A level-set multigrid technique for nonlinear diffusion in the numerical simulation of marble degradation under chemical pollutants

Having in mind the modelling of marble degradation under chemical pollutants, e.g. the sulfation process, we consider governing nonlinear diffusion equations and their numerical approximation. The space domain of a computation is the pristine marble object. In order to accurately discretize it while maintaining the simplicity of finite difference discretizations, the domain is described using a level-set technique. A uniform Cartesian grid is laid over a box containing the domain, but the solution is defined and updated only in the grid nodes that lie inside the domain, the level-set being employed to select them and to impose accurately the boundary conditions. We use a Crank-Nicolson scheme in time, while for the space variables the discretization is performed by a standard Finite-Difference scheme for grid points inside the domain and by a ghost-cell technique on the ghost points (by using boundary conditions). The solution of the large nonlinear system is obtained by a Newton-Raphson procedure and a tailored multigrid technique is developed for the inner linear solvers. The numerical results, which are very satisfactory in terms of reconstruction quality and of computational efficiency, are presented and discussed at the end of the paper

Simulation of incompressible viscous flows on distributed Octree grids

This dissertation focuses on numerical simulation methods for continuous problems with irregular interfaces. A common feature of these types of systems is the locality of the physical phenomena, suggesting the use of adaptive meshes to better focus the computational effort, and the complexity inherent to representing a moving irregular interface. We address these challenges by using the implicit framework provided by the Level-Set method and implemented on adaptive Quadtree (in two spatial dimensions) and Octree (in three spatial dimensions) grids. This work is composed of two sections.In the first half, we present the numerical tools for the study of incompressible monophasic viscous flows. After a study of an alternative grid storage structure to the Quad/Oc-tree data structure based on hash tables, we introduce the extension of the level-set method to massively parallel forests of Octrees. We then detail the numerical scheme developed to attain second order accuracy on non-graded Quad/Oc-tree grids and demonstrate the validity and robustness of the resulting solver. Finally, we combine the fluid solver and the parallel framework together and illustrate the potential of the approach.The second half of this dissertation presents the Voronoi Interface Method (VIM), a new method for solving elliptic systems with discontinuities on irregular interfaces such as the ones encountered when simulating viscous multiphase flows. The VIM relies on a Voronoi mesh built on an underlying Cartesian grid and is compact and second order accurate while preserving the symmetry and positiveness of the resulting linear system. We then compare the VIM with the popular Ghost Fluid Method before adapting it to the simulation of the problem of the electropermeabilization of cells