Numerical Solution of the Inverse Stefan Problem in the Analysis of the Artificial Freezing of a Rock Mass
The problem of parametrization of a heat transfer model in a rock mass in the conditions of its artificial freezing is considered. To clarify the parameters of the model according to the data of measurements of the temperature of the rock mass in the control-thermal wells, it is proposed to solve St...
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| Veröffentlicht in: | Mathematical models and computer simulations Jg. 13; H. 5; S. 877 - 886 |
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Pleiades Publishing
01.09.2021
Springer Nature B.V |
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| Abstract | The problem of parametrization of a heat transfer model in a rock mass in the conditions of its artificial freezing is considered. To clarify the parameters of the model according to the data of measurements of the temperature of the rock mass in the control-thermal wells, it is proposed to solve Stefan’s coefficient inverse problem. In this study, the formulation of the inverse problem is considered and a numerical algorithm for its solution is proposed and implemented. The numerical algorithm is based on the iterative minimization of the smoothing functional of the mismatch between the measured and calculated temperatures in the control-thermal wells. The properties of the smoothing functional in the phase space of the thermophysical properties of the rock mass and the features of the choice of the parameters of the smoothing functional are investigated. |
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| AbstractList | The problem of parametrization of a heat transfer model in a rock mass in the conditions of its artificial freezing is considered. To clarify the parameters of the model according to the data of measurements of the temperature of the rock mass in the control-thermal wells, it is proposed to solve Stefan’s coefficient inverse problem. In this study, the formulation of the inverse problem is considered and a numerical algorithm for its solution is proposed and implemented. The numerical algorithm is based on the iterative minimization of the smoothing functional of the mismatch between the measured and calculated temperatures in the control-thermal wells. The properties of the smoothing functional in the phase space of the thermophysical properties of the rock mass and the features of the choice of the parameters of the smoothing functional are investigated. |
| Author | Semin, M. A. Levin, L. Yu Zaitsev, A. V. |
| Author_xml | – sequence: 1 givenname: M. A. surname: Semin fullname: Semin, M. A. email: seminma@inbox.ru organization: Mining Institute, Ural Branch, Russian Academy of Sciences – sequence: 2 givenname: A. V. surname: Zaitsev fullname: Zaitsev, A. V. organization: Mining Institute, Ural Branch, Russian Academy of Sciences – sequence: 3 givenname: L. Yu surname: Levin fullname: Levin, L. Yu organization: Mining Institute, Ural Branch, Russian Academy of Sciences |
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| Cites_doi | 10.1134/S1062739119015419 10.1007/978-3-642-76436-3 10.1016/j.camwa.2006.12.061 10.1007/s10891-018-1782-3 10.1680/geng.10.00099 10.1007/BF01403678 10.1137/0710047 |
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| Copyright | Pleiades Publishing, Ltd. 2021. ISSN 2070-0482, Mathematical Models and Computer Simulations, 2021, Vol. 13, No. 5, pp. 877–886. © Pleiades Publishing, Ltd., 2021. Russian Text © The Author(s), 2021, published in Matematicheskoe Modelirovanie, 2021, Vol. 33, No. 2, pp. 93–108. |
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| Keywords | inverse Stefan problem artificial ground freezing mathematical model Tikhonov regularization frozen wall model parameterization |
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| References | Man’kovskiiG. I.Special Methods for the Construction of Mine Shafts1965MoscowNauka A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, 1977). O. S. Parshakov, “Development of an automated system for thermometric control of ice walls,” Cand. Sci. (Tech. Sci.) Dissertation (Mining Inst. Ural Branch Russ. Acad. Sci., Perm, 2020) [in Russian]. O. M. Alifanov, Inverse Heat Transfer Problems (Mashinostroenie, Moscow, 1988; Springer, Berlin, 1994). Gol’dmanN. L.Inverse Stefan Problems: Theory and Solution Methods1999MoscowMosk. Gos. Univ. MeyerG. H.Multidimensional Stefan problemsSIAM J. Numer. Anal.19731052253833180710.1137/0710047 LevinL. Yu.SeminM. A.ZaitsevA. V.Solution of an inverse Stefan problem in analyzing the freezing of groundwater in a rock massJ. Eng. Phys. Thermophys.20189161161810.1007/s10891-018-1782-3 SłotaD.Direct and inverse one-phase Stefan problem solved by the variational iteration methodComput. Math. Appl.20075411391146239813610.1016/j.camwa.2006.12.061 Code of Regulations SP 45.13330.2012. Earthworks, Grounds and Footings. Updated Edition of SNiP 3.02.01-87 (Minregion Rossii, Moscow, 2012) [in Russian]. LevinL. Yu.SeminM. A.ZaitsevA. V.Adjustment of thermophysical rock mass properties in modeling frozen wall formation in mine shafts under constructionJ. Min. Sci.20195515716810.1134/S1062739119015419 DolgovO. A.Freezing of Rocks in Mine Shaft Sinking1961MoscowAkad. Nauk SSSR PapakonstantinouS.AnagnostouG.PimentelE.Evaluation of ground freezing data from the Naples subwayProc. Inst. Civ. Eng.: Geotech. Eng.201316628029810.1680/geng.10.00099 TrupakN. G.Freezing of Rocks in Shaft Sinking1954MoscowUgletekhizdat Code of Regulations SP 103.13330.2012. Protection of mines against ground or surface water. Updated Edition of SNiP 2.06.14-85 (Minregion Rossii, Moscow, 2012) [in Russian]. M. S. Zhelnin, O. A. Plekhov, M. A. Semin, and L. Yu. Levin, “Numerical solution for an inverse problem about determination of volumetric heat capacity of rock mass during artificial freezing,” Vestn. Permsk. Nats. Issled. Politekh. Univ. Mekh., No. 4, 56–75 (2017). JochumP.The numerical solution of the inverse Stefan problemNumer. Math.19803441142957740710.1007/BF01403678 4281_CR10 4281_CR11 4281_CR7 4281_CR15 S. Papakonstantinou (4281_CR5) 2013; 166 D. Słota (4281_CR13) 2007; 54 L. Yu. Levin (4281_CR9) 2018; 91 L. Yu. Levin (4281_CR6) 2019; 55 P. Jochum (4281_CR14) 1980; 34 O. A. Dolgov (4281_CR8) 1961 G. H. Meyer (4281_CR16) 1973; 10 N. G. Trupak (4281_CR1) 1954 G. I. Man’kovskii (4281_CR2) 1965 N. L. Gol’dman (4281_CR12) 1999 4281_CR3 4281_CR4 |
| References_xml | – reference: Man’kovskiiG. I.Special Methods for the Construction of Mine Shafts1965MoscowNauka – reference: O. M. Alifanov, Inverse Heat Transfer Problems (Mashinostroenie, Moscow, 1988; Springer, Berlin, 1994). – reference: LevinL. Yu.SeminM. A.ZaitsevA. V.Solution of an inverse Stefan problem in analyzing the freezing of groundwater in a rock massJ. Eng. Phys. Thermophys.20189161161810.1007/s10891-018-1782-3 – reference: JochumP.The numerical solution of the inverse Stefan problemNumer. Math.19803441142957740710.1007/BF01403678 – reference: TrupakN. G.Freezing of Rocks in Shaft Sinking1954MoscowUgletekhizdat – reference: SłotaD.Direct and inverse one-phase Stefan problem solved by the variational iteration methodComput. Math. Appl.20075411391146239813610.1016/j.camwa.2006.12.061 – reference: DolgovO. A.Freezing of Rocks in Mine Shaft Sinking1961MoscowAkad. Nauk SSSR – reference: Code of Regulations SP 45.13330.2012. Earthworks, Grounds and Footings. Updated Edition of SNiP 3.02.01-87 (Minregion Rossii, Moscow, 2012) [in Russian]. – reference: Code of Regulations SP 103.13330.2012. Protection of mines against ground or surface water. Updated Edition of SNiP 2.06.14-85 (Minregion Rossii, Moscow, 2012) [in Russian]. – reference: O. S. Parshakov, “Development of an automated system for thermometric control of ice walls,” Cand. Sci. (Tech. Sci.) Dissertation (Mining Inst. Ural Branch Russ. Acad. Sci., Perm, 2020) [in Russian]. – reference: MeyerG. H.Multidimensional Stefan problemsSIAM J. Numer. Anal.19731052253833180710.1137/0710047 – reference: A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, 1977). – reference: Gol’dmanN. L.Inverse Stefan Problems: Theory and Solution Methods1999MoscowMosk. Gos. Univ. – reference: LevinL. Yu.SeminM. A.ZaitsevA. V.Adjustment of thermophysical rock mass properties in modeling frozen wall formation in mine shafts under constructionJ. Min. Sci.20195515716810.1134/S1062739119015419 – reference: M. S. Zhelnin, O. A. Plekhov, M. A. Semin, and L. Yu. Levin, “Numerical solution for an inverse problem about determination of volumetric heat capacity of rock mass during artificial freezing,” Vestn. Permsk. Nats. Issled. Politekh. Univ. Mekh., No. 4, 56–75 (2017). – reference: PapakonstantinouS.AnagnostouG.PimentelE.Evaluation of ground freezing data from the Naples subwayProc. Inst. Civ. Eng.: Geotech. Eng.201316628029810.1680/geng.10.00099 – ident: 4281_CR15 – volume: 55 start-page: 157 year: 2019 ident: 4281_CR6 publication-title: J. Min. Sci. doi: 10.1134/S1062739119015419 – ident: 4281_CR7 – ident: 4281_CR11 – volume-title: Inverse Stefan Problems: Theory and Solution Methods year: 1999 ident: 4281_CR12 – ident: 4281_CR10 doi: 10.1007/978-3-642-76436-3 – volume: 54 start-page: 1139 year: 2007 ident: 4281_CR13 publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2006.12.061 – volume-title: Freezing of Rocks in Shaft Sinking year: 1954 ident: 4281_CR1 – volume: 91 start-page: 611 year: 2018 ident: 4281_CR9 publication-title: J. Eng. Phys. Thermophys. doi: 10.1007/s10891-018-1782-3 – volume: 166 start-page: 280 year: 2013 ident: 4281_CR5 publication-title: Proc. Inst. Civ. Eng.: Geotech. Eng. doi: 10.1680/geng.10.00099 – volume: 34 start-page: 411 year: 1980 ident: 4281_CR14 publication-title: Numer. Math. doi: 10.1007/BF01403678 – volume: 10 start-page: 522 year: 1973 ident: 4281_CR16 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0710047 – ident: 4281_CR3 – ident: 4281_CR4 – volume-title: Special Methods for the Construction of Mine Shafts year: 1965 ident: 4281_CR2 – volume-title: Freezing of Rocks in Mine Shaft Sinking year: 1961 ident: 4281_CR8 |
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| SubjectTerms | Algorithms Freezing Inverse problems Iterative methods Mathematical Modeling and Industrial Mathematics Mathematical models Mathematics Mathematics and Statistics Numerical analysis Parameterization Parameters Rock masses Simulation and Modeling Smoothing Thermophysical properties |
| Title | Numerical Solution of the Inverse Stefan Problem in the Analysis of the Artificial Freezing of a Rock Mass |
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