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.
References
[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
