Structure-Preserving Physics-Based Model Order Reduction Schemes for Dynamic Contact

Digital twins offer a real-time monitoring and predictive analysis of physical assets. Model order reduction (MOR) is a key enabler of digital twins. Especially challenging is providing MOR schemes for contact mechanics due to additional constraints. In this work, novel MOR schemes are proposed for node-to-node and node-to-segment contact problems in linear elasticity. The former involves linear constraints, allowing contact only in the normal direction, whereas the latter is attached by quadratic inequality constrains, enabling sliding movement at the contact area. A typical application is the wheel-rail contact, which serves as the main motivation for this work. Due to the specific contact condition form, the Karush-Kuhn-Tucker conditions lead to a primal system that is linear in the state variable. The novel primal-dual decoupling technique allows a switch to the adjoint space described by Lagrange multipliers. The resulting systems represent linear (LCP) or nonlinear (NCP) complementary problems, depending on the chosen contact formulation. The LCP is solved by a complementary pivotal method and the NCP is solved using Newton-like iterations with quadratic convergence. These steps are performed in a reduced approximation space, which is obtained by Krylov subspace methods with additional contact treatment by means of the Craig-Bampton method. If the contact area is small compared to the overall structure, both reduction schemes perform very efficiently. The novel MOR methods are validated on various numerical examples and applied to the wheel-rail contact problem, showcasing highly satisfactory results.