Modelling of cracks on different scales in ferroelectric materials

In this work we study contributions to the effective fracture toughness of ferroelectric materials arising from effects on macroscopic and mesoscopic scales of the system. On the macroscopic scale, the crack in a ferroelectric material is modeled taking into account an extended theory of stresses at interfaces in dielectric solids [1-3]. We predict several new effects, such as the “poling effect”, “collinear effect” and the coupling of a Mode-II shear loading and the Mode-I SIF. Further, on the mesoscopic scale, we study the influence of polarization switching limited to the fracture process zone (small scale switching) on the fracture toughness. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ISOMETRIC AND ALLOMETRIC RELATIONSHIPS BETWEEN LARGE AND SMALL‐SCALE TREE SPACING STUDIES

Abstract Basic differences in the relationship between diameter and height have been observed in small and large trees. Small trees (less than 5 m) have little danger of buckling under their own weight, and diameter is proportional to height. Large trees (greater than 5 m) are at risk of buckling under their own weight and are subject to damage from ice and wind. For large trees, diameter cubed is proportional to height squared. This relationship is suggested by the physics of limits to height of cylinders before they buckle under their own weight and has been shown to hold for large trees. Data from large‐scale spacing studies are compared with data from one‐sixteenth scale small spacing studies to determine the validity of this theory. The impact of scaled spacing on scaled diameters at equivalent scaled heights is examined. Results suggest that trees grown at small scales can be “scaled up” to reflect isometric and allometric relationships of trees grown at large scales.     

An averaging theorem for two-point boundary value problems with applications to optimal control

An asymptotic result is obtained for a two-point boundary value problem for a vector system of nonlinear ordinary differential equations involving “fast” and “slow” inputs. The asymptotically limiting system is obtained by an averaging procedure. Using this result, an approximate analysis of the original system may be carried out by considering two lower-order systems each involving only one time scale. It is shown that some optimal control problems for systems with multiple time scales may be analyzed by this method.     

Using self‐dissimilarity to quantify complexity

For many systems characterized as “complex” the patterns exhibited on different scales differ markedly from one another. For example, the biomass distribution in a human body “looks very different” depending on the scale at which one examines it. Conversely, the patterns at different scales in “simple” systems (e.g., gases, mountains, crystals) vary little from one scale to another. Accordingly, the degrees of self‐dissimilarity between the patterns of a system at various scales constitute a complexity “signature” of that system. Here we present a novel quantification of self‐dissimilarity. This signature can, if desired, incorporate a novel information‐theoretic measure of the distance between probability distributions that we derive here. Whatever distance measure is chosen, our quantification of self‐dissimilarity can be measured for many kinds of real‐world data. This allows comparisons of the complexity signatures of wholly different kinds of systems (e.g., systems involving information density in a digital computer vs. species densities in a rain forest vs. capital density in an economy, etc.). Moreover, in contrast to many other suggested complexity measures, evaluating the self‐dissimilarity of a system does not require one to already have a model of the system. These facts may allow self‐dissimilarity signatures to be used as the underlying observational variables of an eventual overarching theory relating all complex systems. To illustrate self‐dissimilarity, we present several numerical experiments. In particular, we show that the underlying structure of the logistic map is picked out by the self‐dissimilarity signature of time series produced by that map. © 2007 Wiley Periodicals, Inc. Complexity 12: 77–85, 2007     

Diffusions in perforated domains with mixed boundary conditions

This article investigates averaging effects associated with a fine-grained boundary. A simple diffusion occurs everywhere except at a large number of small “holes” in the medium, at which an appropriately scaled mixed boundary condition is applied. The scaling considered is fitting for boundary conditions resulting from thin layer approximations in which the layer thickness scales with the diameter of the hole. Probabilistic methods associated with the Feynman-Kac formula are applied to find the limiting behavior, and the perforated domain and complex boundary condition are replaced by a straightforward attenuating term.     

Pattern recognition at different scales: A statistical perspective

In this paper we borrow concepts from Information Theory and Statistical Mechanics to perform a pattern recognition procedure on a set of X-ray hazelnut images. We identify two relevant statistical scales, whose ratio affects the performance of a machine learning algorithm based on statistical observables, and discuss the dependence of such scales on the image resolution. Finally, by averaging the performance of a Support Vector Machines algorithm over a set of training samples, we numerically verify the predicted onset of an “optimal” scale of resolution, at which the pattern recognition is favoured.     

On time‐dependent behaviour of controlled turbulent flow with separation and reattachment

Results from numerical simulations of separating and reattaching turbulent boundary layer flow over a backwardfacing step are analysed with respect to the time‐dependent flow behaviour. Beside the well‐known roll‐up of the separated shear‐layer (“Kelvin‐Helmholtz instability”) and the ejection of subsequently formed large‐scale structures (denoted as “shedding”), another unsteady phenomenon which is commonly called “flapping” can be observed. Additionally, the effects of open‐loop passive and active flow control methods are investigated.     

A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities

We consider a nonlinear parametric Dirichlet problem driven by the anisotropic p-Laplacian with the combined effects of “concave” and “convex” terms. The “superlinear” nonlinearity need not satisfy the Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory and the Ekeland variational principle, we show that for small values of the parameter, the problem has at least two nontrivial smooth positive solutions.     

Choice of the best time scale for preventive maintenance in heterogeneous environments

A principal feature of the model considered in this paper is the presence of several time scales for lifetime measurements when similar objects operate in heterogeneous conditions. A typical example are lifetimes of an aircraft module which can be measured in the total time in air and in the number of flights. A family of lifetimes can be generated by considering a linear combination of the above two principal time scales. We consider a problem of finding an optimal maintenance period for an equipment operating during a single maintenance/replacement cycle, in heterogeneous environmental conditions. Our main concern is finding the time scale which provides the maximal value of the return (cost) functional. We consider two “principal” time scales (the operation time and the total number of shocks) and show numerically that the optimal linear combination of these two scales has also the minimal coefficient of variation (c.v.) of system lifetime. We develop some general theory to connect the optimality in terms of the return functional with the optimality in terms of c.v.     

市场接纳不确定下短周期新产品产能投资研究 ——考虑竞争的投资时机与规模决策

新产品的市场接纳具有很大不确定性,传统投资理论并不适用于新产品投资。针对新产品投资中的产能投资,研究了垄断企业和有成本差异的竞争企业制定短周期新产品的产能投资时机与规模策略。给定企业“早”和“晚”两个投资时机可供选择,定义“早”投资时,企业只知道新产品市场规模的期望和方差;“晚”投资时,企业知道新产品真实的市场规模。垄断企业进入市场之前无法进行销售信息的收集,只会选择“早”投资或者不投资,给出其选择“早”投资的条件、最优产能投资规模及最大期望利润。有成本差异企业竞争的情形可以分为四种,分别给出四种情形下的最优产能投资规模及最大期望利润,并通过比较各情形下两企业的最大期望利润给出最优的产能投资时机策略。     

From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.     

Identification of Microstructural Components of Bitumen by Means of Atomic Force Microscopy (AFM)

Accounting for the large variation of asphalt mixes, resulting from variations of constituents and composition, and from the allowance of additives, a multiscale model for asphalt is currently developed at the Christian Doppler Laboratory for “Performance‐based optimization of flexible road pavements”. The multiscale concept allows to relate macroscopic material properties of asphalt to phenomena and material properties of finer scales of observation. Starting with the characterization of the finest scale, i.e., the bitumen‐scale, Atomic Force Microscopy (AFM) is employed. Depending on the mode of measurement (tapping versus pulsed‐force mode), the AFM provides insight into the surface topography or stiffness and adhesion properties of bitumen. The obtained results will serve as input for upscaling in the context of the multiscale model in order to obtain the homogenized material behavior of bitumen at the next‐higher scale, i.e., the mastic‐scale. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

We provide general conditions for normalized, time-scaled stochastic integrals of independently scattered, Lévy random measures to converge to a limit. These integrals appear in many applied problems, for example, in connection to models for Internet traffic, where both large scale and small scale asymptotics are considered. Our result is a handy tool for checking such convergence. Numerous examples are provided as illustration. Somewhat surprisingly, there are examples where rescaling towards large times scales yields a Gaussian limit and where rescaling towards small time scales yields an infinite variance stable limit, and there are examples where the opposite occurs: a Gaussian limit appears when one converges towards small time scales and an infinite variance stable limit occurs when one converges towards large time scales.      

Axial Compression of a Thin Elastic Cylinder: Bounds on the Minimum Energy Scaling Law

We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression. We focus exclusively on the setting where the radius of the core is greater than or equal to the natural radius of the cylinder. We consider two cases: the “large mandrel” case, where the radius of the core exceeds that of the cylinder, and the “neutral mandrel” case, where the radii of the core and cylinder are the same. In the large mandrel case, our upper and lower bounds match in their scaling with respect to thickness, compression, and the magnitude of pre‐strain induced by the core. We construct three types of axisymmetric wrinkling patterns whose energy scales as the minimum in different parameter regimes, corresponding to the presence of many wrinkles, few wrinkles, or no wrinkles at all. In the neutral mandrel case, our upper and lower bounds match in a certain regime in which the compression is small as compared to the thickness; in this regime, the minimum energy scales as that of the unbuckled configuration. We achieve these results for both the von Kármán–Donnell model and a geometrically nonlinear model of elasticity. © 2017 Wiley Periodicals, Inc.     

Solving Diffusion Problems on Rough Surfaces with a Hierarchical Multiscale FEM

Diffusion on rough surfaces is a basic problem for many applications in engineering and the sciences. Solving these problems with a standard finite element method is often difficult or even impossible, due to the computational work and the amount of memory needed to triangulate the whole surface with a mesh which resolves its oscillations. We discuss in this paper a hierarchical Finite Element Method of “heterogeneous multiscale” type, which only needs to resolve the surface's fine scale on small sampling domains within a macro triangulation of the underlying smooth surface. This method converges, for periodic surface roughness and sufficiently small amplitude, at a robust (i.e. scale independent) rate, to the homogenized solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anomalous diffusion in fast cellular flows at intermediate time scales

It is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a Brownian motion. This paper studies the behaviour on “intermediate” time scales before diffusion sets in. Various heuristics suggest that an anomalous diffusive behaviour should be observed. We prove that the variance on intermediate time scales grows like \(O(\sqrt{t})\). Hence, on these time scales the effective behaviour can not be purely diffusive, and is consistent with an anomalous diffusive behaviour.     

Small and Large Time Scale Analysis of a Network Traffic Model

Empirical studies of the internet and WAN traffic data have observed multifractal behavior at time scales below a few hundred milliseconds. There have been some attempts to model this phenomenon, but there is no model to connect the small time scale behavior with behavior observed at large time scales of bigger than a few hundred milliseconds. There have been separate analyses of models for high speed data transmissions, which show that appropriate approximations to large time scale behavior of cumulative traffic are either fractional Brownian motion or stable Lévy motion, depending on the input rates assumed. This paper tries to bridge this gap and develops and analyzes a model offering an explanation of both the small and large time scale behavior of a network traffic model based on the infinite source Poisson model. Previous studies of this model have usually assumed that transmission rates are constant and deterministic. We consider a nonconstant, multifractal, random transmission rate at the user level which results in cumulative traffic exhibiting multifractal behavior on small time scales and self-similar behavior on large time scales.     

Mean-field games with a major player

We introduce and study mathematically a new class of mean-field-game systems of equations. This class of equations allows us to model situations involving one major player (or agent) and a “large” group of “small” players.     

Life-span of classical solutions to hyperbolic geometry flow equation in several space dimensions

In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with “small” initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with “small” initial data.