Asymptotic Averaging Research Articles

Bayesian analysis of the growth of pears in the cultivate Shinseiki using non-linear models

The pear of the Pyrus genus is the third most consumed fruit in Brazil, however, internal production is low and most demand is obtained through imports. Brazil has favorable conditions for its cultivation and one of its main economic sectors is agriculture, being responsible for generating a large number of jobs and income for the country. In this sense, studies related to this fruit can provide information and encourage its production. Using non-linear models, it is possible to identify growth patterns such as fruit length and diameter, which can help determine the ideal harvest point. Therefore, the goal of this article was to compare the Logistic and Gompertz non-linear models in describing the growth curves in diameter and length of the Asian pear tree using the Bayesian approach. The results indicated the Logistic model as the best to describe these two variables and provided asymptotic averages of 70.0607 and 79.706 for length and diameter, respectively.

Porous Medium Modeling of Catalytic Monoliths Using Volume Averaging

Catalytic monoliths are being explored in conventional catalytic processes for their ability to achieve process intensification. Scientific computing can play an essential role in this exploration. The high computational cost of the first-principles models has led to modeling these reactors as a porous medium. However, this modeling strategy is not performed in a mathematically rigorous manner. We use the volume averaging technique as a mathematical framework to convert the pointwise governing equations for a monolith into averaged equations for a porous domain. These averaged equations require the closure of several unclosed terms, which are neglected in the classical porous medium (CPM) assumption. We discuss these unclosed terms and their impact on model predictions. We show that except in the limit of negligible Damköhler number the treatment of the catalytic reactions in CPM leads to significant errors. We propose a technique to accurately calculate the catalytic reaction rates in the volume-averaging-based porous medium (VAPM) model developed here. This technique is valid for a wide range of Damköhler numbers for both linear and nonlinear kinetics. Moreover, to calculate the effective properties of the porous medium, such as thermal conductivity, we employ asymptotic averaging (numerical homogenization) that can be used for any arbitrary channel shape and size. Predictions of the proposed VAPM model are assessed against three-dimensional multichannel monolith simulations, resolving the solid and fluid phases, for elementary and complex kinetics. In addition, VAPM is validated against the experiments of steam methane reforming in a catalytic monolith. The developed methodology reduces the computational cost by 3 orders of magnitude while maintaining the accuracy of the detailed multichannel simulations.

Asymptotic Study of Plate Bending for a Strongly Orthotropic Materia

The technique of asymptotic averaging has been developed for three-dimensional partial differential equations with rapidly oscillating coefficients. For example, for the equations of elasticity theory. Then it was modified and applied to thin bodies in the form of plates (homogeneous or inhomogeneous, with even front surfaces or not), described by the three-dimensional theory of elasticity. In these cases, asymptotic solutions were constructed with respect to one small parameter, which is usually the ratio of the plate thickness to the characteristic plan size. The averaging technique in this case also reduces the dimension of the problem, i.e. reduces the three-dimensional boundary value problem to some two-dimensional one. In this article, we substantiate the application of the method to a problem with two small parameters in the case of a homogeneous thin, highly orthotropic plate bent by a surface load without taking into account body forces. The second small parameter is the ratio of the transverse elastic moduli to the moduli in the plan of the plate. It is shown that strong orthotropy is equivalent to an increase in the thickness of the equivalent plate. A procedure for obtaining the stress distribution over the plate thickness is described for three approximations. The first approximation gives the classical Kirchhoff theory, also called the Kirchhoff-Love theory, and the third approximation coincides with the Ambartsumian theory and allows one to find transverse shear and normal stresses. The consideration of cylindrical bending makes it possible to find solutions in the framework of classical plate theories in the form of formulas, as well as three approximations of the asymptotic theory, which simplifies comparison. Examples are considered when the averaged orthotropic moduli are taken for a single-layer fibrous composite.

Finite element modeling of integral viscoelastic properties of textile composites

The problem of modeling the effective integral viscoelastic properties of unidirectional composite materials is considered. To calculate the integral properties of viscoelasticity, the Fourier transform and the inverse Fourier transform are used, as well as the method of asymptotic averaging for composites with steady polyharmonic vibrations, and a finite element algorithm for solving local problems of the viscoelasticity theory on the periodicity cell of the composite. To obtain the material constants, a method of approximation of the Fourier images of the relaxation and creep kernels is proposed, which makes it possible to avoid the numerical error of the inverse Fourier transform.

On the Absorbing Problems for Wiener, Ornstein–Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters.

Calculation of effective properties of geocomposites based on computed tomography images

Biot’s parameter is included in the formula for calculating effective stresses and should be taken into account when assessing the stress-strain state of a water-saturated rock mass. A method for calculating Biot’s tensor parameter based on asymptotic averaging of the equilibrium equation for a fluid-saturated porous medium is proposed. Calculations of elastic properties and Biot’s coefficient were carried out on various types of rocks - limestone, dolomite, hyaloclastite, basalt. The calculations were carried out using 3D models of geocomposites built from X-ray computed tomography images. The results of 3D calculations of Young's modulus and Biot’s coefficient coincided with the results of experimental determinations of these properties by the ultrasonic method. This fact shows the expediency of using a computational approach that implements asymptotic averaging to estimate the effective properties using 3D models of the real rock structure. The results of 3D and 2D simulation of effective properties of geocomposites are compared. 2D models were built using photographs of rock sections. It was found that the values of Young's modulus and Biot’s parameter for 2D models differ from the corresponding experimental and 3D calculation results by 20-30 %. Therefore, 2D modeling is not suitable for evaluating the effective properties of porous geomaterials. Based on the results of calculations and experiments, the dependences of Young's modulus and Biot’s coefficient on porosity were obtained and studied. These dependencies are used in non-linear numerical simulation of the consolidation of water-saturated soils. The results of the study showed that Biot’s coefficient does not depend on Young's modulus of the rock skeleton material. The influence of the pore shape on Young's modulus and Biot’s coefficient was studied using 2D calculations. A method for predicting the shape of pores from the values of porosity and Young's modulus using neural networks is proposed. Using this method, a specific algorithm for predicting the shape of pores in hyaloclastites has been implemented and studied.

Pointwise ergodic theorem for locally countable quasi-pmp graphs

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $ L^1 $ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor.The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon–Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

Image-based simulation of dispersed composites and porous rocks

The article presents the results of numerical simulation of the effective elastic properties and stress concentration of the dispersed composite and porous rocks as well as the elastoplastic diagrams of tensile stress of the B4C/2024Al composite using asymptotic averaging method. 2D and 3D simulations of uniaxial tension was carried out using the image-based models of the real structure obtained by X-ray tomography. We also used model structures of the B4C/2024Al composite, consisting of an aluminum matrix with boron carbide inclusions in the form of ellipsoids with a random orientation. It turned out that in the elastic region 3D real structure and 3D ellipsoidal structure lead to the same effective moduli. On contrary, the stress concentrations are much higher in the real structure model than in the model of ellipsoidal inclusions. Also, two types of the models behave completely differently in the plastic region. Comparison of 2D and 3D calculations with experimental data shows that 2D models can be also used for determining the effective elastic moduli of B4C/2024Al composite, but the use of 3D models is preferable for porous geomaterials and dispersed composites.

Asymptotic analysis of the two-layer plates bending with nonlinear interlayer friction

sThe article is devoted to the study of the features of the occurrence of slip between layers of a thin two-layer plate with an initial ideal contact under the influence of critical pressure. A three-dimensional problem of the linear theory of elasticity for a thin two-layer plate with sliding conditions with friction at the interface is considered. To solve this problem, the method of asymptotic averaging is applied. An equation is established for calculating the critical pressure for a two-layer plate at which the layers begin to slip. A numerical example is considered for a two-layer fiberglass for different conditions of fixing at the ends. It is shown that the fixing conditions affect the critical pressure of the beginning of sliding.

Stochastic Wave\u2013Current Interaction in Thermal Shallow Water Dynamics

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

Modeling Elastic Properties of Composites using Asymptotic Averaging Method with Imperfect Interface

The paper presents a modification of the asymptotic averaging method for solving the homogenization problem of elastic properties for composite materials. The elasticity of the phase interface is taken into account. The conditions of a soft imperfect interface are considered, which account for a displacement jump on the phase boundary. A review of the interface-modeling methods in composite materials is presented. The finite element method is used for the numerical implementation of the averaging method. A model of a interface finite element is proposed. Elastic properties averaging is adapted to the presence of a discontinuity in the displacement field. Application limits of the soft imperfect interface are estimated in terms of interphase layer properties. The identification of the interface parameters using experimental data is considered. Computational experiments are conducted for dispersed-reinforced and unidirectional composite with isotropic inclusion.

Plate vibrations with periodically changing parameters

Vibrations of square plate with periodically changing parameters are considered. The averaged fourth-order partial differential equation for the plate deflection w is offered. Solution of the problem is obtained by using the approximate theory. The approximate results are presented by analytical formulas. Asymptotic averaging (realized in Wolfram Mathematica) and Finite Elements Method (ANSYS) are used to get the values of vibrations frequencies. The comparison of numerical and asymptotic results is performed.

Application of the asymptotic homogenization in a parametric space to the modeling of structurally heterogeneous materials

An asymptotic averaging of differential equations with fast oscillating quasi-periodic coefficients so-called “the asymptotic averaging in parametric space” is developed. The system of equations corresponding to structurally heterogeneous thermoelastic media with smoothly varying microstructures is considered. Such materials are usually treated as functionally graded ones. Their description is satisfied by introducing two types of variables into the coefficients of the system of thermoelastic equations: “fast” and “slow”. The fast variables describe the geometry of the inhomogeneities provided their periodic arrangement. The additional variables allow one to describe a smooth spatial variation of a periodic structure of inclusions, i.e. to describe within the proposed approach various materials with functionally graded properties. In addition, the coefficients may depend on the temperature, which also has a given regular distribution in the space. The allocation of this dependence in the separate parameter makes sense, since it is important in practice. It is shown that the asymptotic averaging solves the smooth aperiodic dependencies for the coefficients of the considered equations system parametrically through the functions of rapid variables. A two-level model system for such structurally heterogeneous materials is formulated as a result of averaging, and an algorithm of accurate definition of thermomechanical properties including the functionally graded and thermally depending ones could be found. The combined numerical-analytical block method based on the Papkovich–Neuber​ representation is proposed for low-level problems with arbitrary structures (e.g. for the problem defined on a cell for functions of fast variables; such a problem determines the effective properties of a material). The structures with spherical and cylindrical inclusions and interphase layers are especially considered and the complete function systems allowing one an efficient solution approximation on a cell with exactly satisfied contact conditions on interlayers is constructed.

Numerical investigation of mesoscopic volumetric ablation of 3D braided charring composites

Owing to the complex mesoscopic structures, 3D braided charring composites for the thermal protection system (TPS) have the multi-scale and multi-physical high-temperature responses. The mesoscopic features can significantly affect the volumetric ablative behavior of the braided composites, which will be essential for the TPS design. To take into account the effect of mesoscopic structures on the volume ablation response, the thermo-physical properties of constituents within the braided composites are predicted using asymptotic averaging method. The volumetric ablation model of the braided composites considering the mesoscopic structures is proposed, which is calculated by finite volume method with a tightly coupled solving process. The effect of braid angles on the volumetric ablation responses of the braided composites are investigated. The maximum back temperature and the peak inner pressure of the braided composites decrease with the increase of the braid angle. The isothermal and isobaric surfaces of the braided composites under volumetric ablation exhibit complex characteristics and are not perpendicular to the heating direction, which may induce the complex thermal stress field within the composites. The study on mesoscopic volume ablation may be helpful to accurately predicting the ablation responses and guiding the material design of the braided charring composites.

Surface Finite Element for Imperfect Interface Modeling in Elastic Properties Homogenization

The paper presents a mathematical model of a finite element for modeling imperfect interface conditions for two contacting surfaces. The element is used in the numerical implementation of the Asymptotic Averaging Method (AAM) for the determination of effective elastic properties of composite materials under investigation. Numerical experiments are carried out to calculate the elastic properties taking into account the adhesion layer using a displacements field jump condition at the phase boundary. Results are compared with adhesion modeling using an additional bulk phase.

Investigation of random access computer networks operating in a diffusion environment with an increasing number of subscriber stations

Improving the performance of various types of communication networks in the modern world remains an urgent task, in connection with which research is underway to create hardware that expands the throughput of physical channels, new network protocols are being developed and existing network protocols are being modified, mathematical and computer modeling of data transmission mechanisms in communication networks is being carried out. The speed and reliability of data transmission over networks also depends on a number of factors, the nature of the influence of which is random. The combination of such factors is called a random environment. If the change in the states of the medium is continuous, then we speak of a diffusion medium. The object of the research is communication networks controlled by multiple access protocols and functioning in a random (diffusion) environment. The research tool for multiple access networks is the mathematical apparatus of the theory of finite-difference and differential equations, the theory of random processes and the theory of queuing. The proposed mathematical model of communication networks in a diffusion environment is investigated by an asymptotic method. The scientific novelty of the work lies in the fact that for the first time a mathematical model of a multiple access network operating in a diffusion environment was proposed and an asymptotic study was carried out. The asymptotic average of the normalized number of claims in the orbit (the source of repeated calls) and the deviation from this average are found, and the probability density of the values of the process of changing the number of claims in the orbit is obtained.

Method to improve convergence performance of iterative learning control systems with bounded noise

Iterative learning control (ILC) can significantly improve the tracking performance by learning from previous trials. Recently, kalman-filtering based ILC algorithms (KF-ILC) have been investigated to improve the convergence performance of ILC systems subject to the stochastic noise. However, relevant studies considering the bounded noise have not been reported. Hence, this paper incorporates the set-membership method into the ILC design and analysis for a class of linear systems subject to the bounded noise. Firstly, a new ILC algorithm is designed, i.e., the set-membership based ILC algorithm (SM-ILC), which not only provides an estimate for the desired input but also determines an ellipsoid of confidence for the estimate. Choosing an optimal conclusive parameter by minimizing the size of the confidence ellipsoid under an equality constraint enables SM-ILC monotonic convergence in presence of the bounded noise. The input error under SM-ILC can asymptotically converge to zero if there always exists two worst-case noise vectors of opposite orientations after arbitrary number of trials. Secondly, the convergence performance of KF-ILC when confronted with the bounded noise is revealed with the set-membership method. It is shown that KF-ILC fails to achieve monotonic convergence under the bounded noise. The steady-state input error achievable with KF-ILC is determined by the asymptotic average of the noise. To facilitate the applications of SM-ILC and KF-ILC in practice, the results presented above are generalized to uncertain systems and the frequency-domain counterparts of the lifted algorithms are derived for SISO LTI systems. Finally, a numerical example verifies the theoretical results and an experiment on a linear motor confirms the potential of the proposed ILC algorithms in real-life applications.

Comparison of discrete fiber and asymptotic homogenization methods for modeling deformations of fiber-reinforced materials

The main issue in a numerical simulation of fiber-reinforced materials deformation is high computational complexity. Exact resolution of fibers thickness leads to a huge number of degrees of freedom. In this work, we compare discrete fiber and asymptotic homogenization methods for modeling deformations. The discrete fiber method considers fibers as one-dimensional lines and the asymptotic homogenization method averages elasticity coefficients. We investigate the accuracy of these methods for a large range of models and approximation parameters.

Modeling of thermal stresses in inorganic matrix composite plates based on the asymptotic theory

A mathematical model has been developed for calculating a temperature non-stationary field and thermal stresses in thin multilayer plates of fabric composite materials on an inorganic matrix based on an alum-chromophosphate binder. The model allows to take into account changes in the microstructure in the matrix and composite fibers during high- temperature heating, taking into account the physical and chemical transformations occurring in them, as well as to calculate thermal stresses in thin plates from the tissue inorganic composite taking into account the kinetics of changes in the characteristics of composites during heating. Methods of multiscale homogenization and finite element modeling of microstresses in periodic cells were used to calculate the characteristics of the inorganic composite, and the method of asymptotic averaging was used to calculate the stresses in the plates. Some results of numerical simulation of the change in stresses in the composite plate during heating are given

Rigidity of vector valued harmonic maps of linear growth

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the square of the Jacobian of such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic average, and its large-time heat evolution.