Constrained Delaunay Triangulation Research Articles

Generating grid maps via the snake model

Generating grid maps via the snake model

Collaborative Methods of Resolving Road Graphic Conflicts Based on Cartographic Rules and Generalization Operations

The resolution of road graphic conflicts is a key aspect of map generalization, which involves both scale reduction and the symbolization of map features. This study proposes collaborative methods of road graphic conflict resolution considering different road characteristics. These methods consider both geometric and semantic characteristics, and they incorporate the bend characteristics of roads, the road symbol size, and road semantics. Constrained Delaunay triangulation skeleton lines are used to categorize road graphic conflicts, which are made up of four independent conflict types and four group conflict types. Based on their characteristics, three collaborative methods are designed to deal with the different types of road graphic conflicts: collaboration between deletion and the snake displacement model, collaboration between the snake displacement model and collinearity, and collaboration among simplification, smoothing, and the beam displacement model. Two types of independent conflicts can be processed using only one simple operation. This study summarizes the cartographic rules for resolving road graphic conflicts, and these are used along with geometric features to drive the collaborative methods or one simple operation presented here. The experimental results indicate that the method proposed in this study can effectively resolve road graphic conflicts.

A novel unresolved/semi‐resolved CFD‐DEM coupling method with dynamic unstructured mesh

AbstractThe greatest challenge when performing large deformation simulations using the CFD‐DEM coupling method lies in the dynamical update of the fluid meshes. To address this problem, a novel CFD‐DEM coupling method integrated with the dynamic unstructured grid is proposed in this work. The mesh initialization and reconstruction are performed by the Constrained Delaunay triangulation (CDT) implemented by the tetgen algorithm. Moreover, the equation of state (EOS) is introduced to consider the impact of fluid compressibility. The reliability of this coupling method is preliminarily verified by the particle sedimentation test. Additionally, undrained triaxial shear and one‐dimensional consolidation tests are conducted to examine the applicability of the proposed method in simulating geotechnical cases. From our analysis, it is found that the shear responses in the undrained triaxial shear test obtained from the proposed method with dynamic mesh are in direct accordance with those led by the constant volume (CV) method. However, difficulties arise in accurately describing the moving boundary by the fixed mesh, leading to unreliable results. Moreover, in the one‐dimensional consolidation test, both the dynamic and fixed grids can capture the Mandel—Cryer effect, which is more pronounced under the dynamic grid, mainly due to the dual impact of the loading device and particle skeleton on the fluid. A greater compressibility specified under the dynamic grid led to a weakening of this effect, whereas its duration is increased. Our work provides valuable insights for effectively dealing with dynamic fluid boundaries.

Optimizing Mesh to Improve the Triangular Expansion Algorithm for Computing Visibility Regions

This paper addresses the problem of improving the query performance of the triangular expansion algorithm (TEA) for computing visibility regions by finding the most advantageous instance of the triangular mesh, the preprocessing structure. The TEA recursively traverses the mesh while keeping track of the visible region—the set of all points visible from a query point in a polygonal world. We show that the measured query time is approximately proportional to the number of triangle edge expansions during the mesh traversal. We propose a new type of triangular mesh that minimizes the expected number of expansions assuming the query points are drawn from a known probability distribution. We design a heuristic method to approximate the mesh and evaluate the approach on many challenging instances that resemble real-world environments. The proposed mesh improves the mean query times by 12–16% compared to the reference constrained Delaunay triangulation. The approach is suitable to boost offline applications that require computing millions of queries without addressing the preprocessing time. The implementation is publicly available to replicate our experiments and serve the community.

Boundary constrained quadrilateral mesh generation based on domain decomposition and templates

Compared to triangular or mixed meshes, pure quadrilateral meshes generally offer higher accuracy and computational efficiency in numerical simulations. Recently, the main obstacles in the generation of quadrilateral meshes have revolved around the identification of mesh singularities and the subsequent decomposition of the domain. In this study, a novel quadrilateral mesh generation method based on domain decomposition and templates is presented. The basic idea is to decompose the domain by separating the mesh singularities rather than identifying their precise locations. Surfaces composed of a set of surface patches are meshed independently. For each surface patch, the problem domain is divided into several simple polygonal subregions, including quadrangles, triangles, and pentagons, using constrained Delaunay triangulation (CDT) and recursive bisections. Subsequently, mesh templates with a limited number of singularities are employed to rapidly generate high-quality quadrilateral meshes over those subregions. By constraining the boundary discretization of each surface patch, mesh conformability is automatically achieved. An integer linear programming equation is presented to ensure that the total number of boundary nodes on each surface is an even value. Several experimental results are presented and discussed, demonstrating the reliability and efficiency of the proposed method.

Intelligent mesh generation for crack simulation using graph neural networks

Mesh generation for crack simulation is often the rate-limiting step because of the rapid variations in crack shape. The classical meshing paradigm, place-nodes-and-link, relies on predefined rules and fails to generalize various crack shapes. We proposed a graph neural networks-based method for recovering the missing connection information in the crack meshes. The constrained Delaunay triangulation method created a representative training mesh dataset with different crack shapes. Comprehensive and systematic analyses compare the effectiveness and efficiency of five state-of-art GNNs under different graph representations. Our study shows that the trained GraphSAGE outperforms the state-of-art GNNs on the triangular meshing task efficiently and reveals GNNs' potential to restore the missing information between adjacency vertices or edges. This work pioneers the application of GNNs for intelligent mesh generation and paves the way for complex crack simulation.

Development of a Tetrahedral Mesh Generator Based on Constrained Delaunay Triangulation

The article describes the basic principles of operation of the finite element mesh generator SMCMGrid, which is an integral part of the SMCM software package designed to solve problems in the mechanics of composite materials and structures, which was developed at the Scientific-educational center on supercomputer modeling and software engineering of Bauman Moscow State Technical University (BMSTU SIMPLEX CENTER). The paper describes a software pipeline for constructing high-quality tetrahedral unstructured meshes for arbitrary shape B-rep models based on constrained Delaunay triangulation. We have presented the main algorithms for generating meshes on the edges and surfaces of the model, which make it possible to create the so-called simplicial complex, for which it is then possible to construct a constrained Delaunay triangulation and create the final tetrahedral mesh of the required quality. The mesh is generated in a Riemannian space with a given metric that takes into account the required characteristics of the mesh at each point of the geometric model. We gave examples of finite element meshes for various geometric models of engineering structure elements, including composite materials, and also demonstrated the adequacy of the results of solving the elasticity problem on a mesh built using the SMCMGrid generator. The results of the work led to the conclusion that the presented generator can be used to solve problems of engineering analysis and modeling.

Zero-Watermarking for Vector Maps Combining Spatial and Frequency Domain Based on Constrained Delaunay Triangulation Network and Discrete Fourier Transform

With its lossless properties, zero-watermarking has attracted a lot of attention in the field of copyright protection for vector maps. However, the common zero-watermarking algorithm puts too much emphasis on mining for global features, making it vulnerable to cropping attacks, and the robustness is not comprehensive enough. This study provides a vector map zero-watermarking scheme that utilizes spatial statistical information and frequency domain transformation methods in an effort to solve the aforementioned issue. In order to make the scheme more resistant to cropping and compression, it is constructed on the basis of feature point extraction and point constraint blocking of the original vector map. Within each sub-block, feature points are used to build constraint Delaunay triangulation networks (CDTN), and the angular values within the triangle networks are then extracted as spatial statistics. The angle value sequence is further transformed by discrete Fourier transform (DFT), and the binarized phase sequence is used as the final feature information to build a zero watermark by executing an exclusive disjunction operation with the encrypted copyright watermark image, both of which contribute to the scheme's robustness and security. The results of the attack experiments show that the proposed vector map zero-watermarking can restore identifiable copyright images under common geometric attacks, cropping attacks, and coordinate system transformations, demonstrating a high level of robustness. The theoretical basis for the robustness of this watermarking scheme is the stability of CDTN and the geometric invariance of DFT coefficients, and both theory and experiment validate the method's validity.

Improved Pigeon-Inspired Optimization in an Integrated Obstacle Avoidance Method for Mars UAV Formation

This paper models the Mars UAV formation exploring the surface of Mars, and then the formation obstacle avoidance is brought up with the assumptions of the Mars circumstance and the UAVs. Based on their specialty, constrained Delaunay triangulation, Yen-[Formula: see text] shortest path algorithm, the collaborative function, and the improved pigeon-inspired optimization (PIO) algorithm are integrated to solve the obstacle avoidance for the formation. Since the steering maneuver costs much energy and increases instabilities vulnerable in extraterrestrial exploration, the paper focuses on the route smoothness problem. The PIO is improved to be suitable for smooth routes and is compatible with other PIO variants. The simulation results show that the sum of the steering angle, namely the performance index, is effectively reduced and satisfies the obstacle avoidance requirements for Mars UAV formation.

A Contour Line Group Simplification Method Based on Classified Terrain Features

Contour line group simplification methods can effectively preserve terrain features during map making and producing. This process involves two main steps, namely terrain feature line extraction and contour bend selection. The terrain feature line extraction includes two steps, that is, terrain feature point extraction and classification, and terrain feature line connection. However, to date, many similar studies have not considered the hierarchy of terrain features. Therefore, we proposed a group simplification method for contour lines based on classified terrain features and tested it on a study area with mainly positive landforms. In accordance with geomorphological theory, we divided valleys into either gradually descending or ordinary. Valley points were extracted based on the constrained Delaunay triangulation method and then classified into the two categories. Gradually descending and ordinary valley lines were then connected. The contour bends were grouped based on the valley lines extracted and then selected according to the geometric indicators of the bend group. The results have demonstrated that the valley lines extracted closely matched human perception in integrity and structure. Contour lines simplified by our method achieved effective reduction of map information and adequate retention of the main terrain structures, which are similar to those from manual simplification.

A precise TIN clipping algorithm for the digital mining design of the opencast coal mine.

The triangulated irregular network (TIN) clipping algorithm is one of the vital algorithms for the digital mining design of opencast coal mines based on the geological digital elevation model (DEM) described by TIN. This paper gives a precise TIN clipping algorithm applied in the digital mining design of the opencast coal mine. To improve the algorithm's efficiency, a spatial grid index is built and utilized to embed the Clipping Polygon (CP) into the Clipped TIN (CTIN) by interpolating the CP's vertices' elevation and solving the intersections of the CP and the CTIN. After that, the topology of the triangles situated within (outside of) the CP is reconstructed, and then the boundary polygon of those triangles is obtained based on the reconstructed topology. Finally, a new boundary TIN between the CP and the boundary polygon of the triangles situated within (outside of) the CP is generated by applying the one-time edge-prior constrained Delaunay triangulation (CDT) growth algorithm, and the TIN to be clipped out is separated from the CTIN by topology modification. At that point, CTIN clipping is accomplished with the local details remaining. The algorithm has been programmed in C# and.NET. Additionally, it is also applied to the opencast coal mine digital mining design practice, and it is robust and highly efficient.

Extracting Skeleton Lines from Building Footprints by Integration of Vector and Raster Data

The extraction of skeleton lines of buildings is a key step in building spatial analysis, which is widely performed for building matching and updating. Several methods for vector data skeleton line extraction have been established, including the improved constrained Delaunay triangulation (CDT) and raster data skeleton line extraction methods, which are based on image processing technologies. However, none of the existing studies have attempted to combine these methods to extract the skeleton lines of buildings. This study aimed to develop a building skeleton line extraction method based on vector–raster data integration. The research object was buildings extracted from remote sensing images. First, vector–raster data mapping relationships were identified. Second, the buildings were triangulated using CDT. The extraction results of the Rosenfeld thin algorithm for raster data were then used to remove redundant triangles. Finally, the Shi–Tomasi corner detection algorithm was used to detect corners. The building skeleton lines were extracted by adjusting the connection method of the type three triangles in CDT. The experimental results demonstrate that the proposed method can effectively extract the skeleton lines of complex vector buildings. Moreover, the skeleton line extraction results included a few burrs and were robust against noise.

Struct-MDC: Mesh-Refined Unsupervised Depth Completion Leveraging Structural Regularities From Visual SLAM

Feature-based visual simultaneous localization and mapping (SLAM) methods only estimate the depth of extracted features, generating a sparse depth map. To solve this sparsity problem, depth completion tasks that estimate a dense depth from a sparse depth have gained significant importance in robotic applications like exploration. Existing methodologies that use sparse depth from visual SLAM mainly employ point features. However, point features have limitations in preserving structural regularities owing to texture-less environments and sparsity problems. To deal with these issues, we perform depth completion with visual SLAM using line features, which can better contain structural regularities than point features. The proposed methodology creates a convex hull region by performing constrained Delaunay triangulation with depth interpolation using line features. However, the generated depth includes low-frequency information and is discontinuous at the convex hull boundary. Therefore, we propose a mesh depth refinement (MDR) module to address this problem. The MDR module effectively transfers the high-frequency details of an input image to the interpolated depth and plays a vital role in bridging the conventional and deep learning-based approaches. The Struct-MDC outperforms other state-of-the-art algorithms on public and our custom datasets, and even outperforms supervised methodologies for some metrics. In addition, the effectiveness of the proposed MDR module is verified by a rigorous ablation study.

Resource allocation applied to flexible printed circuit routing based on constrained Delaunay triangulation

Flexible printed circuit (FPC) are widely used as connectors in compact package such as cellular telephones where required high performance. The increasing number of constrains makes manual routing an extremely complicated and time-consuming task. In this paper, we present a resource allocation framework which employs computational geometry considering signal and power integrity (SI/PI). To perform a reasonable resource allocation, we propose an escape-triangle-passage model (ETP model) based on constrained Delaunay triangulation (CDT). Experimental results on some industrial data show our strategy assembled routing channel for each group of nets has good robustness and is suitable for irregular polygonal routing region, significantly improving the routing quality and reducing iteration times.

Recognizing Building Group Patterns in Topographic Maps by Integrating Building Functional and Geometric Information

Recognizing building group patterns is fundamental to numerous fields, such as urban landscape evaluation, social analysis, and map generalization. Despite the increasing number of algorithms available for building group pattern recognition, there is still a lack of satisfactory grouping results due to insufficient information and only geometric features being provided to recognition methods. This study aims to provide a novel building grouping method that combines building function and geometric information. We specifically focus on the process of recognizing building groups in topographic maps as a prerequisite to subsequent map generalization. First, the building functions are inferred using the dynamic time warping (DTW) algorithm based on Tencent user density data and POIs (points of interest). Then, two types of constrained Delaunay triangulations (CDTs) are created for each building block, from which several spatial indices, such as the continuity index (SCI), direction, and distance of every two adjacent buildings, are derived. Finally, each building block is modeled as a graph on the grounds of derived matrices and building function information, and a graph segmentation approach is proposed to extract building groups. A case study is conducted to test the proposed approach. The experimental results indicate that the proposed approach can produce satisfactory results, given that the correctness value is above 81.63% for our study area. Comparative studies reveal that the method without building function information is an ineffective grouping method when buildings with different functions are close to each other. In addition, generalization results derived from the proposed method are more in line with those of maps for daily use, as they provide users with more accurate spatial divisions of urban buildings.

Research on Navigation Safety Evaluation of Coastal Waters Based on Dynamic Irregular Grid

Despite being a minor probability event, marine accidents can cause serious consequences such as casualties, environmental damage, and even massive economic losses. As an important part of the marine traffic safety system, the evaluation of navigation safety in coastal waters is of great significance to ensure the safety of ship navigation. In order to objectively evaluate the safety of navigation, this paper proposes an irregular grid division method that combines influencing factors including seabed topography, ship traffic flow, electronic charts, and marine meteorology. In this work, the navigable boundary was extracted by spatial analysis algorithms of the slope calculation. Based on the alpha-shape algorithm and the Voronoi diagram, the constrained Delaunay triangulation was used to extract the inner and outer boundaries of the ship′s navigation area, and the time-varying factors of marine hydrology and meteorology were integrated to form a dynamic irregular grid. The navigation safety evaluation indicators in coastal waters were divided into five risk levels, namely lower, low, medium, high, and higher. Then, the entropy weight theory was used to calculate the weight of the evaluation index. Finally, a safety evaluation model was constructed to evaluate the risk of navigation safety in coastal waters. Herein, taking the coastal waters of Lianyungang Port as a demonstration area, this paper divided the dynamic irregular grid and conducted the navigation safety analysis and evaluation based on the grid. The experimental results show that our method fully considers the influence of objective factors and the uncertainty of safety evaluation indicators and has favorable adaptability to the evaluation of navigation safety in coastal waters.

Robust and Fast CAD Model Tessellation for Inspection

The digital manufacturing deviation inspection can be used to assess the dimension, form, and position errors, which is more efficient, more consistent, and more robust than manual inspection. CAD models are used as the nominal data and compared with the scanned data of parts to obtain manufacturing errors. CAD model tessellation is a pivotal step in deviation inspection, which directly determines the reliability of analytical results. It requires that the generated triangle mesh has dense vertices, high approximation accuracy, and consistent orientation. A single surface patch tessellation method is proposed, and the adaptive quadtree is first constructed to divide the domain into small spatial grids in the parameter space of the patch; then, a clip-based method is presented to triangulate boundary grids, and inner grids could be triangulated by the constrained Delaunay triangulation method. Furthermore, the mesh patches should be spliced and reoriented consistently, which is very important for deviation analysis. To solve this problem, a robust gap healing method based on merging boundary points of patches is developed, so that all boundary vertices of adjacent patches could completely coincide along the common edges, and all mesh patches would be stitched into a complete mesh without gaps. Finally, the face normal of mesh is adjusted to obtain a consistent orientation. Experiments on industrial CAD models demonstrate that the proposed method could realize faster and more robust mesh generation than the state-of-the-art approaches, including three commercial and two open-source software.

Planning-oriented optimization strategy for visualizing power distribution networks: from global to local

ABSTRACT The spatial distribution of power facilities is uneven, thereby making the topology of geographical wiring diagrams (GWDs) based on the actual coordinates unclear. A single-line diagram has the advantage of a clear topology but it lacks spatial locations. A GWD has the advantage of accurate spatial locations but it lacks a clear topology. Visualizing distribution networks for planning requires both features. We proposed a new planning-oriented method for optimizing the visualization of distribution networks. From the global perspective, we proposed an improved force-directed (FD) algorithm by introducing a space restriction strategy and node–edge repulsion strategy to promote the expansion of the distance between distribution facilities within a limited buffer. We then constructed the constrained Delaunay triangulation to identify the compact districts (CDs) and used a genetic algorithm to optimize the parameters for the improved FD algorithm. A novel visualization evaluation indicator was also proposed for quantitatively assessing the visualizations. From a local perspective, the fisheye algorithm was used to optimize the CDs to further improve the visualization of the distribution network. We verified the proposed methods with real-world data. We used limited spatial displacement in exchange for maximum topology clarity to balance the accurate spatial location and topology clarity.

Chance-constrained path planning in narrow spaces for a dubins vehicle

The problem of optimal path planning through narrow spaces in an unstructured environment is considered. The optimal path planning problem for a Dubins agent is formulated as a chance-constrained optimal control problem (CCOCP), wherein the uncertainty in obstacle boundaries is modelled using standard probability distributions. The chance constraints are transformed to deterministic equivalents using the inverse cumulative distribution function and subsequently incorporated into a deterministic optimal control problem. Due to multiple convex sub-regions introduced by the obstacles, the initial guess provided to optimal control solver is crucial for computation time and optimality of the solution. A constrained Delaunay triangulation mesh based approach is developed that ensures the initial guess to lie in the optimal sub-convex region. Finally, off-the-shelf software is used to transcribe the optimal control problem to a nonlinear program (NLP) using Gaussian quadrature orthogonal collocation and solved to obtain an optimal path that conforms to system dynamics. By varying the upper bound on probability of obstacle collision, a family of solutions is generated, parameterized by the risk associated with each solution. This approach enables discovery of special “keyhole trajectories” that can provide significant cost savings in a tightly-spaced obstacle field. Merits of this approach are illustrated by comparing it with the traditional bounded uncertainty approach.

Recovery of Asteroids from Observations of Too-Short Arcs by Triangulating Their Admissible Regions

The data set collected during the night of the discovery of a minor body constitutes a too-short arc (TSA), resulting in failure of the differential correction procedure. This makes it necessary to recover the object during subsequent nights to gather more observations that will allow a preliminary orbit to be calculated. In this work, we present a recovery technique based on sampling the admissible region (AdRe) by the constrained Delaunay triangulation. We construct the AdRe in its topocentric and geocentric variants, using logarithmic and exponential metrics, for the following near-Earth-asteroids: (3122) Florence, (3200) Phaethon, 2003 GW, (1864) Daedalus, 2003 BH84 and 1977 QQ5; and the main-belt asteroids: (1738) Oosterhoff, (4690) Strasbourg, (555) Norma, 2006 SO375, 2003 GE55 and (32811) Apisaon. Using our sampling technique, we established the ephemeris region for these objects, using intervals of observation from 25 minutes up to 2 hours, with propagation times from 1 up to 47 days. All these objects were recoverable in a field of vision of 95’ × 72’, except for (3122) Florence and (3200) Phaethon, since they were observed during their closest approach to the Earth. In the case of 2006 SO375, we performed an additional test with only two observations separated by 2 minutes, achieving a recovery of up to 28 days after its discovery, which demonstrates the potential of our technique.