Arthur Guittet

Solving elliptic problems with discontinuities on irregular domains - the Voronoi Interface Method

We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solutions gradients are second-order accurate and first-order accurate, respectively, in the L ∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions.

A local level-set method using a hash table data structure

We present a local level-set method based on the hash table data structure, which allows the storage of only a band of grid points adjacent to the interface while providing an O(1) access to the data. We discuss the details of the construction of the hash table data structure as well as the advection and reinitialization schemes used for our implementation of the level-set method. We propose two dimensional numerical examples and compare the results to those obtained with a quadtree data structure. Our study indicates that the method is straightforward to implement but suffers from limitations that make it less efficient than the quadtree data structure.

A stable projection method for the incompressible Navier-Stokes equations on arbitrary geometries and adaptive Quad/Octrees

We present a numerical method for solving the incompressible Navier-Stokes equations on non-graded quadtree and octree meshes and arbitrary geometries. The viscosity is treated implicitly through a finite volume approach based on Voronoi partitions, while the convective term is discretized with a semi-Lagrangian scheme, thus relaxing the restrictions on the time step. A novel stable implementation of the projection step is introduced, making use of the Marker And Cell layout for the data. The solver is validated numerically in two and three spatial dimensions.

Level-set simulations of soluble surfactant driven flows

Abstract We present an approach to simulate the diffusion, advection and adsorption–desorption of a material quantity defined on an interface in two and three spatial dimensions. We use a level-set approach to capture the interface motion and a Quad/Octree data structure to efficiently solve the equations describing the underlying physics. Coupling with a Navier–Stokes solver enables the study of the effect of soluble surfactants that locally modify the parameters of surface tension on different types of flows. The method is tested on several benchmarks and applied to three typical examples of flows in the presence of surfactant: a bubble in a shear flow, the well-known phenomenon of tears of wine, and the Landau–Levich coating problem.

The island dynamics model on parallel quadtree grids

Abstract We introduce an approach for simulating epitaxial growth by use of an island dynamics model on a forest of quadtree grids, and in a parallel environment. To this end, we use a parallel framework introduced in the context of the level-set method. This framework utilizes: discretizations that achieve a second-order accurate level-set method on non-graded adaptive Cartesian grids for solving the associated free boundary value problem for surface diffusion; and an established library for the partitioning of the grid. We consider the cases with: irreversible aggregation, which amounts to applying Dirichlet boundary conditions at the island boundary; and an asymmetric (Ehrlich–Schwoebel) energy barrier for attachment/detachment of atoms at the island boundary, which entails the use of a Robin boundary condition. We provide the scaling analyses performed on the Stampede supercomputer and numerical examples that illustrate the capability of our methodology to efficiently simulate different aspects of epitaxial growth. The combination of adaptivity and parallelism in our approach enables simulations that are several orders of magnitude faster than those reported in the recent literature and, thus, provides a viable framework for the systematic study of mound formation on crystal surfaces.

On Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIM

We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the -norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand.