Constrained Delaunay Triangulation Research Articles

Fast segment insertion and incremental construction of constrained Delaunay triangulations

The most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has a Θ(kn2) worst-case running time, where n is the number of input vertices and k is the number of input segments.We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses. We demonstrate with a performance comparison that for segments that cross many edges, our algorithm is faster than gift-wrapping. We also show that a simple algorithm for segment location, which precedes segment insertion, is fast enough never to be a bottleneck in CDT construction. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(nlog⁡n+nlog2⁡k) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O(nlog⁡n) time, incremental CDT construction is easier to program and competitive in practice.Lastly, we partly extend the analysis (albeit not the linear-time insertion algorithm) to randomized incremental CDT construction in three dimensions.

Single-layer obstacle-aware routing for substrate interconnections

Due to the limit of available layers and the existence of obstacles in a routing region, single-layer obstacle-aware routing becomes an important issue for substrate interconnections. In this paper, based on the terminals of the given nets, the corner of the rectangular obstacles and the routing plane, routability-driven constrained Delaunay triangulation can be firstly constructed. Furthermore, based on the construction of routability-driven constrained Delaunay triangles, a flow-based approach is proposed to assign the global wires of the given nets in a single layer and an efficient approach is proposed to assign the physical paths of the given nets for single-layer obstacle-aware routing. Compared with the 97.35% routability of the routing nets in the modified combination of Liu׳s topological router [13] and our proposed detailed router for substrate routing with the rectangular obstacles, the experimental results show that our proposed flow-based approach can use less CPU time to obtain the 99.90% routability of the routing nets and shorter total wirelength in a single layer on the average for the tested examples.

Efficient tetrahedral mesh generation based on sampling optimization

AbstractWe present a heuristic approach to tetrahedral mesh generation for implicit closed surfaces. It consists of a surface sampling step and a volume sampling step that both work in a unified optimization framework. First, high‐quality isotropic samplings as well as a triangular mesh on the surface are generated. Then uniform volume samplings are determined by optimizing the point distribution inside the closed surface domain. Finally, the tetrahedral mesh is easily obtained by constrained Delaunay triangulation. Experimental results show that the new method can generate ideal tetrahedral meshes for closed implicit surfaces efficiently that are Delaunay based. Our method has the advantage of high efficiency and nice performance at surface boundaries. Copyright © 2015 John Wiley & Sons, Ltd.

Joint Reconstruction Method for the Computational Mesh Using Particles

Meshing geological body with joints is one of the major difficulties of using FEM in slope stability analysis. The traditional method is to divide the body into simply connected domains and triangulation them. However, when the geological body has three groups of joints which are quite common in rock mass, the division and connection are quite complicated. In this article, a new meshing method is introduced. This method uses gravity force instead of geometry constrain to ensure mesh nodes lies on joint faces, and uses non-friction plastic particle to fill the geological body. After get the mesh nodes, constrained Delaunay triangulation (CDT) is used to get triangle or tetrahedron mesh. Joint recovery quality and meshing quality are checked with an example which has crossed joints.

A CDT-Based Heuristic Zone Design Approach for Economic Census Investigators

This paper addresses a special zone design problem for economic census investigators that is motivated by a real-world application. This paper presented a heuristic multikernel growth approach via Constrained Delaunay Triangulation (CDT). This approach not only solved the barriers problem but also dealt with the polygon data in zoning procedure. In addition, it uses a new heuristic method to speed up the zoning process greatly on the premise of the required quality of zoning. At last, two special instances for economic census were performed, highlighting the performance of this approach.

Fast centroidal Voronoi Delaunay triangulation for unstructured mesh generation

A fast unstructured mesh generation algorithm based on conforming centroidal Voronoi Delaunay triangulation (CfCVDT) algorithm (Ju, 2007) is proposed in this paper. In the new algorithm, the constrained Delaunay triangulation (CDT) algorithm is used only for the generation of the initial mesh. The mesh quality shall be continuously improved by updating the positions of vertices and flipping edges in a number of iterations. Since the most time consuming procedure in CfCVDT algorithm is the CDT in each iteration which has been successfully avoided in this new algorithm the efficiency has been significantly improved. Furthermore, the meshes generated by this algorithm have similar high quality features as that generated by CfCVDT. When complex interfaces are involved, our algorithm can keep the mesh conforming to the interfaces very efficiently. By using various density functions, this algorithm can produce high quality non-uniform meshes for potentially many applications.

2D inverse modeling for potential fields on rugged observation surface using constrained Delaunay triangulation

The regular grid discretization is prevalent in the inverse modeling for gravity and magnetic data. However, this subdivision strategy performs lower precision to represent the rugged observation surface. To deal with this problem, we evaluate a non-structured discretization method in which the subsurface with rolling terrain is divided into numbers of Delaunay triangular cells and each mesh has the uniform physical property distributions. The gravity and magnetic anomalies of a complex-shaped anomalous body are represented as the summaries of the single anomaly produced by each triangle field source. When inverting for the potential field data, we specify a minimization objective function composed of data constraints and then use the preconditioned conjugate gradient algorithm to iteratively solve the matrix minimization equations, where the preconditioner is determined by the distances between triangular cells and surface observers. We test our method using synthetic data; all tests return favorable results. In the case studies involving the gravity and magnetic anomalies of the Mengku and Pobei deposits in Xinjiang, northwest China, the inferred magnetite orebodies and ultrabasic rocks distributions are verified by the additional drilling and geological information. The discretization of constrained Delaunay triangulation provides an useful approach of computing and inverting the potential field data on the situations of undulate topography and complicated objects.

Tagging-the-triangle algorithm for partitioning features with inconsistent boundaries

In spatial data sets, gaps or overlaps among features are frequently found in spatial tessellations due to the non-abutting edges with adjacent features. These non-abutting edges in loose tessellations are also called inconsistent boundaries or slivers; polygons containing at least one inconsistent boundary are called inconsistent polygons or sliver polygons. The existing algorithms to solve topological inconsistencies in sliver polygons suffer from one or more of three major issues, namely determination of tolerances, excessive CPU processing time for large data sets and loss of vertex history. In this article, we introduce a new algorithm that mitigates these three issues. Our algorithm efficiently searches the features with inconsistent polygons in a given spatial data set and logically partitions them among adjacent features. The proposed algorithm employs the constrained Delaunay triangulation technique to generate labelled triangles from which inconsistent polygons with gaps and overlaps are identified using label counts. These inconsistent polygons are then partitioned using the straight skeleton method. Moreover, each of these partitioned gaps or overlaps is distributed among the adjacent features to improve the topological consistency of the spatial data sets. We experimentally verified our algorithm using the real land cadastre data set. The comparison results show that the proposed algorithm is four times faster than the existing algorithm for data sets with 200,000 edges.

Performance analysis of fast marching-based motion planning for autonomous mobile robots in ITER scenarios

Operations of transportation in cluttered environments require robust motion planning algorithms. Specially with large and heavy vehicles under hazardous operations of maintenance, such as in the ITER, an international nuclear fusion research project. The load transportation inside the ITER facilities require smooth and optimized paths with safety margin of 30 cm. The transportation is accomplished by large rhombic-like vehicles to exploit its kinematic capabilities. This paper presents the performance analysis of a motion planning algorithm to optimize trajectories in terms of clearance, smoothness and execution time in cluttered scenarios. The algorithm is an upgraded version of a previous one used in ITER, replacing the initialization implemented using Constrained Delaunay Triangulation by the Fast Marching Square. Exhaustive simulated experiments have been carried out in different levels of ITER buildings, comparing the performance of the algorithm using different metrics.

Morphing Linear Features Based on Their Entire Structures

AbstractIn this article, a new morphing method is proposed for two linear features at different scales, based on their entire structures (MLBES in abbreviation). First, the bend structures of the linear features are identified by using a constrained Delaunay triangulation (CDT in abbreviation) model and represented by binary bend‐structure trees. By matching the independent bends represented by the bend‐structure trees, corresponding independent bends are obtained. These corresponding independent bends are further used to match their child bends based on hierarchical bend structures so that corresponding bends are obtained. On this basis, the two linear features are split into pairs of corresponding subpolylines by the start and end points of the corresponding bends. Second, structures of the corresponding subpolylines are identified by the Douglas‐Peucker algorithm and represented by binary line generalization trees (BLG‐trees in abbreviation). The corresponding subpolylines are split into smaller corresponding subpolylines by matching the nodes of the BLG‐trees. Third, the corresponding points are identified by using the linear interpolation algorithm for every pair of corresponding subpolylines. Finally, straight‐line trajectories are employed to generate a family of intermediate‐scale linear features. By comparison with other methods, it is found that MLBES is accurate and efficient.

On triangulation axes of polygons

We propose the triangulation axis as an alternative skeletal structure for a simple polygon P. This axis is a straight-line tree that can be interpreted as an anisotropic medial axis of P, where inscribed disks are line segments or triangles. The underlying triangulation that specifies the anisotropy can be varied, to adapt the axis so as to reflect predominant geometrical and topological features of P. Triangulation axes typically have much fewer edges and branchings than the Euclidean medial axis or the straight skeleton of P. Still, they retain important properties, as for example the reconstructability of P from its skeleton. Triangulation axes can be computed from their defining triangulations in O(n) time. We investigate the effect of using several optimal triangulations for P. In particular, careful edge flipping in the constrained Delaunay triangulation leads, in O(nlog⁡n) overall time, to an axis competitive to ‘high quality axes’ requiring Θ(n3) time for optimization via dynamic programming.

Reconstruction of 3D Strata Model Based on Data from Multiple Sources

Given the multiple-sources feature of geological interface data, in this paper we propose a universal 3D modelling method using data forms presented as dirllhole diagrams, geologic sections, and vectorized contour maps. In our strategy, data from different sources are all treated as discrete property points, upon which cubic interpolation interpolation is performed to make a denser grid. The final step is accomplished through constrained Delaunay triangulation of the top and bottom grids and boundaries to generate the strata model through triangular strips. Our method merges all kinds of data sources available into one single modelling process, thus the most realistic result is guaranteed.

Segmentation and reconstruction of cerebral vessels from 3D rotational angiography for AVM embolization planning.

Diagnosis and computer-guided therapy of cerebral Arterio-Venous Malformations (AVM) require an accurate understanding of the cerebral vascular network both from structural and biomechanical point of view. We propose to obtain such information by analyzing three Dimensional Rotational Angiography (3DRA) images. In this paper, we describe a two-step process allowing 1) the 3D automatic segmentation of cerebral vessels from 3DRA images using a region-growing based algorithm and 2) the reconstruction of the segmented vessels using the 3D constrained Delaunay Triangulation method. The proposed algorithm was successfully applied to reconstruct cerebral blood vessels from ten datasets of 3DRA images. This software allows the neuroradiologist to separately analyze cerebral vessels for pre-operative interventions planning and therapeutic decision making.

제한된 델로네 삼각분할을 이용한 공간 불확실한 영역 탐색 기법

지적 필지를 구성하고 있는 폴리곤 집합은 현실세계의 국토를 반영하는 가장 기반이 되는 데이터 집합이다. 따라서 지적 필지는 서로 간에 겹쳐있거나 공백을 가지지 않는 위상적 무결성이 보장되어야하는 데이터이다. 하지만, 여러 가지 이유로 필지들 간의 겹침과 공백의 문제가 발생하고 있고, 이러한 경우 폴리곤의 경계들은 주변의 폴리곤과 정확하게 인접하고 있지 못하기 때문에 의도하지 않은 겹침 영역과 공백 영역이 생산되고 있다. 이와 같이 정확하게 인접되어 있지 않은 경계가 불확실한 모서리를 하나 이상 포함하고 있는 경우, 이 폴리곤을 불확실한 영역이라고 부른다. 본 논문에서는 이러한 영역을 탐색하기 위한 TTA 기법을 제안하고자 한다. TTA 처리 순서는 우선 폴리곤 데이터 집합으로부터 포인트와 폴리라인을 추출하여 제한된 델로네 삼각분할을 수행한다. 다음으로 각 삼각형마다 데이터 집합과 중첩되는 면의 수를 세어 삼각형에 태깅을 수행한다. 태깅 값이 0 또는 1 이상인 삼각형을 추출한 후 연결성을 가지고 있는 삼각형끼리 병합을 수행하여 위상적 모순이 있는 영역들을 발견한다. 본 실험에서는 제안하는 알고리즘을 자동화하여 실세계에서 경계가 교차하는 지적 데이터에 적용하여 실험을 하였다. Cadastral parcel objects as polygons are fundamental dataset which represent land administration and management of the real world. Thus it is necessary to assure topological seamlessness of cadastral datasets which means no overlaps or gaps between adjacent parcels. However, the problem of overlaps or gaps are frequently found due to non-coinciding edges between adjacent parcels. These erroneous edges are called uncertain edges, and polygons containing at least one uncertain edge are called uncertain polygons. In this paper, we proposed a new algorithm to efficiently search parcels of uncertain polygons between two adjacent cadastral datasets. The algorithm first selects points and polylines around adjacent datasets. Then the Constrained Delaunay Triangulation (CDT) is applied to extract triangles. These triangles are tagged by the number of the original cadastral datasets which intersected with the triangles. If the tagging value is zero, the area of triangles mean gaps, meanwhile, the value is two, the area means overlaps. Merging these triangles with the same tagging values according to adjacency analysis, uncertain edges and uncertain polygons could be found. We have performed experimental application of this automated derivation of partitioned boundary from a real land-cadastral dataset.

An adaptive mesh refinement method for a medium with discrete fracture network: The enriched Persson’s method

Discrete fracture network has significant effects on the thermal–hydraulic–mechanical–chemical coupling analysis in a highly fractured medium. Due to the complex configuration of interior fracture network, generation of a high resolving mesh is crucial in achieving accurate numerical results. An adaptive mesh refinement method is developed for generating adaptive meshes in a domain with discrete fracture networks. An optimized size function is embedded into Persson’s mesh generator. The constrained Delaunay triangulation algorithm is integrated into the analogous force equilibrium method. An assessment method of mesh size quality is proposed. Fractures are pre-meshed with a user-specified length in advance. The whole domain is then adaptively meshed using a constrained Delaunay method in a locally relaxation manner. Varying meshes can be generated with fine meshes around fractures and coarse meshes in the domain. Two case studies are carried out and compared with those obtained from conventional methods. Results show that the present method is advantageous in generating a high resolving mesh. Finer meshes are adaptively generated around fractures and gradually changed to sparser meshes away from fractures. Smoothly and gradually varying meshes are generated with little sacrifice of the size and angle quality. The adaptive mesh refinement method can be used for a wide range of thermal, hydraulic, mechanical, chemical simulations and their numerical coupling system in a fractured medium.

A complete solution of cartographic displacement based on elastic beams model and Delaunay triangulation

Abstract. In map production and generalization, it is inevitable to arise some spatial conflicts, but the detection and resolution of these spatial conflicts still requires manual operation. It is become a bottleneck hindering the development of automated cartographic generalization. Displacement is the most useful contextual operator that is often used for resolving the conflicts arising between two or more map objects. Automated generalization researches have reported many approaches of displacement including sequential approaches and optimization approaches. As an excellent optimization approach on the basis of energy minimization principles, elastic beams model has been used in resolving displacement problem of roads and buildings for several times. However, to realize a complete displacement solution, techniques of conflict detection and spatial context analysis should be also take into consideration. So we proposed a complete solution of displacement based on the combined use of elastic beams model and constrained Delaunay triangulation (CDT) in this paper. The solution designed as a cyclic and iterative process containing two phases: detection phase and displacement phase. In detection phase, CDT of map is use to detect proximity conflicts, identify spatial relationships and structures, and construct auxiliary structure, so as to support the displacement phase on the basis of elastic beams. In addition, for the improvements of displacement algorithm, a method for adaptive parameters setting and a new iterative strategy are put forward. Finally, we implemented our solution on a testing map generalization platform, and successfully tested it against 2 hand-generated test datasets of roads and buildings respectively.

Boundary recovery for 3D Delaunay triangulation

New ideas are presented in this paper for the boundary recovery of 3D Delaunay triangulation. Fully constrained Delaunay triangulations in terms of geometrical and topological integrities on all boundary edges and facets are required in many applications, such as meshing by components, fluid–structure interactions, parallel mesh generation, local remeshing and interface problems. The geometry of boundary edges and facets can be recovered by the introduction of Steiner points. However, for a fully constrained Delaunay triangulation, these Steiner points have to be removed or repositioned towards the interior of the domain to restore the topological integrity of the boundary edges and the facets. It is found that Steiner points on edges could be removed more systematically following a specific sequence in an alternative manner rather than a random selection commonly adopted in practice; whereas for Steiner points on a facet, a weight on the Steiner point adjacency would lead to an optimal order to facilitate their removal. A linear programming technique is also employed to determine the feasible region for the relocation of Steiner points in the interior of the domain. Work examples and industrial applications with details in the boundary recovery are presented to illustrate how the algorithm works on objects with difficult boundary conditions.

Prediction of pedestrians routes within a built environment in normal conditions

Modelling and prediction of pedestrian routing behaviours within known built environments has recently attracted the attention of researchers across multiple disciplines, owing to the growing demand on urban resources and requirements for efficient use of public facilities. This study presents an investigation into pedestrians’ routing behaviours within an indoor environment under normal, non-panic situations. A network-based method using constrained Delaunay triangulation is adopted, and a utility-based model employing dynamic programming is developed. The main contribution of this study is the formulation of an appropriate utility function that allows an effective application of dynamic programming to predict a series of consecutive waypoints within a built environment. The aim is to generate accurate sequence waypoints for the pedestrian walking path using only structural definitions of the environment as defined in a standard CAD format. The simulation results are benchmarked against those from the A∗ algorithm, and the outcome positively indicates the usefulness of the proposed method in predicting pedestrians’ route selection activities.

Wireless Sensor Network Configuration—Part I: Mesh Simplification for Centralized Algorithms

This is the first of a two-part investigation of centralized and decentralized approaches for determining the optimal configuration of a sensor network. In this first part, we present a centralized approach for the generation of mesh (wireless sensor) network configurations that provide complete sensing coverage and communication connectivity of a domain. A challenging problem in deploying wireless sensor networks is maximizing coverage in irregular shaped polygonal areas while maintaining a high degree of node connectivity. The novelties presented in this paper are: 1) a centralized mesh simplification technique, the Iterative Node Removal with Constrained Delaunay Triangulation and Smoothing (INRCDTS) algorithm, and 2) a centralized mesh generation approach with INRCDTS that may be used for any nonintersecting closed polygonal area. Additionally, we provide a comparison of two centralized mesh generation techniques. The INRCDTS was built and tested as an enhancement of two traditional mesh generation techniques: advancing front technique and Matlab partial differential equation toolbox. The INCRCDTS introduces the ability to tune the generated mesh configuration to the number of nodes and nodal spacing. The INRCDTS enhancement has proven to increase the uniformity of the mesh in an irregular shaped polygonal area relative to advancing front and MATLAB partial differential equation algorithms by 23% and 41%, respectively.

Adaptive fracture simulation of multi-layered thin plates

The fractures of thin plates often exhibit complex physical behaviors in the real world. In particular, fractures caused by tearing are different from fractures caused by in-plane motions. In this paper, we study how to make thin-plate fracture animations more realistic from three perspectives. We propose a stress relaxation method, which is applied to avoid shattering artifacts after generating each fracture cut. We formulate a fracture-aware remeshing scheme based on constrained Delaunay triangulation, to adaptively provide more fracture details. Finally, we use our multi-layered model to simulate complex fracture behaviors across thin layers. Our experiment shows that the system can efficiently and realistically simulate the fractures of multi-layered thin plates.