On a new approach to identifying inhomogeneous mechanical properties of~elastic~bodies

A. O. Vatulyan ; V. O. Yurov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 2, pp. 209-221.

Résumé

A new approach to solving the problem of identifying the variable characteristics of an inhomogeneous elastic isotropic body is presented. The most common formulations of problems on determining variable mechanical characteristics (the Lamé parameters and density are functions of coordinates) are presented. The inverse problem of identifying properties, due to its significant nonlinearity, is usually solved iteratively, with each iteration requiring the solution of a direct problem for some initial approximation and a system of the Fredholm integral equations of the first kind with smooth kernels to find corrections. This approach, in turn, requires specifying the displacement field in the area in which the loading occurs. An approach is proposed on the basis of which it is possible to carry out reconstruction by obtaining additional information about the displacement field in an area other than the loading area in a narrower search space. An example of such a reconstruction is presented in the problem of longitudinal vibrations of an inhomogeneous rod, where the amplitude-frequency response is specified at the internal point of the rod, and the loading is implemented at the end. The results of computational experiments on the reconstruction of the elasticity modulus and density in the form of two functions of the longitudinal coordinate are presented.

Bibliographie

[1] Tikhonov A. N., Arsenin V. Ya., Methods for Solving Ill-posed Problems, Nauka, M., 1986, 288 pp. (in Russian)

[2] Levitan B. M., Inverse Sturm – Liouville Problems and Their Applications, Nauka, M., 1984, 240 pp. (in Russian) | MR

[3] Kravchenko V. V., Direct and Inverse Sturm – Liouville Problems. A Method of Solution, Frontiers in Mathematics, Birkhäuser, Cham, 2020, 154 pp. | DOI | MR

[4] Romanov V. G., Inverse Problems of Mathematical Physics, Nauka, M., 1984, 262 pp. (in Russian) | MR

[5] Yakhno V. G., Inverse Coefficient Problems for Differential Elasticity Equations, Nauka, Novosibirsk, 1990, 304 pp. (in Russian) | MR

[6] Bui H. D., Inverse Problems in the Mechanic of Materials: An Introduction, CRC Press, Boca Raton, FL, 1994, 224 pp.

[7] Isakov V., Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, Cham, 2017, 406 pp. | DOI | MR | Zbl

[8] Bonnet M., Constantinescu A., “Inverse problems in elasticity”, Inverse Problem, 21 (2005), 1–50 | DOI | MR

[9] Yurko V. A., Introduction to the Theory of Inverse Spectral Problems, Fizmatlit, M., 2007, 384 pp. (in Russian)

[10] Bal G., Introduction to Inverse Problems, Columbia University, New York, 2012, 205 pp.

[11] Neto F. D. M., Neto A. J. S., An Introduction to Inverse Problems with Applications, Springer, Berlin, 2013, 255 pp. | MR | Zbl

[12] Kabanikhin S. I., Inverse and Ill-posed Problems, Publ. House SB RAS, Novosibirsk, 2018, 512 pp. (in Russian)

[13] Tikhonov A. N., Goncharskiy A. V., Stepanov V. V., Yagola A. G., Numerical Methods for Solving Ill-posed Problems, Nauka, M., 1990, 230 pp. (in Russian) | MR

[14] Samarskiy A. A., Vabishchevich P. N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, Editorial URSS, M., 2004, 480 pp. (in Russian)

[15] Vatulyan A. O., Coefficient Inverse Problems of Mechanics, Fizmatlit, M., 2019, 272 pp. (in Russian)

[16] Vatulyan A. O., Nesterov S. A., Coefficient Inverse Problems of Thermomechanics, Southern Federal University Publ, Rostov-on-Don–Taganrog, 2022, 176 pp. (in Russian)

[17] Bondarenko A. N., Bugueva T. V., Dedok V. A., “Inverse problems of anomalous diffusion theory: An artificial neural network approach”, Journal of Applied and Industrial Mathematics, 10:3 (2016), 311–321 | DOI | DOI | MR | Zbl

[18] Bogachev I. V., Vatulyan A. O., Dudarev V. V., “On the method of property identification of multilayer soft biological tissues”, Russian Journal of Biomechanics, 17:3 (2013), 37–48 (in Russian)

[19] Sinkus R., Lorenzen J., Schrader D., Lorenzen M., Dargatz M., Holz D., “High-resolution tensor MR elastography for breast tumour detection”, Physics in Medicine Biology, 45:6 (2000), 1649–1664 | DOI

[20] Manduca A., Oliphant T. E., Dresner M. A., Mahowald J. L., Kruse S. A., Amromin E., Felmlee J. P., Greenleaf J. F., Ehman R. L., “Magnetic resonance elastography: Non-invasive mapping of tissue elasticity”, Medical Image Analysis, 5:4 (2001), 237–254 | DOI

[21] Sarvazyan A. P., “Low-frequency acoustic characteristics of biological tissues”, Polymer Mechanics, 1975, no. 4, 691–695 (in Russian)

[22] Sarvazyan A., Goukassian D., Maevsky G., “Elasticity imaging as a new modality of medical imaging for cancer detection”, Proceedings of an International Workshop on Interaction of Ultrasound with Biological Media (Valenciennes, France, 1994), 69–81

[23] Sarvazyan A. P., Rudenko O. V., Swanson S. D., Fowlkes J. B., Emelianov S. Y., “Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics”, Ultrasound in Medicine Biology, 24:9 (1998), 1419–1435 | DOI

[24] Arani A., Manduca A., Ehman R. L., Huston III J., “Harnessing brain waves: A review of brain magnetic resonance elastography for clinicians and scientists entering the field”, The British Journal of Radiology, 94 (2021), 20200265 | DOI

[25] Perkowski Z., Czabak M., “Description of behaviour of timber-concrete composite beams including interlayer slip, uplift, and long-term effects: Formulation of the model and coefficient inverse problem”, Engineering Structures, 194 (2019), 230–250 | DOI

[26] Dudarev V. V., Vatulyan A. O., Mnukhin R. M., Nedin R. D., “Concerning an approach to identifying the Lamé parameters of an elastic functionally graded cylinder”, Mathematical Methods in the Applied Sciences, 43:11 (2020), 6861–6870 | DOI | MR | Zbl

[27] Vatulyan A. O., Dudarev V. V., Mnukhin R. M., “Identification of characteristics of a functionally graded isotropic cylinder”, International Journal of Mechanics and Materials in Design, 17:2 (2021), 321–332 | DOI | MR

[28] Vatulyan A. O., Yurov V. O., “On the reconstruction of material properties of a radially inhomogeneous cylindrical waveguide”, Mathematical Methods in the Applied Sciences, 44:6 (2021), 4756–4769 | DOI | MR | Zbl

[29] Vatulyan A. O., Yurov V. O., “On the determination of the mechanical characteristics of rod elements made of functionally graded materials”, Mechanics of Solids, 55:6 (2020), 907–917 | DOI | DOI | MR

[30] Vatulyan A. O., Inverse Problems in the Mechanics of Deformable Solids, Fizmatlit, M., 2007, 223 pp. (in Russian)