An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm

A review of existing augmented Lagrangian methods (ALM) for contact analysis problems reveals that they have not been implemented with automatic penalty updates as intended in their original development. Therefore, although the methods are an improvement over the penalty methods, solution with them still depends on the user-specified penalty values for the contact constraints. To overcome this drawback, an ALM is developed and discussed for contact analysis problems that automatically update the user-specified penalty values to obtain the final appropriate values. Further, to solve the frictional contact analysis problem accurately, a two-phase formulation is proposed. Solution of the Phase 1 problem removes penetration of the contacting nodes and brings them exactly to their initial contact points. In addition, a new contact constraint is introduced which allows determination of the precise friction force at the contacting nodes. Phase 2 of the formulation checks the friction conditions and solves the friction problem to bring the structure to an equilibrium state. Phases 1 and 2 are then combined to provide a general algorithm for multi-node frictional contact problems. The two-phase procedure also removes dependence of the contact solution on the number of load steps for the elastostatic problem. Numerical evaluation of the formulation and the algorithm is presented in Part 2 of the paper.