تجزیه و تحلیل حساسیت و بهینه سازی شکلی برای انتقال حرارت گذرا همراه با تابش

Transient heat conduction problem is stated by the differential heat conduction equation, thermal boundary conditions on the external and internal boundary portions and the initial condition within the domain. Next an arbitrary behavioral functional is defined and its first-order sensitivities are determined using the material derivative concept as well as both the direct and adjoint approaches. The most used shape domain modifications are discussed in order to investigate the effect of design parameters on the integral radiation condition. The shape optimization problem is next formulated applying the obtained sensitivities. The illustration is the simple example of the shape optimization.

Radiative heat transfer is the important fundamental phenomena existing in practical engineering. The examples are the solar radiation in buildings, foundry engineering and solidification processes, die forging, chemical engineering, composite structures applied in industry. The physical analysis demonstrates that the radiative heat transfer problems are encountered as well in textiles (i.e. industrial textiles, textiles designed for use under hermetic protective barrier, multilayer clothing materials, etc.) as in textile structures (i.e. needle heating in heavy industrial sewing). Each of the above-mentioned radiative problems is the particular case characterized by a set of governing equations. Dems and Korycki [4] discuss some of these problems and give a short review of literature. The radiation within the hole is described here by the non-local integral condition according to Bialecki et al. [1] and [2]. The result is an integral equation describing the radiation intensification (caused by the reflected radiation) and absorbing of the radiation within the isothermal and participating medium. These problems can be solved using different methods (cf. [15]). Roche and Sokolowski [14] gave also more information concerning numerical methods applying in optimization practice. The presented paper is an extension of the steady problems stated and discussed by Dems and Korycki [4]. Other best general references here can be Dems and Mróz [3], Dems and Rousselet [6] and [7], and Korycki [10], [11], [12] and [13]. The first-order sensitivities of an arbitrary behavioral functional will be formulated as a function of the transformation velocity field and solutions of primary, direct and adjoint heat transfer problems, cf. [9]. The aim of this paper is to introduce the first-order sensitivities of an arbitrary behavioral functional to the shape design problems associated with the radiative heat transfer. A much more general modeling of transient conduction problems is considered here in view of radiative heat transfer on both the external and internal boundaries described by different conditions. These problems were not yet considered in the analyzed literature for the integral formulation of the radiation condition in transient problems.

The aim of this paper was to present the application of direct and adjoint approaches to sensitivity analysis in the shape optimization problems with radiation on the internal and external boundary portions. The first-order sensitivity vectors were formulated using as well the material derivative concept as direct and adjoint approaches to sensitivity analysis. The direct method is convenient for the low number of functionals, because the number of additional solutions is equal to number of functionals. The disadvantage is in this case the complicated form of the integral radiation condition. The problem is easier to solve for the simplified form of equation (i.e. position independent radiation properties and stationary shape of the hole). The adjoint method is useful for the low number of objective functionals, but is requires the transformation of time. The radiation equation has the same form as for the primary structure and is more convenient to calculate the radiative heat flux density. We introduce also an effective tool for generating the optimal boundary shapes for a wide class of design and redesign problems with radiative heat transfer. Thus, the obtained first-order sensitivity expressions can be also applied to solve the shape identification problems, respectively. Obtained results can be verified by differential numerical methods. The detailed analysis of such implementations and their efficiency and accuracy is beyond the scope of this paper and will be studied in details in consecutive paper.