Generalization of the Nitsche method to apply oblique boundary conditions in regional geodynamic models
Authors: Jourdon Anthony , May Dave A. , Gabriel Alice A. 
Published: February 2023

Regional geodynamic models require to impose boundary conditions that best represent the physical information exchanged between the modelled and a larger, non-modelled domain. Depending on the nature of the physical information exchange, the internal evolution of the regional system may differ. Nevertheless, the first and foremost observation is that the deformation in tectonic plates boundaries is three-dimensional, i.e., non-cylindrical, oblique.To model 3D non-cylindrical deformation, regional geodynamic models mostly use initial conditions through oblique or offset weak zones together with cylindrical boundary conditions implying free slip. However, the problem with the free-slip boundary condition is that it enforces cylindrical behaviours in the vicinity of the boundary, limiting the obliquity of the whole system or forcing to consider very large domains to avoid a too strong influence of the boundary condition.A way to work around this problem is to impose obliquity through boundary conditions. Until now, the main approach to impose oblique boundary conditions involves strong Dirichlet constraints, i.e., directly providing the solution for the velocity (or displacement) along the boundary.However, the choice of velocity values can lead to arbitrarily imposed velocity gradients particularly in the tangential direction of the boundary when the velocity vectors point in different directions. Such boundary effects can then influence the strain localization and produce non-physical results.In this work, we propose a formulation to impose oblique boundary conditions by enforcing the velocity direction but without constraining the magnitude of the velocity vectors. We seek to impose a slip-type boundary condition. The formulation&#160;is a generalszation of&#160;Nitsches&#8217;&#160;method&#160;(Nitsche, 1971)&#160;thereby allowing Navier-slip constraints to be enforced independently of the orientation domain boundary. We refer to this new formulation as the generalised Navier-slip boundary condition..In order to demonstrate that the method works as well as to illustrate the differences it produces on the evolution of a geodynamic system compared to the use of more classical boundary conditions, we show two 3D oblique rift models. The first uses Dirichlet boundary conditions and the second uses the generalised Navier-slip method to enforce an oblique extension at 45&#176;.The models show differences not only along and near their boundaries but also in the centre of the modelled domain during its tectonic evolution in terms of strian localization, basin architecture and topography. Moreover, the model using the generalised Navier-slip method to impose oblique extension shows a more natural evolution of the strain localization and tectonic features as the velocity along and near boundaries can vary in time and space to adapt to the internal evolution of the model.Finally, we show that the generalised Navier-slip method provides a better approach to impose oblique boundary conditions than the classical methods as it does not require to impose an arbitrary velocity function directly into the solution.&#160;Nitsche, J., 1971. &#220;ber ein variationsprinzip zur l&#246;sung von dirichlet-problemen bei verwendung von teilr&#228;umen, die keinen randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universit&#228;t Hamburg 36, 9&#8211;15. doi:10.1007/BF02995904.