×

The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. (English) Zbl 1194.80101

Summary: This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady-state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis, thereby transforming the problem into a fourth-order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L-curve criterion. Numerical results are presented for several two-dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady-state heat conduction.

MSC:

80A23 Inverse problems in thermodynamics and heat transfer
80A20 Heat and mass transfer, heat flow (MSC2010)
80M25 Other numerical methods (thermodynamics) (MSC2010)
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Inverse Source Problems. American Mathematical Society: Providence, Rhode-Island; 1989.
[2] El Badia, Inverse Problems 14 pp 883– (1998)
[3] El Badia, Inverse Problems 21 pp 1487– (2005) · Zbl 1086.35133
[4] Ling, Inverse Problems in Science and Engineering 13 pp 433– (2005)
[5] Kagawa, Inverse Problems in Engineering 1 pp 247– (1995)
[6] Sun, IEEE Transactions on Magnetics 33 pp 1970– (1997)
[7] Trlep, IEEE Transactions on Magnetics 36 pp 1649– (2000)
[8] , . Source identification using boundary element method with dual reciprocity method. In Advances in Boundary Element Techniques IV, (eds). University of London: Queen Mary, 2003; 177–182.
[9] , . Identifications of source distributions using BEM with dual reciprocity method. In Inverse Problems in Engineering Mechanics IV, (ed.). Elsevier Science: Amsterdam, New York, 2003; 127–135. · doi:10.1016/B978-008044268-6/50018-1
[10] Farcas, Inverse Problems in Engineering 11 pp 123– (2003)
[11] El Badia, Inverse Problems in Engineering 8 pp 345– (2000) · Zbl 1028.90053
[12] . The method of fundamental solution for potential, Helmholtz and diffusion problems. In Boundary Integral Methods–Numerical and Mathematical Aspects, (ed.). Computational Mechanics Publications: Southampton, 1998; 103–176.
[13] Fairweather, Advances in Computational Mathematics 9 pp 69– (1998)
[14] Cho, Computers, Materials and Continua 1 pp 1– (2004)
[15] Marin, Computers and Structures 83 pp 267– (2005)
[16] Marin, Applied Mathematics and Computation 165 pp 355– (2005)
[17] Jin, Computer Methods in Applied Mechanics and Engineering 195 pp 2270– (2006)
[18] Jin, International Journal for Numerical Methods in Engineering 65 pp 1865– (2006)
[19] Marin, International Journal of Solids and Structures 41 pp 3425– (2004)
[20] Marin, Computers and Mathematics with Applications 50 pp 73– (2005)
[21] Hon, Engineering Analysis with Boundary Elements 28 pp 489– (2004)
[22] Hon, CMES–Computer Modeling in Engineering and Science 7 pp 119– (2005)
[23] El Badia, Inverse Problems 16 pp 651– (2000) · Zbl 0960.35109
[24] Sober, British Journal of Philosophy Science 32 pp 145– (1981)
[25] Wood, International Communications in Heat and Mass Transfer 22 pp 99– (1995)
[26] Kupradze, USSR Computational Mathematics and Mathematical Physics 4 pp 82– (1964)
[27] Mathon, SIAM Journal on Numerical Analysis 14 pp 638– (1977)
[28] Fairweather, Engineering Analysis with Boundary Elements 27 pp 759– (2003)
[29] Fundamental Solutions for Differential Operators and Applications. Birkhäuser: Boston, 1996. · doi:10.1007/978-1-4612-4106-5
[30] Cheng, Engineering Analysis with Boundary Elements 14 pp 187– (1994)
[31] Poullikkas, Computational Mechanics 21 pp 416– (1998)
[32] Mitic, Engineering Analysis with Boundary Elements 28 pp 143– (2004)
[33] Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion. SIAM: Philadelphia, 1998. · Zbl 0890.65037 · doi:10.1137/1.9780898719697
[34] Hansen, SIAM Journal on Scientific Computing 14 pp 1487– (1993)
[35] Chen, Soil Dynamics and Earthquake Engineering 14 pp 361– (1995)
[36] Hanke, BIT Numerical Mathematics 36 pp 287– (1996)
[37] Vogel, Inverse Problems 12 pp 535– (1996)
[38] . Numerical aspects in locating the corner of the L-curve. In Approximation, Optimization and Mathematical Economics, (ed.). Springer-Verlag: Heidelberg, 2001; 121–131. · Zbl 0989.65053 · doi:10.1007/978-3-642-57592-1_11
[39] Kaufman, IEEE Transactions on Medical Imaging 15 pp 385– (1996)
[40] Castellanos, Applied Numerical Mathematics 43 pp 359– (2002)
[41] Ramachandran, Communications in Numerical Methods in Engineering 18 pp 789– (2002)
[42] Jin, CMES–Computer Modeling in Engineering and Science 6 pp 253– (2004)
[43] Karageorghis, Numerical Methods for Partial Differential Equations 8 pp 1– (1992) · Zbl 0760.65103
[44] Marengo, Journal of Optical Society of America 17 pp 34– (2000)
[45] Hanke, Surveys on Mathematics for Industry 3 pp 253– (1993)
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.