A moving pseudo-boundary method of fundamental solutions for void detection

We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar...

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Veröffentlicht in:	Numerical methods for partial differential equations Jg. 29; H. 3; S. 935 - 960
Hauptverfasser:	Karageorghis, Andreas, Lesnic, Daniel, Marin, Liviu
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.05.2013 Wiley Subscription Services, Inc
Schlagworte:	Algorithms Imaging inverse problem Inverse problems Mathematical models method of fundamental solutions Nonlinearity Numerical analysis Parametrization void detection Voids
ISSN:	0749-159X, 1098-2426
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Abstract	We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least‐squares problem. This approach addresses the major problem of locating the pseudo‐boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013
AbstractList	We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least‐squares problem. This approach addresses the major problem of locating the pseudo‐boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least-squares problem. This approach addresses the major problem of locating the pseudo-boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [PUBLICATION ABSTRACT] We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least-squares problem. This approach addresses the major problem of locating the pseudo-boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. copyright 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013
Author	Lesnic, Daniel Karageorghis, Andreas Marin, Liviu
Author_xml	– sequence: 1 givenname: Andreas surname: Karageorghis fullname: Karageorghis, Andreas email: andreask@ucy.ac.cy organization: Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus – sequence: 2 givenname: Daniel surname: Lesnic fullname: Lesnic, Daniel organization: Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom – sequence: 3 givenname: Liviu surname: Marin fullname: Marin, Liviu organization: Institute of Solid Mechanics, Romanian Academy, Bucharest 010141, Romania
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Cites_doi	10.1016/0307-904X(84)90161-6 10.1016/0021-9991(87)90176-8 10.1007/BF01582221 10.1216/jiea/1181075363 10.1137/0914086 10.1080/17415970802580263 10.1016/0041-5553(64)90006-0 10.1137/0806023 10.1137/1.9780898718836 10.1002/nme.1620141202 10.1002/nme.1763 10.1088/0266-5611/21/3/009 10.1088/0266-5611/15/5/306 10.1016/j.enganabound.2007.05.004 10.1016/0041-5553(64)90092-8 10.1006/jdeq.2000.3987 10.1093/imanum/9.2.231 10.1002/(SICI)1097-0207(19990420)44:11<1653::AID-NME558>3.0.CO;2-1 10.1137/0714043 10.1080/17415977.2011.551830 10.1016/j.enganabound.2010.09.014 10.1016/j.enganabound.2007.10.011 10.1002/num.10500 10.1016/j.cma.2008.02.011
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References	A. Karageorghis and D. Lesnic, The method of fundamental solutions for steady-state heat conduction in nonlinear materials, Commun Comput Phys 4( 2008), 911-928. T. F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J Optim 6( 1996), 418-445. A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric potential problems, Int J Numer Methods Eng 44( 1999), 1653-1669. R. Tankelevich, G. Fairweather, A. Karageorghis, and Y.-S. Smyrlis, Potential field based geometric modelling using the method of fundamental solutions, Int J Numer Methods Eng 68( 2006), 1257-1280. R. L. Johnston and G. Fairweather, The method of fundamental solutions for problems in potential flow, Appl Math Model 8( 1984), 265-270. R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J Numer Anal 14( 1977), 638-650. Matlab, The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA. A. Karageorghis, D. Lesnic, and L. Marin, A survey of applications of the MFS to inverse problems, Inverse Probl Sci Eng 19( 2011), 309-336. T. F. Coleman and Y. Li, On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds, Math Programming 67( 1994), 189-224. P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J Sci Comput 14( 1993), 1487-1503. V. D. Kupradze, Potential methods in the theory of elasticity, Israel Program for Scientific Translations, Jerusalem, 1965. A. Poullikkas, A. Karageorghis, G. Georgiou, and J. Ascough, The method of fundamental solutions for Stokes flows with a free surface, Numer Methods Partial Differential Equations 14( 1998), 667-678. L. Marin, A. Karageorghis, and D. Lesnic, The MFS for numerical boundary identification in two-dimensional harmonic problems, Eng Anal Bound Elem 35( 2011), 342-354. V. D. Kupradze and M. A. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems, Comput Math Math Phys 4( 1964), 82-126. P. C. Hansen, Discrete inverse problems: insight and algorithms, SIAM, Philadelphia, 2010. M. A. Aleksidze, On the question of a practical application of a new approximation method, Differencial'nye Uravnenija 2( 1966), 1625-1629. P. Gorzelańczyk and J. A. Kołodziej, 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 Bound Elem 37( 2008), 64-75. O. Ivanyshyn and R. Kress, Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks, J Integral Equations Appl 18( 2006), 13-38. V. D. Kupradze, On a method of solving approximately the limiting problems of mathematical physics, Comput Math Math Phys 4( 1964), 199-205. A. Karageorghis and G. Fairweather, The method of fundamental solutions for the numerical solution of the biharmonic equation, J Comput Phys 69( 1987), 434-459. A. Karageorghis and D. Lesnic, Detection of cavities using the method of fundamental solutions, Inverse Probl Sci Eng 17( 2009), 803-820. H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems, Inverse Problems 21( 2005), 935-953. G. Alessandrini and L. Rondi, Optimal stability for the inverse problem of multiple cavities, J Diff Equations 176( 2001), 356-386. N. F. M. Martins and A. L. Silvestre, An iterative MFS approach for the detection of immersed obstacles, Eng Anal Bound Elem 32( 2008), 517-524. R. L. Johnston and R. Mathon, The computation of electric dipole fields in conducting media, Int J Numer Methods Eng 14( 1979), 1739-1760. A. Karageorghis and D. Lesnic, Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput Methods Appl Mech Eng 197( 2008), 3122-3137. A. Karageorghis and G. Fairweather, The method of fundamental solutions for the solution of nonlinear plane potential problems, IMA J Numer Anal 9( 1989), 231-242. B. S. Garbow, K. E. Hillstrom, and J. J. Moré, Minpack project, Argonne National Laboratory, Argonne, Illinois, 1980. 1966; 2 1979; 14 2010 1964; 4 1989; 9 1994; 67 2008; 37 1999; 44 2006; 18 2005; 21 2008; 32 2011; 35 2008; 4 2011; 19 1999 1987; 69 2001; 176 1993; 14 1977; 14 2006; 68 1984; 8 1965 1982 1980 2008; 197 1998; 14 1996; 6 2009; 17 e_1_2_8_27_2 e_1_2_8_28_2 e_1_2_8_29_2 e_1_2_8_23_2 e_1_2_8_24_2 e_1_2_8_25_2 e_1_2_8_26_2 e_1_2_8_9_2 Aleksidze M. A. (e_1_2_8_2_2) 1966; 2 Karageorghis A. (e_1_2_8_14_2) 2008; 4 e_1_2_8_3_2 e_1_2_8_6_2 e_1_2_8_5_2 Fairweather G. (e_1_2_8_7_2) 1982 e_1_2_8_8_2 e_1_2_8_20_2 e_1_2_8_21_2 e_1_2_8_22_2 e_1_2_8_16_2 e_1_2_8_17_2 e_1_2_8_18_2 e_1_2_8_19_2 e_1_2_8_12_2 e_1_2_8_13_2 e_1_2_8_15_2 Kupradze V. D. (e_1_2_8_4_2) 1965 Garbow B. S. (e_1_2_8_32_2) 1980 e_1_2_8_31_2 e_1_2_8_30_2 e_1_2_8_10_2 e_1_2_8_11_2
References_xml	– reference: T. F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J Optim 6( 1996), 418-445. – reference: V. D. Kupradze and M. A. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems, Comput Math Math Phys 4( 1964), 82-126. – reference: G. Alessandrini and L. Rondi, Optimal stability for the inverse problem of multiple cavities, J Diff Equations 176( 2001), 356-386. – reference: R. L. Johnston and G. Fairweather, The method of fundamental solutions for problems in potential flow, Appl Math Model 8( 1984), 265-270. – reference: R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J Numer Anal 14( 1977), 638-650. – reference: B. S. Garbow, K. E. Hillstrom, and J. J. Moré, Minpack project, Argonne National Laboratory, Argonne, Illinois, 1980. – reference: H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems, Inverse Problems 21( 2005), 935-953. – reference: P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J Sci Comput 14( 1993), 1487-1503. – reference: N. F. M. Martins and A. L. Silvestre, An iterative MFS approach for the detection of immersed obstacles, Eng Anal Bound Elem 32( 2008), 517-524. – reference: R. L. Johnston and R. Mathon, The computation of electric dipole fields in conducting media, Int J Numer Methods Eng 14( 1979), 1739-1760. – reference: Matlab, The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA. – reference: A. Karageorghis, D. Lesnic, and L. Marin, A survey of applications of the MFS to inverse problems, Inverse Probl Sci Eng 19( 2011), 309-336. – reference: A. Poullikkas, A. Karageorghis, G. Georgiou, and J. Ascough, The method of fundamental solutions for Stokes flows with a free surface, Numer Methods Partial Differential Equations 14( 1998), 667-678. – reference: T. F. Coleman and Y. Li, On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds, Math Programming 67( 1994), 189-224. – reference: R. Tankelevich, G. Fairweather, A. Karageorghis, and Y.-S. Smyrlis, Potential field based geometric modelling using the method of fundamental solutions, Int J Numer Methods Eng 68( 2006), 1257-1280. – reference: V. D. Kupradze, On a method of solving approximately the limiting problems of mathematical physics, Comput Math Math Phys 4( 1964), 199-205. – reference: L. Marin, A. Karageorghis, and D. Lesnic, The MFS for numerical boundary identification in two-dimensional harmonic problems, Eng Anal Bound Elem 35( 2011), 342-354. – reference: P. C. Hansen, Discrete inverse problems: insight and algorithms, SIAM, Philadelphia, 2010. – reference: V. D. Kupradze, Potential methods in the theory of elasticity, Israel Program for Scientific Translations, Jerusalem, 1965. – reference: A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric potential problems, Int J Numer Methods Eng 44( 1999), 1653-1669. – reference: P. Gorzelańczyk and J. A. Kołodziej, 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 Bound Elem 37( 2008), 64-75. – reference: O. Ivanyshyn and R. Kress, Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks, J Integral Equations Appl 18( 2006), 13-38. – reference: A. Karageorghis and D. Lesnic, The method of fundamental solutions for steady-state heat conduction in nonlinear materials, Commun Comput Phys 4( 2008), 911-928. – reference: A. Karageorghis and G. Fairweather, The method of fundamental solutions for the solution of nonlinear plane potential problems, IMA J Numer Anal 9( 1989), 231-242. – reference: A. Karageorghis and G. Fairweather, The method of fundamental solutions for the numerical solution of the biharmonic equation, J Comput Phys 69( 1987), 434-459. – reference: M. A. Aleksidze, On the question of a practical application of a new approximation method, Differencial'nye Uravnenija 2( 1966), 1625-1629. – reference: A. Karageorghis and D. Lesnic, Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput Methods Appl Mech Eng 197( 2008), 3122-3137. – reference: A. Karageorghis and D. Lesnic, Detection of cavities using the method of fundamental solutions, Inverse Probl Sci Eng 17( 2009), 803-820. – article-title: Matlab publication-title: The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA – volume: 4 start-page: 911 year: 2008 end-page: 928 article-title: The method of fundamental solutions for steady‐state heat conduction in nonlinear materials publication-title: Commun Comput Phys – volume: 14 start-page: 1487 year: 1993 end-page: 1503 article-title: The use of the ‐curve in the regularization of discrete ill‐posed problems publication-title: SIAM J Sci Comput – volume: 14 start-page: 1739 year: 1979 end-page: 1760 article-title: The computation of electric dipole fields in conducting media publication-title: Int J Numer Methods Eng – volume: 4 start-page: 199 year: 1964 end-page: 205 article-title: On a method of solving approximately the limiting problems of mathematical physics publication-title: Comput Math Math Phys – volume: 67 start-page: 189 year: 1994 end-page: 224 article-title: On the convergence of interior‐reflective Newton methods for nonlinear minimization subject to bounds publication-title: Math Programming – volume: 19 start-page: 309 year: 2011 end-page: 336 article-title: A survey of applications of the MFS to inverse problems publication-title: Inverse Probl Sci Eng – volume: 44 start-page: 1653 year: 1999 end-page: 1669 article-title: The method of fundamental solutions for axisymmetric potential problems publication-title: Int J Numer Methods Eng – volume: 6 start-page: 418 year: 1996 end-page: 445 article-title: An interior trust region approach for nonlinear minimization subject to bounds publication-title: SIAM J Optim – volume: 14 start-page: 667 year: 1998 end-page: 678 article-title: The method of fundamental solutions for Stokes flows with a free surface publication-title: Numer Methods Partial Differential Equations – volume: 9 start-page: 231 year: 1989 end-page: 242 article-title: The method of fundamental solutions for the solution of nonlinear plane potential problems publication-title: IMA J Numer Anal – year: 2010 – start-page: 349 year: 1982 end-page: 359 – volume: 17 start-page: 803 year: 2009 end-page: 820 article-title: Detection of cavities using the method of fundamental solutions publication-title: Inverse Probl Sci Eng – volume: 68 start-page: 1257 year: 2006 end-page: 1280 article-title: Potential field based geometric modelling using the method of fundamental solutions publication-title: Int J Numer Methods Eng – volume: 37 start-page: 64 year: 2008 end-page: 75 article-title: Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods publication-title: Eng Anal Bound Elem – volume: 2 start-page: 1625 year: 1966 end-page: 1629 article-title: On the question of a practical application of a new approximation method publication-title: Differencial'nye Uravnenija – volume: 4 start-page: 82 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Snippet	We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as... We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as...
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SubjectTerms	Algorithms Imaging inverse problem Inverse problems Mathematical models method of fundamental solutions Nonlinearity Numerical analysis Parametrization void detection Voids
Title	A moving pseudo-boundary method of fundamental solutions for void detection
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