Modelling multiple-material flows on adaptive unstructured meshes.

Abstract

The ability to distinguish between regions with different material properties is essential when numerically modelling many physical systems. Using a dual control volume mesh that avoids the problem of corner coupling, the HyperC face value scheme is extended to multiple dimensions and applied as a device for material advection on unstructured simplex meshes. The new scheme performs well at maintaining sharp interfaces between materials and is shown to produce small advection errors, comparable to those of standard material advection methods on structured meshes. To further minimise numerical diffusion of material interfaces a total variation bounded flux limiter, UltraC, is defined using a normalised variable diagram. Combining the material tracking scheme with dynamically adapting meshes, the use of a minimally dissipative bounded projection algorithm for interpolating fields from the old mesh to the new, optimised mesh is demonstrated that conserves the mass of each material. More generally, material conservation during the advection process is ensured through the coupling of the material tracking scheme to the momentum and mass equations. A new element pair for the discretisation of velocity and pressure is proposed that maintains the stability of the system while conserving the mass of materials. When modelling multiple materials the use of independent advection algorithms for each material can lead to the problem of non-physical material overlap. A novel coupled flux limiter is developed to overcome this problem. This automatically generalises to arbitrary numbers of materials. Using the fully coupled (and rigorously verified) multi-material model, several geophysically relevant simulations are presented examining the generation of waves by landslides. The model is validated by demonstrating close agreement between model predictions and experimental results of wave generation, propagation and run-up. The simulations also showcase the powerful capabilities of an unstructured, adaptive multi-material model in real world scenarios