319 research outputs found
VoroCrust: Voronoi Meshing Without Clipping
Polyhedral meshes are increasingly becoming an attractive option with
particular advantages over traditional meshes for certain applications. What
has been missing is a robust polyhedral meshing algorithm that can handle broad
classes of domains exhibiting arbitrarily curved boundaries and sharp features.
In addition, the power of primal-dual mesh pairs, exemplified by
Voronoi-Delaunay meshes, has been recognized as an important ingredient in
numerous formulations. The VoroCrust algorithm is the first provably-correct
algorithm for conforming polyhedral Voronoi meshing for non-convex and
non-manifold domains with guarantees on the quality of both surface and volume
elements. A robust refinement process estimates a suitable sizing field that
enables the careful placement of Voronoi seeds across the surface circumventing
the need for clipping and avoiding its many drawbacks. The algorithm has the
flexibility of filling the interior by either structured or random samples,
while preserving all sharp features in the output mesh. We demonstrate the
capabilities of the algorithm on a variety of models and compare against
state-of-the-art polyhedral meshing methods based on clipped Voronoi cells
establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed
images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf.
Supplemental materials available on
https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd
TetGen, towards a quality tetrahedral mesh generator
TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation
TetGen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (Version 1.5 â Userâs Manual)
TetGen is a software for tetrahedral mesh generation. Its goal is to generate good quality tetrahedral meshes suitable for numerical methods and scientific computing. It can be used as either a standalone program or a library component integrated in other software. The purpose of this document is to give a brief explanation of the kind of tetrahedralizations and meshing problems handled by TetGen and to give a fairly detailed documentation about the usage of the program. Readers will learn how to create tetrahedral meshes using input files from the command line. Furthermore, the programming interface for calling TetGen from other programs is explained
Higher-quality tetrahedral mesh generation for domains with small angles by constrained Delaunay refinement
Algorithms for generating Delaunay tetrahedral meshes
have difficulty with domains whose boundary polygons meet at small angles.
The requirement that all tetrahedra be Delaunay often forces
mesh generators to overrefine near small domain angles---that is,
to produce too many tetrahedra, making them too small.
We describe a provably good algorithm that generates meshes that
are constrained Delaunay triangulations, rather than purely Delaunay.
Given a piecewise linear domain free of small angles,
our algorithm is guaranteed to construct a mesh in which every tetrahedron has
a radius-edge ratio of or better.
This is a substantial improvement over the usual bound of ;
it is obtained by relaxing the conditions in which
boundary triangles are subdivided.
Given a domain with small angles,
our algorithm produces a mesh in which the quality guarantee is
compromised only in specific places near small domain angles.
We prove that most mesh edges have lengths proportional to
the domain's minimum local feature size;
the exceptions span small domain angles.
Our algorithm tends to generate meshes with fewer tetrahedra than
purely Delaunay methods because
it uses the constrained Delaunay property, rather than vertex insertions,
to enforce the conformity of the mesh to the domain boundaries.
An implementation demonstrates that our algorithm does not overrefine near
small domain angles
Adaptive Mesh Refinement for Electromagnetic Simulation
We consider problems related to initial meshing and adaptive mesh refinement
for the electromagnetic simulation of various structures. The quality of the
initial mesh and the performance of the adaptive refinement are of great
importance for the finite element solution of the Maxwell equations, since they
directly affect the accuracy and the computational time. In this paper, we
describe the complete meshing workflow, which allows the simulation of
arbitrary structures. Test simulations confirm that the presented approach
allows to reach the quality of the industrial simulation software
Decorated discrete conformal equivalence in non-Euclidean geometries
We introduce decorated piecewise hyperbolic and spherical surfaces and
discuss their discrete conformal equivalence. A decoration is a choice of
circle about each vertex of the surface. Our decorated surfaces are closely
related to inversive distance circle packings, canonical tessellations of
hyperbolic surfaces, and hyperbolic polyhedra.
We prove the corresponding uniformization theorem. Furthermore, we show that
on can deform continuously between decorated piecewise hyperbolic, Euclidean,
and spherical surfaces sharing the same fundamental discrete conformal
invariant. Therefore, there is one master theory of discrete conformal
equivalence in different background geometries. Our approach is based on a
variational principle, which also provides a way to compute the discrete
uniformization and geometric transitions.Comment: 41 pages, 10 figures. arXiv admin note: text overlap with
arXiv:2305.1098
Doctor of Philosophy
dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains
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