Mesh Generation and Optimal Triangulation

Bern, Marshall; Eppstein, David

doi:10.1142/9789814355858_0002

Cited by 188 publications

(68 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This subgraph contains a set of edges that must be in every locally minimal triangulation. 3 The algorithm requires O(n 4 ) time and O(n 2 ) space in the worst case. When the algorithm terminates, the LMT-skeleton can then be used to complete the MWT in time O(n 3 ) if the LMT-skeleton is connected, or time O(n k+2 ) if it has k unconnected components [6].…”

Section: A New Subgraph Of the Mwtmentioning

confidence: 99%

See 1 more Smart Citation

A Large Subgraph of the Minimum Weight Triangulation

Dickerson¹,

Keil²,

Montague³

1997

Discrete Comput Geom

View full text Add to dashboard Cite

We present an O(n 4 )-time and O(n 2 )-space algorithm that computes a subgraph of the minimum weight triangulation (MWT) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give a variant called the modified LMTskeleton that is both a more complete subgraph of the MWT and is faster to compute requiring only O(n 2 ) time and O(n) space in the expected case for uniform distributions. Several experimental implementations of both approaches have shown that for moderate-sized point sets (up to 350 points 1 ) the skeletons are connected, enabling an efficient completion of the exact MWT. We are thus able to compute the MWT of substantially larger random point sets than have previously been computed.

show abstract

Section: A New Subgraph Of the Mwtmentioning

confidence: 99%

“…So in the expected case, the initial set of candEdges is of size O(n) and is computed in O(n) time. The number of empty triangles that exist in a set of O(n) vertices and O(n) edges is O(n 1.5 ) [7], [3]. More importantly in the analysis of Step 4 is the following: …”

Section: Complexity Of Modified Skeleton Approachmentioning

confidence: 99%

A Large Subgraph of the Minimum Weight Triangulation

Dickerson¹,

Keil²,

Montague³

1997

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…The region to be meshed is described by a PSLG, P [2]. This is presented to algorithms via the constrained Delaunay triangulation of P, which we denote (0) ([9, p. 85, 10]).…”

Section: Iterative Refinement For Shape Quality Mesh Generationmentioning

confidence: 99%

“…So, in addition to updating the Delaunay refinement algorithm must update S from being the set of bad triangles in to SЈ, the set of bad triangles in Ј. We introduce subsets S del ϭ S ʝ Cav(P) and S add ϭ SЈ ʝ B(P), and express this update as SЈ ϭ S Ϫ S del ϩ S add (2) Iterative Delaunay refinement methods for shape qmg typically take the form of the following Quality-Mesh-Generation procedure. It calls a procedure Delaunay-Insertion, which takes as input a triangle t to be refined in mesh and the minimum angle tolerance.…”

Section: Iterative Refinement For Shape Quality Mesh Generationmentioning

confidence: 99%

“…The study of how to meet a global shape criterion has been undertaken particularly in computational geometry [2][3][4][5][6][7][8]. We will refer to the problem of refining a Constrained Delaunay Triangulation (CDT) on a given Planar Straight Line Graph (PSLG) to meet a minimum angle bound, while respecting the edges of the PSLG as a shape quality mesh generation problem.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Approximate Shape Quality Mesh Generation

Simpson

Hitschfeld

Rivara

2001

EWC

View full text Add to dashboard Cite

We present two techniques for simplifying the list processing required in standard iterative refinement approaches to shape quality mesh generation. The goal of these techniques is to gain simplicity of programming, efficiency in execution, and robustness of termination. 'Shape quality' for a mesh generation method usually means that, under suitable conditions, a mesh with all angles exceeding a prescribed tolerance is generated. The methods introduced in this paper are truncated versions of such methods. They depend on the shape improvement properties of the terminaledge LEPP-Delaunay refinement technique; we refer to them as approximate shape quality methods. They are intended for geometry-based preconditioning of coarse initial meshes for subsequent refinement to meet data representation needs. One technique is an algorithm re-organization to avoid maintaining a global list of triangles to be refined. The reorganization uses a recursive triangle processing strategy. Truncating the recursion depth results in an approximate method. Based on this, we argue that the refinement process can be carried out using a static list of the triangles to be refined that can be identified in the initial mesh. Comparisons of approximate to full shape quality meshes are provided.

show abstract

The natural element method in solid mechanics

Sukumar

Moran

Belytschko

1998

Int. J. Numer. Meth. Engng.

562

470

View full text Add to dashboard Cite

The application of the Natural Element Method (NEM) 1; 2 to boundary value problems in two-dimensional small displacement elastostatics is presented. The discrete model of the domain consists of a set of distinct nodes N , and a polygonal description of the boundary @ . In the Natural Element Method, the trial and test functions are constructed using natural neighbour interpolants. These interpolants are based on the Voronoi tessellation of the set of nodes N . The interpolants are smooth (C ∞ ) everywhere, except at the nodes where they are C 0 . In one-dimension, NEM is identical to linear ÿnite elements. The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. A methodology to model material discontinuities and non-convex bodies (cracks) using NEM is also described. A standard displacement-based Galerkin procedure is used to obtain the discrete system of linear equations. Application of NEM to various problems in solid mechanics, which include, the patch test, gradient problems, bimaterial interface, and a static crack problem are presented. Excellent agreement with exact (analytical) solutions is obtained, which exempliÿes the accuracy and robustness of NEM and suggests its potential application in the context of other classes of problems-crack growth, plates, and large deformations to name a few. ? 1998 John Wiley & Sons, Ltd.

show abstract

Mesh Generation and Optimal Triangulation

Cited by 188 publications

References 18 publications

A Large Subgraph of the Minimum Weight Triangulation

A Large Subgraph of the Minimum Weight Triangulation

Approximate Shape Quality Mesh Generation

The natural element method in solid mechanics

Contact Info

Product

Resources

About