Green’s Function Technique and Global Optimization in Reconstruction of Elliptic Objects in the Regular Triangle

ABSTRACT

The reconstruction problem for elliptic voids located in the regular (equilateral) triangle is studied. A known point source is applied to the boundary of the domain, and it is assumed that the input data is obtained from the free-surface input data over a certain finite-length interval of the outer boundary. In the case when the boundary contour of the internal object is unknown, we propose a new algorithm to reconstruct its position and size on the basis of the input data. The key specific character of the proposed method is the construction of a special explicit-form Green's function satisfying the boundary condition over the outer boundary of the triangular domain. Some numerical examples demonstrate good stability of the proposed algorithm.

The reconstruction problem for elliptic voids located in the regular (equilateral) triangle is studied. A known point source is applied to the boundary of the domain, and it is assumed that the input data is obtained from the free-surface input data over a certain finite-length interval of the outer boundary. In the case when the boundary contour of the internal object is unknown, we propose a new algorithm to reconstruct its position and size on the basis of the input data. The key specific character of the proposed method is the construction of a special explicit-form Green's function satisfying the boundary condition over the outer boundary of the triangular domain. Some numerical examples demonstrate good stability of the proposed algorithm.

KEYWORDS

Reconstruction, Global Optimization, Green's Function, Triangular Domain, Boundary Integral

Reconstruction, Global Optimization, Green's Function, Triangular Domain, Boundary Integral

Cite this paper

nullA. Scalia and M. Sumbatyan, "Green’s Function Technique and Global Optimization in Reconstruction of Elliptic Objects in the Regular Triangle,"*Applied Mathematics*, Vol. 2 No. 3, 2011, pp. 294-302. doi: 10.4236/am.2011.23034.

nullA. Scalia and M. Sumbatyan, "Green’s Function Technique and Global Optimization in Reconstruction of Elliptic Objects in the Regular Triangle,"

References

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[15] M. Corana et al., “Minimizing Multimodal Functions of Continuous Variables with the Simulated Annealing Algorithm,” ACM Transactions on Mathematical Software, Vol. 13, No. 3, 1987, pp. 262-280. doi:10.1145/29380.29864

[1] A. Friedman and M. Vogelius, “Determining Cracks by Boundary Measurements,” Indiana University Mathematics Journal, Vol. 38, No. 3, 1989, pp. 527-556. doi:10.1512/iumj.1989.38.38025

[2] G. Alessandrini, E. Beretta and S. Vessella, “Determining Linear Cracks by Boundary Measurements: Lipschitz Stability,” SIAM Journal on Mathematical Analysis, Vol. 27, No. 2, 1996, pp. 361-375. doi:10.1137/S0036141094265791

[3] A. B. Abda et al., “Line Segment Crack Recovery from Incomplete Boundary Data,” Inverse Problems, Vol. 18, No. 4, 2002, pp. 1057-1077. doi:10.1088/0266-5611/18/4/308

[4] S. Andrieux and A. B. Abda, “Identification of Planar Cracks by Complete Overdetermined Data: Inversion Formulae,” Inverse Problems, Vol. 12, No. 5, 1996, pp. 553-563. doi:10.1088/0266-5611/12/5/002

[5] T. Bannour, A. B. Abda and M. Jaoua, “A Semi-Explicit Algorithm for the Reconstruction of 3D Planar Cracks,” Inverse Problems, Vol. 13, No. 4, 1997, pp. 899-917. doi:10.1088/0266-5611/13/4/002

[6] A. S. Saada, “Elasticity: Theory and Applications,” 2nd Edition, Krieger, Malabar, Florida, 1993.

[7] N. I. Muskhelishvili, “Some Basic Problems of the Mathematical Theory of Elasticity,” Kluwer, Dordrecht, 1975.

[8] R. Courant and D. Hilbert, “Methods of Mathematical Physics,” Interscience Publishing, New York, Vol. 1, 1953.

[9] L. Cremer and H. A. Müller, “Principles and Applications of Room Acoustics,” Applied Science, London, Vol. 1, 2, 1982.

[10] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” 5th Edition, Academic Press, New York, 1994.

[11] H. Hardy, “Divergent Series,” Oxford University Press, London, 1956.

[12] M. Bonnet, “Boundary Integral Equations Methods for Solids and Fluids,” John Wiley, New York, 1999.

[13] A. N. Tikhonov and V. Y. Arsenin, “Solutions of Ill- Posed Problems,” Winston, Washington, 1977.

[14] P. E. Gill, W. Murray and M. H. Wright, “Practical Optimization,” Academic Press, London, 1981.

[15] M. Corana et al., “Minimizing Multimodal Functions of Continuous Variables with the Simulated Annealing Algorithm,” ACM Transactions on Mathematical Software, Vol. 13, No. 3, 1987, pp. 262-280. doi:10.1145/29380.29864