Smooth Interface Reconstruction from Volume Fraction Data Using Variational Techniques and Level Set Methods
The volume fraction data structure describes the location of a given material within space. It partitions a domain into a set of discrete cells, and for each cell stores the ratio of material contained within it to the total volume of the cell. The task of transforming a grid of volume fractions into a surface representation of the boundary is called the Material Interface Reconstruction Problem. We investigate a variational formulation to this problem, implemented via a level set method, that produces a surface representation of the boundary. We start with an initial guess and iteratively refine the surface by computing a surface-area minimizing curvature flow, followed by an approximate L2 projection of that flow onto a volume-preserving space. We find that the method yields satisfactory results for both two-phase and multiphase data. In the cases where there are C0 but not C1 boundaries between only two materials, the exact solution produces oscillations with period equal to the volume fraction grid size, and amplitude exponentially decreasing away from the location of the loss of smoothness. To reduce oscillations, an alternative method is proposed where in place of the volume fraction constraint, we minimize a weighted sum of the total surface area, plus the sum of squared error in reconstruction. We discuss the results and compare with the projection algorithm.