Resolving Localisation Phenomena

Realistic numerical models of the Earth's crust and lithosphere require consitutive models which account for brittle/semi-brittle deformation. These are typically nonlinear, and produce localisation phenomena (e.g. faults, shear bands). The presence of a fault can cause variations in the deformation field over a length of the order of the fault / fault-zone thickness δ. Depending on the overall scale of the numerical model (global or regional), the interpretation of the fault thickness may vary, however it is typically the feature with the smallest length scale in the system. In order to accurately compute the stress and strain rate from our numerical models, we must ensure that we resolve shear bands to the point where characteristic patterns no longer change with resolution. This means using a discretisation which can resolve length scales of size δ. This is computationally difficult as the location of the shear band are not predefined. Furthermore, the length scale δ is typically several orders of magnitude smaller than the size of the domain. We have developed a numerical scheme suitable for modelling crustal/lithosphere deformation which dynamically modifies the spatial resolution in the vicinity of shear bands. The method comprises a mixed eulerian/lagrangian approach employing both finite elements and moving material points. Adaptivity at the grid and material point level is introduced using techniques from the nonlinear, adaptive finite elements. By combining standard element based error estimators and recovery techniques we can obtain continuous solutions for u_i, p and τij which are accurate to within a specified tolerance though out the entire domain. We show that the optimal rate of convergence is obtained for all fields and illustrate the effectiveness of the adaptive material point method via some simple localisation problems.