2,413 research outputs found
On the maximum angle condition for the conforming longest-edge n-section algorithm for large values of n
In this note we introduce the conforming longest-edge -section algorithm and show that for it produces a family of triangulations which does not satisfy the maximum angle condition
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
RTIN-based strategies for local mesh refinement
summary:Longest-edge bisection algorithms are often used for local mesh refinements within the finite element method in 2D. In this paper, we discuss and describe their conforming variant. A particular attention is devoted to the so-called Right-Triangulated Irregular Network (RTIN) based on isosceles right triangles and its tranformation to more general domains. We suggest to combine RTIN with a balanced quadrant tree (QuadTree) decomposition. This combination does not produce hanging nodes within the mesh refinements and could be extended to tetrahedral meshes in 3D
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
Average Interpolation Under the Maximum Angle Condition
Interpolation error estimates needed in common finite element applications
using simplicial meshes typically impose restrictions on the both the
smoothness of the interpolated functions and the shape of the simplices. While
the simplest theory can be generalized to admit less smooth functions (e.g.,
functions in H^1(\Omega) rather than H^2(\Omega)) and more general shapes
(e.g., the maximum angle condition rather than the minimum angle condition),
existing theory does not allow these extensions to be performed simultaneously.
By localizing over a well-shaped auxiliary spatial partition, error estimates
are established under minimal function smoothness and mesh regularity. This
construction is especially important in two cases: L^p(\Omega) estimates for
data in W^{1,p}(\Omega) hold for meshes without any restrictions on simplex
shape, and W^{1,p}(\Omega) estimates for data in W^{2,p}(\Omega) hold under a
generalization of the maximum angle condition which previously required p>2 for
standard Lagrange interpolation
Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees
pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach
Survey of semi-regular multiresolution models for interactive terrain rendering
Rendering high quality digital terrains at interactive rates requires carefully crafted algorithms and data structures able to balance the competing requirements of realism and frame rates, while taking into account the memory and speed limitations of the underlying graphics platform. In this survey, we analyze multiresolution approaches that exploit a certain semi-regularity of the data. These approaches have produced some of the most efficient systems to date. After providing a short background and motivation for the methods, we focus on illustrating models based on tiled blocks and nested regular grids, quadtrees and triangle bin-trees triangulations, as well as cluster-based approaches. We then discuss LOD error metrics and system-level data management aspects of interactive terrain visualization, including dynamic scene management, out-of-core data organization and compression, as well as numerical accurac
Level set based eXtended finite element modelling of the response of fibrous networks under hygroscopic swelling
Materials like paper, consisting of a network of natural fibres, exposed to
variations in moisture, undergo changes in geometrical and mechanical
properties. This behaviour is particularly important for understanding the
hygro-mechanical response of sheets of paper in applications like digital
printing. A two-dimensional microstructural model of a fibrous network is
therefore developed to upscale the hygro-expansion of individual fibres,
through their interaction, to the resulting overall expansion of the network.
The fibres are modelled with rectangular shapes and are assumed to be perfectly
bonded where they overlap. For realistic networks the number of bonds is large
and the network is geometrically so complex that discretizing it by
conventional, geometry-conforming, finite elements is cumbersome. The
combination of a level-set and XFEM formalism enables the use of regular,
structured grids in order to model the complex microstructural geometry. In
this approach, the fibres are described implicitly by a level-set function. In
order to represent the fibre boundaries in the fibrous network, an XFEM
discretization is used together with a Heaviside enrichment function. Numerical
results demonstrate that the proposed approach successfully captures the
hygro-expansive properties of the network with fewer degrees of freedom
compared to classical FEM, preserving desired accuracy.Comment: 27 pages, 22 figures, 4 tables, J. Appl. Mech. June 19, 202
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