Tessellation and interactive visualization of four-dimensional spacetime geometries

TL;DR

The results suggest that visualizing tetrahedra (either those bounding the domain, or extracted from a pentatopal mesh) using a geometry shader achieves the highest frame rate, in the range of $20-30 frames per second for meshes with about $50$ million tetrahedra.

Abstract

This paper addresses two problems needed to support four-dimensional ($3d + t$) spacetime numerical simulations. The first contribution is a general algorithm for producing conforming spacetime meshes of moving geometries. Here, the surface points of the geometry are embedded in a four-dimensional space as the geometry moves in time. The geometry is first tessellated at prescribed time steps and then these tessellations are connected in the parameter space of each geometry entity to form tetrahedra. In contrast to previous work, this approach allows the resolution of the geometry to be controlled at each time step. The only restriction on the algorithm is the requirement that no topological changes to the geometry are made (i.e. the hierarchical relations between all geometry entities are maintained) as the geometry moves in time. The validity of the final mesh topology is verified by ensuring the tetrahedralizations represent a closed 3-manifold. For some analytic problems, the $4d$ volume of the tetrahedralization is also verified. The second problem addressed in this paper is the design of a system to interactively visualize four-dimensional meshes, including tetrahedra (embedded in $4d$) and pentatopes. Algorithms that either include or exclude a geometry shader are described, and the efficiency of each approach is then compared. Overall, the results suggest that visualizing tetrahedra (either those bounding the domain, or extracted from a pentatopal mesh) using a geometry shader achieves the highest frame rate, in the range of $20-30$ frames per second for meshes with about $50$ million tetrahedra.

Figures (13)

• Figure 1: Examples of $3d$ geometries moving in spacetime ($4d$). The wind turbine geometry was adapted from a post on the LaTeX Stack Exchange Wibrow_2014.
• Figure 2: Illustration of how the time-dependent geometry is tessellated at discrete time steps (left) and how a time slab is defined between time steps (right).
• Figure 3: Description of the algorithm for tessellating a moving geometry within a time interval $[t^k, t^{k+1}]$. When a CAD Face ($\mathcal{F}$), Edge ($\mathcal{E}$) or Node $(\mathcal{N}$) is overscored with a tilde ( $\widetilde{}$ ), it should be understood that the entity is in some parametric space, whether it be that of an Edge ($s$) or Face $(u, v)$. A Node $\mathcal{N}$ can be mapped to the parametric spaces of either an Edge $\mathcal{E}$ or Face $\mathcal{F}$. The subscripts ($\alpha,\ \beta,\ \gamma$) denote the unique identifiers assigned to each entity.
• Figure 4: Description of the hyperplane-tetrahedron intersection algorithm. Intersections are listed in the same order as the Marching Tetrahedra algorithm Bourke_1994. Only 8 rows of the look-up table are shown due to symmetry (the symmetric values of $r$ are labelled in parentheses). The intersection region (to be decomposed into rendering primitives) are shown in red. For each case, red vertices are on one side of the hyperplane $\mathcal{H}$ whereas the black vertices are on the other side.
• Figure 5: Convergence of the volume error for analytic geometries.