Volume Conserving Smoothing for Piecewise Linear Curves, Surfaces, and Triple Lines

We present smoothing algorithms for piecewise linear curves, surfaces, and triple lines of intersection of surfaces that are based on the the idea of sequentially relaxing either individual nodes or edges in the mesh. Each relaxation is designed both to smooth the mesh and to conserve down to round-off error the area or volume enclosed by the curve or surface. For the case of smoothing surfaces and lines of intersection of surfaces, each relaxation consists of a pure smoothing component and a volume conserving correction which is chosen to be of minimum norm. Since surfaces and triple intersection lines can be conservatively smoothed, the algorithms are suitable for improving multimaterial grids used by physics simulations where exactly conserving the volume of each individual material may be a requirement or at least highly desirable. The algorithms are also suitable for smoothing piecewise linear functions of one or two variables while simultaneously preserving their integrals. We show examples of the application of the more powerful edge-based algorithms to curve, surface, and multimaterial volume grids and to a thin film simulation.