Generating textures on arbitrary surfaces using reaction-diffusion
Proceedings of the 18th annual conference on Computer graphics and interactive techniques
Curvature Dependent Polygonization of Implicit Surfaces
SIBGRAPI '04 Proceedings of the Computer Graphics and Image Processing, XVII Brazilian Symposium
Anisotropic Centroidal Voronoi Tessellations and Their Applications
SIAM Journal on Scientific Computing
A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant
Journal of Computational Physics
Mesh generation for implicit geometries
Mesh generation for implicit geometries
Finite Volume Methods on Spheres and Spherical Centroidal Voronoi Meshes
SIAM Journal on Numerical Analysis
An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries
Journal of Scientific Computing
Finite element approximation of elliptic partial differential equations on implicit surfaces
Computing and Visualization in Science
Adaptive polygonization of implicit surfaces
Computers and Graphics
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A finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed. The underlying motion is assumed to be described by a fixed, not necessarily normal, velocity field. The ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating simplicial meshes, where grid vertices move on motion trajectories, a consistent finite volume discretization of the induced transport on the simplices, and a proper incorporation of a diffusive flux balance at simplicial faces. The semi-implicit scheme is derived via a discretization of the underlying conservation law, and discrete counterparts of continuous a priori estimates in this geometric setup are proved. The continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces, and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes. Furthermore, numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results. Finally, we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications.