Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

Abstract In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.

[1]  M. Hanke.  Limitations of the L-curve method in ill-posed problems , 1996 .

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

[3]  Quantitative measurement of three-dimensional surface cracks by the electric potential CT method. , 1988 .

[4]  Takahide Sakagami,et al.  Electric Potential CT Method Based on BEM Inverse Analyses for Measurement of Three-Dimensional Cracks , 1986 .

[5]  A. Tikhonov,et al.  Nonlinear Ill-Posed Problems , 1997 .

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

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

[8]  C. Vogel.  Non-convergence of the L-curve regularization parameter selection method , 1996 .

[9]  Weichung Yeih,et al.  On the Modified Tikhonov's Regularization Method for the Cauchy Problem of the Laplace Equation , 2001, Journal of Marine Science and Technology.

[10]  Weichung Yeih,et al.  APPLICATIONS OF THE GENERALIZED SINGULAR-VALUE DECOMPOSITION METHOD ON THE EIGENPROBLEM USING THE INCOMPLETE BOUNDARY ELEMENT FORMULATION , 2000 .

[11]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[12]  S. Kubo.  Inverse Problems Related to the Mechanics and Fracture of Solids and Structures , 1988 .

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

[14]  Peter Rex Johnston,et al.  Computational Inverse Problems in Electrocardiography , 2001 .

[15]  The linear boundary element solution of Laplace's equation with Dirichlet boundary conditions , 1991 .

[16]  A new NDT: Evaluation of plastic strains in bulk from displacements on surfaces , 1985 .

[17]  Marc S. Ingber,et al.  A comparison of the semidiscontinuous element and multiple node with auxiliary boundary collocation approaches for the boundary element method , 1995 .

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

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  Analytical study and numerical experiments for Laplace equation with overspecified boundary conditions , 1998 .

[21]  Application of Cesàro mean and the L-curve for the deconvolution problem , 1995 .

[22]  Rogelio Luck,et al.  Solution to inverse heat conduction problems employing singular value decomposition and model-reduction , 2002 .

[23]  K. Takeuchi,et al.  Back analysis of measured displacements of tunnels , 1983 .

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

[25]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

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