Showing papers on "Constrained Delaunay triangulation published in 2023"

Delaunay Triangulation Voxelization for Print Orientation-Based Topology Optimization in Additive Manufacturing

Abstract: For topology optimization in additive manufacturing, the support structures have a significant impact on production cost and can be minimized by choosing the optimal print orientation, where model voxelization plays an important role on the computing accuracy and efficiency. To overcome the inherent insufficiency of the cube voxelization, we propose a new voxelization modeling method based on the Delaunay Triangulation to match arbitrary shapes with fewer voxels at arbitrary angle, and speed up the computing efficiency by the gradient descent method for the optimal voxelization. Our method can be widely used in the future for accurate modeling and result optimization in the case of solving the optimal orientation.

Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complex

Abstract: In this paper we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). Assigning the Vietoris-Rips weights to simplices, DR experiences speed-up in the persistence calculations by only considering simplices that appear in the Delaunay triangulation of the point cloud. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR for an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.

Stability and machine learning applications of persistent homology using the Delaunay-Rips complex

Abstract: Persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and machine learning modeling, PH has attracted the interest of researchers across scientific disciplines and in many industry applications. However, computational costs may present challenges to effectively using PH in certain data contexts, and theoretical stability results may not hold in practice. In this paper, we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). By only considering simplices that appear in the Delaunay triangulation of the point cloud and assigning the Vietoris-Rips weights to simplices, DR avoids potentially costly computations in the persistence calculations. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR in an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.

Shortest Paths in Portalgons

Abstract: Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric. We analyze the complexity of shortest paths in portalgons. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons. The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons.

OFC-DelaunayTriangulation: A New Efficient Algorithm for Merging Two Adjacent Delaunay Triangulations

Abstract: In this paper, we present an efficient algorithm, namely Orienting and Final Circles (OFC)-Delaunay Triangulation, for constructing the Delaunay triangulation of a finite planar point set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung im Groen mittels Orientierungskurven, Optimization 18(1) (1987) 65–81 and Ein konstruktives Lösungsverfahren für das Problem des Inpolygons kleinsten Umfangs von J. Steiner, Optimization 18(3) (1987) 349–359). The concepts of orienting and final circles are introduced. The Delaunay edges are determined by orienting circles and a final circle. Our algorithm has [Formula: see text]([Formula: see text]) worst-case running time. The algorithm is implemented in Python and some numerical examples are presented.

Development of a Tetrahedral Mesh Generator Based on Constrained Delaunay Triangulation

Abstract: 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.

CircNet: Meshing 3D Point Clouds with Circumcenter Detection

Abstract: Reconstructing 3D point clouds into triangle meshes is a key problem in computational geometry and surface reconstruction. Point cloud triangulation solves this problem by providing edge information to the input points. Since no vertex interpolation is involved, it is beneficial to preserve sharp details on the surface. Taking advantage of learning-based techniques in triangulation, existing methods enumerate the complete combinations of candidate triangles, which is both complex and inefficient. In this paper, we leverage the duality between a triangle and its circumcenter, and introduce a deep neural network that detects the circumcenters to achieve point cloud triangulation. Specifically, we introduce multiple anchor priors to divide the neighborhood space of each point. The neural network then learns to predict the presences and locations of circumcenters under the guidance of those anchors. We extract the triangles dual to the detected circumcenters to form a primitive mesh, from which an edge-manifold mesh is produced via simple post-processing. Unlike existing learning-based triangulation methods, the proposed method bypasses an exhaustive enumeration of triangle combinations and local surface parameterization. We validate the efficiency, generalization, and robustness of our method on prominent datasets of both watertight and open surfaces. The code and trained models are provided at https://github.com/Ruitao-L/CircNet.