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Issue No.03 - March (2013 vol.19)
pp: 527-528
H. Bhatia , Sci. Comput. & Imaging Inst., Univ. of Utah, Salt Lake City, UT, USA
G. Norgard , Numerica, Fort Collins, CO, USA
V. Pascucci , Sci. Comput. & Imaging Inst., Univ. of Utah, Salt Lake City, UT, USA
P. Bremer , Center of Appl. Sci. Comput., Lawrence Livermore Nat. Lab., Livermore, CA, USA
The Helmholtz-Hodge decomposition (HHD) is one of the fundamental theorems of fluids describing the decomposition of a flow field into its divergence-free, curl-free, and harmonic components. Solving for the HHD is intimately connected to the choice of boundary conditions which determine the uniqueness and orthogonality of the decomposition. This article points out that one of the boundary conditions used in a recent paper “Meshless Helmholtz-Hodge Decomposition” [5] is, in general, invalid and provides an analytical example demonstrating the problem. We hope that this clarification on the theory will foster further research in this area and prevent undue problems in applying and extending the original approach.
Boundary conditions, Harmonic analysis, Computational modeling,Helmholtz-Hodge decomposition., Vector fields, boundary conditions
H. Bhatia, G. Norgard, V. Pascucci, P. Bremer, "Comments on the "Meshless Helmholtz-Hodge Decomposition"", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 3, pp. 527-528, March 2013, doi:10.1109/TVCG.2012.62
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[2] F.M. Denaro, "On the Application of the Helmholtz-Hodge Decomposition in Projection Methods for Incompressible Flows with General Boundary Conditions," Int'l J. Numerical Methods in Fluids, vol. 43, pp. 43-69, 2003.
[3] D.J. Griffiths, Introduction to Electrodynamics, second ed. Addison Wesley, 1999.
[4] H. Lamb, Hydrodynamics, sixth ed. Cambridge Univ. Press, 1932.
[5] F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, "Meshless Helmholtz-Hodge Decomposition," IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 2, pp. 338-349, Mar./Apr. 2010.
[6] K. Polthier and E. Preuß, "Variational Approach to Vector Field Decomposition," Proc. Eurographics Workshop Scientific Visualization, 2000.
[7] K. Polthier and E. Preuß, "Identifying Vector Fields Singularities Using a Discrete Hodge Decomposition," Proc. Visualization and Math. III, pp. 112-134, 2003.
[8] Y. Tong, S. Lombeyda, A. Hirani, and M. Desbrun, "Discrete Multiscale Vector Field Decomposition," ACM Trans. Graphics, vol. 22, no. 3, pp. 445-452, 2003.
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