Multiple Scale Approach Research Articles

A direct multiple-scale approach to the parabolic equation method

A new multi-component form of the higher order parabolic approximations to the acoustic Helmholtz equation and the corresponding interface and boundary conditions are derived by the direct use of the multiple-scale approach. These new parabolic approximations have the properties of asymptotic energy conservation and asymptotic absence of the phase error. Numerical examples for some typical environments are presented.

Asymptotic analysis of weakly nonlinear Bessel–Gauß beams

In this paper we investigate the propagation of conical waves in nonlinear media. In particular, we are interested in the effects resulting from applying a Gaussian apodization to an ideal nondiffracting wave. First, we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear Schrödinger type. From these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution. Moreover, we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results.

Modeling of active and passive nonlinear metamaterials

We develop general results for nonlinear metamaterials based on simple circuit models that reflect the elementary nonlinear behavior of the medium. In particular, we consider both active and passive nonlinearities which can lead to gain, harmonic generation and a variety of nonlinear waves depending on circuit parameters and signal amplitude. We show that the medium can exhibit a phase transition to a synchronized state and derive conditions for the transformation based on a widely used multiple time scale approach that leads to the well-known Complex Ginzburg–Landau equation. Further, we examine the variety of nonlinear waves that can exist in such systems, and we present numerical results for both active and passive metamaterial cases.

Output errors of atomic force microscopy in heterogeneous materials

We have used multiple scale approach to obtain a complete close form solution for scanning system of an atomic force microscopy (AFM) and uncover unexpected errors in the AFM system. We find out that the change in the modulus of elasticity of a sample will lead to a phase shift and jump phenomenon in the frequency response. The readout of the AFM probe amplitude will also be affected accordingly. These results indicate that AFM could produce significant errors in the interface regions of heterogeneous materials. In such a case, AFM may mistakenly consider the change in mechanical properties as the change in topography of the surface. The error associated with this condition must be compensated by a post-processing of collected data.

Bifurcation analysis of milling process with tool wear and process damping: regenerative chatter with primary resonance

In this paper, the occurrence of various types of bifurcation including symmetry breaking, period-doubling (flip) and secondary Hopf (Neimark) bifurcations in milling process with tool-wear and process damping effects are investigated. An extended dynamic model of the milling process with tool flank wear, process damping and nonlinearities in regenerative chatter terms is presented. Closed form expressions for the nonlinear cutting forces are derived through their Fourier series components. Non-autonomous parametrically excited equations of the system with time delay terms are developed. The multiple-scale approach is used to construct analytical approximate solutions under primary resonance. Periodic, quasi-periodic and chaotic behavior of the limit cycles is predicted in the presence of regenerative chatter. Detuning parameter (deviation of the tooth passing frequency from the chatter frequency), damping ratio (affected by process damping) and tool-wear width are the bifurcation parameters. Multiple period-doubling and Hopf bifurcations occur when the detuning parameter is varied. As the damping ratio changes, symmetry breaking bifurcation is observed whereas the variation of tool wear width causes both symmetry breaking and Hopf bifurcations. Also, under special damping specifications, chaotic behavior is seen following the Hopf bifurcation.

Dynamics of regenerative chatter and internal resonance in milling process with structural and cutting force nonlinearities

In this paper, internal resonance and nonlinear dynamics of regenerative chatter in milling process is investigated. An extended dynamic model of the peripheral milling process including both structural and cutting force nonlinearities is presented. Closed form expressions for the nonlinear cutting forces are derived through their Fourier series components. In the presence of the large vibration amplitudes, the loss of contact effect is included in this model. Using the multiple-scales approach, analytical approximate response of the delayed nonlinear system is obtained. Considering the internal resonance dynamics (i.e. mode coupling), the energy transfer between the coupled x–y modes is studied. The results show that during regenerative chatter under specific cutting conditions, one mode can decay. Furthermore, it is possible to adjust the rate at which the x-mode (or y-mode) decays by implementation of the internal resonance. Therefore, under both internal resonance and regenerative chatter conditions, it is possible to suppress the undesirable vibration of one mode (direction) in which accurate surface finish is required. Under the steady-state motion, jump phenomenon is investigated for the process with regenerative chatter under various cutting conditions. Moreover, the effects of structural and cutting force nonlinearities on the stability lobes diagram of the process are investigated.

Sustainability issues in agro-ecology: Socio-ecological perspective

In the recent past there has been concern to address the declining trends and deteriorating ecological elements and their functions in productive agricultural landscapes. The efforts to revive the ecological functions needs multiple scale approach, which include scientific understanding, time dependent restorative activities and incorporation of wisdom of the stakeholders. Conservation initiatives linked to precision farming is an apt mechanism to minimize the loss of natural resources of agricultural landscapes to maintain the complex stabilizers of ecological functions. Such an effort must address the scientific evaluation of the farm as “ecologically sustainable unit” with due consideration to social construct at local to regional level. Sustainability indices can be useful tools for evaluating the farms to isolate degradative factors and identify ecologically conservative practices. This review will analyze certain inter-linked concepts relevant to address the sustainability issues in agro-ecology at the interface of socio-ecological perspective.

On the parametric excitation of rotors with rolling element bearings

AbstractIn this approach a 1‐DOF‐model for a rotor in rolling element bearings is derived. The nonlinear equation of motion with a cubic restitution force and harmonic time dependent variation is solved by Multiple Time Scales Approach. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Characterization of precipitative self-healing materials by mechanokinetic modeling approach

A non-deterministic multiple scale approach based on numerical solution of the Monte-Carlo master equation on atomic lattices solved together with a standard finite-element formulation of solid mechanics is discussed. The approach is illustrated in application to long-term evolutionary processes of volume diffusion, precipitation and creep cavity self-healing in nanocrystalline austenite (Fe fcc) samples. A two-way mechanokinetic coupling is achieved through implementation of strain-dependent diffusion rates and dynamic update of the finite element model based on atomic structure evolution. Effect of macroscopic static loading and cavity geometry on the total healing time is investigated. The approach is widely applicable to the modeling and characterization of advanced functional materials with evolutionary internal structure, and emerging behavior in material systems.

Multiple-scale approach for the expansion scaling of superfluid quantum gases

We present a general method, based on a multiple-scale approach, for deriving the perturbative solutions of the scaling equations governing the expansion of superfluid ultracold quantum gases released from elongated harmonic traps. We discuss how to treat the secular terms appearing in the usual naive expansion in the trap asymmetry parameter $\ensuremath{\epsilon}$ and calculate the next-to-leading correction for the asymptotic aspect ratio, with significant improvement over the previous proposals.

Performance of biotic indices in naturally stressed estuarine environments on the Southwestern Atlantic coast (Uruguay): A multiple scale approach

Biotic indices based on benthic communities have become an important tool in the assessment and monitoring of marine pollution. These communities vary at several spatial and temporal scales giving rise to scale-dependent patterns of distribution, being of particular importance because results from the application of ecological indices, and the subsequent classification of communities, could reflect these variations at several scales. We test some of the most widely applied indices for the evaluation of coastal benthic communities, using a hierarchical spatio-temporal sampling design, within two sets of estuarine habitats in the Atlantic coastal zone of Uruguay. Results showed that ecological indices can vary at different spatial scales, with important variation at small scales. So, independently from the used index, an appropriate sampling design should be taken into account considering different scales (both spatial and temporal). At some of the scales studied, indices appear to reflect natural variations in disturbance through currents rather than variation in anthropogenic effects. At the large scale, variation is low consistent with a preliminary classification of sites according to the putative levels of human activity. The low level of similarity between all indices could be denoting some degree of inconsistency in the assignment of the categories to an ecological status.

Flow and geometry induced scattering of high frequency acoustic duct modes

Cut-on cut-off transition of acoustic modes in hard-walled ducts with irrotational mean flow is well understood for Helmholtz numbers of order unity. Previous finite-element simulations of this phenomenon, however, appear to indicate the possibility of energy scattering into neighbouring modes at moderately large Helmholtz numbers. In this paper, such scattering phenomena are explained and predicted in slowly varying aeroengine ducts using a multiple-scales approach. It is found that, for sufficiently high frequencies, two mechanisms exist whereby energy can be scattered into neighbouring modes by an incident propagating mode. One mechanism occurs only when there is a mean flow inside the duct and induces scattering at significantly lower frequencies than the other mechanism which remains present without mean flow. A coupled system of ordinary differential equations is derived and then solved numerically for a number of example cases to obtain the corresponding transmitted and reflected amplitudes of the scattered modes as well as the overall acoustic pressure field. The theory appears to demonstrate that some exchange of energy between the acoustic and mean flow fields occurs during scattering.

Harvesting a logistic population in a slowly varying environment

The classic problem for a logistically evolving single species population being harvested involves three parameters: rate constant, carrying capacity and harvesting rate, which are taken to be positive constants. However, in real world situations, these parameters may vary with time. This paper considers the situation where these vary on a time scale much longer than that intrinsic to the population evolution itself. Application of a multiple time scale approach gives approximate explicit closed form expressions for the changing population, that compare favorably with those generated from numerical solutions.

Introduced brown trout alter native acanthocephalan infections in native fish

1. Native parasite acquisition provides introduced species with the potential to modify native host-parasite dynamics by acting as parasite reservoirs (with the 'spillback' of infection increasing the parasite burdens of native hosts) or sinks (with the 'dilution' of infection decreasing the parasite burdens of native hosts) of infection. 2. In New Zealand, negative correlations between the presence of introduced brown trout (Salmo trutta) and native parasite burdens of the native roundhead galaxias (Galaxias anomalus) have been observed, suggesting that parasite dilution is occurring. 3. We used a multiple-scale approach combining field observations, experimental infections and dynamic population modelling to investigate whether native Acanthocephalus galaxii acquisition by brown trout alters host-parasite dynamics in native roundhead galaxias. 4. Field observations demonstrated higher infection intensity in introduced trout than in native galaxias, but only small, immature A. galaxii were present in trout. Experimental infections also demonstrated that A. galaxii does not mature in trout, although parasite establishment and initial growth were similar in the two hosts. Taken together, these results support the hypothesis that trout may serve as an infection sink for the native parasite. 5. However, dynamic population modelling predicts that A. galaxii infections in native galaxias should at most only be slightly reduced by dilution in the presence of trout. Rather, model exploration indicates parasite densities in galaxias are highly sensitive to galaxias predation on infected amphipods, and to relative abundances of galaxias and trout. Hence, trout presence may instead reduce parasite burdens in galaxias by either reducing galaxias density or by altering galaxias foraging behaviour.

Risk assessment of MUSTANG project experimental site–Methodological development

Abstract One of the work packages of MUSTANG (MUltiple Space and Time scale Approach for the quaNtification of deep saline formations for CO 2 storaGe) EU project is dedicated to developing a generic methodology for risk assessment related to CO 2 in saline aquifers and applying the methodology to an experimental site. This paper presents the work done by OXAND regarding risk assessment and the application of a risk-based approach to the qualification of the storage site ultimately to provide guidelines for further industrial storage projects. The risk assessment process is presented, and illustrated with the data of an experimental site. The eight steps of the risk assessment process highlighted in the ISO 31000 are described in this paper: Risk management policy, establishment of the context, risk identification, risk estimation, risk evaluation, risk treatment, communication and consultation, and monitoring and review.

Stock Market Integration - A Multiple Time Scale Approach

Stock Market Integration - A Multiple Time Scale Approach

Nondeterministic Multiscale Modeling of Biomimetic Crack Self-Healing in Nanocrystalline Solids under Mechanical Loading

ABSTRACTA nondeterministic multiple scale approach based on numerical solution of the Monte-Carlo master equation coupled with a standard finite-element formulation of material mechanics is presented. The approach is illustrated in application to the long-term evolutionary processes of self-diffusion, precipitation and crack/void healing in nanocrystalline fcc and bcc solids. Effect of static and dynamic loading patterns on the crack healing rates are investigated. The approach is widely applicable to the modeling and characterization of advanced functional materials with evolutionary internal structure, as well as emerging behavior in materials systems.

Mechano-kinetic coupling approach for materials with dynamic internal structure

Coupled multiscale approaches for the analysis and simulation of complex multiphysics phenomena in solids have been of great interest during the past decade. These include concurrent MD/FEM models, atomistic-to-continua homogenization techniques and deterministic multiple time scale approaches. In this article we discuss a generic approach to concurrently couple the Monte-Carlo master equation of microscopic kinetic processes, not accessible by the direct particle dynamics, with the continuum mechanics formulation. This approach can be adequate for the modeling and validation of advanced contemporary materials with dynamic internal structure, as well as evolutionary and degradation processes in materials. The approach is illustrated in the application to the models of interstitial and vacancy diffusion in fuel cell catalysts and binary material systems.

Ginzburg-Landau equation bound to the metal-dielectric interface and transverse nonlinear optics with amplified plasmon polaritons

Using a multiple-scale asymptotic approach, we have derived the complex cubic Ginzburg-Landau equation for amplified and nonlinearly saturated surface plasmon polaritons propagating and diffracting along a metal-dielectric interface. An important feature of our method is that it explicitly accounts for nonlinear terms in the boundary conditions, which are critical for a correct description of nonlinear surface waves. Using our model we have analyzed filamentation and discussed the bright and dark spatially localized structures of plasmons.

Density-amplitude formulation of the phase-field crystal model for two-phase coexistence in two and three dimensions

The phase-field crystal approach describes the dynamics of the local atomic probability density on atomic length and diffusive time scales. However, the small grid spacing required to resolve the atomic length scale restricts the applicability to relatively small systems. We present a new amplitude-equation formulation that can be applied to model the evolution of a two-phase system by building upon the multiple-scale approach developed for the phase-field crystal model that explicitly incorporates a miscibility gap in the atomic density field. The new set of equations are less stiff by two orders in the spatial derivative and are found to retain acceptable accuracy even with a significantly larger grid size. Furthermore, the proposed model provides insight into the link between the phase-field-crystal model and a phase-field model in which conserved and nonconserved dynamics are coupled to describe a solid–liquid system. Models and simulation results are presented for two-dimensional hexagonal and three-dimensional body-centered cubic structures. Validation of the two-dimensional model is carried out by examining the Gibbs–Thomson effect. Simulations of growth and coarsening are performed to demonstrate the model's potential for large-scale simulations.