A Numerical-Analytical Approach to the Analysis of Non-Stationary Temperature Fields

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Mechanics, Materials Science & Engineering, March 2016

ISSN 2412-5954

A Numerical-Analytical Approach to the Analysis of Non-Stationary Temperature Fields in Multiply-Connected Solids Roman Kushnir1 & Tetyana Solyar1,a 1 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3-B Naukova St., 79060, Lviv, Ukraine 2

tanya@iapmm.lviv.ua DOI 10.13140/RG.2.1.1167.0165

Keywords: integral Laplace transformation, non-stationary temperature fields, multiply connected solids, plate with heat emission.

ABSTRACT. A technique for the analysis of non-stationary temperature fields in multiply connected solids is presented. This technique rests upon the Laplace integral transformation along with the method of boundary-integral equations. The inversion for the Laplace transformation is performed by making use of the well-known general analytical-numerical -stationary heat-conduction problems to achieve the solution within the controllable accuracy. On this basis, the temperature fields in solids with openings and multiplyconnected plates with heat emission are analyzed.

1. Introduction. It is well known that the non-stationary heat-conduction problems for multiply connected solids present a challenge for the dominant analytical modes of attack and, thus, they are readily attempted by means of the numerical procedures. One of the simplest, from the mathematical standpoint, methods for the solution of such problems is based upon the application of Laplace integral transformation. For the determination of Laplace image-function for the temperature, the method of boundary-integral equations can be efficiently used. Then, the inversion procedures, i.e. the construction of original function, cause certain mathematical complications. There exist a great number of various methods for the numerical Laplace inversion, which, in general, is an ill-posed problem. Due to this reason, a great many of the new developed methods are devoted to the special classes of problems. For instance, the method of decomposition by the Eigen functions is presented in [13] for the analysis of transient phenomena in a porous medium. In [15], the efficiency analysis for the methods of numerical Laplace inversion is presented concerning the homogenization problems in linear viscoelastic materials. A numerical-analytical inversion formula based on the quadrature formulas is suggested in [11] for a class of functions which rapidly vanish for greater values of the time. Among these methods, an inversion formula suggested by Prudnikov [3] presents a great deal of interest. This formula is based on the explicit relationship involving two series. One of these series has a form of the Fourier series with the coefficients presented by the Laplace transformations. The second one is the exponential series, which contains the values of the original-functions. In this paper, the approximate inversion formulas for the Laplace transformation are obtained because of the foregoing explicit relationship in which, by making use of the developed technique, the Fourier series convergence is accelerated and the remainder term is accounted. A key feature of the inversionformula is that it contains the exact values of the original-function at both initial and infinitely distant moments of time. Due to this reason, this formula cannot be regarded as the universal one as it can be used only if the mentioned parameters are given a priori. This makes this inversion-formula to be applicable for certain classes of problems. The employment of the a priori date on the originalfunction in this formula is a kind of regularization of the ill-posed, in general, problem on inversion MMSE Journal. Open Access www.mmse.xyz

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A Numerical-Analytical Approach to the Analysis of Non-Stationary Temperature Fields by MMSE Journal - Issuu