Arbitrary Order Contact Formulation Tailored for Schur Complement Preconditioners
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One of the most established approaches to enforce contact conditions is via the Lagrange Multipliers (LM) field, for which h-refinement approaches have been developed [1]. Recently, p-refinement approaches have gained popularity with the standardisation of hierarchical basis functions along with the choice of appropriate functional spaces. At the same time, LM formulation results in a saddle-point problem with an indefinite matrix, which makes the application of effective iterative solvers difficult. In this work, we present a novel approach to modelling contact problems, where the discrete functional space for Lagrange multipliers defined on contact surfaces emerges from the trace of Raviart-Thomas space defined inside the domain [2]. This approach permits to combine h-refinement with p-refinement using the hierarchical approximation. Moreover, due to the properties of the Raviart-Thomas space, the sub-matrix corresponding to the LM field has a block-diagonal structure, which allows for exact computation of its inverse on the element-by-element basis. Therefore, it becomes straightforward to compute the Schur complement, resulting in a positive-definite matrix for which an efficient iterative solver such as Algebraic Multigrid (AMG) can be applied. This approach is ideal for parallelisation and provides enhanced scalability for large problems. The implementation of the proposed approach in MoFEM [3] and the application to a range of problems, including the modelling of triboelectric nanogenerators and the simulation of crack propagation along contact interfaces in irradiated graphite bricks, will be discussed. References [1] Wohlmuth, Barbara I., “A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier,” SIAM Journal on Numerical Analysis, 38, 989-1012 (2000). [2] D. Boffi et al., “Mixed finite element methods and applications,” Springer, 44, (2013). [3] L. Kaczmarczyk et al., “MoFEM: An open-source, parallel finite element library”, The Journal of Open Source Software, Vol. 5, (2020).
