Homogeneous Problem Research Articles

Sturm-Liouville Problem with Mixed Boundary Conditions for a Differential Equation with a Fractional Derivative and Its Application in Viscoelasticity Models

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative.

Determination of effective properties of porous piezoelectric composite with partially randomly metalized pore boundaries using finite element method

In this study, we investigate the effective characteristics of a porous piezocomposite with randomly distributed pores of arbitrary size and with partly metalized pore boundaries. Using the ANSYS APDL finite element software, we created a novel random representative volume element (RVE), applied the finite element approach to solve boundary value problems of homogenization, and calculated equivalent properties using the Hill–Mandel principle. The results demonstrate that when large random RVE and an appropriate finite element mesh are used, the porous piezocomposite with randomly metalized porosity borders will have the same crystal symmetry as the piezoceramic matrix. Therefore, when there is a significant material contrast between the various phases, a large random RVE is preferred in the modeling of particulate-filled composites. Furthermore, when the areas of metalized surfaces increase, the performance of the investigated composite in transverse sensing and actuation applications improves.

A finite element modeling for anisotropic dielectric elastomer energy harvesters with inhomogeneous displacement fields

In this paper, we are to investigate the effects of imparting anisotropy to dielectric elastomers (DEs) to evaluate the promising capabilities of this phenomenon in energy harvesting (EH) enhancement. To this end, we employ the principle of virtual work to obtain the governing equations and the boundary conditions of dielectric elastomers at finite strains. As for the material law, it incorporates three parts, including an isotropic, an anisotropic, and an electric one with no modes of dissipation. Having to deal with inhomogeneous displacement fields necessitates the employment of finite element (FE) simulations. Moreover, a staggered algorithm has been put into use which is capable of reducing computational expense while producing the same results as that of a fully coupled solution. Further, F-bar method has been selected to alleviate the issue of volumetric locking. The aforementioned finite element implementation is conducted through a computer program. It is noteworthy that all the terms involved in FE modeling, particularly electroelastic moduli and the terms originated from the follower load assumption are fully derived. Three energy harvesting applications, including a cylindrical tube, a diaphragm, and an annular membrane which satisfy the axisymmetric assumptions, are investigated. In the interest of verifying the developed axisymmetric element, the finite element results have been compared with that of analytical ones for a one-dimensional homogenous actuation problem. Intended to reveal the efficacy of imparting anisotropy to DEs in EH, we fix all the other contributing factors in the energy harvesting cycle for each application. As a result, corresponding isotropic and anisotropic cases become possible whose only difference is in the existence or absence of anisotropic fibers. Eventually, through numerical simulations it is proved that imparting anisotropy to dielectric elastomers in most cases culminates in favor of energy harvesting. Particularly, the maximum amount of relative energy harvesting difference of corresponding isotropic and anisotropic cases for the cylindrical tube, the diaphragm, and the annular membrane is achieved to be 27.75%, 10.61%, and 19.76%, respectively, in favor of the anisotropic cases.

Mathematical aspects of quasicrystals

A two-dimensional model of a quasi-crystal is the Penrose tiling (1974), which is an aperiodic “disjoint” covering of the plane generated by two rhombi displaystyle R_{36^{circ }} and displaystyle R_{72^{circ }} with equal side lengths. It is crucial that the areas’ ratio is irrational φ=areaR72∘areaR36∘=1+52(goldenratio)(φ2-φ-1=0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\displaystyle \\varphi = \\frac{\ extrm{area} \\,R_{72^{\\circ }}}{\ extrm{area}\\, R_{36^{\\circ }}} = \\frac{1+\\sqrt{5}}{2}\\, (\\mathrm{golden\\, ratio})\\, (\\varphi ^2 - \\varphi - 1 = 0), \\end{aligned}$$\\end{document}which in turn reveals a local five-fold symmetry, forbidden for crystals. Recent advances on “Wang tiles”, that is square tiles that cover the plane but cannot do it in a periodic fashion, are due to Jeandel and Rao (An aperiodic set of 11 Wang tiles, Advances in Combinatorics, pp 1–37, 2021), giving a definitive answer to the problem raised by Hao Wang in 1961. Other recent applications to variational problems in Homogenization are also mentioned (Braides et al. in C R Acad Sci Paris 347(11–12):697–700, 2009).

Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations

Abstract This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.

MODELING OF NONLINEAR MASS FLOWS DURING WATER SOFTENING IN A MECHANICAL FILTER

A mathematical model of water softening in a mechanical filter based on the balance relations for the mass components of the thermodynamical system is constructed. The initial-boundary value problem of convective diffusion accompanied by a chemical reaction is formulated for one-dimensional case of spatial coordinates and reduced to the equivalent system of integro-differential equations. The solution of this system is constructed by the method of successive iterations by decomposing the functions of the mass concentrations of impurity in water solution and in the body skeleton into series around the solutions of the homogeneous linear initial-boundary value problem. Based on the expression for the impurity concentration in the aqueous solution the mass flows of impurity are obtained and investigated. The nonlinearity of steady-state regime, which is caused by the transfer of mass simultaneously by diffusion and convection mechanisms, is shown.

Boundary-volume Lippmann Schwinger formulation and fast iteration schemes for numerical homogenization of conductive composites. Cases of arbitrary contrasts and Kapitza interface

In this paper, we present a boundary-volume based Lippmann Schwinger integral equation for numerical homogenization problems of heterogeneous conductive materials with arbitrary contrast ratio and imperfect interface behavior. It is shown that the interior temperature gradient within each homogeneous phase is connected to the material dissimilarity quantities at all boundaries between different phases. The Kapitza interface model can also be included in the formulation. The basic FFT iteration schemes converge fast even for infinite contrast in the case of usual interface conditions. In the case of Kapitza’s interface, Conjugate Gradient schemes based on the new formulation converge fast and yield accurate results when compared with standard Finite Element Method.

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide

AbstractIn this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height H. The perforations are periodically placed along the ordinate axis at a distance between them, where ε is a parameter that converges toward zero. Another parameter η, the Floquet‐parameter, ranges in the interval . The boundary conditions are quasi‐periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters ε and η and strongly depend on H. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.

Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

AbstractWe investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

Chemotaxis driven formation of time periodic pattern and steady state pattern

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model.

Generalized initial-boundary problem for the wave equation with mixed derivative

We study an initial-boundary problem for a second-order inhomogeneous hyperbolic equation in a half-strip of the plane containing a mixed derivative with constant coefficients and zero or nonzero potential. This equation is the equation of transverse oscillations of a moving finite string. The case of zero initial velocity and fixed ends (Dirichlet conditions) is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on opposite sides of the origin. The classical solution of the initial-boundary problem is determined. In the case of zero potential, a uniqueness theorem for the classical solution is formulated and a formula for the solution is given in the form of a series consisting of contour integrals containing the initial data of the problem. Based on this formula, the concepts of a generalized initial-boundary value problem and a generalized solution are introduced. The main theorems on finite formulas for the generalized solution in the case of homogeneous and inhomogeneous problems are formulated. To prove these theorems, we apply an approach that uses the theory of divergent series in the sense of Euler, proposed by A.P. Khromov (axiomatic approach). Using this approach, on the basis of formulas for solutions in the form of a series, the formulated main theorems are proved. Further, as an application of the main theorems obtained, we prove a theorem on the existence and uniqueness of a generalized solution of the initial-boundary problem in the presence of a nonzero summable potential and give a formula for the solution in the form of an exponentially convergent series.

Adaptive Hierarchy-Branch Fusion for Online Knowledge Distillation

Online Knowledge Distillation (OKD) is designed to alleviate the dilemma that the high-capacity pre-trained teacher model is not available. However, the existing methods mostly focus on improving the ensemble prediction accuracy from multiple students (a.k.a. branches), which often overlook the homogenization problem that makes student model saturate quickly and hurts the performance. We assume that the intrinsic bottleneck of the homogenization problem comes from the identical branch architecture and coarse ensemble strategy. We propose a novel Adaptive Hierarchy-Branch Fusion framework for Online Knowledge Distillation, termed AHBF-OKD, which designs hierarchical branches and adaptive hierarchy-branch fusion module to boost the model diversity and aggregate complementary knowledge. Specifically, we first introduce hierarchical branch architectures to construct diverse peers by increasing the depth of branches monotonously on the basis of target branch. To effectively transfer knowledge from the most complex branch to the simplest target branch, we propose an adaptive hierarchy-branch fusion module to create hierarchical teacher assistants recursively, which regards the target branch as the smallest teacher assistant. During the training, the teacher assistant from the previous hierarchy is explicitly distilled by the teacher assistant and the branch from the current hierarchy. Thus, the important scores to different branches are effectively and adaptively allocated to reduce the branch homogenization. Extensive experiments demonstrate the effectiveness of AHBF-OKD on different datasets, including CIFAR-10/100 and ImageNet 2012. For example, on ImageNet 2012, the distilled ResNet-18 achieves Top-1 error of 29.28\%, which significantly outperforms the state-of-the-art methods. The source code is available at https://github.com/linruigong965/AHBF.

Heterogeneous-Branch Collaborative Learning for Dialogue Generation

With the development of deep learning, advanced dialogue generation methods usually require a greater amount of computational resources. One promising approach to obtaining a high-performance and lightweight model is knowledge distillation, which relies heavily on the pre-trained powerful teacher. Collaborative learning, also known as online knowledge distillation, is an effective way to conduct one-stage group distillation in the absence of a well-trained large teacher model. However, previous work has a severe branch homogeneity problem due to the same training objective and the independent identical training sets. To alleviate this problem, we consider the dialogue attributes in the training of network branches. Each branch learns the attribute-related features based on the selected subset. Furthermore, we propose a dual group-based knowledge distillation method, consisting of positive distillation and negative distillation, to further diversify the features of different branches in a steadily and interpretable way. The proposed approach significantly improves branch heterogeneity and outperforms state-of-the-art collaborative learning methods on two widely used open-domain dialogue datasets.

Initial conditions problem in cosmological inflation revisited

We present first results from a novel numerical relativity code based on a tetrad formulation of the Einstein-scalar field equations combined with recently introduced gauge/frame invariant diagnostics. The results provide support for the argument that inflation does not solve the homogeneity and isotropy problem beginning from generic initial conditions following a big bang.

Wasserstein convergence rates in the invariance principle for deterministic dynamical systems

AbstractIn this paper, we consider the convergence rate with respect to Wasserstein distance in the invariance principle for deterministic non-uniformly hyperbolic systems. Our results apply to uniformly hyperbolic systems and large classes of non-uniformly hyperbolic systems including intermittent maps, Viana maps, unimodal maps and others. Furthermore, as a non-trivial application to the homogenization problem, we investigate the Wasserstein convergence rate of a fast–slow discrete deterministic system to a stochastic differential equation.

Using of the chemography method for assessing the homogenizati on of graphene nanotubes in the aqueous environment

The problem of effective homogenization of graphene nanotubes in an aqueous environment is considered. Usually, the high activity of nanotubes leads to the formation of various conglomerates in the solution, and with further application of the aqueous solution, such conglomerates can be transferred to the composite materials being created. Currently, there are no effective fast control tools, and therefore in engineering practice, various fairly complex methods that require specific equipment are often used. Chemography as a method of recording oxidative reactions with ultra-low concentrations of active substances can be successfully used as a means of quick and effective control of the homogeneity of graphene solutions (mainly water-based). Recommendations for the application of the method are given, models are proposed for determining the dispersion of the expected concentration on the controlled surface. A conclusion is made regarding the expediency of monitoring the state and activity of graphene nanotubes.

A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

We consider model semilinear elliptic equations of the type{−div(A(x)∇u)=σu−λ,u>0inΩ,u∈H01(Ω), where Ω is a bounded domain in RN, N≥1, A∈L∞(Ω)N×N is a coercive matrix, 0<λ≤1 and σ is a nonnegative function in Lloc1(Ω), or more generally, nonnegative Radon measure on Ω. We discuss H01(Ω)-stability of u under a minimal assumption on σ and apply the result to homogenization problems.

The acceleration of improved Tone's method in advanced reactor lattice neutron spectrum calculation

This paper focuses on the acceleration of solving fixed-source equations in improved Tone's method. Since each equation is independent of the other with respect to the energy variable, the parallel calculation based on energy group domain decomposition was used in the advanced reactor lattice neutron spectrum calculation code TULIP. In order to balance the load of each CPU, the equally-divided strategy and a weighting-based strategy, which were used for the division of energy groups, were proposed and discussed. In the weighting-based strategy, the cross sections were used to calculate the weighting so that the iteration behavior could be caught. Therefore, the CPU load was balanced, and the parallel efficiency was improved. Moreover, the selection of the initial value was also evaluated. To verify the improvement, several tests based on fast and thermal neutron spectrum were calculated. According to the analysis of calculation time and parallel efficiency, using the initial value that was calculated with the homogeneous problem could save more than half of the calculation time. Meanwhile, the weighting-based strategy for parallel calculation could balance the CPU load so that the parallel efficiency was improved about 10%–80%, which is based on different tests.

Homogenization of non-symmetric jump processes

AbstractWe study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.