Estimation of Distributions of Contact Stressess and Displacements Using Regularization Schemes

Abstract

Estimation of tractions and displacements on inaccessible boundaries, such as contact areas of solids, can be regarded as an inverse boundary value problem. In this study finite-element based inverse analysis schemes with regularization were applied to the estimation of the distributions of tractions and displacements on contact areas. The finite element equation was rewritten in terms of unknown boundary values on the contact area using over-prescribed boundary values. This equation was solved for the boundary values on the contact area. Like many other inverse problems, this inverse problem was severely ill-conditioned and the estimated distributions were very sensitive to the over-prescribed boundary values used in the estimation. To overcome the ill-posedness of this boundary value inverse problem, the function expansion method and Tikhonov regularization were introduced in the finite element-based inverse analysis scheme. The number of terms in the function expansion and the smoothing parameter in Tikhonov regularization were regarded as regularization parameters in the inverse analysis. To determine the optimum value of these regularization parameters, the estimated error criterion and the AIC were introduced. The usefulness of the finite element-based inversion scheme was examined by numerical simulations. It was found that the distributions of tractions and displacements can be estimated reasonably even from noisy observations by using the finite-element based inverse analysis schemes with regularization. The optimum value of the regularization parameters can be estimated by the estimated error criterion or by the AIC.