A variational approach is presented for numerical identification of the boundary conditions in solution of the Cauchy problem governed by the Navier equations in two-dimensional elasticity. The Cauchy problem in elastostatics is featured by simultaneously prescribed displacement and traction on a part of the boundary of an elastic body. The boundary data may contain some noise. The problem can be reformulated as a minimization problem of a functional with constraints, then the minimization problem is recast into successive primary and adjoint boundary value problems with no constraints in the corresponding plane elasticity problem. Two variational formulations, i. e., displacement approach and traction approach, are described. For an approximate solution of the elasticity problems, the conventional direct boundary element method is applied. Through simple examples, it is shown that our variational approaches are convergent and the proposed numerical process is stable.

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The purpose of this paper is to present an attempt at a numerical treatment of a kind of under-determined problem of the Laplace equation in two spatial dimensions. A resolution is sought for the problem in which the Dirichlet and Neumann data are arbitrarily imposed on each part of the boundary of the domain. This new problem can be regarded as a boundary inverse problem, in which the proper boundary conditions are to be identified for the rest of the boundary. The solution of this problem is not unique. The treatment is based on the direct variational method, and a functional is minimized by the method of the steepest descent. The minimization problem is recast into successive primary and dual boundary value problems of the Laplace equation. After numerical computations by using the boundary elements, it is concluded that our scheme is stable, but the numerical solutions converge to the nearest local minimum.