Traditional Time-domain Method Research Articles

Research on Electromagnetic Compatibility of High Frequency Switching Power Supply Based on Harmonic Balance Finite Element Method

With the rapid development of electronic technology, high frequency switching power supply has been widely used in various electrical equipment, and its electromagnetic interference has attracted more and more attention. In this paper, the harmonic balance finite element method is used to solve the electromagnetic compatibility of high frequency switching power supply. In this method, the coefficients of sine wave are applied to represent the waveform, and the solution is generated by linear synthesis. We propose to combine it with the finite element method to solve the steady-state response of electromagnetic field in frequency domain. It is proved the method we proposed is superior to the traditional time domain method.

Time-domain extrapolation method for tractor drive shaft loads in stationary operating conditions

The fatigue failure of critical parts of tractors over the whole life cycle has received much attention in recent years. Load extrapolation is a crucial step in load spectrum compilation and can extrapolate the product's life cycle load based on limited load data measurements. The traditional time-domain extrapolation method is limited because it neglects the time interval between the load changes and the load reconstruction within the threshold range. To address this problem, a time-domain extrapolation method for tractor drive shaft load based on the Markov chain Monte Carlo-peak over threshold (MCMC-POT) model is proposed. Firstly, an improved time-domain load reconstruction method based on MCMC that can reproduce the load change interval based on the measured load change characteristics was proposed. Secondly, in the POT model extrapolation process, an improved threshold selection algorithm that is an optimisation of the traditional mean excess function (MEF) method was proposed. This method avoids the errors caused by subjective factors under the premise of ensuring the fitting accuracy. Finally, the dynamic torque spectrum was obtained by extrapolating the tractor drive shaft load data measured under ploughing conditions. Case analysis results showed that the method can obtain extrapolation models with high fitting accuracy. The power spectral density values of the dynamic torque spectrum before and after extrapolation were very consistent with the mean correlation coefficient of 0.909. The spectrum can effectively reproduce the time-frequency characteristics of the load. Furthermore, the precision of the load spectrum is improved. • A time-domain load reconstruction method based on Markov chain Monte Carlo model. • The threshold selection method based on peak over threshold model was improved. • A load extrapolation method that can retain complete time-frequency information. • Tractor transmission shaft testing was implemented under ploughing conditions. • The extrapolated load has the consistent frequency spectrum with the measured.

Atrial Fibrillation Prediction from Critically Ill Sepsis Patients.

Sepsis is defined by life-threatening organ dysfunction during infection and is the leading cause of death in hospitals. During sepsis, there is a high risk that new onset of atrial fibrillation (AF) can occur, which is associated with significant morbidity and mortality. Consequently, early prediction of AF during sepsis would allow testing of interventions in the intensive care unit (ICU) to prevent AF and its severe complications. In this paper, we present a novel automated AF prediction algorithm for critically ill sepsis patients using electrocardiogram (ECG) signals. From the heart rate signal collected from 5-min ECG, feature extraction is performed using the traditional time, frequency, and nonlinear domain methods. Moreover, variable frequency complex demodulation and tunable Q-factor wavelet-transform-based time–frequency methods are applied to extract novel features from the heart rate signal. Using a selected feature subset, several machine learning classifiers, including support vector machine (SVM) and random forest (RF), were trained using only the 2001 Computers in Cardiology data set. For testing the proposed method, 50 critically ill ICU subjects from the Medical Information Mart for Intensive Care (MIMIC) III database were used in this study. Using distinct and independent testing data from MIMIC III, the SVM achieved 80% sensitivity, 100% specificity, 90% accuracy, 100% positive predictive value, and 83.33% negative predictive value for predicting AF immediately prior to the onset of AF, while the RF achieved 88% AF prediction accuracy. When we analyzed how much in advance we can predict AF events in critically ill sepsis patients, the algorithm achieved 80% accuracy for predicting AF events 10 min early. Our algorithm outperformed a state-of-the-art method for predicting AF in ICU patients, further demonstrating the efficacy of our proposed method. The annotations of patients’ AF transition information will be made publicly available for other investigators. Our algorithm to predict AF onset is applicable for any ECG modality including patch electrodes and wearables, including Holter, loop recorder, and implantable devices.

Interior Penalty DGTD Method for Solving Wave Equation in Dispersive Media Described With GDM Model

In this communication, a paralleled interior penalty discontinuous Galerkin time-domain method based on the vector wave equation (IPDG-WE) combining with a generalized dispersive material (GDM) model is introduced. In addition, the auxiliary differential equation (ADE) technique is employed to establish a universal solution for scattering analysis of dispersive structures in a wide frequency range. The proposed method presents great characteristics of energy conservation and the optimal convergence rate even in low-order bases and exhibits promising competitiveness in CPU time and memory usage compared with the traditional Maxwell's equations-based discontinuous Galerkin time-domain (DGTD-ME) method. Numerical simulations are executed to demonstrate the accuracy and efficiency of the proposed method, which realize twice the memory and triple the CPU time reductions.

An improved intrinsic time-scale decomposition method based on adaptive noise and its application in bearing fault feature extraction

This study proposes a new method for detecting and diagnosing fault information for rolling bearings. First, a novel method called complete ensemble intrinsic timescale decomposition with adaptive noise is proposed. Compared with the complete ensemble empirical modes decomposition with adaptive noise (CEEMDAN) method, the proposed method reduces the spurious mode, which interferes with the identification of feature frequency in rolling bearings. In addition, when a typical mode mixing signal is decomposed, the mode mixing problem that appears in the adjacent mode components of the CEEMDAN method is also solved. Second, there is only an intrinsic mode function, which mostly reflects the vibration feature of a rolling bearing. We propose a new method to evaluate these mode functions to determine which mode function contains the most useful information. Our proposed method solves the problem using traditional time-domain analysis methods of a single dimensionless parameter with poor stability and low sensitivity. Our method includes sample entropy, kurtosis, the zero-crossing rate, and a correlation coefficient. The proposed method reduces the limitations of each method to select the sensitive component. Finally, we use a simulation signal and an experimental signal to verify the effectiveness of the proposed method.

Statistical Switching Overvoltage Studies of Optimized Unconventional High Surge Impedance Loading Lines via Numerical Laplace Transform

The development of a revolutionary design for transmission lines was recently introduced by shifting phase configurations and sub-conductors into “unconventional” arrangements that were geometrically optimized to maximize natural power. The objective of this paper is to study the electromagnetic transients that result from the simultaneous and sequential line energization and re-energization of the previously developed unconventional high surge impedance loading (HSIL) lines. Statistical switching overvoltage (SSOV) studies are performed using a frequency domain method based on the inverse numerical Laplace transform. Compared to a conventional line, all unconventional HSIL line designs studied in this paper lead to very similar surges for long line lengths (400 km) and similar values for short line lengths (100 km), indicating additional advantages of these lines. The use of a frequency domain method for SSOV studies offers enhanced accuracy, simulation flexibility, and substantially lower computational burden compared with traditional time-domain methods.

Novel method to operation conditions identification of high-order power converters

IntroductionIntermittent solar energy causes different operational modes of power converters including continuous current modes (CCMs) and discontinuous current modes (DCMs), which need appropriate control strategies and parameters assignment to ensure the functionality of the overall solar energy power generation system. Hence, it is important to identify suitable operation modes for a high-order converter system. However, for a high-order power converter (HOPC), traditional time-domain analysis method and bifurcation analysis are inapplicable, since this requires comprehensive analysis and sophisticated control design. ObjectivesTo improve reliability and reduce mathematical complexity, this paper focuses on the operation mode derivation of HOPCs to well identify its boundary conditions and provide industry standards for converter applications. MethodsWith complex operation modes, 3-Z-network converter is analysed as a typical example and its derivations of boundary conditions are elaborated. In detail, the equilibrium points and boundary conditions of each operation modes are first derived; then with the guidance of boundary conditions, unexpected operation modes can be avoided by parameters reassignment. ResultsSimulations and experimentation on the newly established system prototype are conducted to validate the effectiveness of the proposed approach. It demonstrates that the theoretical and experimental boundary conditions are in good agreement. ConclusionThis paper provides equilibrium points and boundary conditions, and obtains deeper insights into the behaviors of the 3-Z-network converter. The derivations of four operation modes and the boundary condition of each mode has been conducted and provided for the large-signal averaged model of the converter, which provides guidance for engineers to adjust the system parameters so as to realize required operation modes. Simulation and experimentation have verified the accuracy and effectiveness of the proposed identified operation boundaries.

An interpolating scaling functions method with low-storage five-stage fourth-order explicit Runge-Kutta schemes for 3D ground penetrating radar simulation

Accurate and efficient 3D simulation is extremely important for both grasping the features of radar wave propagation in realistic models and the high-resolution inversion. In this paper, we propose a new numerical algorithm called the low-storage five-stage fourth-order explicit Runge-Kutta interpolation scaling function method (LSERK-ISFM) and apply it into the 3D simulation of ground penetrating radar (GPR). Firstly, we derive the auxiliary differential equation (ADE) of the 3D Maxwell's equations based on the complex-frequency shifted perfectly matched layer (CFS-PML), discretize the spatial derivative of the Maxwell equation by ISFM, and integrate the temporal dimension through the LSERK scheme. Compared with the traditional finite-difference time-domain (FDTD) method, the LSERK-ISFM algorithm can effectively improve the precision of space-time dispersion of the Maxwell's equations. Specifically, by the homogeneous vacuum medium model, we compare the forward modeling results of the different number of grids per wavelength, the comparison results prove that a smaller number of grids per wavelength can be selected in the LSERK-ISFM scheme, which can greatly contribute to reducing the consumption of memory. Then we build a simple 3D model to properly compare the proposed algorithm with the coarse-mesh FDTD and the fine-mesh FDTD, the results show that, with the same mesh generation, the LSERK-ISFM algorithm outperforms the conventional FDTD algorithms in terms of both accuracy and efficiency. Finally, we test the algorithm on a 3D complex model by a series of 3D forward profiles and the corresponding slices, both simulated reflections and diffractions clearly depict the shape and location of the anomalous bodies, which shows great potential of application in the field of 3D exploration and interpretation for GPR community.

A modified time domain approach for calculation of noise reduction and acoustic impedance of intake duct system

Calculation of noise reduction and acoustic impedance of the duct system is needed for the one-port source model, which is widely used in the field of duct acoustics for extracting the source characteristics and predicting the radiating noise at the open end. The flow field in the duct influences the acoustic characteristics when the flow velocity is relatively high. Recently, the time domain method has been adopted to study the flow effects on the acoustic response of the duct mufflers. However, the existing method only focuses on the calculation of transmission loss (TL) or transfer matrix. In this paper, a modified time domain approach is proposed for the first time to calculate the noise reduction and acoustic impedance of the intake duct system. In the proposed time domain model, the frequency dependent reflection at the intake port, rather the anechoic boundary used for TL calculation, is simulated by simply adding a correction pipe at the port. This approach has effectively simplified the implementation of the traditional time domain method in dealing with frequency dependent reflection at the open end. The porous sound-absorbing material commonly used in the intake system is modelled by modified equations with the material porosity and flow resistivity. The predicted time-domain sound pressure signals are then post-processed to obtain the noise reduction and the acoustic impedance in a different way from the traditional time domain TL calculation approach. The result of validation has shown that the noise reduction and acoustic impedance predicted with the modified time domain approach agree well with the measured results and the results calculated from frequency domain methods in the cases without and with meanflow.

Laplace domain method for computing the transient response of floating structures to irregular waves

Traditional methods for evaluating the dynamic response of floating structures to irregular waves have been performed in either the frequency or the time domain. Whereas the computed response from the frequency-domain method is merely the steady-state response, implementing the time-domain method is usually time-consuming. In this article, an efficient pole–residue method operated in the Laplace domain to compute transient response of floating structures to irregular waves is developed, given that the frequency-domain data has been obtained from running a standard hydrodynamic package. The proposed method requires both the irregular waves and the transfer functions associated with the wave-structure system be in their pole–residue forms. From those, pole–residue form of the response can be obtained by simple pole–residue operations. Once the poles and residues of the response are known, the response time history is readily obtained. In the numerical studies, this article considers a SDOF model for the heave motion of a spar platform excited by regular and irregular waves, respectively. Numerical studies demonstrated that the proposed method is more efficient and accurate than the traditional time-domain method.

Ultrawideband Discharge Source DOA Estimation Method Using Multiple Baseline Wideband Time-Domain Interferometry with Hilbert Transform

The research interest of ultrawideband (UWB) discharge source location estimation has increased these years. In this paper, a direction of arrival (DOA) estimation method using multiple baseline wideband time-domain interferometry with Hilbert transform for UWB discharge source is proposed based on time-domain and frequency-domain characteristics of radiated RF electromagnetic pulses (EMPs) from discharge sources. Monte Carlo simulations are then carried out; the results indicate that, the proposed method provides a better performance in UWB discharge source DOA estimation than the traditional time-domain method, especially in low signal-to-noise ratio (SNR) conditions. Moreover, the influences of antenna array configurations and incident angles of radiated EMPs on the estimation precision are also studied. It has been shown that, the accuracy of both elevation angle and azimuth angle estimation improves with the increase of the antenna element number and baseline length. As for the influence of incident angles, the estimation accuracy of elevation angle enhances when real elevation angle increases, while that of azimuth angle tends to be opposite. Meanwhile, the real azimuth angle has little effect on the DOA estimation. Finally, an experimental setup for discharge source DOA estimation is introduced and the experiment results are illustrated.

A rapid response calculation method for symmetrical floating structures based on state–space model solving in hybrid time-Laplace domain

The article focuses on proposing a new hybrid time-Laplace domain response calculation method for symmetrical floating structures by solving Cummins equation, which is depending on the state–space model identified from transfer function. Different from a time or frequency domain method, the proposed approach estimates the state–space model of a floating structure system from the transfer function in the Laplace domain and calculates the response by considering the exciting forces as an input of the state–space model. Implementing complex exponential decomposition to the retardation functions, the retardation functions in the Laplace domain are expressed using a series of poles and the corresponding residues, which avoids the numerical integral of the convolution terms in the Cummins equation and greatly improves the computational efficiency during the process of a dynamic response analysis. Three examples are applied to investigate the validity of the proposed method. The first is a simple single degree of freedom mathematical model excited by an irregular wave. Studies have shown that the response calculated by the proposed method matches well with that of a traditional Newmark-β method. Meanwhile, this approach is insensitive to the interval time of the calculation and consumes less calculation time, which means that the proposed method has higher precision and computational efficiency. The last two examples are a spar-type offshore wind turbine and a semi-submersible platform model (SEMI) in SESAM, which extend the proposed method to solve the response estimation problem of marine structures. The results of the spar-type floating structure show that the estimated responses when using the proposed method are in good agreement with the results of the traditional time domain method. By studying the response calculation of SEMI, the following conclusions can be obtained: (1) the estimated responses match well with the traditional time domain method and WASIM code of SESAM, and (2) the calculation time of the new approach is reduced significantly compared with the Newmark-β method especially under a long simulation time.

Estimation of Wave Velocity for Ultrasonic Imaging of Concrete Structures Based on Dispersion Analysis

Abstract Wave velocity is the most critical parameter used by the synthetic aperture focusing technique for the ultrasonic imaging of concrete structures. However, it has been challenging to estimate wave velocity with the traditional time-domain method that requires the picking of direct waves. The dispersion trend of direct waves is theoretically a horizontal line with frequency as the x-axis and phase velocity as the y-axis, and the horizontal line is due to the nondispersive phenomenon of direct waves in an elastic, homogeneous material. To overcome the challenge of picking direct waves in the time domain, this study proposes a frequency-domain method to first obtain the dispersion trend of direct waves based on the nondispersive phenomenon and then estimate the shear-wave velocity from the average phase velocity of the nondispersive trend in the frequency domain. This frequency-domain method was verified with the numerical and experimental ultrasonic data from concrete specimens with embedded delamination. A comparison of the results obtained using the two methods indicates that the frequency-domain method can significantly improve the accuracy of estimating the shear-wave velocity and eventually improve the accuracy of ultrasonic imaging.

Study on electromagnetic characteristics of plasma model-based on the symplectic multiresolution time-domain scheme

A novel numerical computation scheme, which is symplectic multiresolution time-domain (S-MRTD) scheme, for modeling plasma model is proposed. The loss plasma dispersive model is taken into account in S-MRTD scheme, and the detailed formulations of the proposed S-MRTD scheme are also provided. A one-dimensional perfectly matched layers (PML) are used to terminate the computational domain. The analyses of stability and numerical dispersion demonstrate that S-MRTD scheme is more efficient than traditional finite-different time-domain (FDTD) and MRTD methods. The energy conservation characteristics of S-MRTD scheme in electromagnetic simulation are proved by the propagation of pulse in free space for long-term simulation. In the end, the S-MRTD formulations are confirmed by computing the electric field intensity, reflection and the transmission coefficients for the pulse wave through an unmagnetized plasma slab. A favorable agreement between the numerical solutions is demonstrated, and the efficiency of the proposed scheme is verified. Meanwhile, numerical results show that plasma frequency, collision frequency and thickness are important factors affecting reflection and transmission coefficients.

Entropy: A Promising EEG Biomarker Dichotomizing Subjects With Opioid Use Disorder and Healthy Controls.

Electroencephalography (EEG) signals are known to be nonstationary and often multicomponential signals containing information about the condition of the brain. Since the EEG signal has complex, nonlinear, nonstationary, and highly random behaviour, numerous linear feature extraction methods related to the short-time windowing technique do not satisfy higher classification accuracy. Since biosignals are highly subjective, the symptoms may appear at random in the time scale and very small variations in EEG signals may depict a definite type of brain abnormality it is valuable and vital to extract and analyze the EEG signal parameters using computers. The challenge is to design and develop signal processing algorithms that extract this subtle information and use it for diagnosis, monitoring, and treatment of subjects suffering from psychiatric disorders. For this purpose, finite impulse response-based filtering process was employed rather than traditional time and frequency domain methods. Finite impulse response subbands were analyzed further to obtain feature vectors of different entropy markers and these features were fed into a classifier namely multilayer perceptron. The performances of the classifiers were finally compared considering overall classification accuracies, area under receiver operating characteristic curve scores. Our results underline the potential benefit of the introduced methodology is promising and is to be treated as a clinical interface in dichotomizing substance use disorders subjects and for other medical data analysis studies. The results also indicate that entropy estimators can distinguish normal and opioid use disorder subjects. EEG data and theta frequency band have distinctive capability for almost all types of entropies while nonextensive Tsallis entropy outperforms compared with other types of entropies.

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.

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.

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.

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.

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.