An augmented Lagrangian algorithm for contact mechanics based on linear regression

The penalty method for the solution of contact problems yields an approximate satisfaction of the contact constraints. Augmentation schemes, which can be adopted to improve the solution, either include the contact forces as additional unknowns or are strongly affected by the penalty parameter and display a poor conver- gence rate. In a previous investigation, an unconventional augmentation scheme was proposed, on the basis of estimating the ‘exact’ values of the contact forces through linear interpolation of a database extracted by the previous converged states. An enhanced version of this method is proposed herein. With respect to the original method, the enhanced one eliminates some numerical problems and improves the regularity of the convergence path by carrying out the estimate through linear regression methods. The resulting convergence rate is superlinear, and the method is quite insensitive to the penalty parameter. The main underlying con- cept is that, within the iterative solution of a non-linear problem, linear regression techniques may be used as a tool to ‘shoot’ faster to the final solution, on the basis of a set of intermediate data. The generality of this concept makes it potentially applicable to contact problems in more general settings, as well as to other categories of non-linear problems.