Ill-posed problems in mechanics

The notion of ill-posed initial and boundary value problems for partial differential equations was introduced by Hadamard, who also presented the first example of an ill-posed problem for a specific partial differential equation. At the same time, there are numerous examples of ill-posed problems in any field of mechanics.Hadamard and some of his successors believed that any ill-posed problem has no physical meaning and such problems should not be posed.The present paper contains several examples of ill-posed problems. We show that if one deals with an applied problem, then overcoming the ill-posedness mathematically can help one to improve the structure in practice, which justifies the study of ill-posed problems.     

New ill-posed problems in rheology

Many methods of material characterization by rheological methods lead to ill-posed problems. The nature of ill-posedness is discussed. In this light, some such new problems appearing in molecular weight characterization of homopolymers as well as in the characterization of size distribution of dispersed phases in polymer blends by rheological techniques are analyzed. The working capability of a nonlinear regularization method is demonstrated with examples of blend rheology. The role of Professor Meissner in motivating the research in the field of ill-posed problems as well as his important contribution to rheology of polymer blends is outlined.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Maximum entropy principle in relativistic radiation hydrodynamics II: Compton and double Compton scattering

Recently [1], a procedure has been proposed in order to close the set of the moment equations of relativistic radiative fluid dynamics. In particular explicit expressions for the moments of the bremsstrahlung and Thomson scattering source terms have been given. In this work, as anticipated in [1], we shall treat in a systematic way Comptonization and double Compton scattering too. Numerical results relative to the Compton cooling of hot electrons are shown. Received November 14, 2001 / Published online June 4, 2002 RID="a" ID="a" e-mail: mascali@dmi.unict.it Communicated by Ingo Müller, Berlin     

Melt rheology of long-chain-branched polypropylenes

Rheological properties of long-chain-branched isotactic polypropylene (PP) via copolymerization with a very small amount of nonconjugated α,ω-diene monomer using metallocene catalyst system in both linear and nonlinear regions were investigated, comparing with conventional linear and long-chain-branched PP modified at postreactor. Although comonomer incorporation was equal to 0.05 mol% or less, it caused high molecular weight, broad molecular weight distribution, and long-chain branching. A detailed study on the effect of diene incorporation on the polymer properties was conducted, comparing with modified PP in postreactor. Polymer chain microstructures were characterized by gel permeation chromatography with multiangle laser light scattering (MALLS), differential scanning calorimetry, and rheological means: dynamic viscoelasticity, step-strain, uniaxial elongational flow measurements, and large amplitude oscillatory shear. The PP, which incorporated a small amount of diene monomer, showed significantly improved viscoelastic behaviors. The diene-propylene copolymer containing long-chain branches showed extremely long relaxation mode under shear and outstanding viscosity increase under elongational flow, so-called strain hardening. The difference in microstructure of diene-propylene copolymer with modified PP with long-chain branches is investigated by MALLS and rheological characterizations.     

Applications of maximum entropy method in capillary viscometry

Maximum entropy method (MEM) is presented as a technique for processing data obtained from capillary viscometers. The performance of MEM is assessed by comparing the viscosity versus shear rate curves generated by MEM against that obtained by the standard method based on the Weissenberg-Rabinowitsch-Mooney equation. In all the cases in vestigated, MEM proved to be a reliable technique in coping with the experimental noise in the capillary data.     

Generating line spectra from experimental responses. Part I: Relaxation modulus and creep compliance

We describe a recursive computer algorithm which generates line spectra from relaxation modulus or creep compliance data without producing negative spectrum lines. We apply the algorithm here to data read from mathematical models for the relaxation modulus. Since these data were thus free of the usual experimental error, we could use a relatively simple form of the basic algorithm that is applicable also to smoothed data. The spectra faithfully reproduced the input functions and may serve for data storage as well as for predicting other experimental responses.     

Some inverse problems in rheology leading to integral equations

Two inverse problems of the integral type, which are of the general importance to rheology, are discussed. The first of them is the possibility of calculating the molecular weight distribution (MWD) from the flow curve and the second one is the interrelation between creep and relaxation functions. It was shown that the first problem is incorrect and any solution is unstable in respect to minor experimental errors. It means that the general solution of this problem is impossible in principle and only estimations of the width of unimodal MWD can be received from the curvature of the flow curve. The possibility of the correct calculation of the creep function exists in case the relaxation curve being approximated by the sum of exponential members. But the approximation of the relaxation curve within the experimental accuracy is the necessary, but not sufficient condition for the correct solution of this integral inverse problem, because not every mode of approximation leads to the satisfactory prediction of creep function.Delivered as the Courtaulds Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Experimental investigation of a maximum entropy assumption for acceleration terms within a poly‐disperse moment framework

Moment transport methods are being developed to model poly‐dispersed multiphase flows by transporting statistical moments of the particle size–velocity joint probability density function (JPDF). A common feature of these methods is the requirement to reproduce or approximate the form of the JPDF from the transported moments for calculation of body force terms and other source terms. This paper examines the application of a maximum entropy technique against phase Doppler anemometry data sets from an electrostatically charged kerosene spray and also an automotive pressure swirl atomizer. An assessment of which moments are required to reproduce the JPDFs using a maximum entropy assumption to a sufficient level of accuracy is made. It is found that it is possible to reproduce the JPDFs to a high level of accuracy using a large number of moments; however, this incurs large computational overheads. If the moments to be transported are chosen on the basis of physical reasoning (such as the relationship between size and velocity due to drag) it is possible to reduce the number of moments to those which would be conserved via balance equations. This permits an approximation to the JPDF commensurate with the closure level of the moment transport method and thus the closure model method is naturally scalable with the degree of information from available conservation equations. Copyright © 2008 John Wiley & Sons, Ltd.     

General thermostatistical formalisms based on parameterized entropic measures

We revisit the concept of generalized thermostatistical formalisms, based on extremizing parameterized entropic functionals subject to appropriate constraints, in order to incorporate an (effective) temperature dependence of the entropic parameters and of parameters characterizing the relevant constraints. Our main aim is to investigate what kinds of temperature dependence of these parameters are consistent with the Legendre-transform structure of thermodynamics. After discussing this problem in a qui te general context, we discuss in detail the important particular example of the q-nonextensive thermostatistical formalism with a temperature-dependent q-parameter. In this special case, our general formalism implies extremizing the concomitant entropy functional S_q, subject to the constraints imposed by normalization and the presumably known exp ectation values of N relevant observables, for arbitrary variations of both the statistical operator and the parameter q. The ensuing extended variational formalism preserves the usual (Legendre-transform) connection with thermodynamics. For sets of relevant observables that close a Lie semi- algebra with the systems Hamiltonian, we study some features of our approach related to the systems dynamics and show that q is a constant of the motion. Our present developments may be useful for the study of systems whose entropic parameter q is unknown.Received: 20 February 2003, Accepted: 19 September 2003, Published online: 12 December 2003PACS: 05.20.-y, 05.20.Gg, 05.70.-a, 05.90. + mCorrespondence to: A.R. Plastino     

ELEMENT－BY－ELEMENT MATRIX DECOMPOSITION ANDSTEP－BY－STEP INTEGRATION METHOD FOR TRANSIENTDYNAMIC PROBLEMS

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An intermediate model method for obtaining a discrete relaxation spectrum from creep data

A convenient method is described for obtaining a discrete stress relaxation spectrum from linear viscoelastic creep data by means of a three-stage process. In stage one, a discrete retardation spectrum is fitted to the creep data using a least squares procedure, subject to the constraint that the discrete spectrum must be a specified order of polynomial function of the retardation time. In stage two, the resulting generalised Voigt model is solved numerically for an imposed step in strain, to determine the stress relaxation modulus function of time. In stage three, a discrete relaxation spectrum is fitted to the calculated stress relaxation modulus function. Although three stages are involved instead of the usual two, the procedure has been found to have certain practical advantages. These advantages make it suitable for the generation of relaxation spectra needed in viscoelastic stress analyses of solids, for example by the finite element method. In order to illustrate the proposed procedure it is applied to both artificial data and experimental creep data for poly(methyl methacrylate) at 70°C and at the glass transition.     

Determination of the relaxation time spectrum from dynamic moduli using an edge preserving regularization method

For the numerical determination of a continuous relaxation time spectrum from rheological data so-called regularization methods are necessary. The regularization method based on Tikhonov regularization and commonly used in rheology assumes the relaxation time spectrum to be smooth. Accordingly, this method is not able to properly resolve spectra containing edges or at least large curvatures as proposed in the literature. In order to reconstruct such spectra a novel edge preserving regularization (EPR) method was developed which extends the properties of the commonly used method. In order to demonstrate the efficiency of EPR it is compared with the commonly used regularization method by means of simulated and experimental data. This reveals that EPR is able to prove whether or not the theoretically predicted edge containing spectra for high-molecular-weight, monodisperse polymers are compatible with experimental data. Received: 11 June 1999/Accepted: 1 September 1999     

Generating line spectra from experimental responses. Part IV: Application to experimental data

The previously reported algorithms for deriving line spectra (respondance time distributions) from synthetic or smoothed experimental responses is here extended to experimental data. The earlier algorithm was modified to improve performance in the presence of experimental errors. The effect of smoothing the data with the aid of the cubic spline function was examined.The performance of the modified algorithm was studied comprehensively. Auto-predictions and cross-predictions of storage and loss compliances from the generated line spectra were in excellent agreement. In equally good agreement were the line spectra obtained from compliance data and from stress relaxation data obtained on the same material.     

A fundamental relationship between the relaxation spectra and the tangent distribution function in the theory of linear viscoelasticity

The tangent distribution function (TDF) is analyzed within the theory of linear viscoelasticity on mechanical properties. A proof is given that both the relaxation and retardation spectra can be derived from the TDF, through a Fredholm integral equation. Furthermore, the relaxation strength can be calculated as a consequence of this relationship. Finally, as an example, the relationship is applied to discrete spectra.     

Estimation of dynamic stress spectrum distribution in structural fatigue test

The determination of structural dynamic stress spectrum distribution is of great significance in the structural fatigue strength evaluation as well as reliability design. In previous empirical data processing methods, the data grouping and distribution fitting were excessively coarse and contained distinctive defects. This paper proposed an effective approach to statistically group actual measured dynamic stress data and validly extrapolate the combined distribution to fit the dynamic stress spectrum distribution. This approach has been verified its effectiveness through chi-square test, stress spectrum extrapolation and damage calculation in dynamic stress study.     

最大熵法在可靠度计算中的应用

在可靠度计算中,任意对立随机变量往往为不同概率分布,不利于可靠度计算。通常采用正态转换法将其转换成具有统一形式的正态分布,以便进行可靠度计算。但是,正态转换并不能保证原有随机变量概率分布特性,从而一定程度上造成可靠度计算误差。针对上述问题,本文提出在可靠度计算中利用最大熵法将任意随机变量概率密度函数转换成近似精度更高且具有统一标准形式的最大熵概率密度函数(MEPDF),然后提出对应的一阶可靠度计算方法进行可靠度求解,最后通过实例分析证明该方法的有效性。     

随机振动的信息量及最大熵概率估计

本文将信息量等概念及最大熵的概率估计方法引入随机振动信号分析,给出随机振动的信息量及信息熵的概念,讨论了随机振动的信息描述.最后以复合振动信号为例,给出它的最大熵概率密度函数.     

A direct method for obtaining discrete relaxation spectra from creep data

A direct method for obtaining discrete relaxation spectra from creep data is proposed. The conversion of creep data to relaxation data is avoided and the discrete relaxation times are freely adjustable. The nonnegative least square method is used to generate nonnegative discrete relaxation strength. Received: 10 November 1999 Accepted: 22 September 2000     

冲击载荷作用下加筋板架破坏概率分析

取导弹战斗部初始撞击速度、战斗部质量、加筋板架材料的弹性模量和极限强度作为随机变量;采用对较小的样本数量进行拟合的方法,用产生随机数程序得到50组随机变量的初始值,利用ANSYS/LS-DYNA有限元程序进行仿真计算,得到导弹战斗部击穿加筋板架后的末速度值;验证击穿板架后战斗部的末速度是否服从正态分布,若不服从正态分布则采用最大熵法拟合击穿板架后的末速度的分布函数;考虑战斗部撞击点和板架加强筋的相对位置关系,采用速度准则,利用蒙特卡罗方法模拟得到当战斗部入射角度分别为0°、30°、45°时,击穿单层和双层加筋板架的概率。     

随机参数超空泡射弹弹道随机特性分析

超空泡射弹的发射装置及射弹在制造,安装过程中存在各种不确定性.本文采用最大熵法来分析这些不确定性对射弹弹道的影响.介绍了超空泡射弹的动力学模型,对射弹的受力特性和随机参数进行了分析.根据给出的理论编制了基于最大熵法的随机参数超空泡射弹弹道随机特性分析的仿真程序,并对一个模型弹仿真计算.计算表明,与发射装置有关的随机参数...