Far Eastern Mathematical Journal

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An algorithm for solving the boundary value problem of radiation heat transfer without boundary conditions for radiation intensity

Chebotarev A.Yu., Mesenev P.R.

2020, issue 1, P. 114–122 DOI: https://doi.org/10.47910/FEMJ202012

Abstract

An optimization algorithm for solving the boundary value problem for the stationary equations of radiation-conductive heat transfer in the three-dimensional region is presented in the framework of the $ P_1 $ - approximation of the radiation transfer equation. The analysis of the optimal control problem that approximates the boundary value problem where they are not defined boundary conditions for radiation intensity. Theoretical analysis is illustrated by numerical examples.

Keywords: equations of radiative heat transfer, diffusion approximation, optimal control problem, Cauchy type conditions, numerical simulation

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References

[1] R. Pinnau, “Analysis of Optimal Boundary Control for Radiative Heat Transfer Modelled by the SP1-System”, Comm. Math. Sci., 5:4, (2007), 951–969. [2] A. E. Kovtanyuk, A. Yu. Chebotarev, “An iterative method for solving a complex heat transfer problem”, Appl. Math. Comput., 219, (2013), 9356–9362. [3] A. E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, K.-H. Hoffmann, “The unique solvability of a complex 3D heat transfer problem”, J. Math. Anal. Appl., 409:2, (2014), 808–815. [4] A. E. Kovtaniuk, A. Iu. Chebotarev, “ Statsionarnaia zadacha slozhnogo teploobmena”, Zh. vychisl. matem. i matem. fiz., 54:4, (2014), 711–719; Comput. Math. Math. Phys., 54:4, (2014), 719–726. [5] Kovtaniuk A.E., Chebotarev A.Iu., “ Statsionarnaia zadacha svobodnoi konvektsii s radiatsionnym teploobmenom” , Differentsial'nye uravneniia, 50:12, (2014), 1590–1597. [6] Kovtanyuk Andrey E., Chebotarev Alexander Yu., Botkin Nikolai D., and Hoffmann Karl- Heinz, “Theoretical analysis of an optimal control problem of conductive convective radiative heat transfer”, J. Math. Anal. Appl., 412, (2014), 520–528. [7] G.V. Grenkin, A. Iu. Chebotarev, “ Nestatsionarnaia zadacha slozhnogo teploobmena”, Zh. vychisl. matem. fiz., 54:11, (2014), 1806–1816. [8] Kovtanyuk A.E., Chebotarev A.Yu., Botkin N.D., and Hoffman Karl-Heinz, “Unique solv- ability of a steady-state complex heat transfer model”, Commun. Nonlinear Sci. Numer. Simulat., 20, (2015), 776–784. [9] G.V. Grenkin, A. Iu. Chebotarev, “Neodnorodnaia nestatsionarnaia zadacha slozhnogo teploobmena”, Sibirskie elektronnye matematicheskie izvestiia, 12:11, (2015), 562–576. [10] G.V. Grenkin, A. Iu. Chebotarev, “Nestatsionarnaia zadacha svobodnoi konvektsii s radiatsionnym teploobmenom” , Zh. vychisl. matem. fiz., 56:2, (2016), 275–282. [11] Chebotarev A., Kovtanyuk A., Grenkin G., Botkin N., and Hoffman K.-H.“Boundary opti- mal control problem of complex heat transfer model”, J. Math. Anal. Appl., 433:2, (2016), 1243–1260. [12] A.E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, K.-H. Hoffmann, “Optimal boundary control of a steady-state heat transfer model accounting for radiative effects”, J. Math. Anal. Appl., 439, (2016), 678–689. [13] Alexander Yu. Chebotarev, Andrey E. Kovtanyuk, Gleb V. Grenkin, Nikolai D. Botkin, and Karl Heinz Hoffmann, “Nondegeneracy of optimality conditions in control problems for a radiative-conductive heat transfer model”, Applied Mathematics and Computation, 289:10, (2016), 371–380. [14] A.E. Kovtaniuk, A.Iu. Chebotarev, “Nelokal'naia odnoznachnaia razreshimost' statsionarnoi zadachi slozhnogo teploobmena” , Zh. vychisl. matem. fiz., 56:5, (2016), 816–823. [15] A.Yu. Chebotarev, G. V. Grenkin, A. E. Kovtanyuk, “Inhomogeneous steady-state problem of complex heat transfer”, ESAIM Math. Model. Numer. Anal., 51:6, (2017), 2511–2519. [16] Alexander Yu. Chebotarev and Gleb V. Grenkin and Andrey E. Kovtanyuk and Nikolai D. Botkin and Karl-Heinz Hoffmann, “Diffusion approximation of the radiative-conductive heat transfer model with Fresnel matching conditions”, Communications in Nonlinear Sci- ence and Numerical Simulation, 57, (2018), 290–298. [17] A.Yu. Chebotarev, G. V. Grenkin, A. E. Kovtanyuk, N. D. Botkin, K.-H. Hoffmann, “In- verse problem with finite overdetermination for steady-state equations of radiative heat exchange”, J. Math. Anal. Appl., 460:2, (2018), 737–744. [18] A. Yu. Chebotarev, R. Pinnau, “An inverse problem for a quasi-static approximate model of radiative heat transfer”, J. Math. Anal. Appl., 472:1, (2019), 737–744. [19] G. V. Grenkin, A. Iu. Chebotarev, “ Obratnaia zadacha dlia uravnenii slozhnogo teploobmena” , Zh. vychisl. matem. i matem. fiz., 59:8, (2019), 1420–1430. [20] Alexander Yu. Chebotarev and Andrey E. Kovtanyuk and Nikolai D. Botkin, “Problem of radiation heat exchange with boundary conditions of the Cauchy type”, Communications in Nonlinear Science and Numerical Simulation, 75, (2019), 262–269. [21] A. G. Kolobov, T.V. Pak, A.Iu. Chebotarev, “ Statsionarnaia zadacha radiatsionnogo teploobmena s granichnymi usloviiami tipa Koshi”, Zh. vychisl. matem. i matem. fiz., 59:7, (2019), 1258–1263. [22] A. A. Amosov, “ Global'naia razreshimost' odnoi nelineinoi nestatsionarnoi za- dachi s nelokal'nym kraevym usloviem tipa teploobmena izlucheniem” , Differents. uravneniia, 41:1, (2005), 93–104; Differ. Equ., 41:1, (2005), 96–109. doi https://doi.org/10.1007/s10625-005-0139-9. [23] A.A. Amosov, “Stationary nonlinear nonlocal problem of radiative–conductive heat trans- fer in a system of opaque bodies with properties depending on the radiation frequency”, Journal of Mathematical Sciences, 164, (2010), 309–344. [24] A.A. Amosov, “Unique solvability of a nonstationary problem of radiative-conductive heat exchange in a system of semitransparent bodies”, Russian Journal of Mathematical Physics, 23:3, (2016), 309–334. [25] A.A. Amosov, “Unique Solvability of Stationary Radiative-Conductive Heat Transfer Problem in a System of Semitransparent Bodies”, Journal of Mathematical Sciences, 224:5, (2017), 618–646. [26] A.A. Amosov, “Asymptotic Behavior of a Solution to the Radiative Transfer Equation in a Multilayered Medium with Diffuse Reflection and Refraction Conditions”, Journal of Mathematical Sciences, 244:4, (2020), 541–575. [27] A.A. Amosov, N. E. Krymov, “On a Nonstandard Boundary Value Problem Arising in Homogenization of Complex Heat Transfer Problems”, Journal of Mathematical Sciences, 244:3, (2020), 357–377. [28] S. Fucik, A. Kufner, Nonlinear differential equations, Elsevier, Amsterdam–Oxford–New York, 1980. [29] Fursikov A.V., Optimal Control of Distributed Systems. Theory and Applications, American Mathematical Soc., 1999. [30] Martin S. Aln?s and Jan Blechta and Johan Hake and August Johansson and Benjamin Kehlet and Anders Logg and Chris Richardson and Johannes Ring and Marie E. Rognes and Garth N. Wells, “The FEniCS Project Version 1.5”, Archive of Numerical Software, 3:100, (2015), 9-23. [31] Anders Logg and Garth N. Wells and Johan Hake, “DOLFIN: a C++/Python Finite Element Library”, Automated Solution of Differential Equations by the Finite Element Method, Volume 84 of Lecture Notes in Computational Science and Engineering, v. 10, eds. Anders Logg and Kent-Andre Mardal and Garth N. Wells, Springer, 2012, 173–225.

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