Refinement Of Tetrahedral Meshes Research Articles

Lepp-bisection algorithms, applications and mathematical properties

Longest edge (nested) algorithms for triangulation refinement in two dimensions are able to produce hierarchies of quality and nested irregular triangulations as needed both for adaptive finite element methods and for multigrid methods for PDEs. In addition, right-triangle bintree triangulations are multiresolution algorithms used for terrain modeling and real time visualization of terrain applications. These algorithms are based on the properties of the consecutive bisection of a triangle by the median of the longest edge, and can be formulated in terms of the longest edge propagation path (Lepp) and terminal edge concepts, which implies the use of very local refinement operations over fully conforming meshes (where the intersection of pairs of neighbor triangles is either a common edge or a common vertex). In this paper we review the Lepp-bisection algorithms, their properties and applications. To the end we use recent simpler and stronger results on the complexity aspects of the bisection method and its geometrical properties. We discuss and analyze the computational costs of the algorithms. The generalization of the algorithms to 3-dimensions is also discussed. Applications of these methods are presented: for serial and parallel view dependent level of detail terrain rendering, and for the parallel refinement of tetrahedral meshes.

A New Strategy for Reducing Communication Latency in Parallel 3-D Finite Element Tetrahedral Mesh Refinement

A new, efficient pipelined communication strategy that significantly reduces latency for parallel 3-D finite element mesh refinement with tetrahedra is proposed. A Petri Nets-based model is developed in order to simulate the interprocessor communication costs for both the target mesh refinement algorithm and parallel architecture. Benchmark results show that the new pipelined design yields improved communication performance and overall parallel speedup for a range of refinement problem sizes, using different numbers of processors.

Computational aspects of the refinement of 3D tetrahedral meshes

The refinement of tetrahedral meshes is a significant task in many numerical and discretizations methods. The computational aspects for implementing refinement of meshes with complex geometry need to be carefully considered in order to have real-time and optimal results. In this paper we study some computational aspects of a class of tetrahedral refinement algorithms. For local adaptive refinement we give numerical results of the computational propagation cost of a general class of longest edge based refinement and show the implications of the geometry in the global process. Moreover we study the conformity process based on the longest edge bisection and give an algorithm and data structure to efficiently handle tetrahedral refinement.

Embarrassingly parallel mesh refinement by edge subdivision

We have previously proposed a new technique for the communication-free adaptive refinement of tetrahedral meshes that works for all configurations. Implementations of the scheme must deal with all possible geometric configurations, which results in a large number of cases that in turn result in practical programming issues. In this article, we address this issue with a Python script that generates C++ code using the symmetric group $${\mathfrak{S}}_{4}$$ acting over canonical topological and geometric configurations. We then analyze the performance of the technique by characterizing (a) mesh quality, (b) execution time and parallel speedup, and (c) traits of the algorithm that could affect quality or execution time differently for different meshes and different mesh refinement strategies. This article also details the method used to debug the many subdivision templates that the algorithm relies upon. Mesh quality is on par with other similar refinement schemes, and we suggest a more elaborate technique that may substantially improve mesh quality. We show that throughput on modern hardware can exceed 600,000 output tetrahedra per second per processor, and that the method is embarrassingly parallel—assuming the application has partitioned the input properly.

Parallel decoupled terminal-edge bisection method for 3D mesh generation

We present a practical and stable algorithm for the parallel refinement of tetrahedral meshes. The algorithm is based on the refinement of terminal-edges and associated terminal stars. A terminal-edge is a special edge in the mesh which is the longest edge of every element that shares such an edge, while the elements that share a terminal-edge form a terminal star. We prove that the algorithm is inherently decoupled and thus scalable. Our experimental data show that we have a stable implementation able to deal with hundreds of millions of tetrahedra and whose speed is in between one and two order of magnitude higher from the method and implementation we presented (Rivara et al., Proceedings 13th international meshing roundtable, 2004).

Smoothing and local refinement techniques for improving tetrahedral mesh quality

The improvement in the mesh quality without changing its connectivity is bounded. This bound is associated with the topology of the mesh and with the constraints imposed by the boundary of the domain. To solve this problem, we propose in this work to combine the tetrahedral mesh optimisation technique introduced in [Montenegro R, Montero G, Escobar JM, Rodríguez E, González-Yuste JM. Tetrahedral mesh generation for environmental problems over complex terrains. Lect Notes Comput Sci 2002;2329:335–44; Escobar JM, Rodríguez E, Montenegro R, Montero G, González-Yuste JM. Simultaneous untangling and smoothing of tetrahedral meshes. Comput Methods Appl Mech Eng 2003;192:2775–87] with the local mesh refinement algorithm presented in [González-Yuste JM, Montenegro R, Escobar JM, Montero G, Rodríguez E. An object oriented method for tetrahedral mesh refinement. In: Proceedings of the 3rd international conference on engineering computational technology, Praga 2002;1–18]. The main idea consists in increasing the node number, and thus, the degrees of freedom, in the neighbourhood of the regions where the elements have poor quality. Then, we refine all the elements whose quality are below to a certain threshold. Once it is done, we initiate another stage of optimisation until the quality of the mesh reaches a limit.

Communication-Free Streaming Mesh Refinement

This article presents a technique for the adaptive refinement of tetrahedral meshes. What makes this method new is that no neighbor information is required for the refined mesh to be compatible everywhere. Refinement consists of inserting new vertices at edge midpoints until some tolerance (geometric or otherwise) is met. For a tetrahedron, the six edges present 26=64 possible subdivision combinations. The challenge is to triangulate the new vertices (i.e., the original vertices plus some subset of the edge midpoints) in a way that neighboring tetrahedra always generate the same triangles on their shared boundary. A geometric solution based on edge lengths was developed previously, but did not account for geometric degeneracies (edges of equal length). This article provides a solution that works in all cases, while remaining entirely communication-free.

Refinement strategies and approximation errors for tetrahedral elements

Two tetrahedral mesh refinement algorithms proposed by Zhang are compared using the new singular value criterion for the element edge shape matrix. The 'short-edge subdivision' scheme is better than 'labeled edge subdivision'. Validity of the singular value criterion is confirmed and its geometric implications are investigated.

Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision

Let T \mathcal {T} be a tetrahedral mesh. We present a 3-D local refinement algorithm for T \mathcal {T} which is mainly based on an 8-subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron T ∈ T \mathbf {T} \in \mathcal {T} produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, η ( T i n ) ≥ c η ( T ) \eta (\mathbf {T}_i^{n}) \geq c \eta (\mathbf {T}) , where T ∈ T \mathbf {T} \in \mathcal {T} , c c is a positive constant independent of T \mathcal {T} and the number of refinement levels, T i n \mathbf {T}_i^{n} is any refined tetrahedron of T \mathbf {T} , and η \eta is a tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant.

Quality Local Refinement of Tetrahedral Meshes Based on Bisection

We present a three-dimensional (3-D) local refinement algorithm for tetrahedral meshes based on a bisection procedure. The quality of refined meshes is guaranteed in terms of a tetrahedron shape measure $\eta $. Specifically, it is proved that the algorithm creates a finite number, which only depends on the number of tetrahedra in the initial mesh, of classes of similar tetrahedra in all refined tetrahedra. Furthermore, if ${\bf T}$ is a tetrahedron in the original mesh, and ${\bf T}_i^n $ is any refined tetrahedron of ${\bf T}$, then $\eta ({\bf T}_i^n ) \geq c\eta ({\bf T})$, where c is a positive constant independent of ${\bf T}$ and the number of refinements. It is also proved that for any interior face in the final mesh, the absolute value of the difference of the bisection levels of the two adjacent tetrahedra incident on the face is $ \leq 2$, which indicates that local refinements on tetrahedra can be smoothly extended to their neighbors. The expected time complexity of the algorithm is $O(N)$, wh...

A recursive approach to local mesh refinement in two and three dimensions

The newest vertex strategy for local refinement in two dimensions is reviewed and discussed. A recursive algorithm for local refinement of tetrahedral meshes in three dimensions based on similar bisection strategy is described. The recursive approach requires certain restrictions on the initial mesh. On the other hand, under this condition, the refinement process can be kept as local as needed, and it can be fully inverted. Simple data structures and derefinement algorithms are also outlined.

A simple algorithm for adaptive refinement of tetrahedral meshes combined with edge elements

A simple algorithm yielding unique face subdivision patterns of locally refined tetrahedral elements by the Delaunay tessellation process is presented. Due to the numerous advantages of edge elements, the algorithm is combined with the magnetostatic edge element computer code. A comparative analysis of four different error estimates used to locate elements for refinement, including a new one suitable for edge elements, is described. The performance of the proposed algorithm and the effectiveness of the error estimates are demonstrated by means of three-dimensional test problems. >