Generalisation of the Navier-slip boundary condition to arbitrary directions: Application to 3D oblique geodynamic simulations

Abstract

Although boundary conditions are mandatory to solve partial differential equations, they also represent a transfer of information between the domain being modelled and its surroundings. In the case of isolated or closed systems, these can be formulated using free- or no-slip conditions. However, for open systems, the information transferred through the boundaries is essential to the dynamics of the system and can have a first order impact on its evolution. This work addresses regional geodynamic modelling simulating the evolution of an Earth's piece over millions of years by solving non-linear Stokes flow. In this open system, we introduce a new approach to impose oblique boundary conditions generalising the Navier-slip boundary conditions to arbitrary directions in 3D. The method requires defining both slip and stress constraints. The stress constraint is imposed utilising a coordinate transformation to redefine the stress tensor along the boundaries according to the arbitrary direction chosen while for the slip constraint we utilise Nitsche's method in the context of the finite element method, resulting in a symmetrised and penalised weak form. We validate our approach through a series of numerical experiments of increasing complexity, starting with 2D and 3D linear models. Then, we apply those boundary conditions to a 3D non-linear geodynamic model of oblique extension that we compare with a standard model utilising Dirichlet boundary conditions. Our results show that using Dirichlet boundary conditions strongly influences the evolution of the system and generates artefacts near and along the boundaries. In comparison, the model using the generalised Navier-slip boundary conditions behaves closely to a model with an unbounded domain, providing a physically interpretable solution near and along the boundaries.