A regression-based augmented Lagrangian procedure for contact problems

Among the available methods for the numerical solution of contact problems, the penalty method is probably the most widely used, due to its simplicity, clear physical meaning and flexibility. However, its application yields only an approximate satisfaction of the contact constraints, to an extent related to the magnitude of the penalty parameter. The solution can be improved using augmentation schemes. However the efficiency of such schemes is also strongly dependent on the value of the penalty parameter. Moreover, it usually results in a poor rate of convergence to the exact solution. This paper follows up to a previous work [1], which proposed a new method to perform the augmentations based on estimated values of the augmented Lagrangians. In this method, at each augmentation the converged state is used to extract some data. This piece of information updates a database which is then used for the Lagrangian estimation. The prediction is primarily based on the evolution of the constraint violation with respect to the evolution of the contact forces. In the previous version of the method, the estimation of the augmented Lagrangians at each augmentation was performed via the construction of an hyperplane interpolating the updated database of contact forces. The method was shown to exhibit a noticeable efficiency in detecting nearly exact contact forces, and to achieve superlinear convergence for the subsequent minimisation of the residual of constraints. Remarkably, the method was also found relatively insensitive to the penalty parameter. This allows a solution which fulfils the constraints very rapidly, even when using very small penalty values. In this paper, the previous method is applied with a modification in the estimation procedure of the augmented Lagrangians. In place of the aforementioned hyperplane interpolation, techniques of regression analysis are used to predict the updated augmented Lagrangians at each subsequent augmentation. The main objective is to regularize the path of the solution, and thus to improve the number of iterations required for convergence. Several linear and a few non-linear regression analysis techniques are analyzed, pointing out their distinct features and evaluating their relative performance in the context of the proposed method. The most efficient techniques are then used to develop some significant numerical examples.