Nonlinear Wave Propagation Research Articles

Nonlinear bulk wave in a functionally graded symmetric hysteretic material model—A numerical study

A simple spring-mass chain implemented to model nonlinear wave propagation through a functionally graded hysteretic material. Harmonics generation due to single-frequency forward and reflected waves along with the corresponding hysteretic curves discussed.

Revisiting the Lattice Boltzmann Method Through a Nonequilibrium Thermodynamics Perspective

Abstract In this paper, we incorporate a nonequilibrium thermodynamics perspective that is consistent with the Onsager reciprocity principle into the lattice Boltzmann framework to propose a novel regularized lattice Boltzmann formulation for modeling the Navier–Stokes–Fourier equations. The new method is applied to one-dimensional (1D) isothermal situations wherein the advantages of incorporating such a nonequilibrium perspective can be explicitly appreciated. In such situations, the nonequilibrium contribution of the lattice populations obtained by the new method completely vanishes, and the lattice update is entirely reduced to evaluating the equilibrium distribution function. Such a counterintuitive 1D mesoscopic description is not obtained in any other existing lattice Boltzmann scheme. We therefore numerically test the proposed formulation on two complex problems, namely, shockwave and nonlinear wave propagation, and compare results with analytical results along with six existing lattice Boltzmann schemes; it is found that the new method indeed yields results that are more stable and accurate. These results highlight the potency of the nonequilibrium thermodynamics-based approach for obtaining accurate and stable lattice Boltzmann computations, and provide new insights into established lattice Boltzmann simulation methods.

Nonlinear Periodic Ion Acoustic Waves in a Relativistic Plasma with Isothermal Electrons and Cold Ions

Propagation of nonlinear ion acoustic waves in a relativistic electron–ion plasma is studied using the covariant formulation. Ions are assumed cold and a polytropic equation of state of the form p ...

Some comparisons between heterogeneous and homogeneous plates for nonlinear symmetric SH waves in terms of heterogeneous and nonlinear effects

In this article, the propagation of nonlinear shear horizontal waves for some comparisons between the heterogeneous and homogeneous plates is considered. It is assumed that one plate is made of up hyper-elastic, heterogeneous, isotropic, and generalized neo-Hookean materials, and the other consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to the nonlinear Schrodinger equation, these comparisons are made in terms of the heterogeneous and nonlinear effects. All numerical results, based on the asymptotic analyses in which the method of multiple scales is used, are graphically presented for the lowest dispersive symmetric branch of both dispersion relations.

An adaptive harmonic polynomial cell method with immersed boundaries: Accuracy, stability, and applications

AbstractWe present a 2D high‐order and easily accessible immersed‐boundary adaptive harmonic polynomial cell (IB‐AHPC) method to solve fully nonlinear wave‐structure interaction problems in marine hydrodynamics using potential‐flow theory. To reduce the total number of cells without losing accuracy, adaptive quad‐tree cell refinements are employed close to the free‐surface and structure boundaries. The present method is simpler to implement than the existing IB‐HPC alternatives, in that it uses standard square cells both in the fluid domain and at the boundaries, thus without having to use the more complex and expensive overlapping grids or irregular cells. The spurious force oscillations on moving structures, which is a known issue for immersed boundary methods (IBMs), are eliminated in this study by solving a separate boundary value problem (BVP) for a Lagrangian acceleration potential. We also demonstrate that solving a similar BVP for the corresponding Eulerian acceleration potential is far less satisfactory due to the involved second derivatives of the velocity potential in the body‐boundary condition, which are very difficult to calculate accurately in an IBM‐based approach. In addition, we present, perhaps for the first time since the HPC method was developed, a linear matrix‐based stability analysis for the time‐domain IB‐AHPC method. The stability analysis is also used in this study as a general guide to design robust and stable numerical algorithms, in particular related to the treatment of boundary conditions close to the intersection between a Dirichlet and a Neumann boundary, which is essential in time‐domain wave‐structure interaction analyses using IBMs. We confirm theoretically through the stability analysis that square cells have the best stability properties. The present method has been verified and validated satisfactorily by various cases in marine hydrodynamics, including a moving structure in an infinite fluid, fully nonlinear wave generation and propagation, and fully nonlinear diffraction and radiation of a ship cross section.

Nonlinear propagation of fast and slow magnetosonic waves in collisional plasmas

AbstractLow‐frequency fast and slow magnetosonic waves propagating in electron ion plasmas with damping effects through ions and neutral atoms collisions are investigated. Linear wave analysis is performed to obtain dispersion relation. The reductive perturbation method is applied and it is shown that fast and slow modes of nonlinear magnetosonic wave are governed by damped Korteweg‐de Vries (DKdV) equation in the presence of ion neutral collisions in plasmas. The analytical solution of DKdV soliton is presented under the assumption of weak collisional effects and numerical solutions of DKdV equation are also obtained using two‐level finite difference scheme with the help of Runge–Kutta method at different plasma parameters. The damping of nonlinear fast and slow magnetosonic wave structures at different times are discussed in the context of space plasma situations where ions and neutral atoms collisions exist.

Stochastic Wave\u2013Current Interaction in Thermal Shallow Water Dynamics

Holm (Proc R Soc A Math Phys Eng Sci 471(2176):20140963, 2015) introduced a variational framework for stochastically parametrising unresolved scales of hydrodynamic motion. This variational framework preserves fundamental features of fluid dynamics, such as Kelvin’s circulation theorem, while also allowing for dispersive nonlinear wave propagation, both within a stratified fluid and at its free surface. The present paper combines asymptotic expansions and vertical averaging with the stochastic variational framework to formulate a new approach for developing stochastic parametrisation schemes for nonlinear waves in fluid dynamics. The approach is applied to two sequences of shallow water models which descend from Euler’s three-dimensional fluid equations with rotation and stratification under approximation by asymptotic expansions and vertical averaging. In the entire family of nonlinear stochastic wave–current interaction equations derived here using this approach, Kelvin’s circulation theorem reveals a barotropic mechanism for wave generation of horizontal circulation or convection (cyclogenesis) which is activated whenever the gradients of wave elevation and/or topography are not aligned with the gradient of the vertically averaged buoyancy.

Nonlinear Propagation of Electrostatic Waves in Alkali–Fullerene Plasmas

Linear and nonlinear propagation of electrostatic waves is studied in electron-depleted bi-ion plasmas where positive and negative ions have different masses and charge states. The two-fluid model is used. The Korteweg–de Vries equation is obtained using the reductive perturbation technique. The results of the numerical calculations show that in K–fullerene plasma for the experimentally obtained temperature limits of both ions only a negative polarity soliton structure is obtained. Potential structures for some useful alkali–fullerene plasmas are also illustrated. Finally, the exact technique of the Sagdeev pseudopotential is used to consider arbitrary amplitude (large-amplitude) solitons in K–fullerene plasma.

Theoretical elucidation of effect of drag force and translation of bubble on weakly nonlinear pressure waves in bubbly flows

Theoretical investigation of the effects of a translation of bubbles and a drag force acting on bubbles on the wave propagation in bubbly flows has long been lacking. In this study, we theoretically and numerically investigate the weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows that contain uniformly distributed spherical gas bubbles with translation and drag forces. First, we assume that the gas and liquid phases flow at independent velocities. Then, the drag force and virtual mass force are introduced in an interfacial transport across the bubble–liquid interface in the momentum conservation equations. Furthermore, we consider the translation and spherically symmetric oscillations as bubble dynamics and deploy a two-fluid model to introduce the translation and drag forces. Bubbles do not coalesce, break up, extinct, or appear. For simplicity, the gas viscosity, thermal conductivities of the gas and liquid, and phase change and mass transport across the bubble–liquid interface are ignored. The following results are then obtained: (i) Using the method of multiple scales, two types of Korteweg–de Vries–Burgers equations with a correction term due to the drag force are derived. (ii) The translation of bubbles enhances the nonlinear effect of waves, and the drag force acting on bubbles contributes the nonlinear and dissipation effects of waves. (iii) The results of long-period numerical analysis verify that the temporal evolution of the wave (not flow) dissipation due to the drag force differs from that caused by the acoustic radiation.

Rational solutions, and the interaction solutions to the (2 + 1)-dimensional time-dependent Date–Jimbo–Kashiwara–Miwa equation

In this work, the Date–Jimbo–Kashiwara–Miwa (DJKM) equation include time-dependent in -dimensions that characterize the propagation of nonlinear dispersive waves in inhomogeneous media is studied. We construct rational solutions and the interaction physical phenomena between rational solution and multi-soliton wave to the time-dependent DJKM equation. The multi-soliton solutions are revealed by using the Hirota simple method, and then via the long-wave method, the M-lump solutions for two types of time-dependent are derived. Moreover, the interaction phenomena between rational solution and single-, double-soliton solutions are studied.

Oblique propagation of nonlinear ion-acoustic cnoidal waves in magnetized electron–positron–ion plasmas with nonextensive electrons

Theoretical investigation of nonlinear electrostatic ion-acoustic cnoidal waves (IACWs) is presented in magnetized electron–positron–ion plasma with nonextensive electrons and Maxwellian positrons. Using reductive perturbation technique, Korteweg–de Vries equation is derived and its cnoidal wave solution is analyzed. For given plasma parameters, our model supports only positive potential (compressive) IACW structures. The effect of relevant plasma parameters (viz., nonextensive parameter q, positron concentration p, temperature ratio σ, obliqueness l 3) on the characteristics of IACWs is discussed in detail.

A generalized second-order 3D theory for coupling multidirectional wave propagation from a numerical model to a physical model. Part I: Derivation, implementation and model verification

Numerical and physical modeling are the two main tools available for predicting the influence of water waves on coastal or offshore structures. Both models have their strengths and weaknesses. An integrated use of numerical and physical modeling which exploits their advantages can provide an optimal description of full-scale, realistic engineering problems. In this series of two papers, we report on a generalized three-dimensional (3D) deterministic coupling theory for multidirectional nonlinear wave propagation from a numerical model to a physical model up to second order. In this work, the second-order coupling theory developed by [Z. Yang, S. Liu, H.B. Bingham and J. Li, 2014a. Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I Formulation, implementation and numerical properties. Coast. Eng. 92, 48–60] and the ad hoc unified wave generation theory developed by [H. Zhang, H.A., Schäffer, K.P., Jakobsen, 2007. Deterministic combination of numerical and physical coastal wave models. Coast. Eng. 54, 171–186.] have been extended to include the coupling between multidirectional nonlinear waves. In part I of this article series, the full second-order 3D coupling theory for multidirectional nonlinear waves is been described in detail. A novel generalized fully-nonlinear motion boundary equation has been derived, which allows the interface between the numerical and physical wave domains to be a 3D arbitrarily shaped wavemaker system. The corresponding 3D wave coupling equation is given for I-, L-, and O-shaped wavemaker layouts. The new formulation is presented in a unified form for an arbitrarily shaped layout of planar wavemakers, and covers both hinged and piston wavemaker types. For the second-order dispersive correction of the paddle stroke, the super-harmonic and sub-harmonic wave-wave interactions have been taken into account. For practical implementation, a typical discretization method for the 3D coupling equations is derived by considering the case of the I-shaped piston wavemaker. The new 3D coupling equations are solved by a combined five-point Lagrange interpolation and the fourth-order Runge–Kutta scheme. Numerical evaluations of the implementation have been conducted by considering a theoretical second-order unidirectional wave field over a range of spatial and time resolutions. The results thus obtained indicate that the proposed discretization scheme is accurate and effective. In addition, the precision of the discrete scheme is observed to be closely related to the spatial and temporal resolution. A separate experimental validation of the theory is presented in Part II.

Controllable generation of several nonlinear waves in optical fibers with third-order dispersion

We propose a method to controllably generate six kinds of nonlinear waves on continuous waves, including the one- and multi-peak solitons, the Akhmediev, Kuznetsov-Ma, and Taijiri-Watanabe breathers, and stable periodic waves. In the nonlinear fiber system with third-order dispersion, we illustrate their generation conditions by the modified linear stability analysis, and numerically generate them from initial perturbations on continuous waves. We implement the quantitative control over their dynamical features, including the wave type, velocity, periodicity, and localization. Our results may provide an effective scheme for generating optical solitons on continuous waves, and it can also be applied for wave generations in other various nonlinear systems.

A generalized second-order 3D theory for coupling multidirectional wave propagation from a numerical model to a physical model. Part II: Experimental validation using an I-shaped segmented wavemaker

This paper provides the experimental validation of the generalized second-order three-dimensional (3D) coupling theory outlined by Yang et al. [Z. Yang, S. Liu, X. Ji, and H.B. Bingham, 2020. A generalized second-order 3D theory for coupling multidirectional nonlinear wave propagation from a numerical model to a physical model. Part I: Derivation, implementation and model verification, submitted for publication] using multidirectional nonlinear 3D irregular waves. Based on the second-order theory for two-dimensional (2D) irregular waves described by Yang et al. (2014a), this work provides a second-order dispersive correction for the physical wavemaker signal which improves the nonlinear 3D wave information transfer between the numerical and physical models compared to the first-order method of Zhang et al. (2007). The important coupling equation and its discretization schemes were presented in detail in Part I. For further validating the performance of the second-order 3D coupling model under complex waves and bathymetry conditions, in this Part II, a careful experimental validation has been carried out by establishing a flat coupling basin and a concave-convex coupling basin. In addition, a sequence of progressively more complex numerical target waves has been used and an I-shaped segmented piston-type wavemaker system has been considered for verification of the theory. The effectiveness, adaptability, and precision of the second-order 3D coupling model have been discussed in detail. The effects of the wave parameters and its nonlinearity on the coupling simulation have also been evaluated from the discrepancy error and a wave harmonic analysis. By defining a space synchronization error, the overall error of the spatial wave field has been analyzed. Coupling simulation experiments using oblique regular and irregular waves, unidirectional irregular waves, and multidirectional irregular waves show that the traditional first-order coupling model has limitations when coupling complex waves with strong nonlinearity. When controlling the waves using the second-order 3D coupling signal, the higher harmonics underlying the numerical waves are accurately captured and transferred into the physical model, and the unwanted second-order spurious free waves are substantially reduced. Results obtained from the experimental verification described in this paper demonstrate the strengths of the second-order 3D coupling theory, which are more evident with increased wave nonlinearity.

Lump wave phase transition for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation is investigated, which models nonlinear wave propagation in ferromagnetic media. By applying the bilinear method, skillfully setting the polynomial functions in the bilinear form of the equation, the lump wave solution is constructed. Furthermore, this article reports a few interesting findings on the phase patterns and their transition, which can be managed and controlled by the free parameters. The result also demonstrates the lump wave possesses extremely high amplitude, which is essential in optical applications.

On the implementation and validation of a three‐dimensional pressure‐dependent bounding surface plasticity model for soil nonlinear wave propagation and soil‐structure interaction analyses

AbstractNumerous experiments and prior analyses have confirmed that soil inelasticity, which is known to come into effect even at very low strain levels, can significantly affect site response and dynamic soil‐structure interaction (SSI) behavior. To date, only a few studies were able to consider multi‐axial wave propagation problems with appropriate models of soil nonlinearity. Most existing works are limited to either homogeneous soil configurations or equivalent linear soil models. The instances wherein soil nonlinearity is accurately considered have been confined to single element tests and one‐dimensional problems. In this study, an improved pressure‐dependent bounding surface plasticity soil model—with appropriate plastic strain rate direction definition and overshooting correction scheme—is implemented in Abaqus, and validated using recordings from both the Lotung borehole array and centrifuge test data on embedded flexible structures. The implemented model is capable of comprehensively reproducing complex soil behaviors, such as stiffness degradation, damping, dilatancy, and compaction while under a wide strain range, and under general loading conditions using only a few material parameters to be calibrated. Consequently, numerically predicted results are observed to be in better agreement with experimentally measured data, in comparison with linear and another plasticity model, which include accelerations, and bending and hoop strains along the walls of the specimen structures, for low‐ as well as high‐amplitude input motions.

Nonlinear dust acoustic waves in an inhomogeneous magnetized quantum dusty plasma

Propagations of nonlinear quantum dust acoustic waves (QDAWs) in an inhomogeneous magnetized dusty plasma model which is three-dimensional collisionless, and having number density gradients of its components along x-direction is investigated. This model consisting of ions, electrons and charged dust particles in the presence of an external static magnetic field. The reductive perturbation method was adopted to derive the modified Zakhavov Kusnetsov equation (MZK) whose coefficients are space dependent. The incorporation of inhomogeneity caused by nonuniform equilibrium values of particle density leads to a significant modification to the nature of nonlinear QDAWs. Solutions for the MZK equation are obtained by using a generalized expansion method. The solutions have two parts; amplitude factor and the superposition of bell-shaped solitary wave. The soliton amplitude, width, and velocity depends on the plasma inhomogeneities. Findings in these numerical investigations should be useful in understanding the astrophysical plasmas and ultra small micro- and nano-electronic devices.

Parametric study of a new HOS-CFD coupling method

This paper presents a developed new coupled method which combined our in-house CFD solver naoe-FOAM-SJTU and naoe-FOAM-os with a potential theory High Order Spectral method (HOS). A parametric study of nonlinear wave propagation in computational fluid dynamics (CFD) zone is considered. Mesh convergence, time step convergence, time discretization scheme and length of relaxation zone are all carried out. Those parametric studies verify the steady of this new combined method and give better choice for wave propagation. The dissipation in propagation of nonlinear regular wave can be lower than 3% in static mesh, and less than 2% in overset grid mesh. Meanwhile, a LNG FPSO is put into the viscous wave tank to study the suitable size of CFD zone. To achieve a better solution with least calculating resources and best numerical results, the length of CFD zone is discussed. These parametric studies can give reference upon employment of the potential-viscous coupled method and validation of the coupled method.

Generation, reflection and transmission of nonlinear harmonic waves by direct superposition of anharmonic dipoles

It is well known that the linear and nonlinear light reflection and transmission formula can be obtained from the standard macroscopic Maxwell equation. In the past, a microscopic approach based on the Ewald–Oseen extinction theorem to derive the reflection and transmission formula by direct dipole summation has been performed by Fearn et al where they successfully derived the Fresnel formula in linear optics. In this work, we extend their work for the first time to derive the nonlinear reflection and transmission formula by direct summation of anharmonically oscillating dipoles for the case of a normal incidence wave, yielding similar results with coupled-mode-theory (CMT). We demonstrate for a decaying incoming field that dipolar radiation is no longer forbidden inside the bulk of a centrosymmetric material. Using the simplified bond hyperpolarizability model (SBHM), we find that these additional bulk dipole contribution must be accounted along with the surface dipoles and bulk quadrupolar contribution to explain recent rotational anisotropy SHG intensity experimental data of Si(111) with high precision.

Modulational instabilities and chaotic-like behaviors in repulsive lattices

In this paper, we study the dynamics of granular mediums. By comparing each grain of the medium with a ball, and the interaction between various balls like linear elastic interactions, we have established that the dynamics of the system of pendulum used by Fermi–Pasta–Ulam can be governed by a discrete nonlinear Schrodinger equation. Like second advanced, we also established under which conditions this system could be prone to a modulational instability. The long-time dynamics of modulated waves is examined using numerical simulations which are corroborated with analytical results leading to the generation of nonlinear modulated waves which have the shape of a soliton. At the end, we compare our results with what have been carried out previously.