On the third-order bounds of the effective shear modulus of two-phase composites

Formulation of variational bounds for properties of inhomogeneous media constitutes one of the most fundamental parts of mechanics. The earliest work on multiphase media is the so-called Voigt’s upper bound and Reuss’ lower bound, corresponding to the simple rule of mixture or first-order bounds. The second-order bounds were formulated by Hashin and Shtrikman for macroscopically isotropic random composites. The third-order bounds of the bulk modulus were derived by Beran, which contain a pair of third-order bulk parameters. The third-order bounds of the shear modulus first derived by McCoy were improved by Milton and Phan-Thien, which further involve a pair of third-order shear parameters. In this study, by applying the stochastic variational principle of Xu (2009) the third-order bounds of the shear modulus are derived in an analytically most extensive trial function space. By further modifying Milton’s definition of shear parameters, the third-order bounds are finalized into a symmetric form, exactly like the Beran’s bounds of the bulk modulus. Since the bounds of the shear modulus play an essential role in plasticity theory of composites, the finalization of the third-order bounds also paves the way for further formulation of variational principles and bounds of nonlinear composites.     

Computation of the bounds on the elastic moduli of a fiber-reinforced composite by Monte Carlo simulations

Bounds on the effective elastic moduli of a fiber-reinforced composite are studied. The composite is composed of infinitely long, parallel, equal-sized circular cylinders (fibers) randomly embedded in a matrix. Based on the variational principles and by employing the cylindrical inclusion solutions as trial functions, the bounds are determined through Monte Carlo simulations. Comparisons are made with the second-order and third-order perturbation bounds. Because the shape of the inclusion has been explicitly taken into consideration, it is found that the bounds computed by the present approach are the sharpest in terms of the width of the bound pair. Specifically, when the fibers are stiffer than the matrix, the present lower bounds coincide with the second-order lower bounds at low concentrations and are slightly higher than the second-order lower bounds at high concentrations, while the upper bounds are far below the corresponding second-order and third-order perturbation bounds. In many cases, the present bound pair is very close and thus provides accurate prediction on the effective properties of this class of composites.     

Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains

A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain‐based a posteriori error estimate which yields self‐equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux‐free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds. Copyright © 2009 John Wiley & Sons, Ltd.     

Lower bounds for three-dimensional multiple-bin-size bin packing problems

The three-dimensional multiple-bin-size bin packing problem (MBSBPP) is the problem of packing a set of boxes into a set of bins when several types of bins of different sizes and costs are available and the objective is to minimize the total cost of the bins used for packing the boxes. We present a study of lower bounds for this packing problem. We have developed new bounds based on integer programming formulations of some relaxations of the original problem. These formulations are enhanced with logical considerations. The proposed bounds are compared with other existing bounds in an extensive computational study, including two- and three-dimensional instances with up to 100 boxes, some of them taken from the literature and others adapted from the classical Bin Packing Problem. The proposed bounds improve the results of previous bounds by more than 10%, though at a higher computational cost.     

Microstructural effects in capacitive composition measurements

Capacitance-based sensors offer practical advantages for binary composition measurements in various manufacturing processes. The implications of the microstructure of binary mixtures in capacitive composition measurements are explored. Specifically, it is shown that microstructural assumptions can be used to derive measurement uncertainty bounds when inferring composition from measured capacitances. Worst-case bounds are obtained from the well-known series/parallel bounds on the total capacitance of a binary mixture, while tighter bounds are obtained from results of Hashin and Shtrikman for isotropic mixtures     

Computation of upper and lower bounds to the frequencies of elastic systems by the method of Lehmann and Maehly

A method, called the Lehmann-Maehly method, for determining upper and lower bounds for eigenvalues, which was introduced by Bazley and Fox⁴ in 1964, is applied to the determination of bounds to the frequencies of elastic systems. In reviewing the fundamental theory involved, the present paper emphasizes the physical significance of the adjoint operators T and T* employed. An error in the paper by Bazley and Fox is noted and it is pointed out that with the corrected theory it is possible to use the Lehmann-Maehly method as a procedure for computing converging sequences of upper and lower bounds merely by varying a constant, the shifting constant. Thus, sequences of bounds are shown to be easily obtainable without prior knowledge of rough bounds, and the procedure for obtaining the bounds is not significantly more difficult to apply than the familiar Rayleigh-Ritz method for upper bounds. Since displacements and stresses are independently varied, the displacement functions used in the approximation procedure are only required to satisfy prescribed displacement boundary conditions while the stress functions only need satisfy the natural (stress) boundary conditions. Operators are derived for a non- uniform beam based on Timoshenko beam theory. Tables of bounds computed by the Lehmann-Maehly method and also the Rayleigh-Ritz method are given as an illustrative example.     

Bounds on frequency response estimates derived from uncertain stepresponse data

The frequency response of a system can be estimated from measurements of its step response; however, many error sources affect the accuracy of such estimates. This paper investigates the effects of uncertainty in the knowledge of the step response. Methods for establishing uncertainty bounds for the frequency response estimates are developed, based on the corresponding time-domain uncertainties associated with the measured step response. Two methods are described. One method produces bounds that are often very conservative. The other method produces bounds that are more realistic. End effects that influence the bounds are also considered. A simulation example and an application of the bounds are presented     

An Investigation of the Burn-In Problem

A problem involving the derivation of bounds on distributions with a decreasing failure rate (DFR distributions) is presented. Given that an item has a decreasing failure rate, sharp upper and lower bounds on the burn-in time to achieve a specified mean residual life are derived. The bounds rely only on the DFR assumption and a knowledge of the first moment and a percentile of the failure distribution.

In addition, some bounds on the hazard rate and density of a DFR distribution are given.     

A differential approach to microstructure-dependent bounds for multiphase heterogeneous media

We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin–Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.     

非负矩阵Perron根的估计

本文研究非负矩阵Perron根的界值,定义了矩阵有向图的1-path覆盖,利用矩阵的有向图及有向图的1-path覆盖,给出了估计非负矩阵Perron根上、下界的新方法,改进了已有的估计结果,有效提高了估计的精度。     

Empirical evaluation of lower bounding methods for the simple assembly line balancing problem

The simple assembly line balancing problem (SALBP) is a classical problem that arises in the design of assembly lines. The objective of the SALBP is to divide the assembly work among different workstations of the assembly line in order to maximise the efficiency of the entire line. In this paper, we: (a) put forward different families of lower bounds for the SALBP; (b) introduce new lower bounds; (c) generalise previous results; and (d) analyse the performance of these lower bounding techniques in terms of solution quality using a recently proposed set of instances. An analysis of the results of these bounds leads us to: (1) improve the best-known lower bound for 104 out of the 175 open instances from the benchmark set; (2) identify the areas of applicability of different sources of bounds; and (3) identify possible uses of these bounds in other assembly line balancing problems.     

Upper bounds on the error probability of coded space?time block coded systems with diversity reception

New union bounds are derived for space-time block coded systems over Rayleigh fading channels. Both maximal ratio combining and generalised selection combining are considered as combining schemes at the receiver. The union bounds are easy to be evaluated using the transfer function of the code. Furthermore, the bounds are general to any coding scheme with a known weight distribution. Results show that the proposed union bounds are tight to simulation results for wide ranges of diversity orders and signal-to-noise ratio values.     

有限变形下含非完美界面复合材料有效模量的界限

导出了有限变形下含非完美界面两相复合材料的上下界限。在微小变形下, 所导出的界限还原成已知的线性情况下的相应结果。数值上预示了非完美界面特性、有限变形对复合材料有效模量的影响规律, 给出了与实际相符合的结果。      

Computing a-posteriori bounds for stress intensity factors in elastic fracture mechanics

This paper mainly focuses on computing the lower and upper bounds on stress intensity factors in elastic fracture mechanics with an efficient finite element output bound procedure on quantities of interest in engineering. The bounds procedure is obtained by minimizing the quadratic energy functional of output with constraints of equilibrium conditions of mechanics and continuity conditions of finite element space. The computation is based on solving the elemental Neumann residual problems for the bounds on energy norm of error in finite element solutions. The lower and upper bounds on the intensity factors of an open mode and a shear mode elastic fracture problems are computed in this paper.     

Cumulative damage analysis based on fatigue damage bounds

A cumulative damage analysis is presented based on damage curves derived in Ben-Amoz (2003) as bounds on mean fatigue damage in constant amplitude cycling. A detailed analysis is developed for two-stage high-low (H-L) and low-high (L-H) cycling based on single and two-mode bounds. The use of single-mode bounds is questionable as neither bound reflects a crack tip process whereas two-mode bounds reflect the actual process by the presence of both slip and opening-mode damage in each cycle. This leads to results in much closer agreement with experimental data. It is found that to ensure damage irreversibility in L-H cycling the presence of four discrete microstructure barriers obtained in Ben-Amoz (2003) is essential. As a result, two distinct sets of bounds apply: one for H-L another for L-H cycling. The barriers divide the fatigue process into three distinct macro stages while the entire range of lifetimes is divided into five ranges. In L-H cycling different sets of bounds apply to each interval between barriers, hence to each range of lifetimes. A statistical analysis based on an assumed log normal distribution of lifetimes is developed for two-stage H-L and L-H cycling. The bounds lead in a straightforward manner to a median while the variance can only be bracketed by the two-mode and single-mode variances due to exclusion of cyclic hardening as an additional random variable. Predicted residual lifetimes are in close agreement with experimental data in several metals notwithstanding the paucity of data. The two-stage analysis is then generalized to multi stage cycling.     

不可约Z-矩阵最小特征值的数值算法

首先给出了不可约非负矩阵最大特征值的上下界。然后利用相似变换构造了一列相似矩阵,从而得到不可约非负矩阵最大特征值的逐步压缩的一列上下界,其极限为所要求的最大特征值。最后利用Z-矩阵与非负矩阵的关系,给出了计算不可约Z-矩阵最小特征值的一个新算法。理论上给出了收敛性证明。该算法迭代过程简单,不用计算逆矩阵,从而计算量小,占用内存少。数值实验的结果表明该算法具有可行性和有效性。     

Assessment of the local stress state through macroscopic variables

Macroscopic quantities beyond effective elastic tensors are presented that can be used to assess the local state of stress within a composite in the linear elastic regime. These are presented in a general homogenization context. It is shown that the gradient of the effective elastic property can be used to develop a lower bound on the maximum pointwise equivalent stress in the fine-scale limit. Upper bounds are more sensitive and are correlated with the distribution of states of the equivalent stress in the finescale limit. The upper bounds are given in terms of the macrostress modulation function. This function gauges the magnitude of the actual stress. For 1 相似文献   

On Stopping Rules for Facilities Location Algorithms

Two different lower bounds on the objective values of certain location problems have recently been proposed. Such bounds are useful for finite termination of an algorithm. It has been previously established that one of these bounds is superior to the other. Here we present some extensions and prove that a generalized version of the result is also true.     

Static response function in superfluid4He

Rigorous lower and upper bounds for the static response function in superfluid⁴He at zero temperature are derived using a sum rule approach. The ingredients needed to calculate the bounds are taken from recent Monte Carlo simulations for the ground state. The new bounds improve significantly the Feynman approximation and provide estimates of the static response function in agreement with the experimental data at zero pressure.     

Upper and Lower Bounds on Radially Symmetric Optical Transfer Functions

For an optical system under incoherent imaging conditions, the finite aperture and non-negative point-spread function place restrictive bounds on the values of the corresponding optical transfer function. Under the extra condition that the transfer function be radially symmetric, these bounds can be characterized as the eigenvalues of a real symmetric integral operator. In this paper the bounds are calculated first by analytic inequalities that bound the operator norm, then by finding the eigenvalues of a Galerkin approximation to the operator. In contrast to the slit aperture case, the least upper bound and greatest lower bound are found to differ. The pupil functions which achieve these bounds at a given spatial frequency are determined, along with the associated transfer functions.