Statistical Correlation Functions Research Articles

Reconstruction of radial parametric pressure field near ground surface of landing typhoons in Northwest Pacific Ocean

The radial distribution of environmental pressure near the ground surface during typhoons is an initial input condition for height-resolving typhoon modeling widely used as a practical and relatively accurate method for rapid assessment of typhoon-induced natural hazards and wind-resistant design of structures. The radius to maximum wind speed Rmax and radial pressure distribution parameter βs, (usually named the Holland power exponent of atmospheric pressure) are two key pressure field parameters that fundamentally determine the parametric typhoon pressure field and collocated wind velocity field. Observations from the Joint Typhoon Warning Center (JTWC) show that Rmax follows a lognormal distribution and has strong correlation with central pressure difference Δps and typhoon center latitude ψ. However, it is still hard to satisfactorily reproduce the landing typhoon environmental pressure. By employing near-ground environmental pressure data involving several hundreds of meteorological stations distributed in coastline regions of the Northwest Pacific Ocean, evolution of typical environmental pressures during four strong typhoons, Khanun (0515), Wipha (0713), Krosa (0716) and Morakot (0908), have been recorded in Southeast China. Finally, a novel technique for modeling the radial surface pressure model is proposed utilizing an appropriate model for radius to maximum winds Rmax. The variation characteristics for directions, landfall states and probability distributions of βs are compared and discussed, and then probability correlations among βs and Rmax, Δps, ψ are fitted and validated, showing that the statistical correlation function model is obviously different from former formulations and is more suitable for rebuilding a wind velocity field, especially near the ground surface after typhoon landfalls along the southeast coastlines of China, which are typhoon-prone hazard regions during moving typhoons.

Mimicking 3D food microstructure using limited statistical information from 2D cross-sectional image

We used statistical correlation functions (CFs) to describe food microstructure and to reconstruct their 3D complexity by using limited information coming from single 2D microtomographic images. Apple fleshy parenchyma tissue and muffin crumb were chosen to test the ability of the reconstructions to mimic structural diversities. Several metrics based on morphological measures and cluster functions were utilized to analyze the fidelity of reconstructions. For the apple, reconstructions are accurate enough proving that lineal, L2, and two-point, S2, functions sufficiently describe the complexity of apple tissue. Muffin structure is isotropic but statistically inhomogeneous showing at least two different porosity domains which reduced the fidelity of reconstructions. Further improvement could be obtained by using more CFs as input data and by implementation of the techniques dealing with statistical non-stationarity. Novel stochastic reconstruction and CF-based characterization methods could improve the fidelity of reconstruction and future advances of this technology will allow estimating macroscopic food properties based on (limited) 2/3D input information.

Analysis and algebraic­symbolic determination of conditions forsafe motion of a vessel in a non­stationary environment

We have proposed a method for the algebraic formalization of predicativedetermining of zones for safe and dangerous areas of navigation for the criterion Computational stability – continuity. It is not possible tobuild effective mathematical models for all cases of direct application of integrated and differential equations that describe statistical correlation functions of the space-time continuum. At the same time, there are always significant difficulties in operative processing of large volumes of information. Numerical results determine known long-time delays in the work of computers in the systems of navigation and operational control over the motion of a vessel. We have analyzed and obtainedconditions for the effective technology of structuring a safemaneuvering trajectorybased on typical sequences of formalized symbolic elementary zones. Briefpredicative fragments describe effective steps to maneuver in a safe area of navigation. Analytical description of the structural processes that transformthe input informational situational parameters into controlled effective parts and integrated modelsmakes it possible to increase technological speed of operational decision-making. Situational symbolic control over the qualities of adequate safe motion of a vessel is executedunder specific non-stationary changes in theinfluence from a dynamic external environment. Actual effects of the influencefrom external factors in a non-stationary environment that surrounds the hull of a moving vessel were symbolically formulated. In this case, typical estimates of degrees in thenon-stationary environment are algebraically collapsed into integrated safety criteria. We have established the target effect of modeling in terms of computational dynamics of processes for operational rapid control over motion of the vessel. In this case,vector regularized parameters ofevents are continuously entered in real time as corrections for initial data. We have solved the problem on predictive modeling of maneuvering variants for the criteria that guarantee current safety of motion along a planned strategic route. To effectively solve the taskon stable safety of motion, numerical methods to solve integral-differential nonlinear dynamic systems are typically employed. However, they are effective only for the substantiation of strategic routes. Delays in time at these stages are significant and accepted. It is proposed to manage and control algorithmic processes in real time by using alternative methods of symbolic algebra. The modelsproposed would make it possibleto find variants that are guaranteed to be adaptive to a specific current situation based on the transversal trajectory of a vessel motion. It is proven that in the areas where there may occur situations of conflict and risk, local rules that guarantee a safe motion trajectory areimplemented by connecting the boundary conditions for safe navigation area taking into account the presence of adjacent zones with threats, perturbations, obstacles.

Homogeneity and representativeness analyses of solid oxide fuel cell cathode microstructures

Microstructural properties such as relative density, triple phase boundary length, specific surface area, tortuosity factors are often calculated for microstructure-based analysis and modeling of solid oxide fuel cells (SOFCs). A realistic representation of the microstructure is a challenging and important issue for both experiments and modeling. In the present study, statistical correlation functions and a volume expansion method are used to determine the representative volume element size for the analyses of La0.6Sr0.4Co0.2Fe0.8O3-δ cathode microstructures in SOFCs.

Characterizing apple microstructure via directional statistical correlation functions

Our understanding of food is tightly related to the complex food microstructure. We introduce the use of statistical correlation functions to quantitatively describe the spatial distribution of cell and void phases of ‘Braeburn’ and ‘Kanzi’ apples. The lineal-path distribution function, L(r), and the two-point correlation function, S2(r), were measured from bi-dimensional (2D) microtomographic images. While the void fraction of both apples cultivars was similar, 0.840% and 0.853%, the pores of ‘Braeburn’ apples were bigger in size. Pores with an extension of 25μm were found in percentage of 12.6% and 4.3% for ‘Braeburn’ and ‘Kanzi’ respectively. The cell phase of ‘Braeburn’ apples was described by larger clusters as a result of a greater degree of connectivity among the individual cells. ‘Kanzi’ apple tissue was structured by more separated cell clusters due to the presence of small pores with a greater spatial distribution. The clusters showed a good homogeneity in shape for both varieties while the voids of ‘Kanzi’ apples appeared more inhomogeneous and elongated in one of the dimensions. The obtained structural information were employed to model tissue structures of apples. We found that the cell phase could be modeled by overlapping disks having a mean radius of 58 and 53mm for ‘Braeburn’ and ‘Kanzi’ apples, respectively. In this way macroscopic properties of apple tissue could be estimated precisely.

Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations

The classical fluctuation-dissipation theorem predicts the average response of a dynamical system to an external deterministic perturbation via time-lagged statistical correlation functions of the corresponding unperturbed system. In this work we develop a fluctuation-response theory and test a computational framework for the leading order response of statistical averages of a deterministic or stochastic dynamical system to an external stochastic perturbation. In the case of a stochastic unperturbed dynamical system, we compute the leading order fluctuation-response formulas for two different cases: when the existing stochastic term is perturbed, and when a new, statistically independent, stochastic perturbation is introduced. We numerically investigate the effectiveness of the new response formulas for an appropriately rescaled Lorenz 96 system, in both the deterministic and stochastic unperturbed dynamical regimes.

Использование вероятностных методов для оценки токовых нагрузок в элементах тяговой сети

Objective: To develop the method of calculating the electrical loads on the traction circuit components, by using data of train traffic schedule, considering the train weight differences and real-life modes of its traction under the variety of track profile, speed limitations and other parameters of traction and external electric power supply systems. Methods: To determine the mean values and dispersion of current for power supply line of traction circuit, based on the experimental runs and data from movement parameters recorders of locomotives, by type and weight of trains, the methods of probability theory and mathematic statistics were used. The current, consumed by the trains, is considered to be a random variable. Current loads of the trains is put into fixed weight, that allows to obtain statistical expectation and correlation function. Results: The ratios obtained allow to estimate general mean and effective values by sampled values within confidence limits. The paper provides the conditions, under with the current consumption functions have ergodic property cap. It also states the recommendations for train rating, based on its weight. The statistical expectations and mean square deviations of currents of traction power supply components. Practical importance: The load estimations are adjusted, that allows to determine the parameters of power circuits of electric railways in a more substantiated way. Obtained equations allow to determine the possibility of current loads bigger than given value, that should be considered while choosing the capacity of transformers, rectifiers, compensating devices, inverters, power supply and earth leads. Thus, it is possible to reach the required reliability of equipment operation and cost effectiveness of decisions.

Virtual testing of dual-phase steels: Effect of martensite morphology on plastic flow behavior

Dual-phase steels exhibit a moderate plastic anisotropy as a result of the underlying microstructure. A numerical homogenization scheme based on full-field finite element simulations was applied to assess the effect of the martensite morphology on the stress-strain curves and the plastic strain ratios (r-values) of a dual-phase steel under uniaxial tensile loading in different directions. The simulation results show that the martensite morphology affects the macroscopic response and the degree of plastic anisotropy. The mechanical behavior of the dual-phase steel (DP600) investigated in this work could be predicted if the crystallographic texture of the material and a statistically equivalent martensite morphology were considered in the simulations. The corresponding finite element models of the dual-phase microstructure were generated using a reconstruction procedure based on statistical correlation functions. This study demonstrates that the predictive capabilities of the virtual testing approach strongly depend on the level of detail with which the microstructural features are represented in the simulations.

On the inverse problem of the reconstruction of food microstructure from limited statistical information. A study on bread

The possibility to reconstruct food microstructure by limited morphological information has fundamental importance for theoretical and practical applications. We implemented the simulated annealing method proposed by Yeong and Torquato (1998) for reconstructing bread structure through the information contained into the lineal-path distribution function, L(r), and two-point correlation function, S2(r). The method enabled the evolution of two-phase random image toward bread structure. When using the information of lineal-path distribution function, the generated images well captured the main morphological features of bread, although several deviations still existed. This was in accordance with the significant differences between the original and reconstructed images as measured by two-point correlation function. By hybrid reconstruction, based on both correlation functions, a better reconstruction in terms of both number and size of pores was obtained. In the future the use of more several statistical correlation functions could enable further improvement in reconstruction of bread microstructure.

Collapse of ordered spatial pattern in neuronal network

Spatiotemporal systems can emerge some regular spatial patterns due to self organization or under external periodical pacing while external attack or intrinsic collapse can destroy the regularity in the spatial system. For an example, the electrical activities of neurons in nervous system show regular spatial distribution under appropriate coupling and connection. It is believed that distinct regularity could be induced in the media by appropriate forcing or feedback, while a diffusive collapse induced by continuous destruction can cause breakdown of the media. In this paper, the collapse of ordered spatial distribution is investigated in a regular network of neurons (Morris–Lecar, Hindmarsh–Rose) in two-dimensional array. A stable target wave is developed regular spatial distribution emerges by imposing appropriate external forcing with diversity, or generating heterogeneity (parameter diversity in space). The diffusive invasion could be produced by continuous parameter collapse or switch in local area, e.g, the diffusive poisoning in ion channels of potassium in Morris–Lecar neurons causes breakdown in conductance of channels. It is found that target wave-dominated regularity can be suppressed when the collapsed area is diffused in random. Statistical correlation functions for sampled nodes (neurons) are defined to detect the collapse of ordered state by series analysis.

Stochastic Reconstruction and Transport Simulation of PEFC Catalyst Layers

A stochastic method for analysis and reconstruction of PEFC porous media is presented. The method uses statistical correlation function to analyze and parametrize the porous media structure. The improved reconstruction method uses a multigrid hierarchical simulated annealing technique to significantly reduce the reconstruction time and ascertains a long range connectivity. Analysis of the catalyst layers show it to be a homogeneous porous media with strong anisotropy in in-plane and through-plane directions. The stochastic reconstruction procedure is able to reconstruct 3D structures of the catalyst layer with similar statistical correlation functions as of the reference image. Molecular and Knudsen diffusion are simulated on the reconstructions using the open source package OpenFCST. The effective transport properties of the reconstructions is similar to that of the reference structure; however, the accuracy of the reconstructions has been limited by the imaging methods.

3D reconstruction of carbon nanotube networks from neutron scattering experiments

Structure reconstruction from statistical descriptors, such as scattering data obtained using x-rays or neutrons, is essential in understanding various properties of nanocomposites. Scattering based reconstruction can provide a realistic model, over various length scales, that can be used for numerical simulations. In this study, 3D reconstruction of a highly loaded carbon nanotube (CNT)-conducting polymer system based on small and ultra-small angle neutron scattering (SANS and USANS, respectively) data was performed. These light-weight and flexible materials have recently shown great promise for high-performance thermoelectric energy conversion, and their further improvement requires a thorough understanding of their structure-property relationships. The first step in achieving such understanding is to generate models that contain the hierarchy of CNT networks over nano and micron scales. The studied system is a single walled carbon nanotube (SWCNT)/poly (3,4-ethylenedioxythiophene):poly (styrene sulfonate) (PEDOT:PSS). SANS and USANS patterns of the different samples containing 10, 30, and 50 wt% SWCNTs were measured. These curves were then utilized to calculate statistical two-point correlation functions of the nanostructure. These functions along with the geometrical information extracted from SANS data and scanning electron microscopy images were used to reconstruct a representative volume element (RVE) nanostructure. Generated RVEs can be used for simulations of various mechanical and physical properties. This work, therefore, introduces a framework for the reconstruction of 3D RVEs of high volume faction nanocomposites containing high aspect ratio fillers from scattering experiments.

Mathematical Modelling and Experimental Analysis of Thin, Low-Loading Fuel Cell Electrodes

The use of expensive platinum in PEFC remains a major barrier to achieve a low cost fuel cell stack. To reduce stack cost, thin, low loading electrodes have been studied at various laboratories [1]. Experimental data showed that the cell performance is severely limited for cathode electrode loadings below 0.1 mg/cm2. This decreased performance has been attributed to several physical processes [2]. First, the oxygen reduction reaction (ORR) is a multi-step reaction that does not follow Tafel kinetics. Low loading electrodes might operate following a different ORR pathway. Second, due to the higher local mass flux to the Pt sites, mass transport resistances in the ionomer thin film, presumably covering the catalytic sites, might limit oxygen transport to the reaction sites. Finally, water management issues are exacerbated due to the reduced thickness and thereby increased water production rate per unit volume. Experimental and theoretical investigations in thin, low loading fuel cell electrodes are being performed in our laboratory. Thin, low loading fuel cell electrodes of various ionomer and Pt loadings, and Pt/C content were fabricated and tested under a variety of operating conditions. In order to accurately control catalyst/ionomer loading and layer uniformity, the electrodes were deposited on a membrane using a material inkjet printer. Results show that cells with cathode Pt loadings of 0.025 mg/cm2 are severely limited in the kinetic region with cell voltages as low as 0.6V at 200 mA/cm2 under hydrogen/air conditions at ambient pressure. At cell voltages of 0.2 V however, mass activities as high as 50 A/mgPt and current densities above 1 A/cm2were observed [3]. Macro-scale mathematical modelling studies of low loading electrodes have either been based on simplified one-dimensional models, or have relied on simple Tafel kinetics for the ORR. In this work, meso- and macro-scale simulations of thin, low loading electrodes are performed using OpenFCST, an open-source fuel cell simulation software developed by our research group [4]. A multi-dimensional, non-isothermal, two-phase membrane electrode assembly (MEA) model was developed. The Butler-Volmer/Tafel representations of the HOR and ORR were replaced by multi-step reaction kinetic models, i.e., the dual path and double trap models proposed by Wang et al. and modified by our group [5]. To account for thin film mass transport resistances, a pseudo-agglomerate model that accounts only for the ionomer thin film was implemented. The model is justified by microscopy observations and our previous mathematical modelling results that suggest that, based on the expected agglomerate size, the ionomer thin film is the most important micro-scale parameter [6]. The use of multi-step kinetic and ionomer thin film models improves fuel cell predictions for both conventional and low loading electrodes in the kinetic, ohmic and mass transport regions. Water build-up in thin, low loading electrodes is likely to be exacerbated. Several researchers have proposed a variety of pore-size distribution (PSD) based models to account for the layers' micro-structures [7]. These models however have not yet been implemented in a multi-dimensional MEA model. A dual wettability PSD based model was implemented in OpenFCST. The PSD based model has been compared to the common saturation-based two-phase flow model in porous media, e.g., [8]. Preliminary results show that the PSD model is able to identify discontinuities in the saturation profiles inside the MEA as a result of the MPL hydrophobic nature and small PSD. The saturation-based model on the other hand, provides a continuous saturation profile. Focused ion beam-scanning electron microscopy (FIB-SEM) and nano-computed tomography (nanoCT) have been performed on our low loading electrodes. A statistical analysis tool was developed and used to compute multiple correlation functions to assess porosity, active area and pore connectivity. Using the statistical correlation functions and a novel different phase neighbors based simulated annealing strategy, meso-scale reconstructions of the electrode have been created [9]. The electrode reconstructions can be used to estimate the effective transport properties in each phase of the electrode using OpenFCST. [1] J. Wee et al., J. Power Sources, 2007, 165(2) 667. [2] J. Owejan et al., JES, 2013, 160(8), F824. [3] S. Shukla et al., Electrochimica Acta, 2015, 156:289. [4] M. Secanell et al., ECSTrans., 2014, 64(3): 655. [5] J. Wang et al., JES., 2006, 153(9), A1732; J. Wang et al., J. Phys. Chem. A, 2007, 111, 12702; Moore et al., JES, 2013, 160(6), F670. [6] M. Moore et al., JES., 2014, 161(8):E3125. [7] A. Weber. JES., 2004, 151(10), A1715. [8] D. Natarajan, JES, 2001, 148(12), A1324. [9] L. Pant et al., Phys.Rev.E, 2014, 90, 023306. Figure 1

Scattering Studies for Two-Dimensional Exponential Correlation Textured Rough Surfaces Using Small-Slope Approximation Method

Two-dimensional exponential correlation rough surfaces characterized by textures are combined with the small-slope approximation (SSA) method to comparatively study electromagnetic (EM) scattering features of textured surfaces. The normalized copolarized radar cross section from 2-D exponential correlation rough surfaces characterized by stripe texture and block texture, respectively, is analyzed. Several numerical results show the effects of incident angle, texture angle, correlation length, and root-mean-square height on the copolarimetric scattering from the textured rough surface. The validity of the SSA method is verified by comparisons of theoretical value and measured data. Moreover, normalized amplitude distributions of backscattering fields from cells in a scene are studied through its statistical distribution and space correlation function, which are particularly useful for analysis and simulation of remote sensing images. Finally, based on the statistical distribution and space correlation function, the zero memory nonlinear transformation method is utilized to simulate EM scattering from very large scenes. The simulated scene coincides with the original one quite well.

Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions

The reconstruction of microstructure of biological and synthetic materials is of great importance for theoretical and practical application. The obtaining of this purpose through limited statistical information is a challenge at which some pioneering researchers are focusing their efforts. For the first time, a reconstruction method was used to rebuild bread microstructure by using the information of lineal-path distribution function (LPF). The method was a powerful tool to reconstruct bread microstructure; a perfect match between lineal-path function of reference and reconstructed images were obtained. The reconstructed images progressively improve, compared to the reference, increasing the number of LPFs used during reconstruction. The use of LPFs in different direction did not allowed to obtain a perfect match between the bread and the reconstructed images because, LPF did not contain sufficient microstructure information however, the essential features of bread were captured proving the possibility to reconstruct food microstructure.

Use of Lineal-Path Distribution Function as Statistical Descriptor of the Crumb Structure of Bread

The importance of the relation between food microstructure and their quality is recognized as crucial. In terms of bread quality, the precise characterization of the crumb structure is considered essential to evaluate its sensorial properties. A variety of different statistical correlation functions have been theoretically formulated in recent years to quantify the microstructure of general random heterogeneous materials but they have rarely been applied in food science. In this paper, we have developed a method to calculate the lineal-path distribution function L(x) for the void and solid matrix phases from two-dimensional (2D) images of bread. The method has been successfully employed here to characterize the microstructure for two type of commercial breads. Our results reveal that crumb structure may be modeled as a system of polydispersed overlapping disks. Moreover, we have quantified the dimensions of the voids and crumb phase statistically and differences in the homogeneity of crumb structure.

3-D microstructure reconstruction of polymer nano-composite using FIB–SEM and statistical correlation function

Abstract 3-D reconstruction of Halloysite nanotube (HNT) polypropylene composite has been performed using two different methods. In the first method, several slices of the composite material were obtained using focused ion beam (FIB), and scanning electron microscopy (SEM). A representative volume element (RVE) of the real material’s micro/nanostructures was then constructed by stacking these morphological images using VCAT® software. In the second method, SEM images of the nano-composite were used to extract statistical two-point correlation function (TPCF), for reconstruction of an RVE of the nano-composite. The resulting RVEs obtained from both methods were meshed for finite element (FE) simulation of deformation under tension and shear loadings. The FE results were then used to compute the stiffness tensor of the nano-composite. In the statistical approach, the TPCF was obtained from a none-Eigen microstructure which can partially reflect statistical information of the microstructure. The mechanical constants obtained from statistical RVEs using FEM approach shows a 5.7% error compared with those obtained from real RVE, which could be attributed to the approximation using TPCF [1] . It is concluded that the statistical method using TPCF alone can produce an approximate microstructure that should be modified using other statistical descriptor such as two-point cluster function and lineal path function to have better reconstruction of heterogeneous nano-composites [2] .

A Fitting Algorithm for Random Modeling the PLC Channel

The characteristics of the power-line communication (PLC) channel are difficult to model due to the heterogeneity of the networks and the lack of common wiring practices. To obtain the full variability of the PLC channel, random channel generators are of great importance for the design and testing of communication algorithms. In this respect, we propose a random channel generator that is based on the top-down approach. Basically, we describe the multipath propagation and the coupling effects with an analytical model. We introduce the variability into a restricted set of parameters and, finally, we fit the model to a set of measured channels. The proposed model enables a closed-form description of both the mean path-loss profile and the statistical correlation function of the channel frequency response. As an example of application, we apply the procedure to a set of in-home measured channels in the band 2-100 MHz whose statistics are available in the literature. The measured channels are divided into nine classes according to their channel capacity. We provide the parameters for the random generation of channels for all nine classes, and we show that the results are consistent with the experimental ones. Finally, we merge the classes to capture the entire heterogeneity of in-home PLC channels. In detail, we introduce the class occurrence probability, and we present a random channel generator that targets the ensemble of all nine classes. The statistics of the composite set of channels are also studied, and they are compared to the results of experimental measurement campaigns in the literature.

Modeling of two-phase random composite materials by finite element, Mori–Tanaka and strong contrast methods

In this study, finite element, Mori–Tanaka and strong contrast modeling are carried out for the prediction of the effective thermal conductivity and elastic modulus of isotropic random two-phase composite materials with low fillers content. Effects of inclusions geometry (shape), volume fraction (1% and 3%) and properties contrast on the effective thermal conductivity and elastic modulus are analyzed. Our results show that finite element method could capture more details in the prediction of effective properties of the composite materials. On the other hand, Mori–Tanaka method is shown to be a fast as well as a valid alternative for the finite element modeling within a limited range of fillers geometries. Our results reveal that the strong contrast method based on statistical two-point correlation functions could not accurately describe the inclusions geometry effects.

New approximate solution for N-point correlation functions for heterogeneous materials

Statistical N-point correlation functions are used for calculating properties of heterogeneous systems. The strength and the main advantage of the statistical continuum approach is the direct link to statistical information of microstructure. Two-point correlation functions are the lowest order of correlation functions that can describe the morphology and the microstructure-properties relationship. Experimentally, statistical pair correlation functions are obtained using SEM or small x-ray scattering techniques. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded materials) heterogeneous microstructures. Conditional probability functions are used to formulate the proposed theoretical approximation. In this approximation, weight functions are used to connect subsets of ( N−1)-point correlation functions to estimate the full set of N-point correlation function. For the approximation of three and four point correlation functions, simple weight functions have been introduced. The results from this new approximation, for three-point probability functions, are compared to the real probability functions calculated from a computer generated three-phase reconstructed microstructure in three-dimensional space. This three-dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM image) of a three-phase material. This comparison proves that our new comprehensive approximation is capable of describing higher order statistical correlation functions with the needed accuracy.