Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.Comment: Preprint submitted to CAD (26th IMR special issue

Fixed versus random triangulations in 2D simplicial Regge calculus

We study 2D quantum gravity on spherical topologies using the Regge calculus approach with the $dl/l$ measure. Instead of a fixed non-regular triangulation which has been used before, we study for each system size four different random triangulations, which are obtained according to the standard Voronoi-Delaunay procedure. We compare both approaches quantitatively and show that the difference in the expectation value of $R^2$ between the fixed and the random triangulation depends on the lattice size and the surface area $A$. We also try again to measure the string susceptibility exponents through a finite-size scaling Ansatz in the expectation value of an added $R^2$ interaction term in an approach where $A$ is held fixed. The string susceptibility exponent $\gamma_{str}'$ is shown to agree with theoretical predictions for the sphere, whereas the estimate for $\gamma_{str}$ appears to be too negative.Comment: 4 latex pages + 4 ps-figs. + espcrc2.sty, poster presented by W. Janke at LATTICE96(gravity

The F model on dynamical quadrangulations

The dynamically triangulated random surface (DTRS) approach to Euclidean quantum gravity in two dimensions is considered for the case of the elemental building blocks being quadrangles instead of the usually used triangles. The well-known algorithmic tools for treating dynamical triangulations in a Monte Carlo simulation are adapted to the problem of these dynamical quadrangulations. The thus defined ensemble of 4-valent graphs is appropriate for coupling to it the 6- and 8-vertex models of statistical mechanics. Using a series of extensive Monte Carlo simulations and accompanying finite-size scaling analyses, we investigate the critical behaviour of the 6-vertex F model coupled to the ensemble of dynamical quadrangulations and determine the matter related as well as the graph related critical exponents of the model.Comment: LaTeX, 43 pages, 10 figures, 7 tables; substantially shortened and revised version as published, for more details refer to V1, to be found at http://arxiv.org/abs/hep-lat/0409028v

Simplicial Quantum Gravity on a Randomly Triangulated Sphere

We study 2D quantum gravity on spherical topologies employing the Regge calculus approach with the dl/l measure. Instead of the normally used fixed non-regular triangulation we study random triangulations which are generated by the standard Voronoi-Delaunay procedure. For each system size we average the results over four different realizations of the random lattices. We compare both types of triangulations quantitatively and investigate how the difference in the expectation value of the squared curvature, $R^2$, for fixed and random triangulations depends on the lattice size and the surface area A. We try to measure the string susceptibility exponents through finite-size scaling analyses of the expectation value of an added $R^2$-interaction term, using two conceptually quite different procedures. The approach, where an ultraviolet cut-off is held fixed in the scaling limit, is found to be plagued with inconsistencies, as has already previously been pointed out by us. In a conceptually different approach, where the area A is held fixed, these problems are not present. We find the string susceptibility exponent $\gamma_{str}'$ in rough agreement with theoretical predictions for the sphere, whereas the estimate for $\gamma_{str}$ appears to be too negative. However, our results are hampered by the presence of severe finite-size corrections to scaling, which lead to systematic uncertainties well above our statistical errors. We feel that the present methods of estimating the string susceptibilities by finite-size scaling studies are not accurate enough to serve as testing grounds to decide about a success or failure of quantum Regge calculus.Comment: LaTex, 29 pages, including 9 figure

Shaken, but not stirred - Potts model coupled to quantum gravity

We investigate the critical behaviour of both matter and geometry of the three-state Potts model coupled to two-dimensional Lorentzian quantum gravity in the framework of causal dynamical triangulations. Contrary to what general arguments of the effects of disorder suggest, we find strong numerical evidence that the critical exponents of the matter are not changed under the influence of quantum fluctuations in the geometry, compared to their values on fixed, regular lattices. This lends further support to previous findings that quantum gravity models based on causal dynamical triangulations are in many ways better behaved than their Euclidean counterparts.Comment: 19 pages, 9 figure

Introducing Quantum Ricci Curvature

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian manifold by comparing the distance between pairs of spheres with that of their centres. The quantum Ricci curvature is designed for use on non-smooth and discrete metric spaces, and to satisfy the key criteria of scalability and computability. We test the prescription on a variety of regular and random piecewise flat spaces, mostly in two dimensions. This enables us to quantify its behaviour for short lattices distances and compare its large-scale behaviour with that of constantly curved model spaces. On the triangulated spaces considered, the quantum Ricci curvature has good averaging properties and reproduces classical characteristics on scales large compared to the discretization scale.Comment: 43 pages, 27 figure

Three-Dimensional Simplicial Gravity and Degenerate Triangulations

I define a model of three-dimensional simplicial gravity using an extended ensemble of triangulations where, in addition to the usual combinatorial triangulations, I allow degenerate triangulations, i.e. triangulations with distinct simplexes defined by the same set of vertexes. I demonstrate, using numerical simulations, that allowing this type of degeneracy substantially reduces the geometric finite-size effects, especially in the crumpled phase of the model, in other respect the phase structure of the model is not affected.Comment: Latex, 19 pages, 10 eps-figur

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology