| It is often encountered with some problems in engineering that the physical parameters on a part of the boundary are not determined directly, and it can be worked out by means of the known condition on the other boundary. From the viewpoint of mathematics, this is a kind of inverse problem. With regard to complicated inverse problems, numerical methods are always adopted to deal with them, such as finite element method, finite difference method and boundary element method. Boundary element method has a great advantage over others on the inverse problems above in that boundary element method works out the results only based on the boundary datum.In the past, there exists difficulty for calculating nearly singular integrals arising from boundary element analysis of the problems of thin body. However, by now, the difficulty of nearly singular integrals has been resolved by analytical integral algorithm, which has been put into uses in analyzing the conventional structures and thin body structures in the potential problems. Based on this theory, the present paper firstly introduces the fundamental theories of the nearly singular integrals and the singular value decomposition, and quotes the analytical algorithm of the nearly singular integrals of two-dimensional potential boundary element method. Then the potential inverse problems of the conventional structures and thin body structures are regularized by the singular value decomposition. Especially for the problem of thin body, duple regularization of both the nearly singular integrals and ill-posed inverse problems is also successful in the two-dimensional boundary element method. Based on the strategy, the boundary element method is applied to analyzing the inverse problems of the two-dimensional Cauchy potential with isotropic and orthotropic materials for conventional structures and thin-body structures. A good number of numerical examples are given and demonstrate the validity of the present method.
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