Recognizing and Drawing IC-planar Graphs

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph $G$ with $n$ vertices, we present an $O(n)$-time algorithm that computes a straight-line drawing of $G$ in quadratic area, and an $O(n^3)$-time algorithm that computes a straight-line drawing of $G$ with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

GENERATION OF TRIANGULAR MESHES FOR COMPLEX DOMAINS ON THE PLANE

Many physical phenomena can bc modelcd by partial diffcrcntial cąuations. The dcvclopmcnt of numcrical methods bascd on the spatial subdivision of a domain into fmitc clcmcnts immcdiatcly cxtcnded interests to the tasks of generating a mesh. With the availability of vcrsatilc field solv- crs and powerful computcrs, the simulations of cver inereasing gcometrical and physical com- plcxity arc attempted. At somc point the main bottleneck becomcs the mesh generation itsclf.The papcr prcsents a dctailcd description of the triangular mcsh gcneration schcmc on the piane bascd upon the Dclaunay triangulation. A mcsh generator should be fully automatic and simplify input data as much as possible. It should offer rapid gradation from smali to large sizes of elcmcnts. The generated mcsh must be always valid and of good quality. Ali thesc rcquiremcnts were taken into account during the selection and elaboration of utilized algorithms.Successive chapters describe procedures connected with the specification of a modeled domain, gcneration and triangulation of boundary vertices, introducing inner nodes, improving the quality of the crcated mcsh, and renumbering of vertices

Graph-grammar based algorithm for asteroid tsunami simulations

On January 18, 2022, around 1 million kilometers from Earth, five times the distance from Earth to the Moon, a large asteroid passed without harm to the Earth. Theoretically, however, the event of the asteroid falling into Earth, causing the tsunami, is possible since there are over 27,000 near-Earth asteroids [1], and the Earth’s surface is covered in 71 percent by water. We introduce a novel graph-grammar-based framework for asteroid tsunami simulations. Our framework adaptively generates the computational mesh of the Earth model. It is built from triangular elements representing the seashore and the seabed. The computational mesh is represented as a graph, with graph vertices representing the computational mesh element’s interiors and edges. Mesh refinements are often performed by the longest-edge refinement algorithm. We have expressed this algorithm by only two graph-grammar productions. The resulting graph represents the terrain approximating the topography with a prescribed accuracy. We generalize the graph-grammar mesh refinement algorithm to work on the entire Earth model, allowing the generation of the terrain topography, including the seabed. Having the seashore and the seabed represented by a graph, we introduce the finite element method simulations of the tsunami wave propagation. We illustrate the framework with simulations of the disastrous asteroid falling into the Baltic sea