Non-asymptotic Model Research Articles

Mathematical modelling of stability problems for thin transversally graded cylindrical shells

The objects of consideration are thin linearly elastic Kirchhoff–Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new mathematical averaged non-asymptotic model for the analysis of selected stability problems for such shells. As a tool of modelling we shall apply the tolerance averaging technique. The second aim is to derive and discuss a new mathematical averaged asymptotic model. This model will be formulated using the consistent asymptotic modelling procedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love second-order theory of thin elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in circumferential direction, whereas equations of the proposed models have continuous and slowly-varying coefficients. Moreover, some of coefficients of the tolerance model equations depend on a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental cell-independent, but also the new additional cell-dependent critical forces can be derived and analysed.

Non-asymptotic model selection for models of network data with parameter vectors of increasing dimension

Model selection for network data is an open area of research. Using the β-model as a convenient starting point, we propose a simple and non-asymptotic approach to model selection of β-models with and without constraints. Simulations indicate that the proposed model selection approach selects the data-generating model with high probability, in contrast to classical and extended Bayesian Information Criteria. We conclude with an application to the Enron email network, which has 181,831 connections among 36,692 employees.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin uniperiodic cylindrical shells

AbstractThe objects of consideration are thin linearly thermoelastic Kirchhoff–Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selecteddynamic thermoelasticity problemsfor the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a newprocedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selected dynamic thermoelasticity problems for the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a new procedure. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel-Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

Dynamic problems of thin linearly elastic Kirchhoff–Love-type circular cylindrical shells having geometrical, elastic and inertial properties densely and periodically varying in circumferential direction (uniperiodic shells) are studied. In order to take into account the effect of a cell size on the global dynamic behaviour of such shells (the length-scale effect), a new mathematical averaged non-asymptotic model is formulated. This so-called the general tolerance model is derived by applying a certain extended version of the well-known tolerance modelling technique. Governing equations of this averaged model have constant coefficients depending also on a microstructure size, contrary to the starting exact shell equations with periodic, non-continuous and highly oscillating coefficients (the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells). The effect of a cell size on the transversal free vibrations of an uniperiodic shell strip is studied. It will be shown that within this general tolerance model not only fundamental cell-independent, but also the new additional cell-dependent free vibration frequencies can be derived and analysed. The obtained results will be compared with the corresponding results derived from the knownnon-asymptotic standard tolerance model and from the asymptotic one.

The tales that the distribution tails of non-Gaussian autocorrelated processes tell: efficient methods for the estimation of the k-length block-maxima distribution

ABSTRACT The focal point of this work is the estimation of the distribution of maxima without the use of classic extreme value theory and asymptotic properties, which may not be ideal for hydrological processes. The problem is revisited from the perspective of non-asymptotic conditions, and regards the so-called exact distribution of block-maxima of finite-sized k-length blocks. First, we review existing non-asymptotic approaches/models, and introduce an alternative and fast model. Next, through simulations and comparisons (using asymptotic and non-asymptotic models), involving intermittent processes (e.g. rainfall), we highlight the capability of non-asymptotic approaches to model the distribution of maxima with reduced uncertainty and variability. Finally, we discuss an alternative use of such models that concerns the theoretical estimation of the multi-scale probability of obtaining a zero value. This is a useful finding when the scope is the multi-scale modelling of intermittent hydrological processes (e.g. intensity–duration–frequency models). The work also provides step-by-step recipes and an R package.

Non-asymptotic modelling of dynamics and stability for visco-elastic periodic beams on a periodic damping foundation

In this paper mathematical modelling of vibrations and stability problems for slender visco-elastic periodic beams is considered. In order to take into account the effect of microstructure a certain non-asymptotic approach is applied, called the tolerance modelling method. This technique allows to replace the equation with non-continuous, highly oscillating, periodic coefficients by a system of differential equations with constant coefficients. Moreover, the derived equations describe the effect of microstructure on the overall behaviour of the beams under consideration. In the framework of the tolerance modelling governing equations of two different tolerance models can be obtained – the standard and the general, under weakened assumptions. To evaluate the proposed general tolerance model obtained results are compared with results calculated within the known asymptotic homogenization, i.e. by the asymptotic model. In this note mathematical models describing investigated beams are only presented and illustrated by a simple example.

Length-scale effect in stability problems for thin biperiodic cylindrical shells: extended tolerance modelling

Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells of periodically micro-inhomogeneous structure in circumferential and axial directions (biperiodic shells) are investigated. The aim of this contribution is to formulate and discuss a new averaged nonasymptotic model for the analysis of selected stability problems for these shells. This, so-called, general nonasymptotic tolerance model is derived by applying a certain extended version of the known tolerance modelling procedure. Contrary to the starting exact shell equations with highly oscillating, noncontinuous and periodic coefficients, governing equations of the tolerance model have constant coefficients depending also on a cell size. Hence, the model makes it possible to investigate the effect of a microstructure size on the global shell stability (the length-scale effect).

Relationships between Pacific salmon and aquatic and terrestrial ecosystems: implications for ecosystem-based management.

Pacific salmon influence temperate terrestrial and freshwater ecosystems through the dispersal of marine‐derived nutrients and ecosystem engineering of stream beds when spawning. They also support large fisheries, particularly along the west coast of North America. We provide a comprehensive synthesis of relationships between the densities of Pacific salmon and terrestrial and aquatic ecosystems, summarize the direction, shape, and magnitude of these relationships, and identify possible ecosystem‐based management indicators and benchmarks. We found 31 studies that provided 172 relationships between salmon density (or salmon abundance) and species abundance, species diversity, food provisioning, individual growth, concentration of marine‐derived isotopes, nutrient enhancement, phenology, and several other ecological responses. The most common published relationship was between salmon density and marine‐derived isotopes (40%), whereas very few relationships quantified ecosystem‐level responses (5%). Only 13% of all relationships tended to reach an asymptote (i.e., a saturating response) as salmon densities increased. The number of salmon killed by bears and the change in biomass of different stream invertebrate taxa between spawning and nonspawning seasons were relationships that usually reached saturation. Approximately 46% of all relationships were best described with linear or curved nonasymptotic models, indicating a lack of saturation. In contrast, 41% of data sets showed no relationship with salmon density or abundance, including many of the relationships with stream invertebrate and biofilm biomass density, marine‐derived isotope concentrations, or vegetation density. Bears required the highest densities of salmon to reach their maximum observed food consumption (i.e., 9.2 kg/m2 to reach the 90% threshold of the relationship’s asymptote), followed by freshwater fish abundance (90% threshold = 7.3 kg/m2 of salmon). Although the effects of salmon density on ecosystems are highly varied, it appears that several of these relationships, such as bear food consumption, could be used to develop indicators and benchmarks for ecosystem‐based fisheries management.

All in order: Distribution of serially correlated order statistics with applications to hydrological extremes

Classic extreme value theory provides asymptotic distributions of block maxima (BM) Y or peaks over threshold (POT). However, as BM and POT are relatively small subsets of the values assumed by the parent process Z, alternative approaches have been proposed to make inferences on extremes by using intermediate values and non-asymptotic models. In this study, we investigate the finite sample theory of extremes based on order statistics, and present a set of results enabling the analysis of the properties of non-asymptotic distributions of BM in finite-size blocks of data under the assumption that the parent process Z has stationary temporal dependence. In particular, we suggest the beta-binomial distribution (FβB) as a suitable approximation of the marginal distribution of order statistics and BM under dependence, thus generalizing the theoretical results available under independence. We demonstrate the usefulness of the FβB distribution in three conceptual applications of hydrological interest. Firstly, we review the so-called Complete Time-series Analysis (CTA) framework, showing that the differences between FY and FZ are due to the inherent theoretical nature of the processes Y and Z and the different probabilistic failure scenarios described by FY and FZ. Secondly, we provide a theoretical rule to select the number of the largest maxima (LM) and over-threshold exceedances (OT) required to approximate the desired portion of the upper tail of FZ with specified accuracy. Finally, we discuss how the FβB distribution offers an interpretation and generalization of the so-called metastatistical extreme value (MEV) framework and its simplified version (SMEV), avoiding the use of high-dimensional joint distributions and preliminary data thresholding and declustering. All methodological results are validated by Monte Carlo simulations involving widely used stochastic processes with persistence. Real-world stream flow data are also analyzed as proof of concept.

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

On the cell-dependent vibrations and wave propagation in uniperiodic cylindrical shells

The objects of consideration are thin linearly elastic Kirchhoff–Love-type circular cylindrical shells having a periodically micro-heterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to study certain problems of micro-vibrations and of wave propagation related to micro-fluctuations of displacement field caused by a periodic structure of the shells. These micro-dynamic problems will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling includes both the asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves under special conditions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the combined model have constant coefficients depending also on a cell size. Hence, this model takes into account the effect of a microstructure size on the dynamic behaviour of the shells (the length-scale effect). It will be shown that the micro-periodic heterogeneity of the shells leads to cell-depending micro-vibrations and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic structures.

Micro-Vibrations and Wave Propagation in Biperiodic Cylindrical Shells

Abstract The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to study a certain long wave propagation problem related to micro-fluctuations of displacement field caused by a periodic structure of the shells. This micro-dynamic problem will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling applied here includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Both these procedures are conjugated with themselves under special conditions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. It will be shown that the micro-periodic heterogeneity of the shells leads to exponential micro-vibrations and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic structures.

Free and forced large amplitude vibrations of periodically inhomogeneous slender beams

Considered are free and forced transverse vibrations of slender periodic beams of finite length. It is assumed that the vibration amplitude is of the order of cross-section dimensions, still much smaller than the beam length. An averaged non-asymptotic model is proposed as a tool in analysis. The description is based on the tolerance approach to averaging of differential operators, using the concept of weakly slowly-varying function. The resulting differential equations with constant coefficients involve the effect of periodicity cell length. The model is verified by comparison of linear frequencies and mode shapes with Finite Element Method results, and then applied in analysis of free and forced vibrations of beam with variable cross-section. The method employed in obtaining the solution involves Galerkin orthogonalization and Runge–Kutta (RKF45) method. The results of nonlinear vibrations analysis are presented by backbone and amplitude-frequency response curves, time series, Poincare sections and bifurcation diagrams.

Tolerance modelling of free vibrations of medium thickness functionally graded plates

In many works different averaging methods based on the known Hencky-Bolle-type plate assumptions are applied to describe medium thickness functionally graded plates with a microstructure. The effect of the microstructure size plays a role in various thermomechanical problems of similar plates, cf. Brillouin [1], Baron [2], Jędrysiak [3], Cheng et al. [4]. Unfortunately, governing equations of most of averaged models do not include the effect of the microstructure size on the overall behaviour of these plates. This lack of them is corrected by using the tolerance modelling approach, cf. Woźniak and Wierzbicki [5], Woźniak et al. [6,7]. This method leads to model differential equations with slowly-varying, smooth coefficients. Hence, the new non-asymptotic model of microstructured medium thickness functionally graded plates is proposed. An example formulas of free vibration frequencies will be obtained – the fundamental lower frequency of macrovibrations and the first higher frequency of microvibrations related to the plate microstructure. A certain analysis of them will be shown for various distribution functions of material properties of functionally graded plates.

Determining Optimal Rates for Communication for Omniscience

This paper considers the communication for omniscience (CO) problem: A set of users observe a discrete memoryless multiple source and want to recover the entire multiple source via noise-free broadcast communications. We study the problem of how to determine an optimal rate vector that attains omniscience with the minimum sum-rate, the total number of communications. The results cover both asymptotic and non-asymptotic models where the transmission rates are real and integral, respectively. We propose a modified decomposition algorithm (MDA) and a sum-rate increment algorithm (SIA) for the asymptotic and non-asymptotic models, respectively, both of which determine the value of the minimum sum-rate and a corresponding optimal rate vector in polynomial time. For the coordinate saturation capacity (CoordSatCap) algorithm, a nesting algorithm in MDA and SIA, we propose to implement it by a fusion method and show by experimental results that this fusion method contributes to a reduction in computation complexity. Finally, we show that the separable convex minimization problem over the optimal rate vector set in the asymptotic model can be decomposed by the fundamental partition, the optimal partition of the user set that determines the minimum sum-rate, so that the problem can be solved more efficiently.

A new asymptotic-tolerance model of dynamic and stability problems for longitudinally graded cylindrical shells

Thin linearly elastic Kirchhoff-Love-type cylindrical shells of a heterogeneous microstructure which is periodic in the circumferential direction and slowly varying along the axial direction (composite shells with a space-varying periodic microstructure) are considered. Since macroscopic (averaged) properties of such shells are constant in circumferential direction but smoothly slowly varying in the axial one, then we deal with shells of a functionally (longitudinally) graded macrostructure. The aim of the contribution is to formulate and discuss a new mathematical averaged non-asymptotic model for the analysis of selected dynamic and stability problems for these shells. The model is derived by applying both the asymptotic and the tolerance non-asymptotic procedures which are combined together into a new modelling technique. The combined model equations include coefficients constant in periodicity direction and continuously slowly variable along axial coordinate. Since some of these coefficients depend on a cell size, then the model takes into account the effect of a microstructure size on the shells dynamics and stability. Moreover, it makes it possible to separate the macroscopic description of some problems from their microscopic description.

Block-Diagonal Covariance Selection for High-Dimensional Gaussian Graphical Models

ABSTRACTGaussian graphical models are widely used to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To reduce the number of parameters to estimate in the model, we propose a nonasymptotic model selection procedure supported by strong theoretical guarantees based on an oracle type inequality and a minimax lower bound. The covariance matrix of the model is approximated by a block-diagonal matrix. The structure of this matrix is detected by thresholding the sample covariance matrix, where the threshold is selected using the slope heuristic. Based on the block-diagonal structure of the covariance matrix, the estimation problem is divided into several independent problems: subsequently, the network of dependencies between variables is inferred using the graphical lasso algorithm in each block. The performance of the procedure is illustrated on simulated data. An application to a real gene expression dataset with a limited sample size is also presented: the dimension reduction allows attention to be objectively focused on interactions among smaller subsets of genes, leading to a more parsimonious and interpretable modular network. Supplementary materials for this article are available online.

The effect of uncertain material properties on free vibrations of thin periodic plates

Thin periodic plates with uncertain properties in a periodicity cell are investigated. To describe dynamics of these plates the non-asymptotic tolerance modelling method, cf. Woźniak and Wierzbicki (Averaging techniques in thermomechanics of composite solids. Wydawnictwo Politechniki Częstochowskiej, Częstochowa, 15), Woźniak et al. (eds.) (Thermomechanics of microheterogeneous solids and structures. Tolerance averaging approach. Wydawnictwo Politechniki Łódzkiej, Łódź, 16), for those plates is applied. The governing equations of tolerance models based on this method take into account the effect of period lengths on the overall behaviour of the plate. Hence, the additional effects of the periodicity can be analysed, as higher order vibrations. Moreover, properties of the plate in the periodicity cell are determined uncertainly. To analyse an influence of random variables of the properties with a fixed probability distribution on vibrations of the plate the Monte Carlo analysis is applied.

Geometrically nonlinear vibrations of thin visco-elastic periodic plates on a foundation with damping: non-asymptotic modelling

The objects under consideration are thin visco-elastic periodic plates with moderately large deflections. Geometrically nonlinear vibrations of these plates are investigated. In order to take into account the effect of microstructure size on behaviour of these plates a non-asymptotic modelling method is proposed. Using this method, called the tolerance modelling, model equations with constant coefficients involving terms dependent on the microstructure size can be derived. In this paper, only theoretical considerations of the problem of nonlinear vibrations of thin visco-elastic periodic plates resting on a foundation with damping are presented.