An inversion algorithm in two-dimensional elasticity. (English) Zbl 1046.74022
The paper examines an inverse problem which consists in finding the shape of an obstacle from the knowledge of its scattered field. The considered situation is a two-dimensional one, and the scattering body is a bounded, connected and closed subset of \(\mathbb{R}^2\). The field is described in dyadic form which allows a compact representation. The far-field equations are introduced by the use of Herglotz conditions, whereas the Herglotz wave functions allows to derive a pair of integral equations. Some theorems insuring the existence of solution are derived, and numerical experiments are presented (circular and elliptical region). The inversion scheme uses a procedure first given by D. Colton [Proc. Conf. Oberwolfach 1984, ISNM 73, 103–109 (1985; Zbl 0571.76079)] for acoustic field.
Reviewer: Dumitru Stanomir (Bucureşti)
MSC:
| 74J25 | Inverse problems for waves in solid mechanics |
| 74J20 | Wave scattering in solid mechanics |
| 74B05 | Classical linear elasticity |
Citations:
Zbl 0571.76079References:
| [1] | Arens, T., The scattering of plane elastic waves by a one-dimensional periodic surface, Math. Methods Appl. Sci., 22, 55-72 (1989) · Zbl 0918.35101 |
| [2] | Ben-Menahem, A.; Singh, S. J., Seismic Waves and Sources (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0484.73089 |
| [3] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0760.35053 |
| [4] | Colton, D.; Kirsch, A., A simple method for solving the inverse scattering problems in the resonance region, Inverse Problems, 12, 383-393 (1996) · Zbl 0859.35133 |
| [5] | Colton, D.; Monk, P., A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region, SIAM J. Appl. Math., 45, 1039-1053 (1985) · Zbl 0588.76140 |
| [6] | Colton, D.; Piana, M.; Potthast, P., A simple method using Morozov’s discrepancy principle for solving scattering problems, Inverse Problems, 13, 1477-1493 (1999) · Zbl 0902.35123 |
| [7] | Colton, D.; Piana, M., The simple method for solving the electromagnetic inverse scattering problem: The case of TE polarized waves, Inverse Problems, 14, 597-614 (1998) · Zbl 0917.35158 |
| [8] | Dassios, G.; Rigou, Z., On the density of traction traces in scattering by elastic waves, Siam J. Appl. Math., 53, 141-153 (1993) · Zbl 0771.35070 |
| [9] | Dassios, G.; Rigou, Z., Elastic Herglotz functions, Siam J. Appl. Math., 55, 1345-1361 (1995) · Zbl 0844.35128 |
| [10] | Dassios, G.; Rigou, Z., On the reconstruction of a rigid body in the theory of elasticity, Z. Angew. Math. Mech., 77, 911-923 (1997) · Zbl 0910.73017 |
| [11] | Gintides, D.; Kiriaki, K., On the continuity dependence of elastic scattering amplitudes upon the shape of the scatterer, Inverse Problems, 8, 95-118 (1992) · Zbl 0746.35053 |
| [12] | Gintides, D.; Kiriaki, K., The far-field equations in linear elasticity—an inversion scheme, Z. Angew. Math. Mech., 81, 305-316 (2001) · Zbl 1042.74027 |
| [13] | Groetsch, C. W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (1984), Pitman: Pitman Boston · Zbl 0545.65034 |
| [14] | Kirsch, A.; Kress, R., An optimization method in inverse acoustic scattering, Inverse Problems, 77, 93-102 (1986) · Zbl 0607.65089 |
| [15] | Kupradze, V. D., Dynamical problems in elasticity, Progress in Solid Mechanics (1963), North-Holland: North-Holland Amsterdam · Zbl 0115.18702 |
| [16] | Kupradze, V. D., Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity (1979), North-Holland: North-Holland Amsterdam · Zbl 0421.73009 |
| [17] | Morse, P. M.; Feshbach, H., Methods of Theoretical Physics (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.40603 |
| [18] | Phillips, D. L., A technique for the numerical solution of certain integral equations of the first kind, J. Ass. Comp. Math., 9, 84-97 (1962) · Zbl 0108.29902 |
| [19] | V. Sevroglou, and, K. Kiriaki, Basic properties of Herglotz functions in scattering of elastic waves, to appear.; V. Sevroglou, and, K. Kiriaki, Basic properties of Herglotz functions in scattering of elastic waves, to appear. · Zbl 1013.74007 |
| [20] | V. Sevroglou, and, K. Kiriaki, Integral equation methods in obstacle elastic scattering, Greek Bul. Helenic Math. Soc, in press.; V. Sevroglou, and, K. Kiriaki, Integral equation methods in obstacle elastic scattering, Greek Bul. Helenic Math. Soc, in press. · Zbl 1286.74055 |
| [21] | Sevroglou, V.; Kiriaki, K., On Herglotz functions in two dimensional linear elasticity, Scattering Theory and Biomedical Technology Modelling and Applications (1999), World Scientific, p. 151-158 · Zbl 1013.74007 |
| [22] | Tikhonov, A. N.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G., Numerical Methods for the Solution of Ill-Posed Problems (1995), Kluwer: Kluwer Dordrecht · Zbl 0831.65059 |
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