimization algorithms of NNs are useful in many optimization problems. In this paper, we propose and evaluate the tap estimation schemes for the equalizer with traditional structures in optical fiber communications using the optimization algorithms commonly used in the NNs. The experimental results show that adaptive moment estimation algorithm and batch gradient descent method perform well in the tap estimation of equalizer. In conclusion, the optimization algorithms of NNs are useful in the tap estimation of equalizer with traditional structures in optical communications.

Fiber fault detection based on communication signal

This paper presents a practical method of detecting the fault point of optical fiber link based on the communication signal itself in optical network. The method collects the reference signal and the echo signal with the fault point information through the data acquisition device, and then uses the permutation entropy to extract the random part from the communication signals. Finally, the accurate measurement of the fault point of fiber link is detected by correlation operation. The experiment uses the communication signal transmitted in the time-division multiplexing passive optical network as the detection signal, and the detection object is G652 single-mode fiber with simulated fault point. Different from the traditional optical time-domain reflectometry, the proposed method can measure the fault point of the fiber link without disconnecting the normal communication network, achieve a position resolution of 15 cm, and reduce the measurement cost for the fiber link fault detection.

New hybrid FDTD algorithm for electromagnetic problem analysis**Project supported by the National Natural Science Foundation of China (Grant No. 61571348) and the Equipment Pre-Research Foundation of China (Grant No. 61405180202).

Since the time step of the traditional finite-difference time-domain (FDTD) method is limited by the small grid size, it is inefficient when dealing with the electromagnetic problems of multi-scale structures. Therefore, the explicit and unconditionally stable FDTD (US-FDTD) approach has been developed to break through the limitation of Courant–Friedrich–Levy (CFL) condition. However, the eigenvalues and eigenvectors of the system matrix must be calculated before the time iteration in the explicit US-FDTD. Moreover, the eigenvalue decomposition is also time consuming, especially for complex electromagnetic problems in practical application. In addition, compared with the traditional FDTD method, the explicit US-FDTD method is more difficult to introduce the absorbing boundary and plane wave. To solve the drawbacks of the traditional FDTD and the explicit US-FDTD, a new hybrid FDTD algorithm is proposed in this paper. This combines the explicit US-FDTD with the traditional FDTD, which not only overcomes the limitation of CFL condition but also reduces the system matrix dimension, and introduces the plane wave and the perfectly matched layer (PML) absorption boundary conveniently. With the hybrid algorithm, the calculation of the eigenvalues is only required in the fine mesh region and adjacent coarse mesh region. Therefore, the calculation efficiency is greatly enhanced. Furthermore, the plane wave and the absorption boundary introduction of the traditional FDTD method can be directly utilized. Numerical results demonstrate the effectiveness, accuracy, stability, and convenience of this hybrid algorithm.

Frequency‐domain double‐plane‐wave least‐squares reverse time migration

ABSTRACTLeast‐squares reverse time migration is often formulated as an iterative updating process, where estimating the gradient of the misfit function is necessary. Traditional time domain shot‐profile least‐squares reverse time migration is computationally expensive because computing the gradient involves solving the two‐way wave equation several times in every iteration. To reduce the computational cost of least‐squares reverse time migration, we propose a double‐plane‐wave least‐squares reverse time migration method based on a misfit function for frequency‐domain double‐plane‐wave data. In double‐plane‐wave least‐squares reverse time migration, the gradient is computed by multiplying frequency‐domain plane‐wave Green's functions with the corresponding double‐plane‐wave data residual. Because the number of plane‐wave Green's functions used for migration is relatively small, they can be pre‐computed and stored in a computer's discs or memory. We can use the pre‐computed plane‐wave Green's functions to obtain the gradient without solving the two‐way wave equation in each iteration. Therefore, the migration efficiency is significantly improved. In addition, we study the effects of using sparse frequency sampling and sparse plane‐wave sampling on the proposed method. We can achieve images with correct reflector amplitudes and reasonable resolution using relatively sparse frequency sampling and plane‐wave sampling, which are larger than that determined by the Nyquist theorem. The well‐known wrap‐around artefacts and linear artefacts generated due to under‐sampling frequency and plane wave can be suppressed during iterations in cases where the sampling rates are not excessively large. Moreover, implementing the proposed method with sparse frequency sampling and sparse plane‐wave sampling further improves the computational efficiency. We test the proposed double‐plane‐wave least‐squares reverse time migration on synthetic models to show the practicality of the method.

Time domain load extrapolation method for CNC machine tools based on GRA-POT model

The load extrapolation is to obtain load data that may occur during the whole life cycle of product based on the limited measured load data, especially extreme load. Aiming at the two problems of traditional time domain extrapolation: without considering the interval time between extreme adjacent values and the high sensitivity of the peak over threshold (POT) model to the extreme threshold, a time domain load extrapolation method for computerized numerical control (CNC) machine tools based on Gray relational analysis-peak over threshold model (GRA-POT) is proposed. Firstly, for the first problem, an improved Markov chain Monte Carlo (MCMC) time domain load reconstruction method is proposed and analysis of reconstructed time domain load; secondly, according to the mean excess function (MEF) method, the extreme threshold range of POT model is determined and divided into multiple sets. The generalized Pareto distribution (GPD) fitting is performed for multiple sets of candidate load extreme samples. Finally, the Gray relational degrees (GRD) between each group of GPD and the median rank distribution are calculated by the Gray relational analysis (GRA), and the extreme threshold corresponding to the maximum GRD is selected as the optimal threshold of the POT model. The dynamic cutting force signal of CNC machine tool under constant speed cutting condition was reconstructed and extrapolated using the proposed method, and the dynamic cutting force spectrum of CNC machine tool was compiled. Case analysis results show that the POT extrapolation model with high fitting precision can be obtained using the proposed method. Furthermore, the precision of load spectrum of CNC machine tool is improved.

Efficient Aeroelastic Solution Based on Time-Spectral Fluid–Structure Interaction Method

Although the traditional transonic aeroelastic numerical analysis in the time domain is accurate, the high computational cost limits its application. Therefore, in this Paper, an efficient aeroelastic numerical solution based on the time-spectral method, called the time-spectral fluid–structure interaction method, is developed. The periodic aerodynamic forces and structure equations are calculated by using the time-spectral method and the double-iteration method, respectively. The novelty of this method is that in the inner iteration the variable with a strong coupling effect is solved by using the gradient method and the remaining variables are then solved by using the least-square method. Moreover, the computational fluid dynamics solver is not required in the inner iteration. By using the double-iteration method, not only does the computational cost decrease, but the stagnation of the convergence, which occurs frequently when using the one-shot method to search for the critical point, can be avoided. The two-dimensional aeroelastic standard test cases are then used to validate this method. Results show that the flutter boundary and the critical characteristics of the limit cycle oscillation can be captured with a few iterations by the time-spectral fluid–structure interaction method, and the results of this method agree well with those of the traditional time-domain method. This method can improved by one to two orders of magnitude in computational efficiency with a good robustness to initial conditions.

Predicting wrist kinematics from motor unit discharge timings for the control of active prostheses

BackgroundCurrent myoelectric control algorithms for active prostheses map time- and frequency-domain features of the interference EMG signal into prosthesis commands. With this approach, only a fraction of the available information content of the EMG is used and the resulting control fails to satisfy the majority of users. In this study, we predict joint angles of the three degrees of freedom of the wrist from motor unit discharge timings identified by decomposition of high-density surface EMG.MethodsWe recorded wrist kinematics and high-density surface EMG signals from six able-bodied individuals and one patient with limb deficiency while they performed movements of three degrees of freedom of the wrist at three different speeds. We compared the performance of linear regression to predict the observed individual wrist joint angles from, either traditional time domain features of the interference EMG or from motor unit discharge timings (which we termed neural features) obtained by EMG decomposition. In addition, we propose and test a simple model-based dimensionality reduction, based on the physiological notion that the discharge timings of motor units are partly correlated.ResultsThe regression approach using neural features outperformed regression on classic global EMG features (average R2 for neural features 0.77 and 0.64, for able-bodied subjects and patients, respectively; for time-domain features 0.70 and 0.52).ConclusionsThese results indicate that the use of neural information extracted from EMG decomposition can advance man-machine interfacing for prosthesis control.

Short-term multiscale complexity analysis of cardiovascular variability improves low cardiac output syndrome risk stratification after coronary artery bypass grafting

Background: Low cardiac output syndrome (LCOS) is a myocardial dysfunction leading to systemic hypoperfusion, favored by particular conditions of the autonomic nervous system. LCOS is one of the adverse events that might occur after cardiac surgery. Objective: The aim is to test the hypothesis that short-term multiscale complexity (MSC) analysis of heart period (HP) and systolic arterial pressure (SAP) variability series in the frequency bands typical of cardiovascular control could be fruitfully exploited in identifying subjects at risk of developing LCOS after coronary artery bypass graft (CABG). Approach: HP and SAP beat-to-beat series were derived from electrocardiogram (ECG) and invasive arterial pressure (AP) signal acquired in 128 patients scheduled for CABG before (PRE) and after (POST) the induction of general anesthesia with propofol and remifentanil. Subjects were labeled as LCOS (n = 14) and noLCOS (n = 114) according to the LCOS development. MSC markers were calculated as the complement to 1 of the modulus of the average position of the poles dropping in the low-frequency (LF, 0.04–0.15 Hz) and high-frequency (HF, 0.15–0.5 Hz) bands as derived from the autoregressive model of HP and SAP series. Traditional time and frequency domain indexes were also calculated. Main results: Traditional parameters were able to assess the depression of the cardiovascular regulation induced by general anesthesia, but showed weak performances in differentiating LCOS and noLCOS groups. Conversely, HP complexity in LF band and SAP complexity in HF band assessed during POST remained associated with LCOS even after entering a multivariate logistic regression model adjusted for clinical and demographic factors. Significance: The MSC approach can be fruitfully applied to improve risk stratification for LCOS after CABG likely because MSC markers describe the dysfunction of the sympathetic control and the impairment of the mechanical properties of the heart in the LCOS group.

A novel domain decomposition-finite difference time domain method for composite scattering from a target above rough surface

A novel domain decomposition-finite difference time domain (DD-FDTD) method is proposed to improve the computation efficiency of the traditional finite difference time domain (FDTD) method in calculating composite scattering from a target above a rough surface. In this scheme, the target and the rough surface are placed into the target subdomain and the rough surface subdomain, respectively. The incident waves of the target subdomain, including the directly illuminated wave and the nonplanar reflected wave from the rough surface, are introduced by invoking the Huygens’ principle and the equivalence principle. The FDTD method is executed in each subdomain to obtain the scattering fields of the target and the rough surface, and the scattering contribution of the rough surface from the reflected wave of target is taken into account. In comparison with the traditional FDTD method, the proposed method has significant advantages in saving memory usage and in reducing computation time, which is more efficient in solving the composite scattering problems with larger distance between target and rough surface.

Image-guided vibrometry system integrated with spectral- and time-domain optical coherence tomography.

Vibrometry using optical coherence tomography (OCT) can provide valuable information for investigating either the mechanical properties or the physiological function of biological tissues, especially the hearing organs. Real-time imaging of the measured tissues provides structure imaging and spatial guidance for and is thus highly demanded by such vibrometry. However, the traditional time-domain OCT (TD-OCT) systems, although capable of subnanometric vibrometry at large ranges of frequencies, are unable to offer an imaging speed that is high enough to acquire depth-resolved images for guidance. The spectral-domain OCT (SD-OCT) systems, although allowing image-guided vibrometry, are challenged in measuring vibration at high frequencies, particularly for scattering tissue specimens that require longer exposure time to ensure imaging and vibrometry performance. This is because of their limit in the line-scan rate of the CCD, in which the maximum resolvable frequency measured by the SD-OCT is about 1/4 of the CCD line-scan rate in practice. In the present study, we have developed a dual-mode OCT system combining both SD-OCT and TD-OCT modalities for image-guided vibrometry, as the SD-OCT can provide guiding structural images in real-time and, moreover, the TD-OCT can guarantee vibrometry at large ranges of frequencies, including high frequencies. The efficacy of the developed system in image-guided vibrometry has been experimentally demonstrated using both piezoelectric ceramic transducer (PZT) and ex vivo middle-ear samples from guinea pigs. For the vibrometry of PZT, the minimum detectable vibration amplitude was reached at ∼0.01 nm. For the vibrometry of the sound-evoked biological samples, both real-time two-dimensional imaging and subnanometric vibrometry were performed at the frequency ranging from 1 to 40kHz. These results indicate that our dual-mode OCT system is able to act as an excellent vibrometer enabling image-guided high-frequency measurement.

An improved time-domain response estimation method for floating structures based on rapid solution of a state-space model

The proposed time-domain response estimation method aims to improve computational efficiency of the traditional step-by-step techniques by considering convolution item with a state-space model for floating structures. Different from the present three techniques of estimating state-space model matrices, which are the impulse response curve fitting, the realization theory and the regression in the frequency domain, a more accurate and efficient algorithm has been developed by constructing these matrices with poles and residues of retardation functions. One theoretical development is that the poles and residues are estimated by using the state-space model, not directly by solving ordinary differential equations, which implies that the traditional ill-conditioned issue can be reduced significantly. Meanwhile, this algorithm is a direct estimation procedure and also does not require initial values during the process of computing state-state model matrices, which means that a better computational efficiency can be expected. To investigate the performance of the proposed method, three examples are employed. The first example is a numerical retardation function that is based on a purely analytical relationship, and this function satisfies all the properties of the convolution terms in math. By comparing the approach with the three techniques of estimating state-space model matrices, one can draw the conclusion that the approach provides accuracy similar to that of the regression method but outperforms the curve fitting and the realization methods, obviously because of the improperly used initial values. The second example is a single-degree-of-freedom system excited by different forms of external load. Numerical results show that a dynamic response analysis using replacement of convolution with the proposed method is in accordance with the traditional frequency and time domain method. However, the proposed method is insensitive to the step size of the calculation, which means that the proposed method is more stable. To extend the proposed method to a system with multiple degrees of freedom and investigate the capacity for computational efficiency, a semi-submersible platform was used. Conclusions from the numerical and experimental studies can be drawn: (1) the estimated response of the platform by using the proposed method to replace convolution items with a state-space model matches well with the traditional Newmark-β method; (2) the calculation time is reduced significantly using the proposed method when the number of calculation steps is large; and (3) the heave response of semi-submersible physical model calculated by the proposed method is in good agreement with the experimental data.

Fractional Laplacian Viscoacoustic Wave Equation Low-Rank Temporal Extrapolation

Description of seismic wave attenuation is a hot topic in the geophysical area, and it is the basis of the attenuation-compensated seismic imaging technique, which aims to retrieve a high-resolution subsurface image for geological structure analysis and hydrocarbon reservoir prediction. The numerical simulation of viscoacoustic wave equation is an effective way to observe the seismic attenuation in lossy media. The existing study confirms that the seismic-quality-factor (Q) that is used to represent the strength of the viscous behavior of the earth is nearly independent of frequency, which is referred to as the constant-Q (CQ) model. The mathematical concept of fractional Laplacian is recently introduced to the geophysical area to form a compact CQ wave equation to describe seismic wave propagation. However, numerically solving the fractional Laplacian CQ wave equation by the traditional pseudospectral time-domain (PSTD) method suffers from a strict stability condition and great numerical dispersion due to a low-order temporal finite-difference (FD) approximation. To improve the temporal extrapolation accuracy, we derive the analytical wavenumber (k)-space domain propagators underlying the fractional Laplacian wave equation. We regard the k-space operators as mixed-domain matrices in the case of heterogeneous media and adopt a low-rank decomposition to approximate the matrices. With the low-rank approximation, an efficient time-marching formula is obtained for wavefield temporal extrapolation. We formulate the time-marching formula into a first-order equation system in terms of pressure and particle-velocity to welcome the perfectly matched layer (PML) absorbing boundary condition to eliminate the wraparound effects caused by the Fourier transform. A spatial-variable density is also incorporated to simulate more realistic amplitude variation. The numerical examples are carried out to verify the accuracy and stability of the viscoacoustic low-rank extrapolation.

The Study on Near-Field Scattering of a Target Under Antenna Irradiation by TDSBR Method

An efficient time domain shooting and bouncing ray (TDSBR) method is proposed to analyze the transient near-field scattering from an electrically large complex object illuminated by a far-field antenna source. The time-domain far-field incident sources can be derived by a convolution of pulsed excitation with the inverse Fourier transform of frequency domain (FD) far-fields radiated from an antenna. In order to obtain accurate near-field scattering results, time domain physical optics (TDPO) near-field integral representations are proposed and reduced to closed-form expressions to improve computing efficiency by applying locally expanded Green function approximation. Different from the plane wave incident situation, in the case of antenna radiation, the incident fields of each individual facet in the ray path are represented by the radiation fields from the equivalent mirror antenna derived in this paper. Since the radiation fields naturally contains the time delay term, the complicated processing of the time delay problem can be avoided by solving the equivalent mirror antenna radiation fields. It is simpler and more accurate than the traditional time domain geometrical optics (TDGO) method. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Application of variable-time-period harmonic balance method to periodic unsteady vortex shedding

The harmonic balance method (HBM) is an efficient frequency-domain approach to computing periodically unsteady flows. The basic principle of this method is to decompose the flow variables into a Fourier series, and transform the unsteady flow into several steady problems coupled by a spectral time-derivative operator, from which the whole time history of a complete unsteady periodic flow can be reconstructed. In the present work, we investigate the ability of the HBM to be used for modeling the periodic unsteady vortex shedding behind a bluff body at low Reynolds numbers via solving the unsteady incompressible Navier-Stokes equations. For the periodic problem where the time period <i>T</i> of the unsteadiness is unknown, a variable-time-period method based on residual gradients is used to compute the exact time period iteratively starting from an initial guess <i>T</i><sub>0</sub>. By simulating the two-dimensional laminar flows over a circular cylinder and a square cylinder, the accuracy and efficiency of the HBM are investigated and the effects of different parameters on the final results are analyzed. Comparisons with the results of fixed-time-period HBM using a constant time period are also implemented. Three practical methods of optimization are used to iterate the time period, and the values of accuracy and efficiency of different methods are compared with each other. The results show that the HBM can accurately capture the complex nonlinear flow field physics with only three harmonics. The Strouhal frequency and mean drag coefficient each as a function of the Reynolds number agree well with existing experimental and computational data. For both test cases, the computational efficiency of HBM is higher than that from the traditional time-domain method. For the square cylinder test case, the HBM offers speedup rate up to nearly 18 times. The real time period of vortex shedding can be predicted by the gradient based variable-time-period method, and the final result is insensitive to search step <i>λ</i>. The calculation result is sensitive to the initial <i>T</i><sub>0</sub>, and when such a variable is greater than a certain value, the result will converge to an approximate integer multiple of the real one. Therefore, it deserves further exploration on how to specify this initial condition. The shedding time periods computed by different optimization methods are converged to the same value. The computational efficiency from the FR conjugate gradient method and that from Newton method are both equivalent to that from the steepest descent method with the maximum search step <i>λ</i> = 100. Avoiding prescribing parameters such as the search step <i>λ</i>, the Newton method possesses higher application value in engineering calculation than the other two schemes.