An iterative approach to the solution of an inverse problem in linear elasticity

This paper presents an iterative alternating algorithm for solving an inverse problem in linear elasticity. A relaxation procedure is developed in order to increase the rate of convergence of the algorithm and two selection criteria for the variable relaxation factors are provided. The boundary element method is used in order to implement numerically the constructing algorithm. We discuss this implementation, mention the use of Krylov methods to solve the obtained linear algebraic systems of equations and investigate the convergence and the stability when the data is perturbed by noise.

[1]  A. Nachaoui,et al.  Convergence of an Alternating Method to Solve the Cauchy Problem for Poisson's Equation , 2002 .

[2]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[3]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[4]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  Davod Khojasteh Salkuyeh,et al.  Numerical accuracy of a certain class of iterative methods for solving linear system , 2006, Appl. Math. Comput..

[7]  P. Sonneveld CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[8]  Nasser Khalili,et al.  An effective stress elastic–plastic model for unsaturated porous media , 2002 .

[9]  H. D. Bui,et al.  Reciprocity principle and crack identification in transient thermal problems , 2001 .

[10]  A Leitão,et al.  On iterative methods for solving ill-posed problems modeled by partial differential equations , 2001, 2011.14441.

[11]  Abdeljalil Nachaoui,et al.  Iterative solution of the drift-diffusion equations , 1999, Numerical Algorithms.

[12]  Vladimir Maz’ya,et al.  An iterative method for solving the Cauchy problem for elliptic equations , 1991 .

[13]  Cheng-Hung Huang,et al.  An inverse problem in estimating interfacial cracks in bimaterials by boundary element technique , 1999 .

[14]  A. Nachaoui Numerical linear algebra for reconstruction inverse problems , 2004 .

[15]  Weichung Yeih,et al.  An Inverse Problem in Elasticity With Partially Overprescribed Boundary Conditions, Part I: Theoretical Approach , 1993 .