Keywords: inverse problem, boundary element method, nearly singular integral, analytical integral, TSVD, orthotropic potential, undulating-curve method.

It is well known that there are singular integrals and nearly singular integrals arising in the boundary element method (BEM). Unlike singular integrals, nearly singular integrals are not singular in the sense of mathematics. Conventional Gauss numerical quadrature is invalid to evaluate these integrals. Nearly singular integrals usually exist in two situations. One is known as boundary layer effect problem; the other is named as the thin body effect problem.

It is often encountered that all the potentials and fluxes are known on a part of the boundary and no boundary data can be directly measured on the rest of boundary in engineering for potential problems. This is the Cauchy inverse problem. The truncated singular value decomposition (TSVD) [1] and the boundary element method are applied to solve the Cauchy inverse problem in linear elasticity [2].

The completely analytical integral algorithm proposed for 2D direct orthotropic potential problems [3] is applied to evaluate the nearly singular integrals in the inverse problems. Thin body problems are considered. The analytical integral formulas are directly derived with integration by parts. The present algorithm applies the new analytical formulas to deal with the nearly singular integrals.

The system equation is solved by truncated singular value decomposition technique. The condition number and singular value characteristics of the system equation matrix A are discussed. In particular, for some thin body problems, matrix A appears to have a well-conditioned behaviour. However, some thin body Cauchy problems are rank deficient.

For rank deficient problems, the second order physical field variables, namely flux solutions are dominant to determine the optimal number k associated with the useful singular values. If the flux solutions are accurate, the corresponding first order physical field variables potentials are always accurate. As the small singular value leads to the undulation of the numerical solutions, an undulating-curve method is proposed to select the truncated number k. The unknown potential and flux boundary conditions for thin body problems with very small thickness-to-length ratios are accurately calculated.

1

P.C. Hansen, "Rank-Deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion", SIAM, Philadelphia, 1998.

2

L. Marin, D. Lesnic, "Boundary element method for the Cauchy problem in linear elasticity using singular value decomposition", Comput Methods Appl Mech Engrg, 191, 3257-3270, 2002. doi:10.1016/S0045-7825(02)00262-1

3

H.L. Zhou, Z.R. Niu, C.Z. Cheng, Z.W. Guan, "Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies", Engineering Analysis with Boundary Elements, 2007(in press). doi:10.1016/j.enganabound.2007.01.007

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