Publications

This report summarizes the Combinatorial Algebraic Topology: software, applications & algorithms workshop (CAT Workshop). The workshop was sponsored by the Computer Science Research Institute of Sandia National Laboratories. It was organized by CSRI staff members Scott Mitchell and Shawn Martin. It was held in Santa Fe, New Mexico, August 29-30. The CAT Workshop website has links to some of the talk slides and other information, http://www.cs.sandia.gov/CSRI/Workshops/2009/CAT/index.html. The purpose of the report is to summarize the discussions and recap the sessions. There is a special emphasis on technical areas that are ripe for further exploration, and the plans for follow-up amongst the workshop participants. The intended audiences are the workshop participants, other researchers in the area, and the workshop sponsors.

Abstract not provided.

We develop a scheme for providing strong cryptographic authentication on a stream of messages which consumes very little bandwidth (as little as one bit per message) and is robust in the presence of dropped messages. Such a scheme should be useful for extremely low-power, low-bandwidth wireless sensor networks and "smart dust" applications. The tradeoffs among security, memory, bandwidth, and tolerance for missing messages give rise to several new optimization problems. We report on experimental results and derive bounds on the performance of the scheme. © 2008 Springer-Verlag Berlin Heidelberg.

Sandia National Laboratories is investing in projects that aim to develop computational modeling and simulation applications that explore human cognitive and social phenomena. While some of these modeling and simulation projects are explicitly research oriented, others are intended to support or provide insight for people involved in high consequence decision-making. This raises the issue of how to evaluate computational modeling and simulation applications in both research and applied settings where human behavior is the focus of the model: when is a simulation 'good enough' for the goals its designers want to achieve? In this report, we discuss two years' worth of review and assessment of the ASC program's approach to computational model verification and validation, uncertainty quantification, and decision making. We present a framework that extends the principles of the ASC approach into the area of computational social and cognitive modeling and simulation. In doing so, we argue that the potential for evaluation is a function of how the modeling and simulation software will be used in a particular setting. In making this argument, we move from strict, engineering and physics oriented approaches to V&V to a broader project of model evaluation, which asserts that the systematic, rigorous, and transparent accumulation of evidence about a model's performance under conditions of uncertainty is a reasonable and necessary goal for model evaluation, regardless of discipline. How to achieve the accumulation of evidence in areas outside physics and engineering is a significant research challenge, but one that requires addressing as modeling and simulation tools move out of research laboratories and into the hands of decision makers. This report provides an assessment of our thinking on ASC Verification and Validation, and argues for further extending V&V research in the physical and engineering sciences toward a broader program of model evaluation in situations of high consequence decision-making.

Abstract not provided.

Sweeping has become the workhorse algorithm for creating conforming hexahedral meshes of complex models. This paper describes progress on the automatic, robust generation of MultiSwept meshes in CUBIT. MultiSweeping extends the class of volumes that may be swept to include those with multiple source and multiple target surfaces. While not yet perfect, CUBIT's MultiSweeping has recently become more reliable, and been extended to assemblies of volumes. Sweep Forging automates the process of making a volume (multi) sweepable: Sweep Verification takes the given source and target surfaces, and automatically classifies curve and vertex types so that sweep layers are well formed and progress from sources to targets.

In an attempt to automatically produce high-quality all-hex meshes, we investigated a mesh improvement strategy: given an initial poor-quality all-hex mesh, we iteratively changed the element connectivity, adding and deleting elements and nodes, and optimized the node positions. We found a set of hex reconnection primitives. We improved the optimization algorithms so they can untangle a negative-Jacobian mesh, even considering Jacobians on the boundary, and subsequently optimize the condition number of elements in an untangled mesh. However, even after applying both the primitives and optimization we were unable to produce high-quality meshes in certain regions. Our experiences suggest that many boundary configurations of quadrilaterals admit no hexahedral mesh with positive Jacobians, although we have no proof of this.

A new method for lessening skew in mapped meshes is presented. This new method involves progressive subdivision of a surface into loops consisting of four sides. Using these loops, constraints can then be set on the curves of the surface, which will propagate interval assignments across the surface, allowing a mesh with a better skew metric to be generated.

Sweeping algorithms have become very mature and can create a semi-structured mesh on a large set of solids. However, these algorithms require that all linking surfaces be mappable or submappable. This restriction excludes solids with imprints or protrusions on the linking surfaces. The grafting algorithm allows these solids to be swept. It then locally modifies the position and connectivity of the nodes on the linking surfaces to align with the graft surfaces. Once a high-quality surface mesh is formed on the graft surface, it is swept along the branch creating a 2 3/4-D mesh.

This paper presents a new technique for automatically detecting interval constraints for swept volumes with holes. The technique finds true volume constraints that are not necessarily imposed by the surfaces of the volume. A graphing algorithm finds independent, parallel paths of edges from source surfaces to target surfaces. The number of intervals on two paths between a given source and target surface must be equal; in general, the collection of paths determine a set of linear constraints. Linear programming techniques solve the interval assignment problem for the surface and volume constraints simultaneously.

Consider mapping a regular i x j quadrilateral mesh of a rectangle onto a surface. The quality of the mapped mesh of the surface depends heavily on which vertices of the surface correspond to corners of the rectangle. The authors problem is, given an n-sided surface, chose as corners four vertices such that the surface resembles a rectangle with corners at those vertices. Note that n could be quite large, and the length and width of the rectangle, i and j, are not prespecified. In general, there is either a goal number or a prescribed number of mesh edges for each bounding curve of the surface. The goals affect the quality of the mesh, and the prescribed edges may make finding a feasible set of corners difficult. The algorithm need only work for surfaces that are roughly rectangular, particular those without large reflex angles, as otherwise an unstructured meshing algorithm is used instead. The authors report on the theory and implementation of algorithms for this problem. They also given an overview of a solution to a related problem called interval assignment: given a complex of surfaces sharing curves, globally assign the number of mesh edges or intervals for each curve such that it is possible to mesh each surface according to its prescribed quadrilateral meshing algorithm, and assigned and user-prescribed boundary mesh edges and corners. They also note a practical, constructive technique that relies on interval assignment that can generate a quadrilateral mesh of a complex of surfaces such that a compatible hexahedral mesh of the enclosed volume exists.

We give an algorithm for triangulating n-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than π/2. The number of triangles in the triangulation is only O(n), improving a previous bound of O(n 2), and the running time is O(n log2 n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm. © 1995 Springer-Verlag New York Inc.

Occasionally one may be confronted by a hexahedral or quadrilateral mesh containing doublets, two faces sharing two edges. In this case, no amount of smoothing will produce a mesh with agreeable element quality: in the planar case, one of these two faces will always have an angle of at least 180 degrees between the two edges. The authors describe a robust scheme for refining a hexahedral or quadrilateral mesh to separate such faces, so that any two faces share at most one edge. Note that this also ensures that two hexahedra share at most one face in the three dimensional case. The authors have implemented this algorithm and incorporated it into the CUBIT mesh generation environment developed at Sandia National Laboratories.

A popular three-dimensional mesh generation scheme is to start with a quadrilateral of the surface of a volume, and then attempt to fill the interior of volume with hexahedra, so that the hexahedra touch the surface in exactly the given quadrilaterals. Folklore has maintained that there are many quadrilateral meshes for which no such compatible hexahedral mesh exists. In this paper we give an existence proof which contradicts this folklore: A quadrilateral mesh need only satisfy some very weak conditions for there to exist a compatible hexahedral mesh. For a volume that is topologically a ball, any quadrilateral mesh composed of an even number of quadrilaterals admits a compatible hexahedral mesh. We extend this to volumes of higher genus: There is a construction to reduce to the ball case if and only if certain cycles of edges are even.

We consider bounding the cardinality of an arbitrary triangulation with smallest angle {alpha}. We show that if the local feature size (i.e. distance between disjoint vertices or edges) of the triangulation is within a constant factor of the local feature size of the input, then N < O(1/{alpha})M, where N is the cardinality of the triangulation and M is the cardinality of any other triangulation with smallest angle at least {alpha}. Previous results had an O(1/{alpha}{sup 1/{alpha}}) dependence. Our O(1/{alpha}) dependence is tight for input with a large length to height ratio, in which triangles may be oriented along the long dimension.

Given a planar straight-line graph, we find a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. Such a triangulation differs from the usual Steiner triangulation in that we may not add a Steiner vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. As it is possible that no finite covering triangulation is optimal in terms of its maximum angle, we propose an approximation algorithm. Our algorithm produces a covering triangulation whose maximum angle {gamma} is probably close to {gamma}{sub opt}, a lower bound on the maximum angle in any covering triangulation of the input graph. Note that we must have {gamma} {le} 3{gamma}{sub opt}, since we always have {gamma}{sub opt} {ge} {pi}/3 and no triangulation can contain an angle of size greater than {pi}. We prove something significantly stronger. We show that {pi} {minus} {gamma} {ge} ({pi} {minus} {gamma}{sub opt})/6, i.e., our {gamma} is not much closer to {pi} than is {gamma}{sub opt}. This result represents the first nontrivial bound on a covering triangulation`s maximum angle. We require a subroutine for the following problem: Given a polygon with holes, find a Steiner triangulation whose maximum angle is bounded away from {pi}. No angle larger than 8{pi}/9 is sufficient for the bound on {gamma} claimed above. The number of Steiner vertices added by our algorithm and its running time are highly dependent on the corresponding bounds for the subroutine. Given an n-vertex planar straight-line graph, we require O(n + S(n)) Steiner vertices and O(n log n + T(n)) time, where S(n) is the number of Steiner vertices added by the subroutine and T(n) is its running time for an O(n)-vertex polygon with holes.