Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Comparison of finite difference and pseudo-spectral methods in forward modelling based on metal ore model of random media

With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.

Dynamic Coupling Analysis of Semisubmersible Platform Float-over Method for Docking Case

In this paper,the multi-body coupled dynamic characteristics of a semisubmersible platform and an HYSY 229 barge were investigated.First,coupled hydrodynamic analysis of the HYSY 229 barge and the semisubmersible platform was performed.Relevant hydrodynamic parameters were obtained using the retardation function method of three-dimensional frequency-domain potential flow theory.The results of the hydrodynamic analysis were highly consistent with the test findings,verifying the accuracy of the multifloating hydrodynamic coupling analysis,and key hydrodynamic parameters were solved for different water depths and the coupling effect.According to the obtained results,the hydrodynamic influence was the largest in shallow waters when the coupling effect was considered.Furthermore,the coupled motion equation combined with viscous damping,fender system,and mooring system was established,and the hydrodynamics,floating body motion,and dynamic response of the fender system were analyzed.Motion analysis revealed good agreement among the surge,sway,and yaw motions of the two floating bodies.However,when the wave period reached 10 s,the motion of the two floating bodies showed severe shock,and a relative motion was also observed.Therefore,excessive constraints should be added between the two floating bodies during construction to ensure construction safety.The numerical analysis and model test results of the semisubmersible platform and HYSY 229 barge at a water depth of 42 m and sea conditions of 0°,45°,and 90° were in good agreement,and the error was less than 5%.The maximum movement of the HYSY 229 barge reached 2.61 m in the sway direction,whereas that of the semisubmersible platform was 2.11 m.During construction,excessive constraints should be added between the two floating bodies to limit their relative movement and ensure construction safety.

An Accurate Numerical Solution for the Modified Equal Width Wave Equation Using the Fourier Pseudo-Spectral Method

In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L2 and L∞ error norms are computed to study the accuracy and the simplicity of the presented method.

Time-Domain Analysis of Body Freedom Flutter Based on 6DOF Equation

The reduced weight and improved efficiency of modern aeronautical structures result in a decreasing separation of frequency ranges of rigid and elastic modes.Particularly,a high-aspect-ratio flexible flying wing is prone to body freedomflutter(BFF),which is a result of coupling of the rigid body short-periodmodewith 1st wing bendingmode.Accurate prediction of the BFF characteristics is helpful to reflect the attitude changes of the vehicle intuitively and design the active flutter suppression control law.Instead of using the rigid body mode,this work simulates the rigid bodymotion of the model by using the six-degree-of-freedom(6DOF)equation.A dynamicmesh generation strategy particularly suitable for BFF simulation of free flying aircraft is developed.An accurate Computational Fluid Dynamics/Computational Structural Dynamics/six-degree-of-freedom equation(CFD/CSD/6DOF)-based BFF prediction method is proposed.Firstly,the time-domain CFD/CSD method is used to calculate the static equilibrium state of the model.Based on this state,the CFD/CSD/6DOF equation is solved in time domain to evaluate the structural response of themodel.Then combinedwith the variable stiffnessmethod,the critical flutter point of the model is obtained.This method is applied to the BFF calculation of a flyingwing model.The calculation results of the BFF characteristics of the model agree well with those fromthe modalmethod andNastran software.Finally,the method is used to analyze the influence factors of BFF.The analysis results show that the flutter speed can be improved by either releasing plunge constraint or moving the center ofmass forward or increasing the pitch inertia.

Pseudo-Spectral Method for Space Fractional Diffusion Equation

This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.

Calculation of Turbulent Flow with Pseudo-spectral Method

This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and transition of 43×43 computational grids,a coordinate transformation is put up from physical panel to computational panel. Several zero turbulent models are computed comparatively. The results are credible when comparing with the previous methods.

Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

A 1D time-domain method for in-plane wave motions in a layered half-space

A 1D finite element method in time domain is developed in this paper and applied to calculate in-plane wave motions of free field exited by SV or P wave oblique incidence in an elastic layered half-space. First, the layered half-space is discretized on the basis of the propagation characteristic of elastic wave according to the Snell law. Then, the finite element method with lumped mass and the central difference method are incorporated to establish 2D wave motion equations, which can be transformed into 1D equations by discretization principle and explicit finite element method. By solving the 1D equations, the displacements of nodes in any vertical line can be obtained, and the wave motions in layered half-space are finally determined based on the characteristic of traveling wave. Both the theoretical analysis and the numerical results demonstrate that the proposed method has high accuracy and good stability.

Second-Order Wave Diffraction Around 3-D Bodies by A Time-Domain Method

A time-domain method is applied to simulate nonlinear wave diffraction around a surface piercing 3-D arbitrary body. The method involves the application of Taylor series expansions and the use of perturbation procedure to establish the corresponding boundary value problems with respect to a time-independent fluid domain. A boundary element method based on B-spline expansion is used to calculate the wave field at each time step, and the free surface boundary condition is satisfied to the second order of wave steepness by a numerical integration in time. An artificial damping layer is adopted on the free surface for the removal of wave reflection from the outer boundary. As an illustration, the method is used to compute the second-order wave forces and run-up on a surface-piercing circular cylinder. The present method is found to be accurate, computationally efficient, and numerically stable.

Terahertz Time-Domain Spectroscopy method for optical parameter extraction of plastic materials

A method for extracting optical parameters of plastics materials based on terahertz time domain spectroscopy is presented. The transmission-type Terahertz Time-Domain Spectroscopy(THz TDS) system is adopted to detect the refractive index and extinction coefficient on different plastic materials. Then the corresponding spectral information is obtained by Fourier transform of the terahertz time domain waveform of the sampling points, including the corresponding amplitude and phase information of the waveform. The optical parameter extraction model is built. By using the simplex optimization method, the curves of the refractive index and extinction coefficient for the plastic material are obtained. The experimental samples are made of different plastic parallel plate materials. The experimental results show that the optimization of optical parameters can improve their extraction accuracy, and the error of refractive index is ±0.005. Extraction technology with the simplex optimization method of optical parameter based on THz TDS can help to extract the optical parameters of engineering plastics. It is of great significance for the research of terahertz nondestructive testing.

Using a Time-domain Higher-order Boundary Element Method to Simulate Wave and Current Diffraction from a 3-D Body

To study wave-current actions on 3-D bodies a time-domain numerical model was established using a higher-order boundary element method(HOBEM).By assuming small flow velocities,the velocity potential could be expressed for linear and higher order components by perturbation expansion.A 4th-order Runge-Kutta method was applied for time marching.An artificial damping layer was adopted at the outer zone of the free surface mesh to dissipate scattering waves.Validation of the numerical method was carried out on run-up,wave exciting forces,and mean drift forces for wave-currents acting on a bottom-mounted vertical cylinder.The results were in close agreement with the results of a frequency-domain method and a published time-domain method.The model was then applied to compute wave-current forces and run-up on a Seastar mini tension-leg platform.

Stochastic transient analysis of thermal stresses in solids by explicit time-domain method

Stochastic heat conduction and thermal stress analysis of structures has received considerable attention in recent years.The propagation of uncertain thermal environments will lead to stochastic variations in temperature fields and thermal stresses.Therefore,it is reasonable to consider the variability of thermal environments while conducting thermal analysis.However,for ambient thermal excitations,only stationary random processes have been investigated thus far.In this study,the highly efficient explicit time-domain method(ETDM)is proposed for the analysis of non-stationary stochastic transient heat conduction and thermal stress problems.The explicit time-domain expressions of thermal responses are first constructed for a thermoelastic body.Then the statistical moments of thermal displacements and stresses can be directly obtained based on the explicit expressions of thermal responses.A numerical example involving non-stationary stochastic internal heat generation rate is investigated.The accuracy and efficiency of the proposed method are validated by comparison with the Monte-Carlo simulation.

Investigation of three-pulse photon echo in thick crystal using finite-difference time-domain method

This paper investigates the phenomenon of three-pulse photon echo in thick rare-earth ions doped crystal whose thickness is far larger than 0.002 cm which is adopted in previous works.The influence of thickness on the three-pulse photon echo＇s amplitude and efficiency is analyzed with the Maxwell-Bloch equations solved by finite-difference timedomain method.We demonstrate that the amplitude of three-pulse echo will increase with the increasing of thickness and the optimum thickness to generate three-pulse photon echo is 0.3 cm for Tm^（3＋）：YAG when the attenuation of the input pulse is taken into account.Meanwhile,we find the expression 0.09 exp（α＇L）,which is previously employed to describe the relationship between echo＇s efficiency and thickness,should be modified as 1.3 · 0.09 exp（2.4 ·α＇L） with the propagation of echo considered.

Optical simulation of in-plane-switching blue phase liquid crystal display using the finite-difference time-domain method

The finite-difference time-domain method is used to simulate the optical characteristics of an in-plane switching blue phase liquid crystal display.Compared with the matrix optic methods and the refractive method,the finite-difference timedomain method,which is used to directly solve Maxwell＇s equations,can consider the lateral variation of the refractive index and obtain an accurate convergence effect.The simulation results show that e-rays and o-rays bend in different directions when the in-plane switching blue phase liquid crystal display is driven by the operating voltage.The finitedifference time-domain method should be used when the distribution of the liquid crystal in the liquid crystal display has a large lateral change.

Adsorption and Transfer Properties for Toluene and pDichlorobenzene in Dense CO_2/Silica Gel Packed Bed System by Moment Method and Time-Domain Analysis

Based on chromatographic theory, the moment method and the time-domain fitting analysis were applied to measure and evaluate the adsorption equilibrium constant and mass transfer properties (axial dispersion coefficient and effective intra-particle diffusivity) for toluene and p-dichlorobenzene on silica gel adsorbent in the subcritical and supercritical CO2. An apparatus based on supercritical fluid chromatography was established and the experiments were performed at temperatures of 298.15-318.15 K and pressures of 7.5-17.8 MPa. The two methods have been compared. The results show that for the systems studied here the moment method can give reasonable values for both adsorption equilibrium constant and mass transfer properties, but the time-domain analysis only can obtain the adsorption equilibrium constant. The dependence of adsorption equilibrium constant and mass transfer properties on temperature and pressure was investigated.

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain（LOD-CFDTD） method is presented for solving two-dimensional（2D） electromagnetic（EM） scattering problems. The formulation for the 2D transverse-electric（TE） case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit（ADI） CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field（TF/SF） boundary and the perfectly matched layer（PML）, the radar cross section（RCS） of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.

IMPROVED LOCALLY CONFORMAL FINITE-DIFFERENCE TIME-DOMAIN METHOD FOR EDGE INCLINED SLOTS IN A FINITE WALL THICKNESS WAVEGUIDE

An Improved Locally Conformal Finite-Difference Time-Domain (ILC-FDTD) method is presented in this paper, which is used to analyze the edge inclined slots penetrating adjacent broadwalls of a finite wall thickness waveguide. ILC-FDTD not only removes tile instability of the original locally conformal FDTD algorithm, but also improves the computational accuracy by locally modifying magnetic field update equations and the virtual iterative electric fields accordlng to the complexity of tile slot fringe fields. The mutual coupling between two edge inclined slots can also be analyzed by ILC-FDTD effectively.

An Improved Time-Domain Inverse Technique for Localization and Quantifcation of Rotating Sound Sources

The time-domain inverse technique based on the time-domain rotating equivalent source method has been proposed to localize and quantify rotating sound sources. However, this technique encounters two problems to be addressed: one is the time-consuming process of solving the transcendental equation at each time step, and the other is the difculty of controlling the instability problem due to the time-varying transfer matrix. In view of that, an improved technique is proposed in this paper to resolve these two problems. In the improved technique, a de-Dopplerization method in the time-domain rotating reference frame is frst applied to eliminate the Doppler efect caused by the source rotation in the measured pressure signals, and then the restored pressure signals without the Doppler efect are used as the inputs of the time-domain stationary equivalent source method to locate and quantify sound sources. Compared with the original technique, the improved technique can avoid solving the transcendental equation at each time step, and facilitate the treatment of the instability problem because the transfer matrix does not change with time. Numerical simulation and experimental results show that the improved technique can eliminate the Doppler efect efectively, and then localize and quantify the rotating nonstationary or broadband sources accurately. The results also demonstrate that the improved technique can guarantee a more stable reconstruction and compute more efciently than the original one.

Vibration Diagnosis and Optimization of Industrial Robot Based on TPA and EMD Methods

This paper proposedmethod that combined transmission path analysis(TPA)and empirical mode decomposition(EMD)envelope analysis to solve the vibration problemof an industrial robot.Firstly,the deconvolution filter timedomain TPA method is proposed to trace the source along with the time variation.Secondly,the TPA method positioned themain source of robotic vibration under typically different working conditions.Thirdly,independent vibration testing of the Rotate Vector(RV)reducer is conducted under different loads and speeds,which are key components of an industrial robot.The method of EMD and Hilbert envelope was used to extract the fault feature of the RV reducer.Finally,the structural problems of the RV reducer were summarized.The vibration performance of industrial robots was improved through the RV reducer optimization.From the whole industrial robot to the local RV Reducer and then to the internal microstructure of the reducer,the source of defect information is traced accurately.Experimental results showed that the TPA and EMD hybrid methods were more accurate and efficient than traditional time-frequency analysis methods to solve industrial robot vibration problems.