It is essential to Sandia National Laboratory’s continued success in scientific and technological advances and mission delivery to embrace a hybrid workforce culture under which current and future employees can thrive. This report focuses on the findings of the Hybrid Work Team for the Center for Computing Research, which met weekly from March to June 2023 and conducted a survey across the Center at Sandia. Conclusions in this report are drawn from the 9 authors of this report, which comprises the Hybrid Work Team, and 15 responses to a center-wide survey, as well as numerous conversations with colleagues. A major finding was widespread dissatisfaction with the quantity, execution, and tooling surrounding formal meetings with remote participants. While there was consensus that remote work enables people to produce high quality individual and technical work, there was also consensus that there was widespread social disconnect, with particular concern about hires that were made after the onset of the Covid-19 pandemic. There were many concerns about tooling and policy to facilitate remote collaboration both within Sandia and with its external collaborators. This report includes recommendations for mitigating these problems. For problems for which obvious recommendations cannot be made, ideas of what a successful solution might look like are presented.
We propose primal–dual mesh optimization algorithms that overcome shortcomings of the standard algorithm while retaining some of its desirable features. “Hodge-Optimized Triangulations” defines the “HOT energy” as a bound on the discretization error of the diagonalized Delaunay Hodge star operator. HOT energy is a natural choice for an objective function, but unstable for both mathematical and algorithmic reasons: it has minima for collapsed edges, and its extrapolation to non-regular triangulations is inaccurate and has unbounded minima. We propose a different extrapolation with a stronger theoretical foundation, and avoid extrapolation by recalculating the objective just beyond the flip threshold. We propose new objectives, based on normalizations of the HOT energy, with barriers to edge collapses and other undesirable configurations. We propose mesh improvement algorithms coupling these. When HOT optimization nearly collapses an edge, we actually collapse the edge. Otherwise, we use the barrier objective to update positions and weights and remove vertices. By combining discrete connectivity changes with continuous optimization, we more fully explore the space of possible meshes and obtain higher quality solutions.
Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued. Many other approaches perform numerical optimization then convert a floating-point solution into an integer solution, which is slow and error prone. We avoid such steps: we start integer, and stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the meshing constraints, satisfying the integer matrix equation Ax=b. Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. We add vectors from the nullspace to improve the initial solution, maintaining Ax=b. Heuristics find good integer linear combinations of nullspace vectors that provide strict improvement towards variable bounds or goals. IIA always produces an integer solution if one exists. In practice we usually achieve solutions close to the user goals, but there is no guarantee that the solution is optimal, nor even satisfies variable bounds, e.g. has positive intervals. We describe several algorithmic changes since first publication that tend to improve the final solution. The software is freely available.
We show how to sample uniformly within the three-sided region bounded by a circle, a radial ray, and a tangent, called a “chock.” By dividing a 2D planar rectangle into a background grid, and subtracting Poisson disks from grid squares, we are able to represent the available region for samples exactly using triangles and chocks. Uniform random samples are generated from chock areas precisely without rejection sampling. This provides the first implemented algorithm for precise maximal Poisson-disk sampling in deterministic linear time. We prove O(n · M(b) log b), where n is the number of samples, b is the bits of numerical precision and M is the cost of multiplication. Prior methods have higher time complexity, take expected time, are non-maximal, and/or are not Poisson-disk distributions in the most precise mathematical sense. We fill this theoretical lacuna.
We show how to sample uniformly within the three-sided region bounded by a circle, a radial ray, and a tangent, called a “chock.” By dividing a 2D planar rectangle into a background grid, and subtracting Poisson disks from grid squares, we are able to represent the available region for samples exactly using triangles and chocks. Uniform random samples are generated from chock areas precisely without rejection sampling. This provides the first implemented algorithm for precise maximal Poisson-disk sampling in deterministic linear time. We prove O(n · M(b) log b), where n is the number of samples, b is the bits of numerical precision and M is the cost of multiplication. Prior methods have higher time complexity, take expected time, are non-maximal, and/or are not Poisson-disk distributions in the most precise mathematical sense. We fill this theoretical lacuna.
This report details the results of a three-fold investigation of sensitivity analysis (SA) for machine learning (ML) explainability (MLE): (1) the mathematical assessment of the fidelity of an explanation with respect to a learned ML model, (2) quantifying the trustworthiness of a prediction, and (3) the impact of MLE on the efficiency of end-users through multiple users studies. We focused on the cybersecurity domain as the data is inherently non-intuitive. As ML is being using in an increasing number of domains, including domains where being wrong can elicit high consequences, MLE has been proposed as a means of generating trust in a learned ML models by end users. However, little analysis has been performed to determine if the explanations accurately represent the target model and they themselves should be trusted beyond subjective inspection. Current state-of-the-art MLE techniques only provide a list of important features based on heuristic measures and/or make certain assumptions about the data and the model which are not representative of the real-world data and models. Further, most are designed without considering the usefulness by an end-user in a broader context. To address these issues, we present a notion of explanation fidelity based on Shapley values from cooperative game theory. We find that all of the investigated MLE explainability methods produce explanations that are incongruent with the ML model that is being explained. This is because they make critical assumptions about feature independence and linear feature interactions for computational reasons. We also find that in deployed, explanations are rarely used due to a variety of reason including that there are several other tools which are trusted more than the explanations and there is little incentive to use the explanations. In the cases when the explanations are used, we found that there is the danger that explanations persuade the end users to wrongly accept false positives and false negatives. However, ML model developers and maintainers find the explanations more useful to help ensure that the ML model does not have obvious biases. In light of these findings, we suggest a number of future directions including developing MLE methods that directly model non-linear model interactions and including design principles that take into account the usefulness of explanations to the end user. We also augment explanations with a set of trustworthiness measures that measure geometric aspects of the data to determine if the model output should be trusted.
We propose primal-dual mesh optimization algorithms that overcome shortcomings of the standard algorithm while retaining some of its desirable features. “Hodge-Optimized Triangulations” defines the “HOT energy” as a bound on the discretization error of the diagonalized Delaunay Hodge star operator. HOT energy is a natural choice for an objective function, but unstable for both mathematical and algorithmic reasons: it has minima for collapsed edges, and its extrapolation to non-regular triangulations is inaccurate and has unbounded minima. We propose a different extrapolation with a stronger theoretical foundation. We propose new objectives, based on normalizations of the HOT energy, with barriers to edge collapses and other undesirable configurations. We propose mesh improvement algorithms coupling these. When HOT optimization nearly collapses an edge, we actually collapse the edge. Otherwise, we use the barrier objective to update positions and weights. By combining discrete connectivity changes with continuous optimization, we more fully explore the space of possible meshes and obtain higher quality solutions.
Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued, yet many approaches perform floating-point optimization and convert a floating-point solution into an integer solution. We avoid such steps: we start integer, stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the matrix equation Ax = b satisfying the meshing constraints. Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. We add vectors from the nullspace to improve the initial solution. Compared to floating-point optimization approaches, IIA is faster and always produces an integer solution. The potential drawback is that there is no theoretical guarantee that the solution is optimal, but in practice we achieve solutions close to the user goals. The software is freely available.
Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued, yet many approaches perform floating-point optimization and convert a floating-point solution into an integer solution. We avoid such steps: we start integer, stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the matrix equation Ax = b satisfying the meshing constraints. Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. We add vectors from the nullspace to improve the initial solution. Compared to floating-point optimization approaches, IIA is faster and always produces an integer solution. The potential drawback is that there is no theoretical guarantee that the solution is optimal, but in practice we achieve solutions close to the user goals. The software is freely available.
We statistically infer fluid flow and transport properties of porous materials based on their geometry and connectivity, without the need for detailed We summarize structure by persistent homology and then determines the similarity of structures using image analysis and statistics. Longer term, this may enable quick and automated categorization of rocks into known archetypes. We first compute persistent homology of binarized 3D images of material subvolume samples. The persistence parameter is the signed Euclidean distance from inferred material interfaces, which captures the distribution of sizes of pores and grains. Each persistence diagram is converted into an image vector. We infer structural similarity by calculating image similarity. For each image vector, we compute principal components to extract features. We fit statistical models to features estimates material permeability, tortuosity, and anisotropy. We develop a Structural SIMilarity index to determine statistical representative elementary volumes.
This report summarizes the accomplishments and challenges of a two year LDRD effort focused on improving design-to-simulation agility. The central bottleneck in most solid mechanics simulations is the process of taking CAD geometry and creating a discretization of suitable quality, i.e., the "meshing" effort. This report revisits meshfree methods and documents some key advancements that allow their use on problems with complex geometries, low quality meshes, nearly incompressible materials or that involve fracture. The resulting capability was demonstrated to be an effective part of an agile simulation process by enabling rapid discretization techniques without increasing the time to obtain a solution of a given accuracy. The first enhancement addressed boundary-related challenges associated with meshfree methods. When using point clouds and Euclidean metrics to construct approximation spaces, the boundary information is lost, which results in low accuracy solutions for non-convex geometries and mate rial interfaces. This also complicates the application of essential boundary conditions. The solution involved the development of conforming window functions which use graph and boundary information to directly incorporate boundaries into the approximation space.
Interval Assignment (IA) means selecting the number of mesh edges for each CAD curve. IIA is a discrete algorithm over integers. A priority queue iteratively selects compatible sets of intervals to increase in lock-step by integers. In contrast, the current capability in Cubit is floating-point Linear Programming with Branch-and-Bound for integerization (BBIA).
The classical problem of calculating the volume of the union of d-dimensional balls is known as "Union Volume." We present line-sampling approximation algorithms for Union Volume. Our methods may be extended to other Boolean operations, such as setminus; or to other shapes, such as hyper-rectangles. The deterministic, exact approaches for Union Volume do not scale well to high dimensions. However, we adapt several of these exact approaches to approximation algorithms based on sampling. We perform local sampling within each ball using lines. We have several variations, depending on how the overlapping volume is partitioned, and depending on whether radial, axis-aligned, or other line patterns are used. Our variations fall within the family of Monte Carlo sampling, and hence have about the same theoretical convergence rate, 1 /$\sqrt{M}$, where M is the number of samples. In our limited experiments, line-sampling proved more accurate per unit work than point samples, because a line sample provides more information, and the analytic equation for a sphere makes the calculation almost as fast. We performed a limited empirical study of the efficiency of these variations. We suggest a more extensive study for future work. We speculate that different ball arrangements, differentiated by the distribution of overlaps in terms of volume and degree, will benefit the most from patterns of line samples that preferentially capture those overlaps. Acknowledgement We thank Karl Bringman for explaining his BF-ApproxUnion (ApproxUnion) algorithm [3] to us. We thank Josiah Manson for pointing out that spoke darts oversample the center and we might get a better answer by uniform sampling. We thank Vijay Natarajan for suggesting random chord sampling. The authors are grateful to Brian Adams, Keith Dalbey, and Vicente Romero for useful technical discussions. This work was sponsored by the Laboratory Directed Research and Development (LDRD) Program at Sandia National Laboratories. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR), Applied Mathematics Program. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.
Blue noise sampling has proved useful for many graphics applications, but remains underexplored in high-dimensional spaces due to the difficulty of generating distributions and proving properties about them. We present a blue noise sampling method with good quality and performance across different dimensions. The method, spoke-dart sampling, shoots rays from prior samples and selects samples from these rays. It combines the advantages of two major high-dimensional sampling methods: the locality of advancing front with the dimensionality-reduction of hyperplanes, specifically line sampling. We prove that the output sampling is saturated with high probability, with bounds on distances between pairs of samples and between any domain point and its nearest sample. We demonstrate spoke-dart applications for approximate Delaunay graph construction, global optimization, and robotic motion planning. Both the blue-noise quality of the output distribution and the adaptability of the intermediate processes of our method are useful in these applications.
Over the past decade, polyhedral meshing has been gaining popularity as a better alternative to tetrahedral meshing in certain applications. Within the class of polyhedral elements, Voronoi cells are particularly attractive thanks to their special geometric structure. What has been missing so far is a Voronoi mesher that is sufficiently robust to run automatically on complex models. In this video, we illustrate the main ideas behind the VoroCrust algorithm, highlighting both the theoretical guarantees and the practical challenges imposed by realistic inputs.
We study the problem of decomposing a volume with a smooth boundary into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from weighted α-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given a κ-sparse ε-sample, we work with the balls of radius δ times the local feature size centered at each sample. The corners of the union of these balls on both sides of the surface are the Voronoi sites and the interface of their cells is a watertight surface reconstruction embedded in the dual shape of the union of balls. With the surface protected, the enclosed volume can be further decomposed by generating more sites inside it. Compared to clipping-based algorithms, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured or randomly genenerated samples.
Leibniz International Proceedings in Informatics, LIPIcs
Abdelkader, Ahmed; Bajaj, Chandrajit L.; Ebeida, Mohamed S.; Mahmoud, Ahmed H.; Mitchell, Scott A.; Owens, John D.; Rushdi, Ahmad A.
We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from α-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an ϵ-sample on the bounding surface, with a weak σ-sparsity condition, we work with the balls of radius δ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of ϵ, σ and δ, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.
We consider the problem of selecting a small set (mosaic) of sensor images (footprints) whose union covers a two-dimensional Region Of Interest (ROI) on Earth. We take the approach of modeling the mosaic problem as a Mixed-Integer Linear Program (MILP). This allows solutions to this subproblem to feed into a larger remote-sensor collection-scheduling MILP. This enables the scheduler to dynamically consider alternative mosaics, without having to perform any new geometric computations. Our approach to set up the optimization problem uses maximal disk sampling and point-in-polygon geometric calculations. Footprints may be of any shape, even non-convex, and we show examples using a variety of shapes that may occur in practice. The general integer optimization problem can become computationally expensive for large problems. In practice, the number of placed footprints is within an order of magnitude of ten, making the time to solve to optimality on the order of minutes. This is fast enough to make the approach relevant for near real-time mission applications. We provide open source software for all our methods, "GeoPlace."
We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.
We propose a porous materials analysis pipeline using persistent homology. We rst compute persistent homology of binarized 3D images of sampled material subvolumes. For each image we compute sets of homology intervals, which are represented as summary graphics called persistence diagrams. We convert persistence diagrams into image vectors in order to analyze the similarity of the homology of the material images using the mature tools for image analysis. Each image is treated as a vector and we compute its principal components to extract features. We t a statistical model using the loadings of principal components to estimate material porosity, permeability, anisotropy, and tortuosity. We also propose an adaptive version of the structural similarity index (SSIM), a similarity metric for images, as a measure to determine the statistical representative elementary volumes (sREV) for persistence homology. Thus we provide a capability for making a statistical inference of the uid ow and transport properties of porous materials based on their geometry and connectivity.
We present an all-quad meshing algorithm for general domains. We start with a strongly balanced quadtree. In contrast to snapping the quadtree corners onto the geometric domain boundaries, we move them away from the geometry. Then we intersect the moved grid with the geometry. The resulting polygons are converted into quads with midpoint subdivision. Moving away avoids creating any flat angles, either at a quadtree corner or at a geometry–quadtree intersection. We are able to handle two-sided domains, and more complex topologies than prior methods. The algorithm is provably correct and robust in practice. It is cleanup-free, meaning we have angle and edge length bounds without the use of any pillowing, swapping, or smoothing. Thus, our simple algorithm is fast and predictable. This paper has better quality bounds, and the algorithm is demonstrated over more complex domains, than our prior version.
We show how to place a set of seed points such that a given piecewise linear complex is the union of some faces in the resulting Voronoi diagram. The seeds are placed on sufficiently small spheres centered at input vertices and are arranged into little circles around each half-edge where every seed is mirrored across the associated triangle. The Voronoi faces common to the seeds of such arrangements yield a mesh conforming to the input complex. If the input contains sharp angles, then additional seeds are needed, analogous to nonobtuse refinement. Finally, we propose local optimizations to reduce the number of seeds and output facets.
We consider heuristic and optimal solutions to a discrete geometric bin packing problem that arises in a resource allocation problem. An imaging sensor is assigned to collect data over a large area, but some subregions are more valuable than others. To capture these high-value regions with higher fidelity, we can assign some number of non-overlapping rectangular subsets, called “subfootprints.” The sensor image is partitioned into squares called “chips”, and each chip is further partitioned into pixels. Pixels may have different values. Subfootprints are restricted to rectangular collections of chips, but we are free to choose different rectangle heights, widths, and areas. We seek the optimal arrangement over the family of possible rectangle shapes and sizes. We provide a mixed-integer linear program optimization formulation, as well as a greedy heuristic, to solve this problem. For the meta-problem, we have some freedom to align the chip boundaries to different pixels. However, it is too expensive to solve the optimization formulation for each alignment. However, we show that the greedy heuristic can inform which pixel alignments are worth solving the optimization over. We use a variant of k-means clustering to group greedy solutions by their transport shape-similarity. For each cluster, we run the optimization problem over the greedy layout with the highest value. In practice this efficiently explores the geometric configuration space, and produces solutions close to the global optimum. We show a contrived example using surveillance of the Mississippi River. Our software is available as open-source in the Github repository “GeoPlace .
We introduce a novel technique, POF-Darts, to estimate the Probability Of Failure based on random disk-packing in the uncertain parameter space. POF-Darts uses hyperplane sampling to explore the unexplored part of the uncertain space. We use the function evaluation at a sample point to determine whether it belongs to failure or non-failure regions, and surround it with a protection sphere region to avoid clustering. We decompose the domain into Voronoi cells around the function evaluations as seeds and choose the radius of the protection sphere depending on the local Lipschitz continuity. As sampling proceeds, regions uncovered with spheres will shrink, improving the estimation accuracy. After exhausting the function evaluation budget, we build a surrogate model using the function evaluations associated with the sample points and estimate the probability of failure by exhaustive sampling of that surrogate. In comparison to other similar methods, our algorithm has the advantages of decoupling the sampling step from the surrogate construction one, the ability to reach target POF values with fewer samples, and the capability of estimating the number and locations of disconnected failure regions, not just the POF value. We present various examples to demonstrate the efficiency of our novel approach.
Remote sensing systems have firmly established a role in providing immense value to commercial industry, scientific exploration, and national security. Continued maturation of sensing technology has reduced the cost of deploying highly-capable sensors while at the same time increased reliance on the information these sensors can provide. The demand for time on these sensors is unlikely to diminish. Coordination of next-generation sensor systems, larger constellations of satellites, unmanned aerial vehicles, ground telescopes, etc. is prohibitively complex for existing heuristics-based scheduling techniques. The project was a two-year collaboration spanning multiple Sandia centers and included a partnership with Texas A&M University. We have developed algorithms and software for collection scheduling, remote sensor field-of-view pointing models, and bandwidth-constrained prioritization of sensor data. Our approach followed best practices from the operations research and computational geometry communities. These models provide several advantages over state of the art techniques. In particular, our approach is more flexible compared to heuristics that tightly couple models and solution techniques. First, our mixed-integer linear models afford a rigorous analysis so that sensor planners can quantitatively describe a schedule relative to the best possible. Optimal or near-optimal schedules can be produced with commercial solvers in operational run-times. These models can be modified and extended to incorporate different scheduling and resource constraints and objective function definitions. Further, we have extended these models to proactively schedule sensors under weather and ad hoc collection uncertainty. This approach stands in contrast to existing deterministic schedulers which assume a single future weather or ad hoc collection scenario. The field-of-view pointing algorithm produces a mosaic with the fewest number of images required to fully cover a region of interest. The bandwidth-constrained algorithms find the highest priority information that can be transmitted. All of these are based on mixed-integer linear programs so that, in the future, collection scheduling, field-of-view, and bandwidth prioritization can be combined into a single problem. Experiments conducted using the developed models, commercial solvers, and benchmark datasets have demonstrated that proactively scheduling against uncertainty regularly and significantly outperforms deterministic schedulers.
The need to better represent the material properties within the earth's interior has driven the development of higherfidelity physics, e.g., visco-tilted-transversely-isotropic (visco- TTI) elastic media and material interfaces, such as the ocean bottom and salt boundaries. This is especially true for full waveform inversion (FWI), where one would like to reproduce the real-world effects and invert on unprocessed raw data. Here we present a numerical formulation using a Discontinuous Galerkin (DG) finite-element (FE) method, which incorporates the desired high-fidelity physics and material interfaces. To offset the additional costs of this material representation, we include a variety of techniques (e.g., non-conformal meshing, and local polynomial refinement), which reduce the overall costs with little effect on the solution accuracy.
Ebeida, Mohamed; Rushdi, Ahmad A.; Awad, Muhammad A.; Mahmoud, Ahmed H.; Yan, Dong M.; English, Shawn A.; Owens, John D.; Bajaj, Chandrajit L.; Mitchell, Scott A.
We introduce an algorithmic framework for tuning the spatial density of disks in a maximal random packing, without changing the sizing function or radii of disks. Starting from any maximal random packing such as a Maximal Poisson-disk Sampling (MPS), we iteratively relocate, inject (add), or eject (remove) disks, using a set of three successively more-aggressive local operations. We may achieve a user-defined density, either more dense or more sparse, almost up to the theoretical structured limits. The tuned samples are conflict-free, retain coverage maximality, and, except in the extremes, retain the blue noise randomness properties of the input. We change the density of the packing one disk at a time, maintaining the minimum disk separation distance and the maximum domain coverage distance required of any maximal packing. These properties are local, and we can handle spatially-varying sizing functions. Using fewer points to satisfy a sizing function improves the efficiency of some applications. We apply the framework to improve the quality of meshes, removing non-obtuse angles; and to more accurately model fiber reinforced polymers for elastic and failure simulations.
We consider the problem of sampling points from a collection of smooth curves in the plane, such that the CRUST family of proximity-based reconstruction algorithms can rebuild the curves. Reconstruction requires a dense sampling of local features, i.e., parts of the curve that are close in Euclidean distance but far apart geodesically. We show that e < 0:47-sampling is sufficient for our proposed HNN-CRUST variant, improving upon the state-of-the-art requirement of e < 13-sampling. Thus we may reconstruct curves with many fewer samples. We also present a new sampling scheme that reduces the required density even further than e < 0:47-sampling. We achieve this by better controlling the spacing between geodesically consecutive points. Our novel sampling condition is based on the reach, the minimum local feature size along intervals between samples. This is mathematically closer to the reconstruction density requirements, particularly near sharp-angled features. We prove lower and upper bounds on reach r-sampling density in terms of lfs e-sampling and demonstrate that we typically reduce the required number of samples for reconstruction by more than half.
This SAND report summarizes our work on the Sandia National Laboratory LDRD project titled "Efficient Probability of Failure Calculations for QMU using Computational Geometry" which was project #165617 and proposal #13-0144. This report merely summarizes our work. Those interested in the technical details are encouraged to read the full published results, and contact the report authors for the status of the software and follow-on projects.
