Delaunay Refinement Algorithm Research Articles

Experimental study of nonlinear states of oscillating water column in waves

Physical experiments are conducted in a large-scale wave flume to investigate the nonlinear states of an oscillating water column in waves. A four-state classification (sloshing, piston, freak, and trapping states) is proposed to characterize the free-surface behaviors in the cylindrical chamber. The error range of the linearized potential-flow theory for the description of the sloshing- and piston-state internal wave motion is evaluated. This study, for the first time, identifies the phenomenon of freak-state resonance, which features a steep surface hump with wave overturning or water jets in the cylindrical chamber. This special resonance is triggered by the second-order nonlinear component of the incident waves, which cannot be predicted by the analytical solution. The transverse waves observed in the trapping state cannot be analytically predicted, neither. The instantaneous free surface in the cylinder is reconstructed using the Delaunay-refinement algorithm to investigate the volume fluctuation of the water body in the cylinder. A detrending procedure is applied to decompose each volume fluctuation history into trend and oscillatory components. The analytical solution accurately describes the amplitude of the oscillatory component in the sloshing state but overpredicts that in the piston state. The relationship between histories of the volume fluctuation of the internal water body and the measured wave elevation is identified in sloshing and piston states.

Where and how Ruppert’s algorithm fails

Several examples are given to demonstrate non-termination of Ruppert’s Delaunay refinement algorithm for triangular mesh generation exploring the relationship between the minimum angle in the input and the minimum allowable angle in the output. These examples show non-termination despite requiring a minimum output angle of less than 30° which is smaller than the value generally observed in practice. For input containing only points, an example demonstrates failure of the algorithm with a minimum angle of 30° showing that the theoretical analysis in that context is sharp.

Triangulation of Voronoi Faces of Sphere Voronoi Diagram using Delaunay Refinement Algorithm

Triangulation is one of the fundamental problems in computational geometry and computer graphics community, and it has huge application areas such as 3D printing, computer-aided engineering, surface reconstruction, surface visualization, and so on. The Delaunay refinement algorithm is a well-known method to generate quality triangular meshes when point cloud and/or constrained edges are given in two- or three-dimensional space. In this paper, we propose a simple but efficient algorithm to triangulate Voronoi surfaces of Voronoi diagram of spheres in 3-dimensional Euclidean space. The proposed algorithm is based on the Ruppert’s Delaunay refinement algorithm, and we modified the algorithm to be applied to the triangulation of Voronoi surfaces in two ways. First, a new method to deciding the location of a newly added vertex on the surface in 3-dimensional space is proposed. Second, a new efficient but effective way of estimating approximation error between Voronoi surface and triangulation. Because the proposed algorithm generates a triangular mesh for Voronoi surfaces with guaranteed quality, users can control the level of quality of the resulting triangulation that their application problems require. We have implemented and tested the proposed algorithm for random non-intersecting spheres, and the experimental result shows the proposed algorithm produces quality triangulations on Voronoi surfaces satisfying the quality criterion.

Topological Disk Mesh Morphing Based on Area-Preserving Parameterization

Mesh morphing is a technique which gradually deforms a mesh into another one. Mesh parameterization, a powerful tool adopted to establish the one-to-one correspondence map between different meshes, is of great importance in 3D mesh morphing. However, current parameterization methods used in mesh morphing induce large area distortion, resulting in geometric information loss. In this paper, we propose a new morphing approach for topological disk meshes based on area-preserving parameterization. Conformal mapping and Mobius transformation are computed firstly as rough alignment. Then area preserving parameterization is computed via the discrete optimal mass transport map. Features are exactly aligned through radial basis functions. A surface remeshing scheme via Delaunay refinement algorithm is developed to create a new mesh connectivity. Experimental results demonstrate that the proposed method performs well and generates high-quality morphs.

A novel 3D geometrical reconstruction model for porous rocks

In this paper, a novel geometrical reconstruction model is proposed for numerical simulation of pore structure in rocks. The spatial configuration and geometrical information of the microstructure are firstly gathered from Chongqing sandstone by X-ray micro-CT observations, and experimental tests on Chongqing sandstone are applied to obtain the mechanical parameters. Then, on the basis of the gathered geometrical information, the surface geometrical model is established by using the modified Marching cubes algorithm. Next, the established surface geometrical model is extended to 3D tetrahedral model by using Delaunay refinement algorithm. Finally, the 3D tetrahedral model is transformed into finite element (FE) numerical model. Moreover, the relationship between the porosity, fractal dimension of pore structure and mechanical properties of Chongqing sandstone is also investigated. The comparison of the numerical results and the experimental data shows a good consistency.

기하학적 해싱을 이용한 딜러니 개선 알고리듬의 가속화

Delaunay refinement algorithm is a classical method to generate quality triangular meshes when point cloud and/or constrained edges are given in two- or three-dimensional space. It computes the Delaunay triangulation for given points and edges to obtain an initial solution, and update the triangulation by inserting steiner points one by one to get an improved quality triangulation. This process repeats until it satisfies given quality criteria. The efficiency of the algorithm depends on the criteria and point insertion method. In this paper, we propose a method to accelerate the Delaunay refinement algorithm by applying geometric hashing technique called bucketing when inserting a new steiner point so that it can localize necessary computation. We have tested the proposed method with a few types of data sets, and the experimental result shows strong linear time behavior.

Scalable 3D hybrid parallel Delaunay image-to-mesh conversion algorithm for distributed shared memory architectures

In this paper, we present a scalable three-dimensional hybrid parallel Delaunay image-to-mesh conversion algorithm (PDR.PODM) for distributed shared memory architectures. PDR.PODM is able to explore parallelism early in the mesh generation process thanks to the aggressive speculative approach employed by the Parallel Optimistic Delaunay Mesh generation algorithm (PODM). In addition, it decreases the communication overhead and improves data locality by making use of a data partitioning scheme offered by the Parallel Delaunay Refinement algorithm (PDR). PDR.PODM supports fully functional volume grading by creating elements with varying size. Small elements are created near boundary or inside the critical regions in order to capture the fine features while big elements are created in the rest of the mesh. We tested PDR.PODM on Blacklight, a distributed shared memory (DSM) machine in Pittsburgh Supercomputing Center. For the uniform mesh generation, we observed a weak scaling speedup of 163.8 and above for up to 256 cores as opposed to PODM whose weak scaling speedup is only 44.7 on 256 cores. PDR.PODM scales well on uniform refinement cases running on DSM supercomputers. The end result is that PDR.PODM can generate 18 million elements per second as opposed to 14 million per second in our earlier work. The varying size version sharply reduces the number of elements compared to the uniform version and thus reduces the time to generate the mesh while keeping the same fidelity.

Off-centre Steiner points for Delaunay-refinement on curved surfaces

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.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

TetGen is a C++ program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available. This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

Scalable 3D Hybrid Parallel Delaunay Image-to-Mesh Conversion Algorithm for Distributed Shared Memory Architectures

In this paper, we present a scalable three dimensional hybrid parallel Delaunay image-to-mesh conversion algorithm (PDR.PODM) for distributed shared memory architectures. PDR.PODM is able to explore parallelism early in the mesh generation process because of the aggressive speculative approach employed by the Parallel Optimistic Delaunay Mesh generation algorithm (PODM). In addition, it decreases the communication overhead and improves data locality by making use of a data partitioning scheme offered by the Parallel Delaunay Refinement algorithm (PDR). PDR.PODM utilizes an octree structure to decompose the initial mesh and to distribute the bad elements to different octree leaves (subregions). A set of independent subregions are selected and refined in parallel without any synchronization among them. In each subregion, a group of threads is assigned to insert or delete multiple points based on the refinement rules offered by PODM. We tested PDR.PODM on Blacklight, a distributed shared memory (DSM) machine in the Pittsburgh Supercomputing Center, and observed a weak scaling speedup of 163.8 and above for up to 256 cores as opposed to PODM whose weak scaling speedup is only 44.7 on 256 cores. The end result is that we can generate 18 million elements per second as opposed to 14 million per second in our earlier work. To the best of our knowledge, PDR.PODM exhibits the best scalability among parallel guaranteed quality Delaunay mesh generation algorithms running on DSM supercomputers.

Voronoi-based Point-placement for Three-dimensional Delaunay-refinement

An extension of the restricted Delaunay-refinement algorithm for three-dimensional tetrahedral mesh generation is described, in which an off-centre type point-placement scheme is utilised. It is shown that the use of generalised Steiner points, positioned along edges in the associated Voronoi complex, typically leads to improvements in the overall size, quality and grading of the resulting tetrahedral meshes. 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 new method is shown to inherit many of the best features of classical Delaunay-refinement and advancing-front type algorithms, combining good practical performance with theoretical robustness. Experimental comparisons show that the new method outperforms classical Delaunay-refinement techniques for a number of three-dimensional benchmark problems.

4D space–time Delaunay meshing for medical images

In this paper, we present a Delaunay refinement algorithm for 4-dimensional ($$\hbox {3D}+t$$3D+t) segmented images. The output mesh is proved to consist of sliver-free simplices. Assuming that the hyper-surface is a closed smooth manifold, we also guarantee faithful geometric and topological approximation. We implement and demonstrate the effectiveness of our method on publicly available segmented cardiac images. Finally, we devise a tightly coupled parallelization technique to boost the performance of our 4-dimensional mesher, thereby taking advantage of the multi-core and many-core platforms already available in the market.

Surface Meshing with Curvature Convergence.

Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

Reprint of: Delaunay refinement algorithms for triangular mesh generation

Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. A compromise is necessary. A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30° or greater and no angle is smaller than arcsin[(3/2)sin(ϕ/2)]∼(3/4)ϕ, where ϕ⩽60° is the smallest angle separating two segments of the input domain. New angles smaller than 30° appear only near input angles smaller than 60°. In practice, the algorithm's performance is better than these bounds suggest.Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms. Chew proved that his algorithm produces no angle smaller than 30° (barring small input angles), but without any guarantees on grading or number of triangles. He conjectures that his algorithm offers such guarantees. His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 26.5°, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal.

Guaranteed quality tetrahedral Delaunay meshing for medical images

In this paper, we present a Delaunay refinement algorithm for meshing 3D medical images. Given that the surface of the represented object is a smooth 2-manifold without boundary, we prove that (a) all the tetrahedra of the output mesh have radius–edge ratio less than 3+2(≈1.93), (b) all the boundary facets have planar angles larger than 30 degrees, (c) the symmetric (2-sided) Hausdorff distance between the object surface and mesh boundary is bounded from above by a user-specified parameter, and (d) the mesh boundary is ambient isotopic to the object surface. The first two guarantees assure that our algorithm produces elements of bounded radius–edge ratio. The last two guarantees assure that the mesh boundary is a good geometric and topological approximation of the object surface. Our method also offers control over the size of tetrahedra in the final mesh. Experimental evaluation of our algorithm on synthetic and real medical data illustrates the theory and shows the effectiveness of our method.

Fatigue Multi-Cracks Growths in Plates Using J-Integral Approach with a Developed Home FEM Software

In this paper multiple fatigue cracks propagation are simulated in two-dimensional plates. Since re-meshing the cracked bodies in each increment of crack extension is a time-consuming and complicated procedure, numerical simulation of mixed-mode crack propagation with FEM is a difficulty. For this purpose, a FEM software is programmed and mesh refinement in each increment of crack is performed by Delaunay Refinement Algorithm. Using different refinement methods, complex boundaries such as multiple cracks and discontinuities which are closed together are easily refined by this algorithm. Crack propagation path is predicted using domain form of J-integral. Modified tensile stress (MTS) criterion is used to predict the crack propagation path in each increment. Different numerical examples illustrate the validation and reliability of present software.

Fracture Response of Reinforced Concrete Deep Beams Finite Element Investigation of Strength and Beam Size

This article presents a finite element analysis of reinforced concrete deep beams using nonlinear fracture mechanics. The article describes the development of a numerical model that includes several nonlinear processes such as compression and tension softening of concrete, bond slip between concrete and reinforcement, and the yielding of the longitudinal steel reinforcement. The development also incorporates the Delaunay refinement algorithm to create a triangular topology that is then transformed into a quadrilateral mesh by the quad-morphing algorithm. These two techniques allow automatic remeshing using the discrete crack approach. Nonlinear fracture mechanics is incorporated using the fictitious crack model and the principal tensile strength for crack initiation and propagation. The model has been successful in reproducing the load deflections, cracking patterns and size effects observed in experiments of normal and high-strength concrete deep beams with and without stirrup reinforcement.

DELAUNAY REFINEMENT ALGORITHMS FOR ESTIMATING LOCAL FEATURE SIZE IN 2D AND 3D

We present Delaunay refinement algorithms for estimating local feature size on the input vertices of a 2D piecewise linear complex and on the input vertices and segments of a 3D piecewise linear complex. These algorithms are designed to eliminate the need for a local feature size oracle during quality mesh generation of domains containing acute input angles. In keeping with Ruppert's algorithm, encroachment in these algorithms can be determined through only local information in the current Delaunay triangulation. The algorithms are practical to implement and several examples are given.

CONSTRUCTING CONSTRAINED DELAUNAY TETRAHEDRALIZATIONS OF VOLUMES BOUNDED BY PIECEWISE SMOOTH SURFACES

This article presents an algorithm to construct constrained Delaunay tetrahedralizations of geometric domains bounded by piecewise smooth surfaces. Meshes are built from the bottom-up by first discretizing the boundary curves and then by sampling the smooth surfaces. The sampling procedure refines the Delaunay triangulation restricted to these surfaces, targeting topological violations and poor quality triangles. Unlike previously published algorithms adopting a similar approach, we propose to sample each smooth surface patch independently. This obviates the need for a boundary protection scheme around small dihedral angles in the input and can also lead to coarser constraining triangulations. Starting from a Delaunay tetrahedralization of the point samples, a combination of mesh reconfigurations and vertex insertions is then used to obtain a tetrahedralization constrained to the boundary surfaces. The algorithm is designed to produce tetrahedralizations that can be used in conjunction with a Delaunay refinement algorithm implemented on a Bowyer-Watson framework.

Meshing interfaces of multi-label data with Delaunay refinement

In medical imaging, the generation of surface representations of anatomical objects obtained by labeling images from various modalities, is a critical component for visualization, simulation, and analysis. The interfaces between labeled regions can meet at arbitrary angles and with complex topologies, causing most automatic meshing algorithms to fail. We apply a recent Delaunay refinement algorithm to generate high quality triangular meshes that approximate the interface surfaces. This algorithm has proven guarantees for meshing piecewise-smooth shapes and its implementation overhead is low. Consequently, the approach is applicable to labeled datasets generated from binary segmentations as well as from probabilistic segmentation algorithms. We show the effectiveness of this technique on data from a variety of medical fields and discuss its ability to control the quality and size of the output meshes. The same algorithm can be used to generate tetrahedral meshes of the segmentation space.