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An algorithm for solving the boundary value problem of radiation heat transfer without boundary conditions for radiation intensity. (Russian. English summary) Zbl 1454.35149

Summary: An optimization algorithm for solving the boundary value problem for the stationary equations of radiation-conductive heat transfer in the threedimensional 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.

MSC:

35J61 Semilinear elliptic equations
35Q79 PDEs in connection with classical thermodynamics and heat transfer

Software:

FEniCS; DOLFIN
PDFBibTeX XMLCite
Full Text: MNR

References:

[1] R. Pinnau, “Analysis of Optimal Boundary Control for Radiative Heat Transfer Modelled by the \(SP_1\)-System”, Comm. Math. Sci., 5:4 (2007), 951-969 · Zbl 1145.49015 · doi:10.4310/CMS.2007.v5.n4.a11
[2] A. E. Kovtanyuk, A. Yu. Chebotarev, “An iterative method for solving a complex heat transfer problem”, Appl. Math. Comput., 219 (2013), 9356-9362 · Zbl 1290.80013
[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 · Zbl 1343.35230 · doi:10.1016/j.jmaa.2013.07.054
[4] Comput. Math. Math. Phys., 54:4 (2014), 719-726 · Zbl 1313.80005 · doi:10.1134/S0965542514040095
[5] Kovtanyuk A.E., Chebotarev A.Yu., “Statsionarnaya zadacha svobodnoi konvektsii s radiatsionnym teploobmenom”, Differentsialnye uravneniya, 50:12 (2014), 1590-1597 · Zbl 1316.35241 · doi:10.1134/S0374064114120036
[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-/1/2014 · Zbl 1337.49006 · doi:10.1016/j.jmaa.2013.11.003
[7] G. V. Grenkin, A. Yu. Chebotarev, “Nestatsionarnaya zadacha slozhnogo teploobmena”, Zh. vychisl. matem. fiz., 54:11 (2014), 1806-1816 · Zbl 1331.80004 · doi:10.7868/S0044466914110064
[8] Kovtanyuk A.E., Chebotarev A.Yu., Botkin N.D., and Hoffman Karl-Heinz, “Unique solvability of a steady-state complex heat transfer model”, Commun. Nonlinear Sci. Numer. Simulat., 20 (2015), 776-784 · Zbl 1308.80002 · doi:10.1016/j.cnsns.2014.06.040
[9] G. V. Grenkin, A. Yu. Chebotarev, “Neodnorodnaya nestatsionarnaya zadacha slozhnogo teploobmena”, Sibirskie elektronnye matematicheskie izvestiya, 12:11 (2015), 562-576 · Zbl 1342.35146
[10] G. V. Grenkin, A. Yu. Chebotarev, “Nestatsionarnaya zadacha svobodnoi konvektsii s radiatsionnym teploobmenom”, Zh. vychisl. matem. fiz., 56:2 (2016), 275-282 · Zbl 1344.35100 · doi:10.7868/S0044466916020101
[11] Chebotarev A., Kovtanyuk A., Grenkin G., Botkin N., and Hoffman K.-H. “Boundary optimal control problem of complex heat transfer model”, J. Math. Anal. Appl., 433:2 (2016), 1243-1260 · Zbl 1326.35412 · doi:10.1016/j.jmaa.2015.08.049
[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 · Zbl 1337.80004 · doi:10.1016/j.jmaa.2016.03.016
[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 · Zbl 1410.49004
[14] A. E. Kovtanyuk, A. Yu. Chebotarev, “Nelokalnaya odnoznachnaya razreshimost statsionarnoi zadachi slozhnogo teploobmena”, Zh. vychisl. matem. fiz., 56:5 (2016), 816-823 · Zbl 1366.35191 · doi:10.7868/S0044466916050112
[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 · Zbl 1387.35122 · doi:10.1051/m2an/2017042
[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 Science and Numerical Simulation, 57 (2018), 290-298 · Zbl 1478.35235 · doi:10.1016/j.cnsns.2017.10.004
[17] A. Yu. Chebotarev, G. V. Grenkin, A. E. Kovtanyuk, N. D. Botkin, K.-H. Hoffmann, “Inverse problem with finite overdetermination for steady-state equations of radiative heat exchange”, J. Math. Anal. Appl., 460:2 (2018), 737-744 · Zbl 1478.35235 · doi:10.1016/j.jmaa.2017.12.015
[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 · Zbl 1427.35348 · doi:10.1016/j.jmaa.2018.11.026
[19] G. V. Grenkin, A. Yu. Chebotarev, “Obratnaya zadacha dlya uravnenii slozhnogo teploobmena”, Zh. vychisl. matem. i matem. fiz., 59:8 (2019), 1420-1430 · Zbl 1427.80017 · doi:10.1134/S0044466919080088
[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 · Zbl 1508.80003 · doi:10.1016/j.cnsns.2019.01.028
[21] A. G. Kolobov, T. V. Pak, A. Yu. Chebotarev, “Statsionarnaya zadacha radiatsionnogo teploobmena s granichnymi usloviyami tipa Koshi”, Zh. vychisl. matem. i matem. fiz., 59:7 (2019), 1258-1263 · Zbl 1433.35058 · doi:10.1134/S004446691907010X
[22] Differ. Equ., 41:1 (2005), 96-109 · Zbl 1081.35052 · doi:10.1007/s10625-005-0139-9
[23] A. A. Amosov, “Stationary nonlinear nonlocal problem of radiative{\textendash}conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency”, Journal of Mathematical Sciences, 164 (2010), 309-344 · Zbl 1402.80001 · doi:10.1007/s10958-009-9750-2
[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 · Zbl 1362.35302 · doi:10.1134/S106192081603002X
[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 (201), 618-646 · Zbl 1458.35124 · doi:10.1007/s10958-017-3440-2
[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 · Zbl 1441.78010 · doi:10.1007/s10958-019-04633-y
[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 · Zbl 1436.35031 · doi:10.1007/s10958-019-04623-0
[28] S. Fučik, A. Kufner, Nonlinear differential equations, Elsevier, Amsterdam-Oxford-New York, 1980 · Zbl 0426.35001
[29] Fursikov A. V., Optimal Control of Distributed Systems. Theory and Applications, American Mathematical Soc., 1999 · Zbl 0938.93003
[30] Martin S. Alnæs 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, v. 10, Lecture Notes in Computational Science and Engineering, 84, eds. Anders Logg and Kent-Andre Mardal and Garth N. Wells, Springer, 2012, 173-225 · Zbl 1247.65105 · doi:10.1007/978-3-642-23099-8_10
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