## Identifying Vector Field Singularities Using a Discrete Hodge Decomposition (2002)

Citations: | 52 - 4 self |

### BibTeX

@INPROCEEDINGS{Polthier02identifyingvector,

author = {Konrad Polthier and Eike Preuß},

title = {Identifying Vector Field Singularities Using a Discrete Hodge Decomposition},

booktitle = {},

year = {2002},

pages = {112--134},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper we use a slightly more general definition of the spaces S h respectively S # h , namely we include functions which are only defined at vertices respectively at edge midpoints. For example, the (total) Gau curvature is defined solely at vertices. Here for a given vector field # we will have div h # S h (respectively div # h # S # h ) to be defined solely at a vertex. The motivation of this generalization is two-fold: first, a simplified notation of many statements, and, second, the fact that for visualization purposes one often extends these point-based values over the surface. For example, barycentric interpolation allows to color the interior of triangles based on the discrete Gauss curvature at its vertices. Caution should be taken if integral entities are derived

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Citation Context ...we have rot ∗ h∇u =0which implies rot h∇u =0,weobtain rot hv(p) =roth(ξ − δ(wω)) − rot h∇u =0.sIdentify Singularities using a Hodge Decomposition 15 Theorem 5 was stated without proof a=-=s Theorem 2 in [17]-=- where it was not made clear enough that the remaining component v is harmonic with respect to the vertex based operators div h and rot h. 6 Decomposition Algorithm and Detecting Vector Field Singular... |

20 | Computational aspects of discrete minimal surfaces
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Citation Context ...o emphasize the geometric interpretation of the discrete differentials, and to relate them with the discrete Hodge operator which also played a role in the discrete minimal surface theory in Polthier =-=[15]-=-. In Sect. 7 we apply our method to several test cases with artificial and simulated flows which are accurately analyzed. The simulated flow in the Bay of Gdansk reproduces similar results of Post and... |

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Citation Context ...of this approach depends on the quality of the underlying grid and the accuracy of the vector field. For practical problems of vortex identification we refer to the case study of Kenwright and Haimes =-=[12]-=-, and the eigenvector method in Sujudi and Haimes [22]. Another class of methods follows a geometric approach where geometric properties of streamlines and pathlines are investigated and put in relati... |

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Citation Context ...ujudi and Haimes [22]. Another class of methods follows a geometric approach where geometric properties of streamlines and pathlines are investigated and put in relation to properties of the flow [19]=-=[20]-=-. Tittgemeyer et al. [23] use a contraction mapping to detect singularities of displacement fields in magnetic resonance imaging. This helps in the understanding of pathological processes in a brain. ... |

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Citation Context ...er-order methods try to overcome this problem [18]. The detection and visualization of higher-order singularities is an active research area where rather heavy mathematical methods have been employed =-=[21]. The Jac-=-obian ∇ξ of a differentiable vector field ξ in R 2 or R 3 can be decomposed into a stretching tensor S and a vorticity matrix Ω, the symmetric and anti-symmetric parts of ∇ξ. The eigenvalues ... |

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Citation Context ...other class of methods follows a geometric approach where geometric properties of streamlines and pathlines are investigated and put in relation to properties of the flow [19][20]. Tittgemeyer et al. =-=[23]-=- use a contraction mapping to detect singularities of displacement fields in magnetic resonance imaging. This helps in the understanding of pathological processes in a brain. Their method is applicabl... |

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Citation Context ...gnetic fields. All these singularities must be detected and analyzed in order to understand the physical behavior of a flow or in order to use them as an ingredient for many topology-based algorithms =-=[24]-=-[26]. Although feature analysis is an important area, only a few technical tools are available for the detection of singularities and their visualization. Methods for direct vortex detection are often... |

2 |
JavaView homepage
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Citation Context ...ly separates the source from the two vortices. The centers of the potentials may be varied at interactive speed since on a smaller grid the decomposition is done in real-time as shown at the web site =-=[14]-=-. In Fig. 4 the decomposition is applied to an incompressible flow around a cylinder from a CFD simulation. The rotation-free component of the incompressible flow vanishes as expected. The harmonic co... |