Homogeneous Part Research Articles

Fourier Variant Homogenization Treatment of Single Impulse Boundary Effect Behaviour

Abstract Boundary effect behavior understood as near-boundary suppression of boundary fluctuation loads is described in various ways depending on the mathematical representation of solutions and the type of the center. In the case of periodic composites, the homogenization method is decisive here. In the framework of the Tolerance Averaging Approach, developed by prof. Cz. Woźniak leading to an approximate model of phenomena related to periodic composites this effect is described by a homogeneous part of differential equation for fluctuation amplitudes and usually this approximate description of the boundary effect behavior is restricted to a single fluctuation. In this paper, contrary to the previous elaborations, the boundary effect is developed in the variant of the tolerance thermal conductivity model in which the temperature field is represented by the Fourier expansions composed by an average temperature with infinite number of Fourier terms imposed on the average temperature as tolerance fluctuation suppressed in the framework of the boundary effect.

Further-Adjusted Long-Term Temperature Series in China Based on MASH

A set of homogenized monthly mean surface air temperature (SAT) series at 32 stations in China back to the 19th century had previously been developed based on the RHtest method by Cao et al., but some inhomogeneities remained in the dataset. The present study produces a further-adjusted and updated dataset based on the Multiple Analysis of Series for Homogenization (MASH) method. The MASH procedure detects 33 monthly temperature records as erroneous outliers and 152 meaningful break points in the monthly SAT series since 1924 at 28 stations. The inhomogeneous parts are then adjusted relative to the latest homogeneous part of the series. The new data show significant warming trends during 1924–2016 at all the stations, ranging from 0.48 to 3.57°C (100 yr)−1, with a regional mean trend of 1.65°C (100 yr)−1; whereas, the previous results ranged from a slight cooling at two stations to considerable warming, up to 4.5°C (100 yr)−1. It is suggested that the further-adjusted data are a better representation of the large-scale pattern of climate change in the region for the past century. The new data are available online at https://doi.org/www.dx.doi.org/10.11922/sciencedb.516.

Significance and behaviour of the homogeneous and inhomogeneous components of linearized viscoelastic reflection coefficients

In a recent paper, seismic amplitude-variation-with-offset (AVO) equations describing P-to-P and P-to-S reflections from boundaries separating low-loss viscoelastic media, taking into account variations in the attenuation angle, have been derived. We find that opportunities now present themselves to use these equations to expose a range of relationships between measured amplitudes and subsurface elastic and anelastic properties. This has significant applicability in the quantitative interpretation of seismic data in, for instance, reservoir characterization. To facilitate the analysis, we decompose the equations into three parts: elastic, homogeneous and inhomogeneous. We show that, for PP modes, the elastic part is sensitive to changes across a reflecting boundary in density and P- and S-wave velocities; the homogeneous part is sensitive to changes in density, S-wave velocity and the P- and S-wave quality factors; and the inhomogeneous part is sensitive to changes in density, and P-and S-wave velocities. The latter term is seen to vanish when the attenuation angle vanishes. For PP modes, elastic and homogeneous terms are linear with respect to , where is the P-wave incidence angle; however, the inhomogeneous term is similarly linear only if normalized by dividing by tan θ P. For PS modes, the elastic part is sensitive to changes in density and S-wave velocity; the homogeneous part is sensitive to changes in density, S-wave velocity and the S-wave quality factor; and the inhomogeneous part is sensitive to changes in density and S-wave velocity. This term also vanishes for the zero attenuation angle, i.e. in the homogenous limit. For PS modes, the inhomogeneous terms are linear with respect to ; however the elastic and homogeneous terms are first and third order in . A further and key result of this expansion of the wave types allowable in AVO analysis is that, for inhomogeneous PS scattering, the viscoelastic AVO equations predict a non-zero reflectivity at normal incidence. This is a significant deviation from common models of converted wave amplitude analysis.

On a coupling between the Finite Element (FE) and the Wave Finite Element (WFE) method to study the effect of a local heterogeneity within a railway track

Railway rolling noise is the main source of noise from trains at speeds below about 300 km/h. At the wheel–rail contact area, the track and the wheel are dynamically excited and vibrate together to emit the well-known rolling noise. The rail lies on elastomeric pads, which are essential parts concerning wave propagation. Stiff pads are advantageous as they increase the track decay rate and perform better acoustically.The Wave Finite Element Method (WFEM), based on the theory of periodicity, has been widely used to compute the response of infinite periodically supported railway tracks. However, the track should not always be regarded as a perfect infinite periodic system: a breakdown of the periodicity can be due to a local increase of the stiffness of the supports due to the wheel load, a transition area, geometric irregularities of the track, or to the use anti-vibration supports.The goal of this paper is to compute the dynamic response of an infinite railway track including a heterogeneous part. This zone is made of supports whose stiffness is different from the ones in the homogeneous part. To compute the dynamic response of the track, an extended Finite Element (FE) – Wave Finite Element (WFE) coupling method is proposed. It consists in dividing the heterogeneous part into several FE different supports linked with internal WFE wave-guides. This part is then coupled with semi-infinite WFE wave-guides. The proposed method is validated with experimental data of a track on uniform supports. Then, the dynamic response of the track with different supports is computed. Finally, the performances of an anti-vibration support is evaluated and discussed in terms of the drive point mobility and the track decay rates.

Moment Problem for Symmetric Algebras of Locally Convex Spaces

It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows the application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of $\overline{\tau}$-continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(\sum S(V)^{2d}) \subseteq [0,\infty)$ (more generally, $L(M) \subseteq [0,\infty)$ for some $2d$-power module $M$ of $S(V)$) as integrals with respect to uniquely determined Radon measures $\mu$ supported by special sorts of closed balls in the dual space of $V$. The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and \v Sifrin. It is more general because $V$ can be any lc topological space (not just a separable nuclear space), the result holds for arbitrary $2d$-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that $L : S(V) \rightarrow \mathbb{R}$ is $\overline{\tau}$-continuous (not just continuous on each homogeneous part of $S(V)$).

Solution of the Extended Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface(s) to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

A special purpose Trefftz-element for mode III crack tip loading

A special purpose hybrid Trefftz-element for mode III cracks in thin structures is presented. The element is an extension of well-established mode I/II elements to simulate arbitrary mixed mode loading of a crack. The element formulation is based on analytical solutions of the Kirchhoff plate and exact fulfillment of boundary conditions at the crack edges. Using complex analysis, a T-complete set of homogeneous and particular Ansatz functions is derived. With the particular basis functions a general strip yield model is established and solutions for different distributions of the crack closing force are calculated. Numerical tests of the homogeneous part of the solution are in good accordance with results from high resolution FE simulation. Finally, further extension of the presented element with a kinked crack and a crack propagation algorithm for explicit FE simulations are outlined.

Approximation of the convective flux in the incompressible Navier–Stokes equations using local boundary-value problems

We present the spatial discretization of the nonlinear convective flux in the incompressible Navier–Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier–Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection–diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations.

Seismic b-values and its correlation with seismic moment and Bouguer gravity anomaly over Indo-Burma ranges of northeast India: Tectonic implications

b-value is one of the most significant seismic parameters for describing the seismicity of a given region at a definite time window. In this study, high-resolution map of the Gutenberg-Richter b-value, seismic moment-release, Bouguer gravity anomaly and fault-plane solutions containing faulting styles are analyzed in the Indo-Burma ranges of northeast India using the unified and homogeneous part of the seismicity record in the region (January 1964–December 2016). The study region is subdivided into few square grids of geographical window size 1° × 1° and b-values are calculated in each square grid. Our goal is to explore the spatial correlations and anomalous patterns between the b-value and parameters like seismic moment release, Bouguer gravity anomaly and faulting styles that can help us to better understand the seismotectonics and the state of present-day crustal stress within the Indo-Burma region. Most of the areas show an inverse correlation between b-value and seismic moment release as well as convergence rates. While estimating the b-value as a function of depth, a sudden increase of b-value at a depth of 50–60 km was found out and the receiver function modeling confirms that this depth corresponds to the crust-mantle transition beneath the study region. The region is also associated with negative Bouguer gravity anomalies and an inverse relation is found between Gravity anomaly and b-value. Comparing b-values with different faulting styles, reveal that the areas containing low b-values show thrust mechanism, while the areas associated with intermediate b-values show strike-slip mechanism. Those areas, where the events show thrust mechanism but containing a strike-slip component has the highest b-value.

Renormalization of photon dyadic Green function by finite element method and its applications in the study of spontaneous emission rate and energy level shift

The spontaneous emission rate and the energy level shift of a quantum dot in any micro-nanostructures can be expressed by the classical dyadic Green's function. However, the real part of the dyadic Green's function is divergent, when the source point and the field point are at the same position. This leads to an unphysical divergent level shift. Theoretically, the dyadic Green's function can be decomposed into a homogeneous part and a scattering part. Traditionally, the homogeneous field contribution is introduced into the definition of the transition frequency and the only need is to consider the effect of the scattering part which is non-divergent. Another renormalization method is to average the Green tensor over the volume of the quantum dot. In this work, a finite element method is proposed to address this problem. The renormalized dyadic Green function is expressed by the averaged radiation field of a point dipole source over the quantum dot volume. For the vacuum case, numerical results of the renormalized Green tensor agree well with the analytical ones. For the nanosphere model, the renormalized scattering Green tensor, which is the difference between the renormalized Green tensor and the analytical renormalized one in homogeneous space, agrees well with the analytical scattering Green tensor in the center of the quantum dot. Both of the above models clearly demonstrate the validity and accuracy of our method. Compared with the previous scattering Green function method where two different finite element runs are needed for one frequency point, our renormalization method just needs one single run. This greatly reduces the computation burden. Applying the theory to a gap plasmonic nano-cavity, we find extremely large modifications for the spontaneous emission rate and the energy level shift which are independent of the size of the quantum dot. For frequency around the higher order mode of the nano-cavity, spontaneous emission enhancement is about Г/Г0 2.02106 and the energy level shift is about △ 1000 meV for a dipole moment 24D. These findings are instructive in the fields of quantum light-matter interactions.

Homogeneous Hamiltonian Control Systems Part II: Application to thermodynamic systems

In this paper, we give several examples of the formulation of the dynamics of open physical systems in terms of homogeneous Hamiltonian control systems. Therefore we have systematically used Gibbs-Massieu equations in the entropy form for the definition of the Lagrangian submanifold associated with the thermodynamical properties of the system. Accordingly we have used the total entropy function of the system to define the generating function of this Lagrange submanifold and have used the canonical lifting of systems of balance equations of the remaining independent extensive variables. In the first part we consider two examples of thermodynamic systems: two cells exchanging a heat flow and the Continuous Stirred Tank Reactor (CSTR). In the second part, we consider an elementary dissipative mechanical, the mass-spring-damper system and then embed it into the gas-piston system.

On the regularization of the Cauchy problem for a system of linear difference equations

The article proposes unusual regularization conditions as well as a scheme for finding solutions of the linear Cauchy problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology. The problem posed in the article continues the study of the regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by S.G. Krein, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Cauchy problem has no inverse. In the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Cauchy problem for a system of difference equations in the critical case is found. The proposed regularization conditions, as well as the scheme for finding solutions to linear Cauchy problems for a system of difference equations in the critical case, are illustrated in details with examples. In contrast to the earlier articles of the authors, the regularization problem for a linear Cauchy problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a solution to the regularization problem.

Radiation of a Charge Intersecting a Boundary Between a Bilayer Area and a Homogeneous One in a Circular Waveguide

This paper is devoted to the analysis of electromagnetic field excited by a charge moving in a circular waveguide, which consists of two semi-infinite parts. The charge moves with a constant velocity along the waveguide axis from the partially dielectric section into the homogeneous one. It is assumed that Cherenkov radiation is generated in the bilayer section. The total field is represented as a sum of a known forced field that is the field of the charge in the infinite regular waveguide and a free field resulting from the presence of the transverse boundary. The infinite system of algebraic equations for amplitudes of the free-field modes is derived. The further analysis is conducted using numerical calculations. Special attention is given to the study of the so-called Cherenkov-transition radiation (CTR) within the homogeneous part of the waveguide volume. Typical distributions of CTR over frequencies and modes are presented for the case when the channel and the homogeneous area are free of medium. Analytical results obtained for the point charge are generalized for the charged bunch with an arbitrary profile and verified using direct simulations for the case of Gaussian bunch.

Seismic b-value anomalies prior to the 3rd January 2016, Mw = 6.7 Manipur earthquake of northeast India

Spatial variation of seismic b-value is estimated in the Indo-Myanmar subduction zone of northeast (NE) India using the homogeneous part of earthquake catalogue (1996–2015), recorded by International Seismological Center (ISC), consisting of 895 events of magnitude MW ≥ 3.9. The study region is divided into 1° × 1° square grids and b-values are estimated at each grid by maximum likelihood method. In this study, the b-value varies from 0.75 to 1.54 in the region. Significant variation of low b-value in the respective location may indicate high stress accumulation in that region. Spatial variation reveals intermediate b-value anomalies around the epicenter of the Mw = 6.7 Manipur earthquake which occurred on 3rd January at 23:05 UTC (4 January 2016 at 04:35 IST). The variations of b-values are also estimated with respect to depth. The low b-value associated with the depth range ∼15–55 km, which may imply crustal homogeneity and high stress accumulation in the crust. Since, NE India lies in the seismic zone V of the country; this study can be helpful to understand seismotectonics in the region.

Effects of geometric modulation and surface potential heterogeneity on electrokinetic flow and solute transport in a microchannel

A numerical investigation is performed on the electroosmotic flow (EOF) in a surface-modulated microchannel to induce enhanced solute mixing. The channel wall is modulated by placing surface-mounted obstacles of trigonometric shape along which the surface potential is considered to be different from the surface potential of the homogeneous part of the wall. The characteristics of the electrokinetic flow are governed by the Laplace equation for the distribution of external electric potential; the Poisson equation for the distribution of induced electric potential; the Nernst–Planck equations for the distribution of ions; and the Navier–Stokes equations for fluid flow simultaneously. These nonlinear coupled set of governing equations are solved numerically by a control volume method over the staggered system. The influence of the geometric modulation of the surface, surface potential heterogeneity and the bulk ionic concentration on the EOF is analyzed. Vortical flow develops near a surface modulation, and it becomes stronger when the surface potential of the modulated region is in opposite sign to the surface potential of the homogeneous part of the channel walls. Vortical flow also depends on the Debye length when the Debye length is in the order of the channel height. Pressure drop along the channel length is higher for a ribbed wall channel compared to the grooved wall case. The pressure drop decreases with the increase in the amplitude for a grooved channel, but increases for a ribbed channel. The mixing index is quantified through the standard deviation of the solute distribution. Our results show that mixing index is higher for the ribbed channel compared to the grooved channel with heterogeneous surface potential. The increase in potential heterogeneity in the modulated region also increases the mixing index in both grooved and ribbed channels. However, the mixing performance, which is the ratio of the mixing index to pressure drop, reduces with the rise in the surface potential heterogeneity.

Asymptotic behaviour of the solutions to a virus dynamics model with diffusion

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the $L^1$ boundedness of the orbit. Then the pre-compactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number $R_0$ is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if $R_0>1$, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

Time Lapse Visualization of Catalyst Ink Degradation by X-ray CT

The catalyst layer (CL) is one of critical component for determining the performance of proton exchange membrane fuel cells (PEMFCs), and the production process of CL is important issue. In order to form the CL, catalyst ink mixed with platinum-supported carbon (Pt/C), ionomer solution, and a solvent is necessary. When the catalyst ink is coated on the sheet and dried, the CL is formed. By transferring the catalyst layer to the membrane, catalyst coated membrane can be made [1]. The process from the initial characteristics of the catalyst ink to the drying process determines the structure of CL and dominates the characteristics of the fuel cell. Thus, quality control of the catalyst ink is important to form a high performance and uniform CL [2], but it is difficult to observe how the characteristics of the catalyst ink change with time from the appearance. In this study, time lapse visualization of inside the catalyst ink was performed by X-ray CT. Catalyst ink was prepared by using the mixture of water and NPA as a dispersant. We used platinum catalyst supported on carbon black and Nafion dispersion D2020 (Chemours). Pre-mixing of catalyst ink was performed using ultrasonicaton to break large Pt/C agglomerates into smaller units. As a main mixing process, a high-speed rotary type mixer Filmix (Primix) utilizing shearing force, specialized for producing nanometer slurry was used. The prepared ink was immediately transferred to a glass sample container and sealed tightly. Thereafter, it was placed in the X-ray CT (Shimadzu) immediately and time lapse three-dimensional visualization was performed. Figure 1 shows sliced image of catalyst ink. The contrast of the image corresponds to the amount of absorption of X-rays, and it is possible to determine the state of the substance. In the black region, X-rays are easy to transmit, which is a light substance. In the initial state, micro to millimeter scale bubbles (several black dots) were dispersed, however, the other part was uniform light gray, which shows that the catalyst ink was homogeneously mixed. On the other hand, drastic state change inside the catalyst ink with the passage of time was observed. It was found that the volume of bubbles increased with the passage of time. Different color regions (dark gray) were also emerged between the bubbles. Changes with time of the volume of the homogeneous part of the ink, bubbles, and the emerged phase are shown in the graph (Figure 2). Observed changes in the catalyst ink are thought to be due to alcohol decomposition by the catalyst. For comparison, pseudo catalyst ink using carbon black not carrying platinum was visualized, but no change was observed. Analysis of the gas phase in the upper part of the glass sample container showed that the CO2 concentration was increased. From these results, NPA was decomposed by the platinum catalyst, and it is considered that generation of gas and formation of different liquid components were occurred. These experimental results suggest the importance of management of catalyst ink preparation, storage, and coating processes. Acknowledgment: This study is based on the results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO).

A fast numerical scheme for the Godunov–Peshkov–Romenski model of continuum mechanics

A new second-order numerical scheme based on an operator splitting is proposed for the Godunov–Peshkov–Romenski model of continuum mechanics. The homogeneous part of the system is solved with a finite volume method based on a WENO reconstruction, and the temporal ODEs are solved using some analytic results presented here. Whilst it is not possible to attain arbitrary-order accuracy with this scheme (as with ADER-WENO schemes used previously), the attainable order of accuracy is often sufficient, and solutions are computationally cheap when compared with other available schemes. The new scheme is compared with an ADER-WENO scheme for various test cases, and a convergence study is undertaken to demonstrate its order of accuracy.

On the internal interfaces in finite-difference schemes

Implementing sharp internal interfaces in finite-difference schemes with high spatial accuracy is challenging. The propagation of fields in a locally homogeneous part of a model can be performed with spectral accuracy. The implementations of interfaces are generally considered accurate to, at best, second order. This situation can be improved by a proper band limitation of the simulation grid. Interfaces can be located anywhere on the grid; however, the fine detail information regarding the interface location must be imprinted correctly in the coarse simulation grid. This can be done by starting out with a representation of a sharp material jump in the wavenumber domain and limiting the highest wavenumber to the maximum wavenumber allowed for the simulation grid. The resulting wavenumber representation is then transformed to the space domain. An alternative procedure is to create a fine grid model that is low-pass filtered to remove wavenumbers above the maximum wavenumber allowed for the coarse simulation grid. The fine grid is thereafter sampled at the required coordinates for the coarse simulation grid. An accurate and flexible interface implementation is a requisite for reducing staircase diffractions in higher dimensional finite-difference simulations. Our strategy achieves this. The frequency content of the source must be constrained to a level in which the spatial sampling is at approximately four to five grid points per shortest wavelength. Simulation results indicate that the implementation of the interface is accurate to at least the sixth order for large contrasts. Our method can be used for all systems of partial differential equations that formally can be expressed as a material parameter times a dynamic field on one side of the equal sign and with spatial derivatives on the other side of the equal sign. For geophysical simulations, the most important cases will be the Maxwell equations and the acoustic and elastic wave equations.

Evaluation of Efficient Closed-Form Green’s Function in a Cylindrically Stratified Medium

An efficient technique is proposed for the removal of the spatial singularity for a z-directed current source in a cylindrically layered structure by decomposing the spectral domain Green’s function into the particular and homogeneous parts. The particular part with the singularity is converted into the spatial domain using Sommerfeld’s identity, while the DCIM with general pencil of function technique is applied on the spectral domain homogeneous part. The use of the ratio form of the reflection/transmission matrices, the Debye approximation, and the subsequent envelope extraction for the treatment of the spatial domain singularity is thus avoided. In addition, a very significant reduction in the number of modes required for convergence is also achieved. The technique is applied for evaluating the input impedance of several antennas in the cylindrical multilayer structure together with a coupling configuration for the excitation of the multilayer structure.