763 research outputs found
Frame Fields for Hexahedral Mesh Generation
As a discretized representation of the volumetric domain, hexahedral meshes have been a popular choice in computational engineering science and serve as one of the main mesh types in leading industrial software of relevance. The generation of high quality hexahedral meshes is extremely challenging because it is essentially an optimization problem involving multiple (conflicting) objectives, such as fidelity, element quality, and structural regularity. Various hexahedral meshing methods have been proposed in past decades, attempting to solve the problem from different perspectives. Unfortunately, algorithmic hexahedral meshing with guarantees of robustness and quality remains unsolved.
The frame field based hexahedral meshing method is the most promising approach that is capable of automatically generating hexahedral meshes of high quality, but unfortunately, it suffers from several robustness issues. Field based hexahedral meshing follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a feature-aligned frame field and (2) generation of an integer-grid map that best aligns with the frame field. The main robustness issue stems from the fact that smooth frame fields frequently exhibit singularity graphs that are inappropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The thesis aims at analyzing the gap between the topologies of frame fields and hexahedral meshes and developing algorithms to realize a more robust field based hexahedral mesh generation.
The first contribution of this work is an enumeration of all local configurations that exist in hexahedral meshes with bounded edge valence and a generalization of the Hopf-Poincaré formula to octahedral (orthonormal frame) fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large non-linear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field.
In the collaboration work with colleagues [BRK+22], the dataset HexMe consisting of practically relevant models with feature tags is set up, allowing a fair evaluation for practical hexahedral mesh generation algorithms. The extendable and mutable dataset remains valuable as hexahedral meshing algorithms develop. The results of the standard field based hexahedral meshing algorithms on the HexMesh dataset expose the fragility of the automatic pipeline.
The major contribution of this thesis improves the robustness of the automatic field based hexahedral meshing by guaranteeing local meshability of general feature aligned smooth frame fields. We derive conditions on the meshability of frame fields when feature constraints are considered, and describe an algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite the fact that local meshability is only a necessary but not sufficient condition for the stronger requirement of meshability, our algorithm increases the 2% success rate of generating valid integer-grid maps with state-of-the-art methods to 57%, when compared on the challenging HexMe dataset
Fast Topological Signal Identification and Persistent Cohomological Cycle Matching
Within the context of topological data analysis, the problems of identifying
topological significance and matching signals across datasets are important and
useful inferential tasks in many applications. The limitation of existing
solutions to these problems, however, is computational speed. In this paper, we
harness the state-of-the-art for persistent homology computation by studying
the problem of determining topological prevalence and cycle matching using a
cohomological approach, which increases their feasibility and applicability to
a wider variety of applications and contexts. We demonstrate this on a wide
range of real-life, large-scale, and complex datasets. We extend existing
notions of topological prevalence and cycle matching to include general
non-Morse filtrations. This provides the most general and flexible
state-of-the-art adaptation of topological signal identification and persistent
cycle matching, which performs comparisons of orders of ten for thousands of
sampled points in a matter of minutes on standard institutional HPC CPU
facilities
Inference in particle tracking experiments by passing messages between images
Methods to extract information from the tracking of mobile objects/particles
have broad interest in biological and physical sciences. Techniques based on
simple criteria of proximity in time-consecutive snapshots are useful to
identify the trajectories of the particles. However, they become problematic as
the motility and/or the density of the particles increases due to uncertainties
on the trajectories that particles followed during the images' acquisition
time. Here, we report an efficient method for learning parameters of the
dynamics of the particles from their positions in time-consecutive images. Our
algorithm belongs to the class of message-passing algorithms, known in computer
science, information theory and statistical physics as Belief Propagation (BP).
The algorithm is distributed, thus allowing parallel implementation suitable
for computations on multiple machines without significant inter-machine
overhead. We test our method on the model example of particle tracking in
turbulent flows, which is particularly challenging due to the strong transport
that those flows produce. Our numerical experiments show that the BP algorithm
compares in quality with exact Markov Chain Monte-Carlo algorithms, yet BP is
far superior in speed. We also suggest and analyze a random-distance model that
provides theoretical justification for BP accuracy. Methods developed here
systematically formulate the problem of particle tracking and provide fast and
reliable tools for its extensive range of applications.Comment: 18 pages, 9 figure
Efficient Exact Inference in Planar Ising Models
We give polynomial-time algorithms for the exact computation of lowest-energy
(ground) states, worst margin violators, log partition functions, and marginal
edge probabilities in certain binary undirected graphical models. Our approach
provides an interesting alternative to the well-known graph cut paradigm in
that it does not impose any submodularity constraints; instead we require
planarity to establish a correspondence with perfect matchings (dimer
coverings) in an expanded dual graph. We implement a unified framework while
delegating complex but well-understood subproblems (planar embedding,
maximum-weight perfect matching) to established algorithms for which efficient
implementations are freely available. Unlike graph cut methods, we can perform
penalized maximum-likelihood as well as maximum-margin parameter estimation in
the associated conditional random fields (CRFs), and employ marginal posterior
probabilities as well as maximum a posteriori (MAP) states for prediction.
Maximum-margin CRF parameter estimation on image denoising and segmentation
problems shows our approach to be efficient and effective. A C++ implementation
is available from http://nic.schraudolph.org/isinf/Comment: Fixed a number of bugs in v1; added 10 pages of additional figures,
explanations, proofs, and experiment
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