One-dimensional Domain Research Articles

Nonlinear Eigenvalue Methods for Linear Pointwise Stability of Nonlinear Waves

We propose an iterative method to find pointwise exponential growth rates in linear problems posed on essentially one-dimensional domains. Such pointwise growth rates capture pointwise stability and instability in extended systems and arise as spectral values of a family of matrices that depends on a spectral parameter, obtained via a scattering-type problem. Different from methods in the literature that rely on computing determinants of this nonlinear matrix pencil, we propose and analyze an inverse power method that allows one to locate robustly the closest spectral value to a given reference point in the complex plane. The method finds branch points, eigenvalues, and resonance poles without a priori knowledge.

One-dimensional turbulence modeling of compressible flows: II. Full compressible modification and application to shock–turbulence interaction

One-dimensional turbulence (ODT) is a simulation methodology that represents the essential physics of three-dimensional turbulence through stochastic resolution of the full range of length and time scales on a one-dimensional domain. In the present study, full compressible modifications are incorporated into ODT methodology, based on an Eulerian framework and a conservative form of the governing equations. In the deterministic part of this approach, a shock capturing scheme is introduced for the first time. In the stochastic part, one-dimensional eddy events are modeled and sampled according to standard methods for compressible flow simulation. Time advancement adjustments are made to balance comparable time steps between the deterministic and stochastic parts in compressible flows. Canonical shock–turbulence interaction cases involving Richtmyer–Meshkov instability at Mach numbers 1.24, 1.5, and 1.98 are simulated to validate the extended model. The ODT results are compared with available reference data from large eddy simulations and laboratory experiments. The introduction of a shock capturing scheme significantly improves the performance of the ODT method, and the results for turbulent kinetic energy are qualitatively improved compared with those of a previous compressible Lagrangian ODT method [Jozefik et al., “Simulation of shock–turbulence interaction in non-reactive flow and in turbulent deflagration and detonation regimes using one-dimensional turbulence,” Combust. Flame 164, 53 (2016)]. For the time evolution of profiles of the turbulent mixing zone width, ensemble-averaged density, and specific heat ratio, the new model also yields good to reasonable results. Furthermore, it is found that the viscous penalty parameter Z of the ODT model is insensitive to compressibility effects in turbulent flows without wall effects. A small value of Z is appropriate for turbulent flows with weak wall effects, and the parameter Z serves to suppress extremely small eddy events that would be dissipated instantly by viscosity.

The emergence of spatial patterns for compartmental reaction kinetics coupled by two bulk diffusing species with comparable diffusivities.

Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns can typically only occur from a spatially uniform state when a slowly diffusing 'activator' species reacts with a much faster diffusing 'inhibitor' species. However, from a modelling perspective, this large diffusivity ratio requirement for pattern formation is often unrealistic in biological settings since different molecules tend to diffuse with similar rates in extracellular spaces. As a result, one key long-standing question is how to robustly obtain pattern formation in the biologically realistic case where the time scales for diffusion of the interacting species are comparable. For a coupled one-dimensional bulk-compartment theoretical model, we investigate the emergence of spatial patterns for the scenario where two bulk diffusing species with comparable diffusivities are coupled to nonlinear reactions that occur only in localized 'compartments', such as on the boundaries of a one-dimensional domain. The exchange between the bulk medium and the spatially localized compartments is modelled by a Robin boundary condition with certain binding rates. As regulated by these binding rates, we show for various specific nonlinearities that our one-dimensional coupled PDE-ODE model admits symmetry-breaking bifurcations, leading to linearly stable asymmetric steady-state patterns, even when the bulk diffusing species have equal diffusivities. Depending on the form of the nonlinear kinetics, oscillatory instabilities can also be triggered. Moreover, the analysis is extended to treat a periodic chain of compartments. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Simulating the effect of subsurface drainage on the thermal regime and ground ice in blocky terrain in Norway

Abstract. Ground temperatures in coarse, blocky deposits such as mountain blockfields and rock glaciers have long been observed to be lower in comparison with other (sub)surface material. One of the reasons for this negative temperature anomaly is the lower soil moisture content in blocky terrain, which decreases the duration of the zero curtain in autumn. Here we used the CryoGrid community model to simulate the effect of drainage on the ground thermal regime and ground ice in blocky terrain permafrost at two sites in Norway. The model set-up is based on a one-dimensional model domain and features a surface energy balance, heat conduction and advection, as well as a bucket water scheme with adjustable lateral drainage. We used three idealized subsurface stratigraphies, blocks only, blocks with sediment and sediment only, which can be either drained (i.e. with strong lateral subsurface drainage) or undrained (i.e. without drainage), resulting in six scenarios. The main difference between the three stratigraphies is their ability to retain water against drainage: while the blocks only stratigraphy can only hold small amounts of water, much more water is retained within the sediment phase of the two other stratigraphies, which critically modifies the freeze–thaw behaviour. The simulation results show markedly lower ground temperatures in the blocks only, drained scenario compared to other scenarios, with a negative thermal anomaly of up to 2.2 ∘C. For this scenario, the model can in particular simulate the time evolution of ground ice, with build-up during and after snowmelt and spring and gradual lowering of the ice table in the course of the summer season. The thermal anomaly increases with larger amounts of snowfall, showing that well-drained blocky deposits are less sensitive to insulation by snow than other soils. We simulate stable permafrost conditions at the location of a rock glacier in northern Norway with a mean annual ground surface temperature of 2.0–2.5 ∘C in the blocks only, drained simulations. Finally, transient simulations since 1951 at the rock glacier site (starting with permafrost conditions for all stratigraphies) showed a complete loss of perennial ground ice in the upper 5 m of the ground in the blocks with sediment, drained run; a 1.6 m lowering of the ground ice table in the sediment only, drained run; and only 0.1 m lowering in the blocks only, drained run. The interplay between the subsurface water–ice balance and ground freezing/thawing driven by heat conduction can at least partly explain the occurrence of permafrost in coarse blocky terrain below the elevational limit of permafrost in non-blocky sediments. It is thus important to consider the subsurface water–ice balance in blocky terrain in future efforts in permafrost distribution mapping in mountainous areas. Furthermore, an accurate prediction of the evolution of the ground ice table in a future climate can have implications for slope stability, as well as water resources in arid environments.

Neutron angular flux reconstruction in slab geometry using multigroup discrete ordinates transport models

In this article, we present an application of the coarse-mesh Deterministic Spectral Method (SDM) to generate multigroup angular fluxes in one-dimensional spatial domains using the neutron transport stationary equation, in the formulation of discrete ordinates (SN), considering isotropic scattering source. After obtaining the analytical solution of the SN equations, we replace the integral term of the scattering source in the original neutron transport equation. Thus, we obtain analytically two expressions for angular fluxes in the multigroup formulation, considering the neutron propagation in the positive ( ) and negative ( ) directions, presenting a meaningful reduction in the computational time of simulations of typical neutron shielding problems.

Charge–velocity correlation transport equations in gas–solid flow with triboelectric effects

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

Existence of a weak solution for a nonlinear parabolic problem with fractional derivates

The primary objective of this study was to demonstrate the existence and uniqueness of a weak solution for a nonlinear parabolic problem with fractional derivatives for the spatial and temporal variables on a one-dimensional domain. Using the Nehari manifold method and its relationship with the Fibering maps, the existence of a weak solution for the stationary case was demonstrated. Finally, using the Arzela-Ascoli theorem and Banach's fixed point theorem, the existence and uniqueness of a weak solution for the nonlinear parabolic problem were shown.

Spin thermoelectric effects of skyrmions in ferromagnetic topological insulators

The thermoelectric effects of ferromagnetic topological insulators with either two-dimensional circular or one-dimensional domain wall skyrmions are studied theoretically. It is found that the topological spin-textures play a significant role in the manipulation of spin-dependent thermoelectric properties. In the vicinity of the charge neutrality point, spin Seebeck coefficients possess finite values whose sign and magnitude can be tuned by temperature in spite of vanishing charge Seebeck coefficients. The majority of the effects of circular skyrmions occurs in the edge-state transport regime by generating Fano antiresonances. While the domain wall skyrmion primarily influences the thermoelectric behaviors near the boundary between the edge-state and bulk-state transport regimes with the resonant tunneling mechanism. Both types of skyrmions which function effectively in distinct transport regimes have potential applications in thermoelectrics.

Sharp decay rates of degenerate hyperbolic-parabolic coupled system: Rectangular domain vs one-dimensional domain

In this work, we study the long time behavior of a coupled PDEs' system consisting of one wave equation and one degenerate heat equation in two connected regions. The coupling between these two different components occurs at the interface with certain transmission conditions. For the system on rectangular domain, by frequency domain analysis, the explicit polynomial decay rate of the solutions to the system is obtained, which only depends on the degree of degeneration of the diffusion coefficient of the heat equation near the interface line. Specifically, a clear relationship between the decay rate and the degeneration parameter of the diffusion coefficient is identified. Moreover, the explicit decay rate of the solutions to such a system on one-dimensional domain is also further estimated. Especially, these two obtained decay rates are consistent with the optimal ones for the systems on rectangular domain and one-dimensional domain with constant-diffusion coefficient, obtained by Batty et al. [7] and Zhang and Zuazua [23], respectively. From this view, the obtained decay rates can all be considered as sharp ones for such kind of degenerate systems.

Design of cost effective variable bandwidth 2D low-pass, high-pass and band-pass filters with improved circularity

This paper discusses two-dimensional (2D) finite impulse response (FIR) filter structures to achieve improved circularity and variability in bandwidth without restructuring. The P3 transformation employed in this work for the design of circular filters helps to achieve better circularity and less complexity. The variability is obtained by using two different approaches. In Method 1, the bandwidths of the 2D circular filters are changed by adjusting a parameter in the one-dimensional (1D) domain. In contrast to Method-1, Method-2 directly adjusts the bandwidths of the 2D filters using a parameter in the 2D domain. Low-pass, high-pass, and band-pass variable bandwidth filters are designed using both methods to illustrate the benefits of the two approaches. Comparing their implementation complexity, contour approximation error, and normalized root mean square error to those of other multiplier-less transformations demonstrates their efficacy. It can be seen that Method 1 and 2 respectively achieve around 5% and 20% reduction in the number of multipliers with a maximum normalized root mean square errors of 0.3129 and 0.3532 when compared to that of the 2D filters designed using the latest existing transformation.

An alternative understanding of the skyrmion Hall effect based on one-dimensional domain wall motion

A moving magnetic skyrmion exhibits transverse deflection. This so-called skyrmion Hall effect has been explained by the Thiele equation. Here, we provide an alternative interpretation of the skyrmion Hall effect based on the dynamics of domain walls enclosing the skyrmion. We relate the spin-torque-induced local rotation of the domain wall segments to the shift of the skyrmion core, explaining the skyrmion Hall effect at the micromagnetic level. Based on our intuitive interpretation, we also show that the skyrmion Hall effect can be suppressed by combining the spin-transfer and spin–orbit torques, thereby removing the major obstacle to utilizing skyrmions in devices.

Rigorous numerical inclusion of the blow-up time for the Fujita-type equation

Multiple studies have addressed the blow-up time of the Fujita-type equation. However, an explicit and sharp inclusion method that tackles this problem is still missing due to several challenging issues. In this paper, we propose a method for obtaining a computable and mathematically rigorous inclusion of the L^2(varOmega ) blow-up time of a solution to the Fujita-type equation subject to initial and Dirichlet boundary conditions using a numerical verification method. More specifically, we develop a computer-assisted method, by using the numerically verified solution for nonlinear parabolic equations and its estimation of the energy functional, which proves that the concerned solution blows up in the L^2(varOmega ) sense in finite time with a rigorous estimation of this time. To illustrate how our method actually works, we consider the Fujita-type equation with Dirichlet boundary conditions and the initial function u(0,x)=frac{192}{5}x(x-1)(x^2-x-1) in a one-dimensional domain varOmega and demonstrate its efficiency in predicting L^2(varOmega ) blow-up time. The existing theory cannot prove that the solution of the equation blows up in L^2(varOmega ). However, our proposed method shows that the solution is the L^2(varOmega ) blow-up solution and the L^2(varOmega ) blow-up time is in the interval (0.3068, 0.317713].

Generalized lattices over one-dimensional noetherian domains

We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Drawing inspiration from the representation theory of orders in separable algebras, we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.

Research on fault diagnosis method of planetary gearbox based on dynamic simulation and deep transfer learning

To address the issue of not having enough labeled fault data for planetary gearboxes in actual production, this research develops a simulation data-driven deep transfer learning fault diagnosis method that applies fault diagnosis knowledge from a dynamic simulation model to an actual planetary gearbox. Massive amounts of different fault simulation data are collected by creating a dynamic simulation model of a planetary gearbox. A fresh deep transfer learning network model is built by fusing one-dimensional convolutional neural networks, attention mechanisms, and domain adaptation methods. The network model is used to learn domain invariant features from simulated data, thereby enabling fault diagnosis on real data. The fault diagnosis experiment is verified by using the Drivetrain Diagnostics Simulator test bench. The validity of the proposed means is evaluated by comparing the diagnostic accuracy of various means on various diagnostic tasks.

Numerical analysis of the limits of effective medium theory for nonlinearly propagating dispersive waves

Acoustic metameterials are an increasingly popular approach to control both linear and nonlinear sound propagation. In this approach, heterogeneous structures are engineered to behave as predetermined continuous media. A key requirement for a system to be accurately approximated as a homogeneous material for wave motion is that the smallest length scale of the waveform be much larger than any length scale associated with the microscale structure of the system. However, the fact that the characteristic length scales of a finite-amplitude waveform can change due to nonlinear effects means that care must be taken when analyzing such signals in an acoustic metamaterial. This talk presents recent work toward understanding the breakdown of effective medium theory as finite-amplitude waves propagate in a heterogeneous system and weak shocks or solitons begin to form. A finite-difference time-domain code was used to model the propagation of a pulse of sound in a one-dimensional propagation domain using both a full description of the system and an effective-medium description accounting for effective mass density, compressibility, coefficient of nonlinearity, and dispersion. The difference between the solutions of the two models provides a method by which one can estimate the error associated with using effective properties.

Radar range profile processing using Green’s function

In this study, a useful method for synthesizing range profiles to be used in ultra-wideband radar operations is discussed. One-dimensional transmission-reflection geometry of the radar scenario, which consists of a single receiver and single transmitter, is formulated by using the Green’s function. A one-dimensional time domain Green’s function is employed in the formulation and the impulsive characteristic of this function, which provides focus on the target position, has been determined. This characteristic represents the impulse response of the propagating medium to the electromagnetic wave. The resulting derivation offers a framework that can also be used in higher dimensional and complex radar problems by taking the advantage of its flexibility. Proposed method has been detailed and tested on an exemplary implementation and successful results that provide good focus on the target positions are shown in the conclusions. Finally, outcomes and benefits are discussed.

An analytical solution for the advection-dispersion equation inversely in time for pollution source identification

Researchers have long explored inverse solution methods for identifying unknown pollutant sources in water resources. More often than rivers, these methods are used to study groundwater; therefore, employing efficient and up-to-date methods to determine the characteristics of unknown pollutants in rivers has been the cause for concern to water and environmental researchers. This study was thus designed to solve the pollutant transport equation in rivers as an inverse problem to reconstruct pollutant source intensity functions. A unique analytical method, based on the quasi-reversibility (QR) method and the Fourier transform tool, was developed to solve the advection-dispersion equation (ADE) in rivers inversely in time and a one-dimensional domain for various pollutant loading patterns. In this method, a stability term is incorporated into the original transport equation to prevent the problem from becoming ill-posed during the inverse solution process. The analytical solution results demonstrate its computational cost-effectiveness, high accuracy and potential capability for practical modes (particularly when applied to observational concentration data with errors).

Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments

We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equations and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable x/t and in which the time and space derivatives are coupled together. We first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s=x/t. In terms of the standard Fisher-KPP equation of reaction-diffusion type, we give explicit formulas of the spreading speed when the environment has one or two shifting speeds. As a byproduct, we also introduce a novel class of “asymptotically homogeneous” environments which share the same spreading speed with the corresponding homogeneous environments.

Nonparametric bagging clustering methods to identify latent structures from a sequence of dependent categorical data

Nonparametric bagging clustering methods are studied and compared to identify latent structures from a sequence of dependent categorical data observed along a one-dimensional (discrete) time domain. The frequency of the observed categories is assumed to be generated by a (slowly varying) latent signal, according to latent state-specific probability distributions. The bagging clustering methods use random tessellations (partitions) of the time domain and clustering of the category frequencies of the observed data in the tessellation cells to recover the latent signal, within a bagging framework. New and existing ways of generating the tessellations and clustering are discussed and combined into different bagging clustering methods. Edge tessellations and adaptive tessellations are the new proposed ways of forming partitions. Composite methods are also introduced, that are using (automated) decision rules based on entropy measures to choose among the proposed bagging clustering methods. The performance of all the methods is compared in a simulation study. From the simulation study it can be concluded that local and global entropy measures are powerful tools in improving the recovery of the latent signal, both via the adaptive tessellation strategies (local entropy) and in designing composite methods (global entropy). The composite methods are robust and overall improve performance, in particular the composite method using adaptive (edge) tessellations.