Yu. A. Basistov

Linear inverse problems in viscoelastic continua and a minimax method for Fredholm equations of the first kind

A new theory based on an extensively modified version of the minimax method is proposed to estimate the cause from the result, that is, the characteristic functions of viscoelastic media from experimentally obtained material functions through the solution of Fredholm integral equations of the first kind. The method takes into account the non-Gaussian outliers, and does not require the assumption of a priori error bounds as in other smoothing techniques which may lead to instability or to a stable solution not representative of the true solution. The algorithm is applied to several hypothetical test problems to show the excellent performance of the method in extreme severe conditions. The shortcomings of the Tikhonovs regularization and other smoothing techniques are discussed. It is shown that the solution via these methods may not represent the real solution in any norm. The new method is applied to linear viscoelasticity to obtain the relaxation spectrum from experimental material functions. The relaxation spectra of some materials obtained via the proposed adaptive-robust minimax algorithm and experiments run in a rotary viscometer are presented.

A new method of calculation of polymer media relaxation functions from rheological experimental data

Abstract The linear theory of viscoelasticity is based on one function which completely characterizes the material: the relaxation spectrum. In principle, from this one pivotal function all other material functions may be deduced. Although this seems rather straightforward in theory, in practice serious difficulties arise to start with, in obtaining a reliable and unique relaxation spectrum from experimental measurements. The problem of finding the relaxation spectrum, as the kernel of an integral Fredholm equation of the first kind, is an ill-posed problem. Tikhonovs regularization method has been applied to the inverse problem of finding the kernel of the Fredholm equation. This method allows unique determination of the kernel by introducing a priori error estimates and optimizing a regularization function. Introducing error estimates and assuming a certain error distribution makes the solution unique, but not necessarily the correct solution as it depends on the a priori estimates. The new algorithm based on the minimax method we develop in this paper does not require a priori error estimates and the set of outliers are taken into account. We construct the algorithm, give several hypothetical examples to test the reliability and exactness of the method, and finally end the paper with a concrete example of the relaxation spectrum we determine for a polybutadiene.

Comparison of image recognition efficiency of Bayes, correlation, and modified Hopfield network algorithms

The statistical estimates of the probability of correct recognition of the images, noisy reference by an additive handicap, for Bayes, correlation, and modified Hopfield network algorithms are compared. It is shown that, in the case of complete a priori probability concerning a handicap, the modified Hopfield network algorithm reaches the quality of the Bayes algorithm. At a deviation a priori probability on a handicap, the quality of the Bayes algorithm is worse than that of the modified Hopfield network algorithm. The correlation algorithm is worse than the modified Hopfield network algorithm, in general.

Neurodynamic model of viscoelastic materials with associative memory

1. The simulation of viscoelastic media may be divided into two stages. At the first stage, we synthe� sized the mathematical model of the medium; at the second stage, we identified the model from the exper� imental data obtained by a device. Nowadays, as a rule, the Maxwell, Jeffrey, and Voigt–Kelvin elements [1] are used for the synthesis of models. The Maxwell element is the series connection of elastic and viscous components. From the parallel set of such elements, the mathematical model is formed, for example, the Wagner model [2]:

Application of minimax method for calculation of viscoelastic material functions and relaxation spectra of polymer melts and solutions

Many publication were devoted to the problem of finding methods for calculating the distribution function of relaxation time spectra h(τ) for viscoelastic media. Direct calculations of this function on the basis of known relations of the linear theory of viscoelasticity connected with the solution of fredholm integral equations of the first kind involve seriuous difficulties. At the same time, numerical approaches to determined the function H(τ) based on the use of experimentaly determined material functions were proposed in a number of recent publications. Since the accuracy of calculating the functions depends directly on the confidence of determining the primary experimental data that generally always contain errors of a statistical nature, a number of authors employ smoothing numerical methods of determining the material functions, or ones that fail to react to the appearance of rough errors. These methods include methods of regularisation and maximum of entropy. However, they have a number of serious shortcommings limiting the possibilities of their successful use. In the present work, we propose to employ a different numerical method for these purposes, namely, the minimax technique. In our opinion, it is mor accurate, rapid, and universal. We consider a scheme of constructing algorithm, give examples of solvingtest problems showing how the algorithm functions in various extreme hypothetic situations. The results of the numerical calculations of the function h(τ) for a real object, viz. polybutadiene are based on experiments run in a rotary viscometer Rheotron of the firm BRabender

A modified regularization method for calculating viscoelastic material relaxation functions

Conclusions1.The accuracy of solving incorrectly stated problems for the Tikhonov regularization method depends markedly on the accuracy of prescribing additional a priori information with respect to starting data, and in the present case to error parameters.2.If in the case in question error parameters and their statistics are determined with marked errors, then any accurate conformity of the regularization parameter with error parameters given in [16] does not make it possible to obtain a satisfactory regularized solution.3.An estimate of the maximum probability of error parameters for data may also lead to an unsatisfactory regularized solution if it contains gross overshoots.4.In our view the adaptive-robust algorithm for correcting gross experimental anomalies is a more reliable method for improving the accuracy of the regularized solution in the problem of calculating the relaxation spectrum for viscoelastic material from experimentally determined material functions without using a priori information about error parameters. The work was carried out with the support and sponsorship of the Russian Fund for Basic Research.