Fading regularization MFS algorithm for inverse boundary value problems in two-dimensional linear elasticity

Abstract We investigate the numerical reconstruction of the missing displacements (Dirichlet data) and tractions (Neumann data) on an inaccessible part of the boundary in the case of two-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved using the fading regularization method, originally proposed by Cimetiere et al. (2000, 2001) for the Laplace equation, in conjunction with a meshless method, namely the method of fundamental solutions (MFS). The stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to Morozov’s discrepancy principle (Morozov, 1966).

[1]  Gene H. Golub,et al.  Generalized cross-validation as a method for choosing a good ridge parameter , 1979, Milestones in Matrix Computation.

[2]  Božidar Šarler,et al.  Non-singular Method of Fundamental Solutions for anisotropic elasticity , 2014 .

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

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

[5]  P. Bailly,et al.  An iterative method for the Cauchy problem in linear elasticity with fading regularization effect , 2010 .

[6]  An inversion method for harmonic functions reconstruction , 2002 .

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

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

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

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

[11]  D. Lesnic,et al.  A survey of applications of the MFS to inverse problems , 2011 .

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

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

[14]  An iterative boundary element method for Cauchy inverse problems , 2002 .

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

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

[17]  Mohamed Jaoua,et al.  Solution of the Cauchy problem using iterated Tikhonov regularization , 2001 .

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

[19]  V. D. Kupradze,et al.  The method of functional equations for the approximate solution of certain boundary value problems , 1964 .

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

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

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

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

[24]  Peter Bauer,et al.  An Extension of the MOSUM Technique for Quality Control , 1980 .

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

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

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

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

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

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

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

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

[33]  B. Tomas Johansson,et al.  Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity , 2010 .

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

[35]  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 .

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