A new numerical method is presented with the purpose to calculate the Lamé coefficients, associated with an elastic material, through eigenvalues of the elasticity operator. The finite element method is used to solve repeatedly, using different Lamé coefficients values, the direct problem by training a direct radial basis neural network. A map of eigenvalues, as a function of the Lamé constants, is then obtained. This relationship is later inverted and refined by training an inverse radial basis neural network, allowing calculation of mentioned coefficients. A numerical example is presented to prove the effectiveness of this novel method.
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Sebastián Ossandón 1 ; Camilo Reyes 2
@article{CRMECA_2016__344_2_113_0, author = {Sebasti\'an Ossand\'on and Camilo Reyes}, title = {On the neural network calculation of the {Lam\'e} coefficients through eigenvalues of the elasticity operator}, journal = {Comptes Rendus. M\'ecanique}, pages = {113--118}, publisher = {Elsevier}, volume = {344}, number = {2}, year = {2016}, doi = {10.1016/j.crme.2015.10.004}, language = {en}, }
TY - JOUR AU - Sebastián Ossandón AU - Camilo Reyes TI - On the neural network calculation of the Lamé coefficients through eigenvalues of the elasticity operator JO - Comptes Rendus. Mécanique PY - 2016 SP - 113 EP - 118 VL - 344 IS - 2 PB - Elsevier DO - 10.1016/j.crme.2015.10.004 LA - en ID - CRMECA_2016__344_2_113_0 ER -
%0 Journal Article %A Sebastián Ossandón %A Camilo Reyes %T On the neural network calculation of the Lamé coefficients through eigenvalues of the elasticity operator %J Comptes Rendus. Mécanique %D 2016 %P 113-118 %V 344 %N 2 %I Elsevier %R 10.1016/j.crme.2015.10.004 %G en %F CRMECA_2016__344_2_113_0
Sebastián Ossandón; Camilo Reyes. On the neural network calculation of the Lamé coefficients through eigenvalues of the elasticity operator. Comptes Rendus. Mécanique, Volume 344 (2016) no. 2, pp. 113-118. doi : 10.1016/j.crme.2015.10.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.10.004/
[1] Inverse scattering for elastic plane cracks, Inverse Probl., Volume 15 (1999), pp. 91-97
[2] Sur l'identification de fissures planes via le concept d'écart à la réciprocité en élasticité, C. R. Acad. Sci. Paris, Ser. II, Volume 324 (1997), pp. 1431-1438
[3] Ultrasonic Lamb wave tomography, Inverse Probl., Volume 18 (2002), pp. 1795-1808
[4] Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elast., Volume 67 (2002), pp. 97-129
[5] On the inversion of subsurface residual stresses from surface stress measurements, J. Mech. Phys. Solids, Volume 42 (1994), pp. 1767-1788
[6] Inversion gaussienne appliquée à la correction paramétrique de modèles structuraux, École polytechnique, Paris, France, 1995 (Ph.D. thesis)
[7] Properties of elastic materials using contacting and non-contacting acoustic spectroscopy, Colorado School of Mines, Golden, Colorado, USA, 2005 (Ph.D. thesis)
[8] Regularization theory and neural networks architectures, J. Neural Comput., Volume 7 (1995), pp. 219-269
[9] Approximation of nonlinear systems with radial basis function neural networks, IEEE Trans. Neural Netw., Volume 12 (2001) no. 1, pp. 1-15
[10] Eigenvalue problems (P.G. Lions, ed.), Finite Element Methods (Part 1), Handbook of Numerical Analysis, vol. II, North-Holland, Amsterdam, 1991, pp. 641-787
[11] Finite element approximation of eigenvalue problems, Acta Numer., Volume 19 (2010), pp. 1-120
[12] Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, Germany, 2013
[13] Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, 1991
[14] The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978
[15] Eigenvalue approximation by mixed and hybrid methods, Math. Comput., Volume 36 (1981), pp. 427-453
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