Numerical solution of the Cauchy problem in plane elastostatics

Abstract

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.

A numerical method for a Cauchy problem for elliptic partial differential equations

“…The theory developed in sections 2 and 3 is valid also for Cauchy problems of elliptic systems. In this example, we consider the following Cauchy problem in plane elastostatics studied in [20]: find u ∈ (H 1 ( )) 2 such that for i = 1, 2,…”

“…To further compare our method and the method studied in [20], we consider the noise effect. We perturb g and ϕ to get the noisy data g δ and ϕ δ in the form (This figure is in colour only in the electronic version) Table 3.…”

“…We perturb g and ϕ to get the noisy data g δ and ϕ δ in the form (This figure is in colour only in the electronic version) Table 3. Numerical comparison between our method and that in [20]. with h = 1/2 and of the method studied in [20] at different noise level and regularization parameters.…”

“…Numerical comparison between our method and that in [20]. with h = 1/2 and of the method studied in [20] at different noise level and regularization parameters. Observe that our method performs better than that studied in [20].…”

“…with h = 1/2 and of the method studied in [20] at different noise level and regularization parameters. Observe that our method performs better than that studied in [20]. When δ = 5% and h = 1/4, the numerical results of u 1 (3, x 2 ) with different are shown in figure 2, from which we may find that = 10 −2 is the best choice.…”