An integral equations method combined minimum norm solution for 3D elastostatics Cauchy problem

Abstract

In this paper, we establish new density results for the equilibrium equations. Based on the denseness result of the elastic potential functions, the Cauchy problem for the equilibrium equations is investigated. For this ill-posed problem, we construct a regularizing solution using the single-layer potential function. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. The method combines minimum norm solution with Morozov discrepancy principle to solve an inverse problem. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the discretisation about the integral equations on pseudo-boundary and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples.