Quasi-Periodic Attractors in Celestial Mechanics

We prove that KAM tori smoothly bifurcate into quasi-periodic attractors in dissipative mechanical models, provided external parameters are tuned with the frequency of the motion. An application to the dissipative spin–orbit model of celestial mechanics (which actually motivated the analysis in this paper) is presented.     

Models in the Engineering Mechanics of Polymer-Matrix Composite Systems

This paper discusses the complete system of structural, thermochemical, and mechanical-mathematical models that describe all the phenomena accompanying the formation of polymer-matrix composite materials (PCMs) and structures made of them. The issues of optimizing design engineering and modeling the postprocess behavior of PCM structures are addressed     

Global Attractors of Evolutionary Systems

An abstract framework for studying the asymptotic behavior of a dissipative evolutionary system with respect to weak and strong topologies was introduced in Cheskidov and Foias (J Differ Equ 231:714–754, 2006) primarily to study the long-time behavior of the 3D Navier-Stokes equations (NSE) for which the existence of a semigroup of solution operators is not known. Each evolutionary system possesses a global attractor in the weak topology, but does not necessarily in the strong topology. In this paper we study the structure of a global attractor for an abstract evolutionary system, focusing on omega-limits and attracting, invariant, and quasi-invariant sets. We obtain weak and strong uniform tracking properties of omega-limits and global attractors. In addition, we discuss a trajectory attractor for an evolutionary system and derive a condition under which the convergence to the trajectory attractor is strong.     

On Existence of Attractors in Dissipative Three-Dimensional Systems

International Applied Mechanics - The theorems on the birth of attractors from homoclinic loops are proved. Applications of the theorems are considered.     

Attractors of Reaction Diffusion Systems on Infinite Lattices

In this paper, we study global attractors for implicit discretizations of a semilinear parabolic system on the line. It is shown that under usual dissipativity conditions there exists a global (Z _u ,Z _ρ )-attractor $A$ in the sense of Babin-Vishik and Mielke-Schneider. Here Z _ρ is a weighted Sobolev space of infinite sequences with a weight that decays at infinity, while the space Z _u carries a locally uniform norm obtained by taking the supremum over all Z _ρ norms of translates. We show that the absorbing set containing $A$ can be taken uniformly bounded (in the norm of Z _u ) for small time and space steps of the discretization. We establish the following upper semicontinuity property of the attractor $A$ for a scalar equation: if $A$ _N is the global attractor for a discretization of the same parabolic equation on the finite segment [?N,N] with Dirichlet boundary conditions, then the attractors $A$ _N (properly embedded into the space Z _u ) tend to $A$ as N→∞ with respect to the Hausdorff semidistance generated by the norm in Z _ρ . We describe a possibility of “embedding” certain invariant sets of some planar dynamical systems into the global attractor $A$ . Finally, we give an example in which the global attractor $A$ is infinite-dimensional.     

Design Models in Linearized Solid Mechanics

The paper is devoted to design models in linearized solid mechanics     

Exponential Attractors for the 3D Fractional-order Bardina Turbulence Model With Memory and Horizontal Filtering

Journal of Dynamics and Differential Equations - We consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term...     

3D Dynamic Modelling of Spatial Geared Systems

In this paper, a nonlinearly dynamic model for a spatial geared systemwith intersecting-axes, which consists of bevel gears, bearings, andshafts, is proposed based on a specific finite element theory. A newtype of spatial gear element which is consistent with the specific finiteelement theory is developed and completely describes all thedeformations that exist for the spatial motions of a pair of bevel gears,the time-variant meshing stiffness, and various types of gear errors inmanufacturing and assembly. The 3D motions of the spatial geared systemin axial, lateral, and torsional directions are coupled in the model. Theproposed approach has been coded into a software system and a dynamicanalysis for the spatial geared system is carried out. The nonlinearinfluences of axial, lateral and torsional stiffnesses of the shaft onthe vibration of the spatial geared system are especially investigated.The lateral stiffness changes the resonance peak frequencies of thespatial geared system and the torsional stiffness greatly affects thesize of the dynamic load and vibration amplitude.     

$${\mathcal{A}}$$-Stability of Global Attractors of Competition Diffusion Systems

We study the structural stability of global attractors (A{\mathcal{A}}-stability) for two-species competition diffusion systems with Morse-Smale structure. Such systems generate semiflows on positive cones of certain infinite-dimensional Banach spaces (e.g., fractional order spaces). Our main result states that a two species competition diffusion system with Morse-Smale structure is structurally A{\mathcal{A}}-stable, which implies that the set of nonlinearities for which the system possesses Morse-Smale structure is open in an appropriate space under the topology of C ²-convergence on compacta. Moreover, we provide a sufficient condition under which a system has Morse-Smale structure and provide some examples which satisfy the sufficient condition.     

Finite Element Modeling of 3D Fluid Dynamics in Crystal Growth Systems

We describe massively parallel finite element compulations of fluid dynamics for several crystal growth systems. Examples are presented of how large-scale numerical simulations have been used to gain insights to the workings of several processes, specifically the melt growth of oxide crystals by the Czochralski process and the solution growth of nonlinear optical crystals. These systems are characterized by nonlinear interactions between field and interfacial phenomena-the transport of momentum, heat, and mass coupled with solidification kinetics. Modern finite element methods show great potential to provide the understanding needed to optimize these processes.     

Mathematical Models and Methods of the Mechanics of Stochastic Composites

The basic approaches used in mathematical models and general methods for solution of the equations of the mechanics of stochastic composites are generalized. They can be reduced to the stochastic equations of the theory of elasticity of a structurally inhomogeneous medium, to the equations of the theory of effective elastic moduli, to the equations of the theory of elastic mixtures, or to more general equations of the fourth order. The solution of the stochastic equations of the elastic theory for an arbitrary domain involves substantial mathematical difficulties and may be implemented only rather approximately. The construction of the equations of the theory of effective moduli is associated with the problem on the effective moduli of a stochastically inhomogeneous medium, which can be solved by the perturbation method, by the method of moments, or by the method of conditional moments. The latter method is most appropriate. It permits one to determine the effective moduli in a two-point approximation and nonlinear deformation properties. In the structure of equations, the theory of elastic mixtures is more general than the theory of effective moduli; however, since the state equations have not been strictly substantiated and the constants have not been correctly determined, theoretically or experimentally, this theory cannot be used for systematic designing composite structures. A new model of the nonuniform deformation of composites is more promising. It is constructed by performing strict mathematical transformations and averaging the output stochastic equations, all the constants being determined. In the zero approximation, the equations of the theory of effective moduli follow from this model, and, in the first approximation, fourth-order equations, which are more general than those of the theory of mixtures, follow from it     

Spatially Localized Models of Extended Systems

We investigate the construction of low-dimensional spatiallylocalized models of extended systems. Specifically, theKuramoto–Sivashinsky (KS) equation on large one-dimensional domainsdisplays spatiotemporally complex dynamics that are remarkablywell-localized in both real and Fourier space, as demonstrated by a(spline) wavelet representation. We show how wavelet projectionsmay be used to construct various localized, relativelylow-dimensional models of KS spatiotemporal chaos. There ispersuasive evidence that short, periodized systems, internally forcedat their largest scales, form minimal models for chaotic dynamics inarbitrarily large domains. Such models assist in the understandingof extended systems.     

基底上薄膜结构的非线性屈曲力学进展

基底上薄膜结构中的过大残余压应力常常通过屈曲不稳定性诱发薄膜结构和功能的失效.屈曲不稳定性、演化与斑图形成是近年来非线性力学研究的热点.此类屈曲不稳定性受薄膜基底的力学性质以及界面相互作用影响,进而呈现出复杂的屈曲模式如褶皱、翘曲和折痕等.论文简要综述褶皱、翘曲和折痕等屈曲模式的形成机制、影响因素和后屈曲形貌相关方面的进展.褶皱部分,重点介绍了褶皱的形成、多级褶皱结构、局域化的褶皱、各向异性褶皱和曲面上的褶皱.翘曲部分,介绍了翘曲结构包括一维翘曲结构、“电话线”屈曲泡、网络状屈曲泡等的形成与生长过程,并讨论了曲面几何、界面滑移、开裂等因素的影响.折痕及其它复杂屈曲模式部分,介绍了折痕、叠痕及隆起失稳的形成机制与临界条件.     

Attraction in Nonmonotone Planar Systems and Real-Life Models

Journal of Dynamics and Differential Equations - Let $$h:Vsubset {mathbb {R}}^{2}longrightarrow {mathbb {R}}^{2}$$ be an embedding. The aim of this paper is to analyze the dynamical behavior of...     

Contributions of 3D Printed Fracture Networks to Development of Flow and Transport Models

Transport in Porous Media - Conventional experiments using natural rock samples have trouble in observing rock structures and controlling fracture properties. Taking advantage of 3D printing...     

Discrete Models of a Class of Isolation Systems

In this paper, a class of isolation systems with rigid limiters has been considered. For this class of systems, some general discrete-time models described by means of some impact Poincaré maps have been established. Two examples: a simple isolation system of one-stage and a real isolation system of two-stages have been investigated. The calculated results show that those models can reveal complex nonlinear behaviors. And even a small random perturbation may change the dynamical character of the system.     

Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media*

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927l... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 – the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks.     

Micromechanisms and Mechanics of Damage and Fracture in Thin Film/Substrate Systems

The purpose of this paper is to review the mechanisms and available theoretical methods for modeling the strength and failure of thin film/substrate systems