Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T15:06:16.607Z Has data issue: false hasContentIssue false
  • English
  • Français

A Moving Pseudo-Boundary MFS for Three-Dimensional Void Detection

Published online by Cambridge University Press:  03 June 2015

Andreas Karageorghis ,
Daniel Lesnic  and
Liviu Marin
Andreas Karageorghis*
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Daniel Lesnic*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Liviu Marin*
Affiliation:
Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, P.O.Box 1-863, 010141 Bucharest, and Centre for Continuum Mechanics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014 Bucharest, Romania
*
Corresponding author. Email: [email protected]
Email: [email protected]
Email: [email protected]
Get access

Abstract

We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a three-dimensional void (rigid inclusion or cavity) within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary. The algorithm for imaging the interior of the medium also makes use of radial spherical parametrization of the unknown star-shaped void and its centre in three dimensions. We also include the contraction and dilation factors in selecting the fictitious surfaces where the MFS sources are to be positioned in the set of unknowns in the resulting regularized nonlinear least-squares minimization. The feasibility of this new method is illustrated in several numerical examples.

Keywords

65N3565N2165N38Void detectioninverse problemmethod of fundamental solutions
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 510 - 527
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alessandrini, G. and DiBenedetto, E., Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability, Indiana Univ. Math. J., 46 (1997), pp. 1–82.Google Scholar
[2]Alves, C. J. S., On the choice of source points in the method of fundamental solutions, Eng. Anal. Boundary Elements, 33 (2009), pp. 1348–1361.Google Scholar
[3]Alves, C. J. S. and Martins, N. F. M., Reconstruction of inclusions or cavities in potential problems using the MFS, In: The Method of Fundamental Solutions–A Meshless Method, (Chen, C.S., Karageorghis, A. and Smyrlis, Y. S., eds.), Dynamic Publishers Inc., Atlanta, 2008, pp. 51–73.Google Scholar
[4]Borman, D., Ingham, D. B., Johansson, B. T., and Lesnic, D., The method of fundamental solutions for detection of cavities in EIT, J. Integral Equations Appl., 21 (2009), pp. 381–404.Google Scholar
[5]Coleman, T. F. and Li, Y., On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds, Math. Programming, 67 (1994), pp. 189–224.CrossRefGoogle Scholar
[6]Coleman, T. F. and Li, Y., An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optimiz., 6 (1996), pp. 418–445.Google Scholar
[7]Das, S. and Mitra, A., An algorithm for the solution of inverse Laplace problems and its application in flaw identification in materials, J. Comput. Phys., 99 (1992), pp. 99–105.Google Scholar
[8]Divo, E. A., Kassab, A. J., and Rodríguez, F., An efficient singular superposition technique for cavity detection and shape optimization, Numer. Heat Transfer, Part B Fundamentals, 46 (2004), pp. 1–30.Google Scholar
[9]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.CrossRefGoogle Scholar
[10]Fairweather, G. and Johnston, R. L., The method of fundamental solutions for problems in potential theory, In: Treatment of Integral Equations by Numerical Methods, (Baker, C.T.H. and Miller, G. F., eds.), Academic Press, London, 1982, pp. 349–359.Google Scholar
[11]Gorzelańczyk, P. and Kolodziej, J.A., Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods, Eng. Anal. Boundary Elements, 32 (2008), pp. 64–75.Google Scholar
[12]Haddar, H. and Kress, R., Conformal mappings and inverse boundary value problems, Inverse Problems, 21 (2005), pp. 935–953.CrossRefGoogle Scholar
[13]Hansen, P. C. and O’leary, D.P., The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), pp. 1487–1503.CrossRefGoogle Scholar
[14]Hansen, P. C., Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, 2010.CrossRefGoogle Scholar
[15]Hsieh, C. K. and Kassab, A. J., A general method for the solution of inverse heat conduction problems with partially unknown system geometries, Int. J. Heat Mass Transfer, 29 (1985), pp. 47–58.Google Scholar
[16]Huang, C. H. and Chao, B.-H., Inverse geometry problem in identifying irregular boundary configurations, Int. J. Heat Mass Transfer, 40 (1997), pp. 2045–2053.CrossRefGoogle Scholar
[17]Johnston, R. L. and Fairweather, G., The method of fundamental solutions for problems in potential flow, Appl. Math. Modelling, 8 (1984), pp. 265–270.Google Scholar
[18]Karageorghis, A. and Lesnic, D., Detection of cavities using the method of fundamental solutions, Inverse Problems Sci. Eng., 17 (2009), pp. 803–820.CrossRefGoogle Scholar
[19]Karageorghis, A. and Lesnic, D., The method of fundamental solutions for the inverse conductivity problem, Inverse Problems Sci. Eng., 18 (2010), pp. 567–583.Google Scholar
[20]Karageorghis, A., Lesnic, D., and Marin, L., A moving pseudo-boundary MFS for void detection, Numer. Methods Partial Differential Equations, 29 (2013), pp. 935–960.CrossRefGoogle Scholar
[21]Karageorghis, A., Lesnic, D., and Marin, L., The MFS for inverse geometric problems, In: Inverse Problems and Computational Mechanics (Marin, L., Munteanu, L., and Chiroiu, V., eds.), vol. 1, Editura Academiei Române, Bucharest, 2011, pp. 191–216.Google Scholar
[22]Karageorghis, A., Lesnic, D., and Marin, L., A survey of applications of the MFS to inverse problems, Inverse Problems Sci. Eng., 19 (2011), pp. 309–336.Google Scholar
[23]Kassab, A. J. and Hsieh, C. K., Application of infrared scanners and inverse heat conduction problems to infrared computerized axial tomography, Rev. Scientific Instruments, 58 (1987), pp. 89–95.Google Scholar
[24]Kassab, A. J. and Pollard, J., Automated algorithm for the nondestructive detection of subsurface cavities by the IR–CAT method, J. Nondestructive Evaluation, 12 (1993), pp. 175–187.Google Scholar
[25]Kassab, A. J. and Pollard, J., Automated cubic spline anchored grid pattern algorithm for the high resolution detection of subsurface cavities by the IR–CAT method, Numer. Heat Transfer, Part B Fundamentals, 26 (1994), pp. 63–78.CrossRefGoogle Scholar
[26]Mathon, R. and Johnston, R. L., The approximate solution of elliptic boundary–value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), pp. 638–650.Google Scholar
[27]The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA, Matlab.Google Scholar
[28]Serranho, P., A hybrid method for inverse scattering for sound-soft obstacles in R3, Inverse Problems Imaging, 1 (2007), pp. 691–712.CrossRefGoogle Scholar