Weighted Reproducing Kernel Collocation Method Based on Error Analysis for Solving Inverse Elasticity Problems

For inverse problems equipped with incomplete boundary conditions, a simple solution strategy to obtain approximations remains a challenge in the fields of engineering and science. Based on our previous study, the weighted reproducing kernel collocation method (W-RKCM) shows optimal convergence in solving inverse Cauchy problems. As such, this work further introduces the W-RKCM to solve inverse problems in elasticity. From mathematical error estimate and numerical convergence study, it is shown that the weighted least-squares formulation can properly balance the errors in the domain and on the boundary. By comparing the approximations obtained by W-RKCM with those obtained by the direct collocation method, the reproducing kernel shape function can retain the locality without using a large support size, and the corresponding approximations exhibit extremely high solution accuracy. The stability of the W-RKCM is demonstrated by adding noise on the boundary conditions. This work shows the efficacy of the proposed W-RKCM in solving inverse elasticity problems as no additional technique is involved to reach the desired solution accuracy in comparison with the existing methods in the literature.