169,058 research outputs found
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes
The Modified Direct Method (MDM) is an iterative mesh smoothing method for
smoothing planar and surface meshes, which is developed from the non-iterative
smoothing method originated by Balendran [1]. When smooth planar meshes, the
performance of the MDM is effectively identical to that of Laplacian smoothing,
for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian
smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular,
quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh
elements always increases and the mean square error (MSE) decreases during
smoothing; For tri-dominant mixed mesh, the quality of triangles always
descends while that of quads ascends. Test examples show that the MDM is
convergent for both planar and surface triangular, quadrilateral and tri-quad
meshes.Comment: 18 page
Connectivity Compression for Irregular Quadrilateral Meshes
Applications that require Internet access to remote 3D datasets are often
limited by the storage costs of 3D models. Several compression methods are
available to address these limits for objects represented by triangle meshes.
Many CAD and VRML models, however, are represented as quadrilateral meshes or
mixed triangle/quadrilateral meshes, and these models may also require
compression. We present an algorithm for encoding the connectivity of such
quadrilateral meshes, and we demonstrate that by preserving and exploiting the
original quad structure, our approach achieves encodings 30 - 80% smaller than
an approach based on randomly splitting quads into triangles. We present both a
code with a proven worst-case cost of 3 bits per vertex (or 2.75 bits per
vertex for meshes without valence-two vertices) and entropy-coding results for
typical meshes ranging from 0.3 to 0.9 bits per vertex, depending on the
regularity of the mesh. Our method may be implemented by a rule for a
particular splitting of quads into triangles and by using the compression and
decompression algorithms introduced in [Rossignac99] and
[Rossignac&Szymczak99]. We also present extensions to the algorithm to compress
meshes with holes and handles and meshes containing triangles and other
polygons as well as quads
Efficient Representation of Computational Meshes
We present a simple yet general and efficient approach to representation of
computational meshes. Meshes are represented as sets of mesh entities of
different topological dimensions and their incidence relations. We discuss a
straightforward and efficient storage scheme for such mesh representations and
efficient algorithms for computation of arbitrary incidence relations from a
given initial and minimal set of incidence relations. The general
representation may harbor a wide range of computational meshes, and may also be
specialized to provide simple user interfaces for particular meshes, including
simplicial meshes in one, two and three space dimensions where the mesh
entities correspond to vertices, edges, faces and cells. It is elaborated on
how the proposed concepts and data structures may be used for assembly of
variational forms in parallel over distributed finite element meshes.
Benchmarks are presented to demonstrate efficiency in terms of CPU time and
memory usage
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