Optimal Angle Bounds for Quadrilateral Meshes

International Journal of …, 2005

a We should point out that the term constrained triangular mesh has been used in several other papers with a different meaning.

Lecture Notes Series on Computing, 1992

We survey the computational geometry relevant to nite element mesh generation. We especially focus on optimal triangulations of geometric domains in two-and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a xed set of vertices and for the placement of new vertices (Steiner points). We brie y survey the heuristic algorithms used in some practical mesh generators.

International Journal for Numerical Methods in Engineering, 2000

This work is devoted to the description of an algorithm for automatic quadrilateral mesh generation. The technique is based on a recursive decomposition of the domain into quadrilateral elements. This automatically generates meshes composed entirely by quadrilaterals over complex geometries (there is no need for a previous step where triangles are generated). A background mesh with the desired element sizes allows to obtain the preferred sizes anywhere in the domain. The ÿnal mesh can be viewed as the optimal one given the objective function is deÿned. The recursive algorithm induces an e cient data structure which optimizes the computer cost. Several examples are presented to show the e ciency of this algorithm. E cient automated meshing techniques are expected to have certain features in order to ensure its applicability in a wide scope of cases, which can range from regular domains with uniform element sizes to non-singly connected domains with large boundary curvatures and non-uniform element sizes. Haber et al. present an excellent discussion of such features: precise modelling of the boundaries; good correlation between the interior mesh and the information prescribed at the boundary; minimal input e ort; broad range of applicability; general topology; automatic topology generation; and favorable element shapes. Some of these features can be easily implemented; for instance, BÃ ezier or B-splines interpolation curves allow a precise modelling of the boundaries. Others, such as minimal input e ort and broad range of applicability are much more di cult to obtain. Therefore, all the developed techniques for mesh generation should include most of the previous features and this is the goal of the proposed algorithm.

International Journal for Numerical Methods in Engineering, 2012

A new indirect way of producing all-quad meshes is presented. The method takes advantage of a well known algorithm of the graph theory, namely the Blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time. The new Blossom-Quad algorithm is compared with standard indirect procedures. Meshes produced by the new approach are better both in terms of element shape and in terms of size field efficiency.

Procedia Engineering, 2015

There are many automatic quadrilateral mesh generators that can produce high quality mesh with low distortion. However, they typically generate a large number of singularities that could be detrimental to downstream applications. This paper introduces Minimum Singularity Templates (MST) to reduce the number of singularities in an existing pure quad mesh. These templates are easy to encode with high-level grammar rules for complete automation, or interactive control. The MST exploits two important properties of quadrilateral meshes: (1) every submesh has even number of quad edges on its boundary, and (2) every submesh with 3, 4 or 5 topological convex corners on its boundary has at most two interior singularities. The MST (1) does not change the boundary edges of the patch, (2) avoids corner picking on a patch and solving NP hard internal matching algorithm to select divisions, (3) is extremely fast with time complexity of O(1) in template creation, and (4) has low memory footprint and is robust. To illustrate the concepts, we consider quadrilateral meshes generated using Abaqus, Gmsh, and Cubit, and reduce the singularities within these meshes.

Computers & Graphics, 2011

Generating quadrilateral meshes is a highly non-trivial task, as design decisions are frequently driven by specific application demands. Automatic techniques can optimize objective quality metrics, such as mesh regularity, orthogonality, alignment and adaptivity; however, they can not make subjective design decisions. There are a few quad meshing approaches that offer some mechanisms to include the user in the mesh generation process; however, these techniques either require a large amount of user interaction or do not provide necessary or easy to use inputs. Here, we propose a template-based approach for generating quad-only meshes from triangle surfaces. Our approach offers a flexible mechanism to allow external input, through the definition of alignment features that are respected during the mesh generation process. While allowing user inputs to support subjective design decisions, our approach also takes into account objective quality metrics to produce semi-regular, quad-only meshes that align well to desired surface features.

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2022

For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an N-gon are computable in time O(N), even though the number of triangles needed to attain these bounds has no bound in terms of N alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. For example, we show that any polygon with minimal interior angle θ has a triangulation with all angles in the interval I = [θ, 90 • − min(36 • , θ)/2], and for θ ≤ 36 • both bounds are best possible. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver.

Handbook of Computational Geometry, 2000

1997

Conformal mesh re nement has gained much attention as a necessary preprocessing step for the nite element method in the computer-aided design of machines, vehicles, and many other technical devices. For many applications, such as torsion problems and crash simulations, it is important to have mesh re nements into quadrilaterals. In this paper, we consider the problem of constructing a minimum-cardinality conformal mesh re nement into quadrilaterals. However, this problem is NP{hard, which motivates the search for good approximations. The previously best known performance guarantee has been achieved by a linear-time algorithm with a factor of 4. We give improved approximation algorithms. In particular, for meshes without so-called folding edges, we now present a 2{approximation algorithm. This algorithm requires O(n 2 log n) time, where n is the number of polygons in the mesh. The asymptotic complexity of the latter algorithm is dominated by solving a minimum{cost perfect b{matching problem in a certain variant of the dual graph of the mesh.

Proceedings of the 20th International Meshing Roundtable, 2011

We propose an approach for automatically generating isotropic 2D quadrangle meshes from arbitrary domains with a fine control over sizing and orientation of the elements. At the heart of our algorithm is an optimization procedure that, from a coarse initial tiling of the 2D domain, enforces each of the desirable mesh quality criteria (size, shape, orientation, degree, regularity) one at a time, in an order designed not to undo previous enhancements. Our experiments demonstrate how well our resulting quadrangle meshes conform to a wide range of input sizing and orientation fields.

International Journal of Computational Geometry & Applications, 2005

We present an algorithm to compute a Delaunay mesh conforming to a polyhedron possibly with small input angles. The radius-edge ratio of most output tetrahedra are bounded by a constant, except possibly those that are provably close to small angles. Furthermore, the mesh is not unnecessarily dense in the sense that the edge lengths are at least a constant fraction of the local feature sizes at the edge endpoints. This algorithm is simple to implement as it eliminates most of the computation of local feature sizes and explicit protective zones. Our experimental results validate that few skinny tetrahedra remain and they lie close to small acute input angles.

2020

A significant amount of research in Architectural Geometry has dealt with skins and structures which follow a quadrilateral layout with double curvature. In many cases, such quad networks are computationally accessed by quad meshes which obey various constraints. These may concern planarity of faces, supporting structures which follow specific curvature paths, conditions on node angles, static equilibrium and others. In this paper we draw the attention to a new way of computing such constrained quad meshes. The new methodology is based on the diagonal meshes of a quad mesh and the checkerboard pattern of parallelograms one obtains by subdividing a quad mesh at its edge midpoints. The new approach is easy to understand and implement. It simplifies the transfer from the familiar theory of smooth surfaces to the discrete setting of quad meshes. This is illustrated with planar quad meshes and asymptotic nets, in particular with those exhibiting a constant node angle. The application of ...

1994

We give an algorithm for triangulating n-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than 7r/2. The number of triangles in the triangulation is only O(n), improving a previous bound of 0(n2), and the worst-case running time is O(n logz n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm. Throughout the application areas named above, it is generally true that large angles (that is, angles close to m) are undesirable. Babu5ka and Aziz [2] justi

International Meshing Roundtable, 2000

A new technique for automatically generating anisotropic quadrilateral meshes is presented in this paper. The inputs are (1) a 2-D geometric domain and (2) a desired anisotropy -defined as a metric tensor over the domain -specifying mesh sizing in two independent directions. Node locations are obtained by closely packing rectangles in accordance with the inputs. The centers of the rectangles, or node, are then connected using anisotropic Delaunay triangulation that takes into account the desired anisotropy. The obtained triangular mesh is converted into a quadrilateral mesh using mesh conversion templates. The novelty of the method is that closely packed rectangles resemble a pattern of Voronoi polygons corresponding to a well-shaped quadrilateral mesh. The result is a high quality mesh that conforms well to the input. As a sample application, this method was used to generate a mesh to solve a steady state heat transfer problem.