Marin, Liviu Landweber-Fridman algorithms for the Cauchy problem in steady-state anisotropic heat conduction. (English) Zbl 1482.74065 Math. Mech. Solids 25, No. 6, 1340-1363 (2020). Summary: We investigate the numerical reconstruction of the missing thermal boundary conditions on an inaccessible part of the boundary in the case of steady-state heat conduction in anisotropic solids from the knowledge of over-prescribed noisy data on the remaining accessible boundary. This inverse problem is tackled by employing a variational formulation that transforms it into an equivalent control problem; four such approaches are discussed thoroughly. The numerical implementation is realised for the 2D case via the boundary element method for perturbed Cauchy data, whilst the numerical solution is stabilised/regularised by stopping the iterative procedure according to V. A. Morozov’s discrepancy principle [Sov. Math., Dokl. 7, 414–417 (1966; Zbl 0187.12203); translation from Dokl. Akad. Nauk SSSR 167, 510–512 (1966)]. Cited in 7 Documents MSC: 74F05 Thermal effects in solid mechanics 74G75 Inverse problems in equilibrium solid mechanics 74E10 Anisotropy in solid mechanics 74S15 Boundary element methods applied to problems in solid mechanics 80A23 Inverse problems in thermodynamics and heat transfer Keywords:anisotropic heat conduction; inverse problem; variational formulation; gradient method; regularisation; boundary element method Citations:Zbl 0187.12203 PDFBibTeX XMLCite \textit{L. Marin}, Math. Mech. Solids 25, No. 6, 1340--1363 (2020; Zbl 1482.74065) Full Text: DOI References: [1] Özişik, MN. Heat conduction. New York: John Wiley & Sons, 1993. [2] Keyhani, M, Polehn, RA. Finite difference modeling of anisotropic flows. J Heat Transfer 1995; 117: 458-464. [3] Mera, NS, Elliott, L, Ingham, DB, et al. The boundary element solution for the Cauchy steady heat conduction problem in an anisotropic medium. Int J Numer Methods Eng 2000; 49: 481-499. · Zbl 0974.80004 [4] Mera, NS, Elliott, L, Ingham, DB, et al. A comparison of boundary element formulations for steady state anisotropic heat conduction problems. Eng Anal Boundary Elem 2001; 25: 115-128. · Zbl 0982.80009 [5] Knabner, P, Agermann, L. Numerical methods for elliptic and parabolic partial differential equations. New York: Springer-Verlag, 2003. [6] Szabó, B, Babuška, I. Introduction to finite element analysis. New York: John Wiley & Sons, 2011. · Zbl 1410.65003 [7] Fairweather, G, Karageorghis, A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 1998; 9: 69-95. · Zbl 0922.65074 [8] Marin, L. An alternating iterative MFS algorithm for the Cauchy problem in two-dimensional anisotropic heat conduction. CMC Comput Mater Con 2009; 12: 71-100. [9] Hadamard, J. Lectures on Cauchy’s problem in linear partial differential equations. New Haven: Yale University Press, 1923. · JFM 49.0725.04 [10] Kozlov, VA, Mazya, VG, Fomin, AV. An iterative method for solving the Cauchy problem for elliptic equations. Zh Vychisl Mat Mat Fiz 1991; 31: 64-74, 1991. English translation: USSR Comput Math Math Phys 1991; 31: 45-52. [11] Jin, B, Zheng, Y, Marin, L. The method of fundamental solutions for inverse boundary value problems associated with the steady-state heat conduction in anisotropic media. Int J Numer Methods Eng 2006; 65(11): 1865-1891. · Zbl 1124.80400 [12] Marin, L. Stable boundary and internal data reconstruction in two-dimensional anisotropic heat conduction Cauchy problems using relaxation procedures for an iterative MFS algorithm. CMC Comput Mater Con 2010; 17(3): 233-274. [13] Landweber, L. An iteration formula for Fredholm integral equations of the first kind. Am J Math 1951; 73: 615-624. · Zbl 0043.10602 [14] Fridman, VM. Method of successive approximation for a Fredholm integral equation of the first kind. Usp Mat Nauk 1956; 11: 233-234. [15] Johansson, T. Reconstruction of a stationary flow from boundary data. PhD Thesis, Linköping University, Sweden, 2000. [16] Marin, L, Lesnic, D. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA J Appl Math 2005; 70: 323-340. · Zbl 1073.74050 [17] Lions, JL, Magenes, E. Non-homogeneous boundary value problems and their applications. vol. 1. Berlin: Springer-Verlag, 1972. · Zbl 0227.35001 [18] Engl, HW, Hanke, M, Neubauer, A. Regularization of inverse problems. Boston: Kluwer, 1996. · Zbl 0859.65054 [19] Aliabadi, MH. The boundary element method (Applications in Solids and Structures, vol. 2). London: John Wiley & Sons, 2002. · Zbl 0994.74003 [20] Morozov, VA. On the solution of functional equations by the method of regularization. Dokl Math 1966; 7: 414-417. · Zbl 0187.12203 [21] Hanke, M, Hansen, PC. Regularization methods for large-scale problems. Surv Math Ind 1993; 3(4): 253-315. · Zbl 0805.65058 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.