Uniform refinable 3D grids of regular convex polyhedrons and balls

Abstract

We construct a simple volume-preserving map from the cube [0,b]3 to the tetrahedron $${\{(x,y,z)\in \mathbb R^3}$$ , $${x \geq0}$$ , $${y \geq 0}$$ , $${z \geq 0}$$ , $${x+y+z\leq a\},}$$ with $${a=b\sqrt[3]6}$$ . This map will allow us to construct equal-volume subdivisions of arbitrary tetrahedrons and arbitrary convex polyhedrons into polyhedral cells. Moreover, mapping the regular octahedron onto the ball using a volume-preserving map previously constructed by the authors, one can obtain uniform and refinable grids on the 3D ball by a simple procedure, starting from appropriate grids on the cube.