51,503 research outputs found
Visualization of 2-D and 3-D Tensor Fields
In previous work we have developed a novel approach to visualizing second order symmetric 2-D tensor fields based on degenerate point analysis. At degenerate points the eigenvalues are either zero or equal to each other, and the hyper-streamlines about these points give rise to tri-sector or wedge points. These singularities and their connecting hyper-streamlines determine the topology of the tensor field. In this study we are developing new methods for analyzing and displaying 3-D tensor fields. This problem is considerably more difficult than the 2-D one, as the richness of the data set is much larger. Here we report on our progress and a novel method to find , analyze and display 3-D degenerate points. First we discuss the theory, then an application involving a 3-D tensor field, the Boussinesq problem with two forces
Visualization of 2-D and 3-D Tensor Fields
In previous work we have developed a novel approach to visualizing second order symmetric 2-D tensor fields based on degenerate point analysis. At degenerate points the eigenvalues are either zero or equal to each other, and the hyperstreamlines about these points give rise to trisector or wedge points. These singularities and their connecting hyperstreamlines determine the topology of the tensor field. In this study we are developing new methods for analyzing and displaying 3-D tensor fields. This problem is considerably more difficult than the 2-D one, as the richness of the data set is much larger. Here we report on our progress and a novel method to find, analyze and display 3-D degenerate points. First we discuss the theory, then an application involving a 3-D tensor field, the Boussinesq problem with two forces
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Atomic Bifurcations of Degenerate Curves in 3D Linear Symmetric Tensor Fields
Tensor field topology is of importance to research areas of medicine, science, and engineering. Degenerate curves are one of the crucial topological features that provide valuable insights for tensor field visualization. In this thesis, we study the atomic bifurcations of degenerate curves in 3D linear second-order symmetric tensor fields, and present the findings of sub-scenarios generated by changing the number of degenerate curves, by reconnecting degenerate curves, and by varying the number of transition points on different degenerate curves in such fields
A spectral solver for evolution problems with spatial S3-topology
We introduce a single patch collocation method in order to compute solutions
of initial value problems of partial differential equations whose spatial
domains are 3-spheres. Besides the main ideas, we discuss issues related to our
implementation and analyze numerical test applications. Our main interest lies
in cosmological solutions of Einstein's field equations. Motivated by this, we
also elaborate on problems of our approach for general tensorial evolution
equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy
symmetry here.Comment: 29 pages, 11 figures, uses psfrag and hyperref, large parts rewritten
in order to match to the requirements of the journal, conclusions unchanged;
J. Comput. Phys. (2009
Genericity of nondegenerate geodesics with general boundary conditions
Let M be a possibly noncompact manifold. We prove, generically in the
C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on
M do not possess any degenerate geodesics satisfying suitable boundary
conditions. This extends a result of Biliotti, Javaloyes and Piccione for
geodesics with fixed endpoints to the case where endpoints lie on a compact
submanifold P of the product MxM that satisfies an admissibility condition.
Such condition holds, for example, when P is transversal to the diagonal of
MxM. Further aspects of these boundary conditions are discussed and general
conditions under which metrics without degenerate geodesics are C^k-generic are
given.Comment: LaTeX2e, 21 pages, no figure
Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
We introduce a new method, the Local Monge Parametrizations (LMP) method, to
approximate tensor fields on general surfaces given by a collection of local
parametrizations, e.g.~as in finite element or NURBS surface representations.
Our goal is to use this method to solve numerically tensor-valued partial
differential equations (PDE) on surfaces. Previous methods use scalar
potentials to numerically describe vector fields on surfaces, at the expense of
requiring higher-order derivatives of the approximated fields and limited to
simply connected surfaces, or represent tangential tensor fields as tensor
fields in 3D subjected to constraints, thus increasing the essential number of
degrees of freedom. In contrast, the LMP method uses an optimal number of
degrees of freedom to represent a tensor, is general with regards to the
topology of the surface, and does not increase the order of the PDEs governing
the tensor fields. The main idea is to construct maps between the element
parametrizations and a local Monge parametrization around each node. We test
the LMP method by approximating in a least-squares sense different vector and
tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply
the LMP method to two physical models on surfaces, involving a tension-driven
flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP
method thus solves the long-standing problem of the interpolation of tensors on
general surfaces with an optimal number of degrees of freedom.Comment: 16 pages, 6 figure
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