A b str a c tT he grid generation m ethod based on th e m inim ization of th e discrete barrier functional w ith feasible set consisting of quasi-isom etric grids is suggested. T he deviation from isom etry for given grid connectivity an d fixed boundary nodes is m inim ized via th e contraction of th e feasible set into small vicinity of th e optim al grid. Form ulation of functional w ith given m etrics in b o th physical and logical spaces allows to consider th e adaptive grid generation in term s of quasi isom etric grids and cover m any practical applications. T he fast an d reliable procedure for finding feasible solution based on th e penalty-like reform ulation of barrier functional and th e continuation technique is described. T he relations between different barrier approxim ations to quasi-isom etric functional in 2-D and 3-D cases are investigated. N um erical experim ents have confirm ed th a t th e suggested functional allows to obtain high quality grids.

Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers. In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to be defined in the map space and the geometric domain is not explicit, supposing that each element is regular. In this paper, we propose a mapping method inspired by the untangling problem and compare its performance to the state of the art. The main advantage of our method is that the untangling aims at producing locally injective maps, which is the major challenge of mapping. In practice, our method produces locally injective maps in very difficult settings, both in 2D and 3D. We demonstrate it on a large reference database as well as on more difficult stress tests. For a better reproducibility, we publish the code in Python for a basic evaluation, and in C++ for more advanced applications.

No abstract

No abstract

No abstract

No abstract

SUMMARYDistortion measures for polylinear mappings are investigated. It is shown that certain distortion measures satisfy the maximum principle which allows us to obtain upper bounds on the distortion measures for hexahedral cells and other types of elements widely used in the ÿnite element method. These estimates allow to apply a maximum-norm optimization technique for spatial mappings in the case of ÿnite element grids consisting of hexahedra. A global hexahedral grid untangling procedure suggested earlier was tested on hard 3-D examples demonstrating its ability to work in a black box mode and its high level of robustness.

No abstract