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Register a new userSize-optimal Steiner points for Delaunay-refinement on curved surfaces
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Added by
Darren Engwirda
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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An extension of the restricted Delaunay-refinement algorithm for surface mesh generation is described, where a new point-placement scheme is introduced to improve element quality in the presence of mesh size constraints. Specifically, it is shown that the use of off-centre Steiner points, positioned on the faces of the associated Voronoi diagram, typically leads to significant improvements in the shape- and size-quality of the resulting surface tessellations. The new algorithm can be viewed as a Frontal-Delaunay approach -- a hybridisation of conventional Delaunay-refinement and advancing-front techniques in which new vertices are positioned to satisfy both element size and shape constraints. The performance of the new scheme is investigated experimentally via a series of comparative studies that contrast its performance with that of a typical Delaunay-refinement technique. It is shown that the new method inherits many of the best features of classical Delaunay-refinement and advancing-front type methods, leading to the construction of smooth, high quality surface triangulations with bounded radius-edge ratios and convergence guarantees. Experiments are conducted using a range of complex benchmarks, verifying the robustness and practical performance of the proposed scheme.
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Large-scale finite element simulations of complex physical systems governed by partial differential equations crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a new problem for RL in that both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence scale to arbitrarily large and complex simulations. We demonstrate in comprehensive experiments on static function estimation and the advection of different fields that RL policies can be competitive with a widely-used error estimator and generalize to larger, more complex, and unseen test problems.
Lightness and sparsity are two natural parameters for Euclidean $(1+varepsilon)$-spanners. Classical results show that, when the dimension $din mathbb{N}$ and $varepsilon>0$ are constant, every set $S$ of $n$ points in $d$-space admits an $(1+varepsilon)$-spanners with $O(n)$ edges and weight proportional to that of the Euclidean MST of $S$. Tight bounds on the dependence on $varepsilon>0$ for constant $din mathbb{N}$ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a $(1+varepsilon)$-spanner. They gave upper bounds of $tilde{O}(varepsilon^{-(d+1)/2})$ for the minimum lightness in dimensions $dgeq 3$, and $tilde{O}(varepsilon^{-(d-1))/2})$ for the minimum sparsity in $d$-space for all $dgeq 1$. They obtained lower bounds only in the plane ($d=2$). Le and Solomon (ESA 2020) also constructed Steiner $(1+varepsilon)$-spanners of lightness $O(varepsilon^{-1}logDelta)$ in the plane, where $Deltain Omega(sqrt{n})$ is the emph{spread} of $S$, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner $(1+varepsilon)$-spanners. Using a new geometric analysis, we establish lower bounds of $Omega(varepsilon^{-d/2})$ for the lightness and $Omega(varepsilon^{-(d-1)/2})$ for the sparsity of such spanners in Euclidean $d$-space for all $dgeq 2$. We use the geometric insight from our lower bound analysis to construct Steiner $(1+varepsilon)$-spanners of lightness $O(varepsilon^{-1}log n)$ for $n$ points in Euclidean plane.
We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.
We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.
We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced provided the transition functions are bi-Lipschitz with a constant close to 1, and the sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the ouput complex is also a Delaunay triangulation of its vertices with respect to this piecewise flat metric.
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