Mesh Modification Under Local Domain Changes

Coll, Narcís; Guerrieri, Marité; Sellarès, J. Antoni

doi:10.1007/978-3-540-34958-7_3

Cited by 6 publications

(3 citation statements)

References 13 publications

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“…Recent years witnessed an inflation in the development and use of Delaunay refinement algorithms [5,8,13,14,18,17,25,24,29]. Some of these research studies mainly concentrate on the theoretical bounds [13,14,18], while others emphasize the benefits on the practical side [25,24,29].…”

Section: Introductionmentioning

confidence: 99%

Quality Triangulations with Locally Optimal Steiner Points

Erten¹,

Üngör²

2009

SIAM J. Sci. Comput.

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We propose two novel ideas to improve the performance of Delaunay refinement algorithms which are used for computing quality triangulations. The first idea is an effective use of the Voronoi diagram and unifies previously suggested Steiner point insertion schemes (circumcenter, sink, off-center) together with a new strategy. The second idea is the integration of a new local smoothing strategy into the refinement process. These lead to two new versions of Delaunay refinement, where the second is simply an extension of the first. For a given input domain and a constraint angle α, Delaunay refinement algorithms aim to compute triangulations that have all angles at least α. The original Delaunay refinement algorithm of Ruppert is proven to terminate with size-optimal quality triangulations for α ≤ 20.7 • . In practice, the original and the consequent Delaunay refinement algorithms generally work for α ≤ 34 • and fail to terminate for larger constraint angles. Our algorithms provide the same theoretical guarantees as the previous Delaunay refinement algorithms. The second of the proposed algorithms generally terminates for constraint angles up to 42 • . Experiments also indicate that our algorithm computes significantly (about by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms. Moreover, the new algorithms are experimentally shown to outperform the previous algorithms even in the existence of additional constraints, such as the maximum area triangle constraint which is commonly used for computing uniform triangulations. Introduction.Triangulations are geometric discretization tools essential in many scientific applications, such as engineering simulations, medical imaging, visualizations, geological analysis, and meteorological forecasting. The shape, the size, as well as the number of triangles in a triangulation are key in its efficient and effective use in these and other applications. While the preferred shape of a triangle is not exactly the same for all applications, theoretical and experimental analysis (of numerical methods that are used in conjunction with triangulations) suggests that triangles with no large angles and/or small angles serve well in most applications [2]. In general, the better the shape of the triangles, the smaller the interpolation and approximation errors are in their use. A triangle is usually considered to have a good shape if its smallest angle is bounded from below (by a user-specified constraint angle α), which implies an upper bound on its largest angle also (π − 2α). Unless new vertices/points are added into the domain, even using Delaunay triangulations, which are optimum in maximizing the smallest angle [10], may result in output that involve bad quality triangles. Hence, we are allowed to introduce points (called the Steiner points) in addition to input points in order to compute good quality triangulations. This, however, might dramatically increase the number of points and triangles in a triangulation, which is a key fact...

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Section: Introductionmentioning

confidence: 99%

Quality Triangulations with Locally Optimal Steiner Points

Erten¹,

Üngör²

2009

SIAM J. Sci. Comput.

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“…The one we use in our study is by measuring the angles of the triangulated surfaces, a strategy commonly used in the mesh generation and smoothing community (Zhou and Shimada, 2000; Xu and Newman 2005). The mesh quality may be improved by a combination of three major techniques: inserting/deleting vertices (Shewchuk, 2003; Coll et al, 2006), swapping edges/faces (Gooch, 2002; Yamakawa and Shimada, 2009), and moving the vertices without changing the mesh topology (Field, 1988; Wang and Yu, 2009). The last one, also known as mesh smoothing, is the strategy we will explore in the present paper.…”

Section: Introductionmentioning

confidence: 99%

Quality mesh smoothing via local surface fitting and optimum projection

Wang

2011

Graphical Models

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The smoothness and angle quality of a surface mesh are two important indicators of the “goodness” of the mesh for downstream applications such as visualization and numerical simulation. We present in this paper a novel surface mesh processing method not only to reduce mesh noise but to improve angle quality as well. Our approach is based on the local surface fitting around each vertex using the least square minimization technique. The new position of the vertex is obtained by finding the maximum inscribed circle (MIC) of the surrounding polygon and projecting the circle’s center onto the analytically fitted surface. The procedure above repeats until the maximal vertex displacement is less than a pre-defined threshold. The mesh smoothness is improved by a combined idea of surface fitting and projection, while the angle quality is achieved by utilizing the MIC-based projection scheme. Results on a variety of geometric mesh models have demonstrated the effectiveness of our method.

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“…There has been relatively little progress on solving the dynamic problem. Existing solutions either do not produce size-optimal outputs (e.g., [NvdS04]) or they are asymptotically no faster than running a static algorithm from scratch [LTU99,MBF04,CGS06].…”

Section: Introductionmentioning

confidence: 99%

Dynamic well-spaced point sets

Acar¹,

Cotter

Hudson

et al. 2010

Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry

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In a well-spaced point set the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Well-spaced point sets satisfy some important geometric properties and yield quality Voronoi or simplicial meshes that can be important in scientific computations. In this paper, we consider the dynamic well-spaced point-sets problem, which requires computing the well-spaced superset of a dynamically changing input set, e.g., as points are inserted or deleted. We present a dynamic algorithm that allows inserting/deleting points into/from the input in worst-case O(log ∆) time, where ∆ is the geometric spread, a natural measure that is bounded by O(log n) when input points are represented by log-size words. We show that the runtime of the dynamic update algorithm is optimal in the worst case by showing that there exists inputs and modifications that require Ω(log ∆) Steiner points to be inserted to the output. Our algorithm generates size-optimal outputs: the resulting output sets are never more than a constant factor larger than the minimum size necessary. A preliminary implementation indicates that the algorithm is indeed fast in practice. To the best of our knowledge, this is the first time-and size-optimal dynamic algorithm for well-spaced point sets.

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Mesh Modification Under Local Domain Changes

Cited by 6 publications

References 13 publications

Quality Triangulations with Locally Optimal Steiner Points

Quality Triangulations with Locally Optimal Steiner Points

Quality mesh smoothing via local surface fitting and optimum projection

Dynamic well-spaced point sets

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