Nonstationary Wave Research Articles

Physics-statistics-based hybrid simulation scheme of coupled nonstationary hurricane wind and wave fields for long-span floating bridges

Abstract Long-span bridges with floating towers have recently drawn great attention from the engineering community. Due to their sensitivity to the aerodynamic and hydrodynamic loads during extreme storms, accurate and efficient simulation tools for hurricane winds and waves are needed for improved understanding of the complex dynamics of the fully coupled wind–wave–structure interaction system. Conventional simulation schemes usually generate winds and waves separately, and hence cannot capture the intense wind–wave interactions under hurricanes. In this study, a physics-statistics-based hybrid simulation scheme of nonstationary hurricane wind and wave fields is presented, where the winds and waves are coupled in both large and small scales. To simulate the large-scale winds and waves, a height-resolving hurricane wind model is coupled with a parametric hurricane wave model through a dependence between sea surface roughness and surface wind speed. In the small-scale simulations, the nonstationary wind fluctuations are statistically obtained by a Hilbert-wavelet-based scheme in which the target parameters (e.g. fluctuation intensity) are estimated based on the local sea state, while the nonstationary sea surface elevations are physically acquired by solving the governing equation of the nonlinear wave evolution under the action of winds. The simulation fidelity of the proposed physics-statistics-based hybrid scheme is demonstrated by generating the coupled nonstationary wind and wave fields approaching to a hypothetical long-span bridge with floating towers under a hurricane event.

High-fidelity simulations for Turing pattern formation in multi-dimensional Gray–Scott reaction-diffusion system

In this study, the authors present high-fidelity numerical simulations for capturing the Turing pattern in a multi-dimensional Gray–Scott reaction-diffusion system. For this purpose, an explicit mixed modal discontinuous Galerkin (DG) scheme based on multi-dimensional structured meshes is employed. This numerical scheme deals with high-order derivatives in diffusion, and highly nonlinear functions in reaction terms. Spatial discretization is accomplished using hierarchical basis functions based on scaled Legendre polynomials. A new reaction term treatment is also detailed, showing an intrinsic property of the DG scheme and preventing incorrect solutions due to excessively nonlinear reaction terms. The system is reduced to a set of time-dependent ordinary differential equations, which are solved with an explicit third-order Total Variation Diminishing (TVD) Runge–Kutta method. The Saddle-Node, Hopf, and Turing bifurcations’ boundary conditions are also examined for the system, and subsequently, Turing space is identified. The developed numerical scheme is applied to various multidimensional Gray–Scott systems to assess its ability to capture Turing pattern formation. Several test problems, such as stationary and non-stationary waves, pulse-splitting waves, self-replicating waves, and different types of spots, are chosen from the literature to demonstrate the different Turing patterns.

A hybrid VMD-LSTM/GRU model to predict non-stationary and irregular waves on the east coast of China

Accurate wave forecasting is essential for the safety of port and offshore structure operations and ship navigation. Computational fluid dynamics (CFD) and traditional time series models are ineffective in dealing with non-linearities and non-smoothness. However, long short-term memory (LSTM) and gate recurrent units (GRU) have strong non-linear handling capabilities but are deficient in non-stationary situations. Variational mode decomposition (VMD) can effectively separate the non-linearity and non-smoothness in data. In this report, a VMD-LSTM/GRU model is proposed by combining the advantages of the LSTM model, the GRU model, and the VMD technique. Based on the one-tenth maximum wave height at three locations on the east coast of China, the error of the VMD-LSTM/GRU model is shown to be lower than that of the LSTM/GRU model, and significantly lower than that of the single LSTM and single GRU models. By analyzing different forecast durations, it was found that the correlation of the VMD-LSTM/GRU model improved from 10.75% to 20.99% compared to the corresponding LSTM/GRU model. The RMSE and relative errors of the VMD-LSTM/GRU model were reduced by 96.77% and 95.52%, respectively, for the most difficult forecast result 10 h ahead. Thus, this model has proved to be superior in predicting non-linear and non-stationary waves.

The anti-localization of non-stationary linear waves and its relation to the localization. The simplest illustrative problem

We introduce a new wave phenomenon, which can be observed in continuum and discrete systems, where a trapped mode exists under certain conditions, namely, the anti-localization of non-stationary linear waves. This is zeroing of the non-localized propagating component of the wave-field in a neighbourhood of an inclusion. In other words, it is a tendency for non-stationary waves to propagate avoiding a neighbourhood of an inclusion. The anti-localization is caused by a destructive interference of the harmonics involved into the representation of the solution in the form of a Fourier integral. The anti-localization is associated with the waves from the pass-band, whereas the localization related with a trapped mode is due to poles inside the stop-band. In the framework of a simple illustrative problem considered in the paper, we have demonstrated that the anti-localization exists for all cases excepting the boundary of the domain in the parameter space where the wave localization occurs. Thus, the anti-localization can be observed in the absence of the localization as well as together with the localization. We also investigate the influence of the anti-localization on the wave-field in whole.

Analytical H∞ and H2 optimization for negative-stiffness inerter-based systems

Two novel dynamic vibration absorbers containing inerter and negative stiffness (NI-DVA) are presented to reduce the vibration of a primary system which is subjected to the acceleration excitation from the base. The NI-DVAs which are called N-TID and N-TVMD, are composed of a parallel inerter and negative stiffness spring that connected with a spring and a dashpot in different layouts. The analytical expressions of optimal stiffness ratio, damping ratio and negative stiffness ratio for both H∞ and H2 optimization design are derived based on the fixed-points theory and extreme value theory under the Routh-Hurwitz stability condition. These analytically optimal design parameters of NI-DVAs are expressed as functions of the inertance mass ratio. The effectiveness of the analytical solutions of optimal parameters for two NI-DVAs is verified by comparing them with the results obtained from the numerical searching optimization method. The vibration reduction performance of the negative stiffness element in the N-TID and N-TVMD is evaluated by comparing it with TID and TVMD harmonic excitation, white noise, and non-stationary earthquake waves. The numerical studies suggest that the presence of the negative stiffness element in NI-DVAs possesses superior vibration reduction performance compared with their original counterparts. Though higher improvement in vibration reduction performance is observed from N-TID due to the existence of a negative stiffness element, N-TVMD outperforms N-TID in terms of vibration reduction under both harmonic and random acceleration excitation.

Machine learning control of complex nonlinear dynamics in fibre lasers - INVITED

We review our recent work on the use of genetic algorithms to control non-stationary nonlinear wave dynamics in ultrafast fibre lasers, including the generation of breathing-soliton dynamics with controlled characteristics, the disclosure of the fractal dynamics of breathers, and the generation of rogue waves with controlled intensity.

Propagation of skew-symmetric unsteady shear waves from thick-walled shell in elastic space

Problems of propagation and diffraction of nonstationary waves in elastic bodies are of great theoretical and practical importance in such fields of science and technology as aircraft construction, shipbuilding, seismic exploration of minerals, seismic resistance of structures, and many others. The paper considers the problem of the propagation of skew-symmetric unsteady shear waves from a thick-walled spherical shell in elastic space. To solve the problem, the integral Laplace transforms in dimensionless time, and the method of incomplete separation of variables was used. In the image space, the problem is reduced to an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Formulas for the displacement vector components and the stress tensor are obtained. The transition to the originals is carried out using the theory of residues. Numerical experiments have been carried out, the results of which are presented in graphs. The obtained results of the work can be used in geophysics, seismology, and design organizations in the construction of structures and in the design of underground reservoirs.

On the Problem of Determining the Position of the Source of Internal Waves

When bodies move in a continuously stratified fluid, the steady wave field moves along with the body and form a field of so-called associated internal waves. The flow incident on the body is usually assumed to be constant; non-stationary waves generated at the initial stage of motion are neglected. In this case, the body is modeled by point mass sources, and the wave field is found using the Green’s function method, followed by the use of asymptotic expansions based on the stationary phase method [1]. The inverse problem of determining the position of the source is solved from the wave field.

Transient Wave Propagation in Functionally Graded Viscoelastic Structures

Transient wave processes in viscoelastic structures built from functionally graded material (FGM) still remain almost unexplored. In this article, the problem of the propagation of nonstationary longitudinal waves in an infinite viscoelastic layer of a FGM with plane–parallel boundaries is considered. The physical and mechanical parameters of the FGM depend continuously on the transverse coordinate, while the wave process propagates along the same coordinate. The viscoelastic properties of the material are taken into account employing the linear integral Boltzmann–Volterra relations. The viscoelastic layer of the FGM is replaced by a piecewise-homogeneous layer consisting of a large number of sub-layers (a package of homogeneous layers), thus approximating the continuous inhomogeneity of the FGM. A solution of a non-stationary dynamic problem for a piecewise-homogeneous layer is constructed and, using a specific example, the convergence of the results with an increase in the number of sub-layers in the approximating piecewise-homogeneous layer is shown. Furthermore, the problem of the propagation of nonstationary longitudinal waves in the cross section of an infinitely long viscoelastic hollow FGM cylinder, whose material properties continuously change along the radius, is also considered. The cylinder composed of the FGM is replaced by a piecewise-homogeneous one, consisting of a large number of coaxial layers, for which the solution of the non-stationary dynamic problem is constructed. For both the layer and the cylinder composed of a viscoelastic FGM, the results of calculating the characteristic parameters of the wave processes for the various initial data are presented. The influence of the viscosity and inhomogeneity of the material on the dynamic processes is demonstrated.

Vibration isolation performance and optimization design of a tuned inerter negative stiffness damper

A tuned inerter negative stiffness damper (TINSD) is a vibration isolation and energy dissipation device providing a negative stiffness effect, which has the dual damping enhancement effect of negative stiffness and inertance. To date, there have been several studies on TINSDs; however, these studies are incomplete in terms of optimization, isolation system parameter analysis and application in seismic engineering. Therefore, a systematic study of a TINSD in an isolation system is needed. This paper presents a theoretical and numerical study of a SDOF-TINSD system and a realistic structure-TINSD subjected to ground motion. The absolute transmissibility and stability condition of the SDOF-TINSD are derived, and the absolute stiffness value of the negative stiffness element should be less than the primary structural stiffness. The TINSD parameters are optimized based on the H∞ and H2 norms, and analytical solutions for the optimal parameters are obtained. The parameter and sensitivity analyses of the TINSD are carried out. For a known inertance-mass ratio, the vibration isolation performance can be improved by the negative stiffness. The TINSD is very sensitive to deviations of the positive stiffness ratio, and its optimal parameters require special attention. Finally, the vibration control effect of the optimal TINSD is evaluated for harmonic excitation, stationary random excitation and seismic excitation, and an optimization method based on nonstationary random seismic waves (NRSW) for isolators in realistic structures is proposed. The control effect of the TINSD is better than that of the TID and TVMD with the same inertance-mass ratio. Moreover, the TINSD optimized by NRSW has a better seismic response control effect than the H∞ and H2 optimization methods.

Dynamic Problem for a Viscoelastic Hollow Cylinder with Coaxial Elastic Inclusion


 
 
 A problem of non-stationary longitudinal wave propagation in the cross-section of an infinite hollow linear-viscoelastic cylinder with coaxial elastic inclusion is considered. On the contact surfaces between the layers, continuity of displacements and stresses is assumed. The solution of the problem is constructed using the integral Laplace transform in time with subsequent reversal. For the case when each of the relaxation kernels is a finite sum of exponentials, the original is presented as a series of residues. Thus, the process of constructing a non-stationary solution is reduced to the search for the roots of the characteristic transcendental equation. The derived solution allowed to investigate the transient wave propagation in a viscoelastic cylinder with coaxial elastic inclusion with specified initial data.
 
 

Magnetised and unmagnetized axisymmetric particle-in-cell simulations of ion energy distributions in cathodic vacuum arcs

Cathodic arcs are electrical discharges consisting of a succession of discrete pulses of energetic plasma travelling from the surface of a cathode toward an anode. Currently, there are no vacuum arc simulations where spots are continuously generated with a set frequency, including the far-field plasma jet, with the inclusion of kinetic behaviour for both ions and electrons. The VSim 11 particle-in-cell software was used to simulate specific vacuum arcs as axisymmetric, electrostatic, and fully kinetic, from the initial generation of each cathode spot to the streaming plasma discharge at a far field, validating the predictions against experimental data. The models were configured to match the experimental arc gun of Zohrer et al and the Mevva V experiment with the cathode materials Al and Nb. The ion and electron velocity data were collected at the far edge of the simulation domain, analogous to a physical energy detector. The simulations successfully predicted the evolution of ion charge state energy distributions, showing peak unmagnetized ion energies that agree with prior experimental data, resulting in a mean error of 3% for Al and Nb. A peak in the electrostatic potential is observed above the cathode surface, supporting the potential hump theory as the cause of the higher-than-expected ion energies observed in cathodic arc discharges. Lower than expected relative energies between ion charge states are observed, matching prior experimental results, with this coupling of ion charge states attributed to non-stationary electrostatic wave–particle interactions, as the use of collisionless simulations rules out Coulombic ion friction. Magnetised simulations incorporated a statically powered short solenoid equivalent to the coil used in the Mevva V experiment to create a diverging magnetic nozzle. The magnetised simulation results demonstrate an annular jet of magnetically confined plasma and indicate an increase in nonstationary electrostatic effects including wave–particle interactions.

Компьютерное моделирование нестационарных упругих волн напряжений в консоли и десятиэтажном здании при фундаментальном воздействии в виде функции

The aim of the work is to consider the problems of numerical simulation of seismic safety of a console and a ten-storey building with a base in the form of an elastic half-plane under non-stationary wave influences. Modeling of tasks of the transition period is an actual fundamental and applied scientific task. Methodology. To solve a two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements is used. Based on the finite element method in displacements, an algorithm and a set of programs have been developed for solving linear planar two-dimensional problems that allow solving problems with non-stationary wave effects on complex systems. The algorithmic language Fortran-90 was used in the development of the software package. Результаты. Results. The problem of the effect of a plane longitudinal wave in an elastic half-plane in the form of four trapezoids and in the form of two half-periods of a sinusoid is considered to assess the physical reliability and mathematical accuracy. A system of equations consisting of 8016008 unknowns is solved. The problem of the effect of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. A system of equations consisting of 16016084 unknowns is solved. The problem of the effect of a plane longitudinal elastic wave in the form of a Heaviside function on a ten-storey building with a base in the form of a half-plane is considered. A system of equations consisting of 16202276 unknowns is solved. Contour stresses and components of the stress tensor are obtained in the characteristic areas of the problem under study. Based on the conducted research, the following conclusions can be drawn. Elastic contour stress on the sides of the console and a ten-story building are almost a mirror image of each other, that is, antisymmetric. The console and supporting structures of the building work like a beam during seismic action, that is, if there are tensile stresses on one side, then compressive stresses on the other. Bending waves mainly prevail on the contours of the console and supporting structures of the building under seismic influence.

The Kinks, the Solitons and the Shocks in Series‐Connected Discrete Josephson Transmission Lines

The localized running waves in the discrete Josephson transmission lines (JTL), constructed from Josephson junctions (JJ) and capacitors, are analytically studied. The quasicontinuum approximation reduces calculation of the running wave properties to the problem of equilibrium of an elastic rod in the potential field. Making additional approximation, the problem is reduced to the motion of the fictitious Newtonian particle in the potential well. It is shown that there exist running waves in the form of supersonic kinks and solitons and their velocities and profiles are calculated. It is shown that the nonstationary smooth waves which are small perturbations on the homogeneous nonzero background are described by Korteweg–de Vries equation and those on zero background by modified Korteweg–de Vries equation. The effect of dissipation on the running waves in JTL is also studied and it is found that in the presence of the resistors, shunting the JJ and/or in series with the ground capacitors, the only possible stationary running waves are the shock waves, whose profiles are also found. Finally, in the framework of Stocks expansion, the nonlinear dispersion and modulation stability in the discrete JTL are studied.

Nonlinear interference between solitons and nonstationary dispersive waves in a passively mode-locked fiber laser

We numerically investigated the interaction of a soliton with nonstationary dispersive waves as well as the soliton interaction induced by them, in a fiber ring laser mode locked by a nonlinear polarization rotation technique. In low-loss mode-locked states, a resonant frequency shifting effect and two time-jittering motions of solitons are found during the interaction. An analysis reveals that an indirect phase-matching interference between a pulsed dispersive wave and a soliton upon cross-phase modulation breaks the conservation of soliton momentum, while the subsequent soliton reconstruction leads to a recovery of the conservation. The interference mechanism, together with the conservation of soliton momentum, accounts for some basic behaviors of soliton laser dynamics.

Defect Detection using Depth Resolvable Statistical Post Processing in Non-Stationary Thermal Wave Imaging

Defects that are generated during various phases of manufacturing or transporting limit the future applicability and serviceability of materials. In order to detect these defects a non-destructive testing modality is required. Depth resolvable subsurface anomaly detection in non-stationary thermal wave imaging is a vital outcome for a reliable prominent investigation of materials due to its fast, remote and non-destructive features. The present work solves the 3-Dimensional heat diffusion equation under the stipulated boundary conditions using green’s function based analytical approach for recently introduced quadratic frequency modulated thermal wave imaging (with FLIR SC 655A as infrared sensor with spectral range of 7.5-14µm and 25 fps) to explore the subsurface details with improved sensitivity and resolution. The temperature response obtained by solving the 3-Dimensional heat diffusion equation is used along with random projection-based statistical post-processing approach to resolve the subsurface details by imposing a band of low frequencies (0.01-0.1 Hz) over a carbon fiber reinforced polymer for experimentation and extracting orthonormal projection coefficients to improve the defect detection with enhanced depth resolution. Orthonormal projection coefficients are obtained by projecting the orthonormal features of the random vectors that are extracted by using Gram-Schmidt algorithm, on the mean removed dynamic thermal data. Further, defect detectability of random projection-based post-processing approach is validated by comparing the full width at half maxima (FWHM) and signal to noise ratio (SNR) of the processed results of the conventional approaches. Random projection provides detailed visualization of defects with 31% detectability even for deeper and small defects in contrast to conventional post processing modalities. Additionally, the subsurface anomalies are compared with their sizes based on full width at half maxima (FWHM) with a maximum error of 0.99% for random projection approach.

A modal decomposition imaging algorithm for ultrasonic detection of delamination defects in carbon fiber composite plates using air-coupled Lamb waves

In this paper, a modal decomposition imaging (MDI) algorithm for detecting delamination defects in carbon fiber composite plates using air-coupled Lamb waves was proposed. Compared with the traditional time-domain imaging method using amplitude difference, the proposed MDI algorithm contained modal decomposition process and rotating scanning defect probability imaging method, which was more suitable for analyzing the nonlinear and non-stationary leakage Lamb waves signal. The cross-correlation coefficient of the instantaneous energy was constructed to be the damage index, which was obtained by the pretreatment of the relatively pure A0 mode Lamb waves. The effectiveness of MDI algorithm proposed in this paper for realizing the delamination defects of carbon fiber composite plates was verified by qualitative imaging and quantitative characterization comparing with the defect probability imaging (DPI) algorithm. It would be suitable for achieving accurate characterization of defects in situ testing and large-area rapid scanning of aerospace composite plates after quickly scanning.

Time-frequency analysis of pupil size modulated by global luminance, arousal, and saccade preparation signals using Hilbert-Huang transform

Pupil size changes constantly and is mainly determined by global luminance signals. In addition, the pupil responds to various cognitive and arousal processes, with larger pupil dilation observed in higher levels of cognitive or arousal processing. Although these task-evoked pupillary responses are extensively used in the pupil research, pupil analysis focusing on the frequency domain, particularly in the context of arousal and cognitive modulations, is less established. Fourier Transform method (FFT) has been used to understand the modulation of task difficulty on pupil oscillations. However, physiological signals are often characterized as non-linear and non-stationary waves, and the conventional spectral analytical method with linearity presumption is less appropriate to reveal modulation dynamics between time and frequency. Here, we used Hilbert-Huang Transform (HHT) to examine the time-frequency modulations on pupil size regulated by arousal, cognitive, and global luminance signals. Consistent with previous research, using FFT, higher spectral densities were obtained with lower luminance background. Moreover, higher spectral densities were found in the high emotional arousal condition. With HHT, we further demonstrated temporal changes on amplitude spectrum and inter-trial phase coherence (ITPC) in each intrinsic mode function (IMF), with stronger amplitudes in higher IMFs (i.e., low frequencies). Moreover, although global luminance, arousal and saccade preparation modulated pupil oscillatory responses, the modulation pattern in different IMFs was different. Together, our results demonstrated dynamics between the time and frequency domain on pupil oscillatory responses, highlighting the importance of examining the time-frequency interactions in the context of various pupil modulations.

Saddle Point Method for Transient Processes in Waveguides

A modification of the saddle point method is proposed for computation of nonstationary wave processes (pulses) in waveguides. The dispersion diagram of the waveguide is continued analytically. A set of possible saddle points on the dispersion diagram is introduced. A method of checking whether the particular saddle points contribute terms to the field decomposition is proposed. A classification of the waveguides based on the topology of the set of possible saddle points is outlined.

Nonstationary spin waves in a single rectangular permalloy microstrip under uniform magnetic excitation

Ferromagnetic resonance modes in a single rectangular ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ microstrip were directly imaged using time-resolved scanning transmission x-ray microscopy combined with a phase-locked ferromagnetic resonance excitation scheme, and the findings were corroborated by micromagnetic simulations. Although under uniform excitation in a single confined microstructure typically standing spin waves are expected, all imaged spin waves showed a nonstationary character both at and off resonance, the latter being additionally detected with microantenna-based ferromagnetic resonance. The effect of the edge quality on the spin waves was observed in micromagnetic simulations.