Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity

We investigate two numerical procedures for the Cauchy problem in linear elasticity, involving the relaxation of either the given boundary displacements (Dirichlet data) or the prescribed boundary tractions (Neumann data) on the over-specified boundary, in the alternating iterative algorithm of Kozlov et al. (1991). The two mixed direct (well-posed) problems associated with each iteration are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method, while the optimal value of the regularization parameter is chosen via the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The MFS-based iterative algorithms with relaxation are tested for Cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the proposed method.

[1]  Thouraya Baranger,et al.  An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity , 2008 .

[2]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[3]  Daniel Lesnic,et al.  Regularized boundary element solution for an inverse boundary value problem in linear elasticity , 2002 .

[4]  Enayat Mahajerin,et al.  A comparison of the boundary element and superposition methods , 1984 .

[5]  Daniel Lesnic,et al.  Boundary element-Landweber method for the Cauchy problem in linear elasticity , 2004 .

[6]  Emilio Turco,et al.  An effective algorithm for reconstructing boundary conditions in elastic solids , 2001 .

[7]  Daniel Lesnic,et al.  The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity , 2004 .

[8]  D. Schnur,et al.  Spatially regularized solution of inverse elasticity problems using the BEM , 1989 .

[9]  Piotr Gorzelańczyk,et al.  Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods , 2008 .

[10]  E. Schnack,et al.  Computational identification of interface delaminations in layered composites based on surface measurements , 2006 .

[11]  Emilio Turco,et al.  A boundary element approach to identify static boundary conditions in elastic solids from stresses at internal points , 1999 .

[12]  B. Johansson,et al.  A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity , 2010 .

[13]  A. Nachaoui,et al.  An iterative approach to the solution of an inverse problem in linear elasticity , 2008, Math. Comput. Simul..

[14]  Nicholas Zabaras,et al.  Investigation of regularization parameters and error estimating in inverse elasticity problems , 1994 .

[15]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[16]  M. Golberg.  Boundary integral methods : numerical and mathematical aspects , 1999 .

[17]  T. J. Martin,et al.  An inverse method for finding unknown surface tractions and deformations in elastostatics , 1995 .

[18]  D. Ingham,et al.  Boundary Element Regularisation Methods for Solving the Cauchy Problem in Linear Elasticity , 2002 .

[19]  Elasticity problems with partially overspecified boundary conditions , 1991 .

[20]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[21]  Boundary element method for the Cauchy problem in linear elasticity , 2001 .

[22]  Ulrich Heise,et al.  Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves , 1978 .

[23]  R. Gallego,et al.  An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity , 2007 .

[24]  Cheng-Hung Huang,et al.  A boundary element based solution of an inverse elasticity problem by conjugate gradient and regularization method , 1997 .

[25]  Daniel Lesnic,et al.  Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition , 2000 .

[26]  D. Lesnic,et al.  Conjugate Gradient–Boundary Element Method for the Cauchy Problem in Elasticity , 2002 .

[27]  L. Marin.  The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity , 2009 .

[28]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[29]  M. Aliabadi,et al.  The Boundary Element Method , 2002 .

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

[31]  Lawrence E. Payne,et al.  Uniqueness Theorems in Linear Elasticity , 1971 .

[32]  M. H. Aliabadi,et al.  Applications in solids and structures , 2002 .

[33]  D. Schnur,et al.  Finite element solution of two‐dimensional inverse elastic problems using spatial smoothing , 1990 .

[34]  G. Wahba.  Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy , 1977 .

[35]  Graeme Fairweather,et al.  The method of fundamental solutions for scattering and radiation problems , 2003 .

[36]  Liviu Marin,et al.  A meshless method for solving the cauchy problem in three-dimensional elastostatics , 2005 .

[37]  E. M. Lifshitz,et al.  Course in Theoretical Physics , 2013 .

[38]  R. Mathon,et al.  The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions , 1977 .

[39]  Xin Li,et al.  Trefftz Methods for Time Dependent Partial Differential Equations , 2004 .

[40]  Kim Stelson,et al.  Finite Element Analysis of Some Inverse Elasticity Problems , 1989 .

[41]  G. K. Ananthasuresh,et al.  On computing the forces from the noisy displacement data of an elastic body , 2008 .

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

[43]  Eckart Schnack,et al.  Inverse Identification of Delamination Regions in Layered CFRP‐Composites , 2006 .

[44]  Marc Bonnet,et al.  Inverse problems in elasticity , 2005 .

[45]  Weichung Yeih,et al.  An Inverse Problem in Elasticity With Partially Overprescribed Boundary Conditions, Part II: Numerical Details , 1993 .

[46]  D. Ingham,et al.  An iterative boundary element algorithm for a singular Cauchy problem in linear elasticity , 2002 .

[47]  Thouraya Baranger,et al.  An optimization approach for the Cauchy problem in linear elasticity , 2008 .

[48]  Emilio Turco,et al.  A numerical study on the solution of the Cauchy problem in elasticity , 2009 .

[49]  K. Kunisch,et al.  Iterative choices of regularization parameters in linear inverse problems , 1998 .

[50]  S. Timoshenko,et al.  Theory of elasticity , 1975 .