Arbitrary-Order Contact Formulation Using Tensorial Lagrange Multipliers in H(div) Space Tailored for an Exact Schur Complement Preconditioner
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One of the most established approaches to enforcing contact conditions uses the Lagrange multipliers (LM) field, for which h-refinement strategies exist, including discontinuous (dual) approximations (Wohlmuth, 2000). More recently, p-refinement has gained popularity with the use of hierarchical basis functions and suitable functional spaces (Fuentes et al., 2015). Such bases improve approximation without remeshing and produce matrices well suited to multigrid preconditioners. However, the construction and implementation of higher-order dual shape functions are technically challenging. At the same time, LM formulations lead to saddle-point problems with indefinite matrices, complicating the use of effective iterative solvers. In this work, we propose a novel approach to solving contact problems, in which the LM field, rather than being scalar and equivalent to contact pressure, is approximated as a tensorial field in elements adjacent to the contact surface. Using the Raviart–Thomas H(div) space (Boffi et al., 2013), the normal trace of the LM field (contact traction) is approximated in the same space as for the dual shape functions. However, constructing the LM in the Raviart–Thomas space allows straightforward use of hierarchical shape functions of arbitrary order, enabling higher-order convergence. Furthermore, defining LM only on contact boundary elements eliminates shared LM degrees of freedom between adjacent domain elements. This yields a block-diagonal LM “mass” matrix, with each contact element represented by an isolated dense block. This structure enables the use of a Schur complement preconditioner with exact element-wise inversion, supporting high scalability, and is particularly well suited to GPU acceleration. Moreover, the Schur complement block has the same structure as the standard stiffness matrix and can be efficiently inverted using algebraic multigrid (AMG). The implementation of the proposed approach in MoFEM, an open-source parallel finite element library (Kaczmarczyk et al., 2020), together with demonstrations of higher-order convergence and strong and weak scalability, will be discussed. In addition, applications to a range of problems, including modelling of triboelectric nanogenerators requiring simulation of contact between rough surfaces, will be presented.
