Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity

We investigate the stable numerical reconstruction of an unknown portion of the boundary of a two-dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements (i.e. Cauchy data) on the remaining known portion of the boundary. This inverse geometric problem is approached by combining the method of fundamental solutions (MFS) and the Tikhonov regularization method, whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle. Various geometries are considered, i.e. convex and non-convex domains with a smooth or piecewise smooth boundary, in order to show the numerical stability, convergence, consistency and computational efficiency of the proposed method.

[1]  Cheng-Hung Huang,et al.  A transient inverse two-dimensional geometry problem in estimating time-dependent irregular boundary configurations , 1998 .

[2]  Youssef F. Rashed,et al.  The method of fundamental solutions applied to 3D elasticity problems using a continuous collocation scheme , 2009 .

[3]  Philip E. Gill,et al.  Practical optimization , 1981 .

[4]  M. Aliabadi,et al.  Boundary‐Element Method , 2009 .

[5]  Bangti Jin,et al.  The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction , 2007 .

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

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

[8]  Jin Cheng,et al.  Stability for an inverse boundary problem of determining a part of a boundary , 1999 .

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

[10]  Y. C. Hon,et al.  A computational method for inverse free boundary determination problem , 2008 .

[11]  Hubert Maigre,et al.  Inverse Problems in Engineering Mechanics , 1994 .

[12]  Y. Hon,et al.  A fundamental solution method for inverse heat conduction problem , 2004 .

[13]  C. Fu,et al.  The method of fundamental solutions for the inverse heat source problem. , 2008 .

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

[15]  Y. Hon,et al.  A numerical computation for inverse boundary determination problem , 2000 .

[16]  Alain J. Kassab,et al.  An inverse boundary element method/genetic algorithm based approach for retrieval of multi-dimensional heat transfer coefficients within film cooling holes/slots , 2005 .

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

[18]  Liviu Marin.  Boundary Reconstruction in Two-Dimensional Functionally Graded Materials Using a Regularized MFS , 2009 .

[19]  M. Arab,et al.  The method of fundamental solutions for the inverse space-dependent heat source problem , 2009 .

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

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

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

[23]  Daniel Lesnic,et al.  The MFS for numerical boundary identification in two-dimensional harmonic problems , 2011 .

[24]  Bangti Jin,et al.  A meshless method for some inverse problems associated with the Helmholtz equation , 2006 .

[25]  H. Park,et al.  Shape identification for natural convection problems , 2001 .

[26]  L. Marin.  The Method of Fundamental Solutions for Inverse Problems Associated with the Steady-State Heat Conduction in the Presence of Sources , 2008 .

[27]  Liviu Marin,et al.  A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations , 2005, Appl. Math. Comput..

[28]  Michael Pidcock,et al.  The boundary inverse problem for the Laplace equation in two dimensions , 1996 .

[29]  Satya N. Atluri,et al.  Stability analysis for inverse heat conduction problems , 2006 .

[30]  D. L. Young,et al.  Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points , 2008, J. Comput. Phys..

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

[32]  Daniel Lesnic,et al.  The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations , 2005 .

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

[34]  Graeme Fairweather,et al.  Potential field based geometric modelling using the method of fundamental solutions , 2006 .

[35]  L. Marin.  Numerical boundary identification for Helmholtz-type equations , 2006 .

[36]  Cheng-Hung Huang,et al.  An inverse geometry problem in identifying irregular boundary configurations , 1997 .

[37]  N. S. Mera,et al.  A three-dimensional boundary determination problem in potential corrosion damage , 2005 .

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

[39]  Elena Beretta,et al.  Stable determination of boundaries from Cauchy data , 1999 .

[40]  D. L. Young,et al.  The method of fundamental solutions for inverse 2D Stokes problems , 2005 .

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

[42]  Derek B. Ingham,et al.  The method of fundamental solutions for a biharmonic inverse boundary determination problem , 2008 .

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

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

[45]  N. S. Mera.  The method of fundamental solutions for the backward heat conduction problem , 2005 .

[46]  Liviu Marin,et al.  Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials , 2005 .

[47]  Alain J. Kassab,et al.  A general method for the solution of inverse heat conduction problems with partially unknown system geometries , 1986 .

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

[49]  Daniel Lesnic,et al.  A Boundary Element Regularization Method for the Boundary Determination in Potential Corrosion Damage , 2002 .

[50]  D. Lesnic,et al.  BEM first-order regularisation method in linear elasticity for boundary identification , 2003 .

[51]  Daniel Lesnic,et al.  The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation , 2005, Math. Comput. Model..

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

[53]  B. Meng,et al.  A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. , 2007 .

[54]  Youssef F. Rashed,et al.  Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources , 2004 .