Boundary Integrals for Data Reconstruction on an Elastostatic Crack
Tóm tắt
The elastostatic Cauchy problem of fracture mechanics is studied in a two-dimensional bounded domain containing a crack. Given Cauchy data on the boundary of the domain, the displacement and normal stress (traction) are reconstructed on the crack. The reconstruction is done by reducing the original problem, via the elastostatic potential, to a system of integral equations to be solved for densities over the boundary of the domain and the crack. Discretization is carried out by the Nyström method using quadrature formulas adjusted for singularities manifesting at the endpoints of the crack. Tikhonov regularization is applied for the stable solution of the discretized system. The results of numerical experiments for different input data and parameters are given showing that relevant physical quantities on the crack can be stably reconstructed.
Tài liệu tham khảo
Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Prob. 25, 123004 (2009)
Andrieux, S., Ben Abda, A., Bui, H.D.: Reciprocity principle and crack identification. Inverse Prob. 15, 59 (1999)
Andrieux, S., Baranger, T.N.: Emerging crack front identification from tangential surface displacements. Comptes Rendus Mécanique 340, 565–574 (2012)
Andrieux, S., Baranger, T.N.: Three-dimensional recovery of stress intensity factors and energy release rates from surface full-field displacements. Int. J. Solids Struct. 50, 1523–1537 (2013)
Bonnet, M., Constantinescu, A.: Inverse problems in elasticity. Inverse Prob. 21, R1–R50 (2005)
Broek, D.: Elementary Engineering Fracture Mechanics. Springer-Verlag, Berlin (2012)
Cakoni, F., Kress, R.: Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems Imaging 1, 229–245 (2007)
Carpinteri, A., Paggi, M.: Asymptotic analysis in linear elasticity: From the pioneering studies by Wieghardt and Irwin until today. Eng. Fract. Mech. 76, 1771–1784 (2009)
Chapko, R., Johansson, B.T.: An alternating potential-based approach for a Cauchy problem for the Laplace equation in a planar domain with a cut. Comput. Methods Appl. Math. 8, 315–335 (2008)
Chapko, R., Johansson, B.T.: Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation. Ukr. Math. J. 68, 1665–1682 (2016)
Chapko, R., Johansson, B.T., Shtoyko, M.: A double-layer potential approach for planar Cauchy problems for the Laplace equation, submitted to Visnyk of the Lviv University
Chapko, R., Johansson, B.T., Shkolyk, M.: On a boundary integral equation method for elastostatic Cauchy problems in annular planar domains. J. Numer. Appl. Math. 126, 16–32 (2017)
Chapko, R., Johansson, B.T., Sobeyko, O.: On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion. CMES Comput. Model. Eng. Sci. 62, 57–75 (2010)
Chapko, R., Kress, R., Moench, L.: On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack. IMA J. Numer. Anal. 29, 601–619 (2000)
Chapko, R., Johansson, B.T., Vavrychuk, V.: A projected iterative method based on integral equations for inverse heat conduction in domains with a cut. Inverse Prob. 29, 065003 (2013)
Costabel, M., Dauge, M.: Crack singularities for general elliptic systems. Math. Nachr. 235, 29–49 (2002)
Eldén, L.: Algorithms for regularization of ill-conditioned least-squares problems. BIT 17, 134–145 (1977)
Faverjon, B., Puig, B., Baranger, T.N.: Identification of boundary conditions by solving Cauchy problem in linear elasticity with material uncertainties. Comput. Math. Appl. 73, 494–504 (2017)
Hansen, P.C.: The L-curve and its use in the numerical treatment of inverse problems, Ed. P. Johnston, in Computational Inverse Problems in Electrocardiology, WIT Press, Southampton, 119–142 (2001)
Helsing, J., Jonsson, A.: A seventh-order accurate and stable algorithm for the computation of stress inside cracked rectangular domains. Internat. J. Multiscale Comput. Engrg. 2, 47–68 (2004)
Hsiao, G.C., Stephan, E.P., Wendland, W.L.: On the Dirichlet problem in elasticity for a domain exterior to an arc. J. Comput. Appl. Math. 34, 1–19 (1991)
Ikehata, M., Itou, H.: Reconstruction of a linear crack in an isotropic elastic body from a single set of measured data. Inverse Prob. 23, 589–607 (2007)
Isakov, V.: Inverse Problems for Partial Differential Equations, 3rd edn. Springer-Verlag, Cham (2017)
Ferrier, R., Kadri, M., Gosselet, P.: Crack identification with incomplete boundary data in linear elasticity by the reciprocity gap method. Comput. Mech. 6, (2021). https://doi.org/10.1007/s00466-021-02006-4
Kress, R.: Linear Integral Equation, 3rd edn. Springer-Verlag, New York (2014)
Kubo, S.: Inverse problems related to the mechanics and fracture of solids and structures. JSME Int. J. 31, 157–166 (1988)
Kupradze, V.D.: Potential Methods in the Theory of Elasticity, Jerusalem: Israel Program for Scientific Translations, (1965)
Marin, L., Hào, D.N., Lesnic, D.: Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quart. J. Mech. Appl. Math. 55, 227–247 (2002)
Marin, L., Karageorghis, A., Lesnic, D.: Regularized MFS solution of inverse boundary value problems in three-dimensional steady-state linear thermoelasticity. Int. J. Solids Struct. 91, 127–142 (2016)
Zehnder, A.T.: Fracture Mechanics. Springer-Verlag, London (2012)
Zhang, A., Gu, Y., Hua, Q., Chen, W., Zhang, C.: A regularized singular boundary method for inverse Cauchy problem in three-dimensional elastostatics. Adv. Appl. Math. Mech. 10, 1459–1477 (2018)