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Image-Based Triangular and Tetrahedral Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
Given n points, its d-dimensional Delaunay triangulation has the following properties. The Delaunay triangulation contains at most O(nd/2) simplices, and the union of all simplices forms the convex hull of the given points. Suppose there are m vertices on the convex hull in the 2D plane, then any triangulation of these vertices has at most 2n – 2 – m triangles with one exterior face. According to the Delaunay criterion, Delaunay triangulation maximizes the minimum angle, and any circumcircle does not contain any other input points in its interior (they can be on the perimeter). If a circle passing through two input points is empty, then the line segment of these two points is an edge of a Delaunay triangulation. The Delaunay triangulation in d-dimensional spaces is the projection of the convex hull of point projections onto a (d + 1)-dimensional paraboloid. For a point p inside the convex hull of a Delaunay triangulation, the closest vertex to p does not need to be the corners of the triangle containing p.
Experimental analysis of typical hollow clay fictile tubules of cultural heritage of Mediterranean area
Published in Koen Van Balen, Els Verstrynge, Structural Analysis of Historical Constructions: Anamnesis, Diagnosis, Therapy, Controls, 2016
R.S. Olivito, C. Scuro, R. Codispoti
To perform a finite element analysis, it was created a model using a pseudo three-dimensional approach axisymmetric. The modeling of the samples, NS and ES, was performed under the assumption of polar symmetry respect to the transverse axis. The finite element models were created using the "COMSOL multiphysics" program, generated with a triangular mesh distributed in a averagely dense way over the entire body, and more thickened near the corners rounded (Figure 19). To ensure accurate results, the triangles and tetrahedral must be "well shaped", having small aspect ratios or bounds on their smallest and largest angles. Mesh generation algorithms based on Delaunay refinement, are effective both in theory and in practice. Delaunay refinement algorithms operate by maintaining a constrained Delaunay triangulation, which is refined by inserting carefully placed vertices until the mesh meets constraints on triangle quality and size. The criterion used is that of Willam-Warnke (William et al., 1974) which is based on to predict failure in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. Just as the BreslerPister criterion, it depends only on three parameters, fc that is the uniaxial compressive strength, ft that is uniaxial tensile strength and fb that is obtained from the biaxial compressive test. All parameters are positive. The octahedral normal and shear stresses are defined
Domain-Modeling Techniques
Published in Alexandru Telea, Data Visualization, 2014
The best-known triangulation method is the Delaunay algorithm [de Berg et al. 00]. This method generates triangular cells ci for a set of 2D points pi ∈ ℝ2 and tetrahedra for a set of 3D points pi ∈ ℝ3. A Delaunay triangulation of a point set consists of a set of triangles that covers the convex hull of the point set. An important property of a Delaunay triangulation is that no point from the input point set {pi} lies in the circumscribed circle of any triangle in the triangulation. Triangulations that obey this property are called conforming Delaunay triangulations. Given a set of scattered points with data values recorded at the point locations, using the Delaunay triangulation is the most “natural” way to create a 𝒞1, piecewise linear, interpolation of the data values over the convex hull of the points. To do this, we define piecewise linear basis functions over the triangles contained in the unstructured grid generated by the Delaunay triangulation, and use these functions to interpolate the vertex data values, as explained in Section 3.3. Figure 8.3(a) shows a Delaunay triangulation of a random point cloud containing 600 points. The point density is higher in the center, which causes the creation of smaller triangles in that area. Another example of Delauney triangulation is shown in Figure 3.12 (middle).
‘Turbulent’ shear flow of solids under high-pressure torsion
Published in Philosophical Magazine, 2023
Yan Beygelzimer, Alexander Filippov, Yuri Estrin
In addition to the velocity distributions, it is also instructive to use the Voronoi and Delaunay diagrams. A Voronoi diagram for a set of points (particles) on a plane is a tessellation of the plane into regions where each point is closer to all points within that region than to any point of the set outside of it. A Delaunay diagram is a triangulation of the same set of points such that for any triangle all points, except those at its apexes, are situated outside of its circumcircle. The Voronoi diagram permits identification of the symmetry of the neighbourhood of any point, while Delaunay triangulation, which is dual to the Voronoi mesh, enables visual construction of a lattice at any step of a numerical experiment. Figure 1 displays a typical instantaneous snapshot of a system represented by the Voronoi and Delaunay constructions. One can readily recognize a subdivision of the system into domains and chains of defects with fivefold and sevenfold symmetry axes located at domain boundaries (coloured blue and orange, respectively). A blow-up in the insert presents an enlarged view of the Delaunay lattice where some of the defects are highlighted by pentagons for clarity.
Improving template-based CT data evaluation by integrating CMM reference data into a CAD model-based high fidelity triangle mesh
Published in Nondestructive Testing and Evaluation, 2022
Andreas Michael Müller, Tino Hausotte
A second example demonstrating the parametrisation for the STEP entity ‘cylindrical surface’ is shown in Figure 5. The upper image shows the vertices defining the sub-mesh edges in red and the CMM measurement data in blue for the large cylindrical surface of the specimen. The result of the parametrisation into the interpolation space is shown below, with edge points having by design. It is important to note that both and are given in mm although the angular parameter is given in radian (3). In order to ensure interpolation in an isotropic space, the parameter is therefore converted to arc length to have the same unit as . The Delaunay triangulation is created, such that no other vertex is inside the circumcircle of a triangle. If space itself is deformed by choosing different units, those circles become ellipses and the interpolation is biased. The space enclosed by the parameters and is now exactly the area of the cylinder surface.
An efficient workflow for meshing large scale discrete fracture networks for solving subsurface flow problems
Published in Petroleum Science and Technology, 2022
The core module of the DFN meshing tool is geometry creation module, which provides all required classes and API for geometry operations. It is the base module on which all other modules are dependent. It also provides its functionality to service layer module DFN I/O. DFN Meshing tool contains all the algorithmic implementation for meshing a surface. It implements Delaunay triangulation, and different triangle refinement using different methods of edge refinement, triangle refinement, and area refinement. DFN mesher provides API to create domain boundary and add constraint line, use these inputs to generate constraint mesh. DFN data structure includes data structure classes to store DFN entities such as Fracture, Grid, Grid cell, and Wells. DFN algorithm module contains implementation of the algorithms required specific to DFN. DFN I/O module is the connector between core modules and GUI. It provides DFN Core Application class as starting point of application. It provides all required API to run application.