Meshing Deforming Spacetime for Visualization and Analysis

We introduce a novel paradigm that simplifies the visualization and analysis of data that have a spatially/temporally varying frame of reference. The primary application driver is tokamak fusion plasma, where science variables (e.g., density and temperature) are interpolated in a complex magnetic field-line-following coordinate system. We also see a similar challenge in rotational fluid mechanics, cosmology, and Lagrangian ocean analysis; such physics implies a deforming spacetime and induces high complexity in volume rendering, isosurfacing, and feature tracking, among various visualization tasks. Without loss of generality, this paper proposes an algorithm to build a simplicial complex -- a tetrahedral mesh, for the deforming 3D spacetime derived from two 2D triangular meshes representing consecutive timesteps. Without introducing new nodes, the resulting mesh fills the gap between 2D meshes with tetrahedral cells while satisfying given constraints on how nodes connect between the two input meshes. In the algorithm we first divide the spacetime into smaller partitions to reduce complexity based on the input geometries and constraints. We then independently search for a feasible tessellation of each partition taking nonconvexity into consideration. We demonstrate multiple use cases for a simplified visualization analysis scheme with a synthetic case and fusion plasma applications

Symbolic integration of polynomial functions over a linear polyhedron in euclidean three-dimensional space

The paper concerns analytical integration of polynomial functions over linear polyhedra in three-dimensional space. To the authors' knowledge this is a first presentation of the analytical integration of monomials over a tetrahedral solid in 3D space. A linear polyhedron can be obtained by decomposing it into a set of solid tetrahedrons, but the division of a linear polyhedral solid in 3D space into tetrahedra sometimes presents difficulties of visualization and could easily lead to errors in nodal numbering, etc We have taken this into account and also the linearity property of integration to derive a symbolic integration formula for linear hexahedra in 3D space. We have also used yet another fact that a hexahedron could be built up in two, and only two, distinct ways from five tetrahedral shaped elements These symbolic integration formulas are then followed by an illustrative numerical example for a rectangular prism element, which clearly verifies the formulas derived for the tetrahedron and hexahedron elements

On decomposition of embedded prismatoids in R3R^3 without additional points

This paper considers three-dimensional prismatoids which can be embedded in ℝ³ A subclass of this family are twisted prisms, which includes the family of non-triangulable Scönhardt polyhedra [12, 10]. We call a prismatoid decomposable if it can be cut into two smaller prismatoids (which have smaller volumes) without using additional points. Otherwise it is indecomposable. The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa. In this paper we prove two basic facts about the decomposability of embedded prismatoid in ℝ³ with convex bases. Let P be such a prismatoid, call an edge interior edge of P if its both endpoints are vertices of P and its interior lies inside P. Our first result is a condition to characterise indecomposable twisted prisms. It states that a twisted prism is indecomposable without additional points if and only if it allows no interior edge. Our second result shows that any embedded prismatoid in ℝ³ with convex base polygons can be decomposed into the union of two sets (one of them may be empty): a set of tetrahedra and a set of indecomposable twisted prisms, such that all elements in these two sets have disjoint interiors

On Indecomposable Polyhedra and the Number of Steiner Points

The existence of indecomposable polyhedra, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. However, the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we investigate the structure of some well-known examples, the so-called Schönhardt polyhedron [10] and the Bagemihl's generalization of it [1], which will be called Bagemihl's polyhedra. We provide a construction of an additional point, so-called Steiner point, which can be used to decompose the Schönhardt and the Bagemihl's polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schönhardt's and Bagemihl's polyhedra, but they may need more than one Steiner point to be decomposed. Given such a polyhedron with n ≥ 6 vertices, we show that it can be decomposed by adding at most interior Steiner points. We also show that this number is optimal in theworst case

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

The existence of triangulations of non-convex polyhedra without new vertices

It is well known that a simple three-dimensional non-convex polyhedron may not be triangulated without using new vertices (so-called {\it Steiner points}). In this paper, we prove a condition that guarantees the existence of a triangulation of a non-convex polyhedron (of any dimension) without Steiner points. We briefly discuss algorithms for efficiently triangulating three-dimensional polyhedra

On indecomposable polyhedra and the number of interior Steiner points

The existence of 3d {\it indecomposable polyhedra}, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. While the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we first investigate the geometry of some well-known examples, the so-called {\it Sch\"on\-hardt polyhedron}~\cite{Schonhardt1928} and the Bagemihl's generalization of it~\cite{Bagemihl48-decomp-polyhedra}, which will be called {\it Bagemihl polyhedra}. We provide a construction of an interior point, so-called {\it Steiner point}, which can be used to tetrahedralize the Sch\"on\-hardt and the Bagemihl polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Sch\"onhardt and Bagemihl polyhedra, but they may need more than one interior Steiner point to be tetrahedralized. Given such a polyhedron with $n \ge 6$ vertices, we show that it can be tetrahedralized by adding at most $\left\lceil \frac{n - 5}{2}\right\rceil$ interior Steiner points. %, is sufficient to decompose it. We also show that this number is optimal in the worst case

ON THE GENERATION OF HIERARCHICAL MESHES FOR MULTILEVEL FEM AND BEM SOLVERS FROM CAD DATA

As numerical techniques for solving PDE or integral equations become more sophisticated, treatments of the generation of the geometric inputs should also follow that numerical advancement. This document describes the preparation of CAD data so that they can later be applied to hierarchical BEM or FEM solvers. For the BEM case, the geometric data are described by surfaces which we want to decompose into several curved foursided patches. We show the treatment of untrimmed and trimmed surfaces. In particular, we provide prevention of smooth corners which are bad for diffeomorphism. Additionally, we consider the problem of characterizing whether a Coons map is a diffeomorphism from the unit square onto a planar domain delineated by four given curves. We aim primarily at having not only theoretically correct conditions but also practically efficient methods. As for FEM geometric preparation, we need to decompose a 3D solid into a set of curved tetrahedra. First, we describe some method of decomposition without adding too many Steiner points (additional points not belonging to the initial boundary nodes of the boundary surface). Then, we provide a methodology for efficiently checking whether a tetrahedral transfinite interpolation is regular. That is done by a combination of degree reduction technique and subdivision. Along with the method description, we report also on some interesting practical results from real CAD data