Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves

The classical overlapping Schwarz algorithm is here extended to the spectral element discretization of linear elastic problems, for both homogeneous and heterogeneous compressible materials. The algorithm solves iteratively the resulting preconditioned system of linear equations by the conjugate gradient or GMRES methods. The overlapping Schwarz preconditioned technique is then applied to the numerical approximation of elastic waves with spectral elements methods in space and implicit Newmark time advancing schemes. The results of several numerical experiments, for both elastostatic and elastodynamic problems, show that the convergence rate of the proposed preconditioning algorithm is independent of the number of spectral elements (scalability), is independent of the spectral degree in case of generous overlap, otherwise it depends inversely on the overlap size. Some results on the convergence properties of the spectral element approximation combined with Newmark schemes for elastic waves are also presented.     

The Application of the Theory of the Price Elasticity of Demand to the EMS Management

1　Introduction　ExpressMailService (EMS )wasopenedupbyChinaPostintheearly 80stomeettheneedsofeco nomicdevelopment.It sthemostcompetitiveoneofthepostservices.There recivilaviation ,Chinarail wayandotherprivateexpressdeliverycompaniesdomesticallyandUPS ,TNT ,DHLandTEDEXcompetingwithEMSintheinternationalmarket.OneofthemostpowerfulcompeteweaponofthesecompaniesisthelowerpricethanEMS[1 ] .BecauseofthefiercecompetitionEMSdeclinedfromthemo nopolistindomesticandinternationalmarketoftheear…     

Network topology and the theory of rubber elasticity

The earliest investigations on rubber elasticity, commencing in the 19th century, were necessarily limited to phenomenological interpretations. The realisation that polymers consist of very long molecular chains. commencing c. 1930, gave impetus to the molecular theory of rubber elasticity (1932-). according to which the high deformability of an elastomer, and the elastic force generated by deformation, stem from the configurations accessible to long molecular chains. Theories of rubber elasticity put forward from 1934-1946 relied on the assumption that the junctions of the rubber network undergo displacements that are affine in macroscopic strain. The theory of James and Guth (1947) dispensed with this premise, and demonstrated instead that the mean positions of the junctions of a ‘phantom’ network consisting of Gaussian chains devoid of material properties are affine in the strain. The vital significance of the distinction between the actual distribution of chain vectors in a network and their distribution if the junctions would be fixed at their mean positions went unnoticed for nearly 30 years. Experimental investigations, commencing with the incisive work of Gee in 1946. revealed large departures from the relationship of stress to strain predicted by the theories cited. This discrepancy prompted extensive studies, theoretical and experimental, during succeeding years. Inquiry into the fundamentals of polymer networks, formed for example by interlinking very long polymer molecules, exposed the need to take account of network imperfections, typically consisting of chains attached at only one end to a network junction. Various means were advocated to make corrections for these imperfections. The cycle rank ζ of the network has been shown (1976) to be the fundamental measure of its connectivity, regardless of the junction functionality and pattern of imperfections. Often overlooked is the copious interpenetration of the chains comprising typical elastomeric networks. Theories that attempt to represent such networks on a lattice are incompatible with this universal feature. Moreover, the dense interpenetration of chains may limit the ability of junctions in real networks to accommodate the fluctuations envisaged in the theory of phantom networks. It was suggested in 1975 that departures from the form predicted for the elastic equation of state are due to constraints on the fluctuations of junctions whose effect diminishes with deformation and with dilation. Formulation of a self-consistent theory based on this suggestion required recognition of the non-affine connection between the chain vector distribution function and the macroscopic strain in a real network, which may partake of characteristics of a phantom network in some degree. Implementation of the idea was achieved through postulation of domains of constraint affecting the equilibrium distribution of fluctuations of network junctions from their mean positions. This led in due course to a theory that accounts for the relationship of stress to strain virtually throughout the ranges of strain accessible to measurement. The theory establishes connections between structure and elastic properties. This is achieved with utmost frugality in arbitrary parameters.     

Disorder effects on the strain response of model polymer networks

Molecular dynamics simulations are used to investigate polymer networks made by either end-linking or randomly crosslinking a melt of linear precursor chains. The resulting network structures are very different, since end-linking leads to nearly ideal monodisperse networks, while random crosslinking leads to polydisperse networks, characterized by an exponential strand length distribution. Networks with average strand length 20 and 100 were generated. These networks were used to study the effects of disorder in the network connectivity on observables averaged either over the entire network or selected sub-structures. Heterogeneities in the randomly crosslinked networks cause significant differences in the localization of monomers, however, neither the localization of crosslinks nor the microscopic strain response are significantly affected. Compared to end-linked networks, randomly crosslinked networks have a slightly increased tube diameter, and as a result a slightly decreased shear modulus, but otherwise identical stress-strain behavior. For the investigated systems, we conclude that the microscopic strain response, tube diameter, and stress-strain relation are all insensitive to the heterogeneities due to the linking process by which the network were made.     

On a three-dimensional axisymmetric boundary-value problem of nonlinear elastic deformation: Asymptotic solution and exponentially small error

In this paper, we study a three-dimensional axisymmetric boundary-value problem of a slender cylinder composed of a nonlinearly elastic material subjected to an axial force. Starting from the field equations, after a transformation and proper scalings, we identify a small variable and two small parameters, which characterize the present problem. Then, by an approach involving compound series-asymptotic expansions, a nonlinear ODE is derived, which governs the axial strain (the first-term in the series expansion). By imposing the zero radial displacement conditions at two ends, we manage to get the analytical solution of the axial strain, from which all other physical quantities can be deduced and thus the three-dimensional displacement field can be determined. Graphical results are presented, which show that there are two boundary layers near the two ends while the middle part is in a state of almost uniform extension. The asymptotic structure of the analytical solution is derived, which offers clear explanations to the structure of the deformed configuration and shows that the thickness of both boundary layers is of the order of the radius. We also point out the relevance of the present results to the St. Venant’s problem. In particular, we obtain the explicit uniformly-valid exponentially small error term, when the obtained deformed configuration is compared to the configuration of a uniform extension.     

Significance of the elastic peak stress evaluated by FE analyses at the point of singularity of sharp V-notched components

The paper presents an expression useful to estimate the notch stress intensity factor (NSIF) from finite element analyses carried out by using a mesh pattern with a constant element size. The evaluation of the NSIF from a numerical analysis of the local stress field usually requires very refined meshes and then large computational effort. The usefulness of the presented expression is that (i) only the elastic peak stress numerically evaluated at the V‐notch tip is needed and no longer the whole stress–distance set of data; (ii) the adopted meshes are rather coarse if compared to those necessary for the evaluation of the whole local stress field. The proposed expression needs the evaluation of a virtual V‐notch tip radius, i.e. the radius which would produce the same elastic peak stress than that calculated by FEM at the sharp V‐notch tip by means of a given mesh pattern. Once such a radius has been theoretically determined for a given geometry, the expression can be applied in a wide range of notch depths and opening angles.     

F-X域粘弹性波动方程保幅偏移

提出了一种利用粘弹性声波波动方程进行偏移的新方法。其基本思路是,修改成像条件,使修改后的成像方程中考虑振幅补偿,然后利用粘弹性、单程波动方程在F-X域对震源和接收点波场进行延拓,进而完成散射、透射等补偿。理论数据处理和实际地震资料处理表明,该方法理论基础可靠,处理效果明显,能较好地解决实际地质问题。     

圆杆弹性和弹—塑性扭转公式的探讨

对圆杆在线弹性阶段，弹－塑性阶段和全塑性阶段的扭转剪应力的计算公式作了分析，并推导了在材料的应力－应变关系（τ－γ关系）不服从虎克定律时，扭转真实剪应力的计算公式。文中对扭转强度计算公式的来源，物理含义及其都作了详细的说明。这对执行和理解国家ＧＢ１０１２８－８８《金属室温扭转试验方法》是有益的，而且对高校中材料力学的教学具有一定的参考价值。     

Testing viability of cross subsidy using time-variant price elasticities of industrial demand for electricity: Indian experience

Indian electric tariffs are characterized by very high rates for industrial and commercial classes to permit subsidized electric consumption by residential and agricultural customers. We investigate the viability of this policy using monthly data for 1997–2003 on electric consumption by a few large industrial customers under the aegis of a small distribution company in the state of Uttar Pradesh. For a given price/cost ratio, it can be shown that if the cross-subsidizing class’ electricity demand is sufficiently elastic, increasing the class’ rates fail to recover incremental cross-subsidy necessary to support additional revenues for subsidized classes. This suboptimality is tested by individually estimating time-variant price-elasticities of demand for these industrial customers using Box-Cox and linear regressions. We find that at least for some of these customers, cross-subsidy was suboptimal prior to as late as October 2001, when rates were changed following reforms.