Redistribution of Meshes’ Nodes Using Moving Surfaces

Surface processing tools based on Partial Differential Equations (PDEs) are emerging recently in computer graphics, digital animation, computer aided modelling, and computer vision. In this work we propose an algorithm to move and redistribute the nodes of a mesh representing a surface in R3. This is obtained by a surface evolution process according to normal and tangential velocities with two aims: • to obtain a more homogeneous surface representation and a more, numerically easy to process, mesh • to move the surface in space avoiding mesh nodes’ collision. The evolution of the surface is formulated in a Lagrangian framework using a couple of PDEs applied to the surfaces’ spatial characteristics. In the first equation a scalar field which follows the spatial features of the mesh, i.e. mean curvature or local elements’ areas, is determined solving an intrinsic Poisson equation. In the second equation the actual surface evolution is computed using the field. Numerical schemes based on finite element discretization in space will be considered, together with several strategies for tangential velocities. The numerical results illustrate how this framework is able both to improve the uniformity of the mesh nodes, and to control the surface evolution avoiding nodes’ collision. Finally, to stress the importance of our approach in a computer graphics context, we consider simple applications to surface smoothing and remeshing in morphing processes. This work has been implemented integrating COMSOL functions into a MATLAB environment. A COMSOL Multiphysics solution is also tested.