Numerical solution of the Cauchy problem in plane elastostatics (Q2709826)

The Cauchy problem for the Navier equation in elastoststics with unknown boundary condition is considered. The problem consists in identifying either unknown displacements or unknown tractions on a part of the boundary of the elastic material, when displacements and tractions are simultaneously prescribed as Cauchy data an the rest of the boundary. This problem is reformulated as a minimization problem for a functional with constraints. Based on the displacement and traction approaches two variational formulations of an initial value problem are described, supposed that the unknown displacement or unknown traction is uniquely determined. NEWLINENEWLINENEWLINEThe considered problem belongs to the class of ill-posed problems and for its solution a regularization of Tikhonov type is suggested. By application of the gradient method the variational problem is iteratively recast into successive direct primary and adjoint mixed boundary value problems with no constraints in the corresponding plane elasticity problem. The iterative process yields a boundary displacement or traction, at which the objective functional attains its minimum. For numerical implementation of the iterative process, the boundary element method with linear interpolation functions is used in the approximation of both the primary and adjoint problems. Convergence and stability of the suggested numerical algorithm are based on numerical experiments, the results of which are summarized in the tables and graphics. It is shown that the displacement approach is superior to the traction approach on account of the fact that a priori information is required for the traction approach to be feasible.