The Community for Technology Leaders
RSS Icon
Issue No.08 - Aug. (2013 vol.19)
pp: 1386-1404
H. Bhatia , Lawrence Livermore Nat. Lab., Univ. of Utah, Livermore, CA, USA
G. Norgard , Numerica, Fort Collins, CO, USA
V. Pascucci , Sci. Comput. & Imaging Inst. (SCI), Univ. of Utah, Salt Lake City, UT, USA
Peer-Timo Bremer , Lawrence Livermore Nat. Lab., Univ. of Utah, Livermore, CA, USA
The Helmholtz-Hodge Decomposition (HHD) describes the decomposition of a flow field into its divergence-free and curl-free components. Many researchers in various communities like weather modeling, oceanology, geophysics, and computer graphics are interested in understanding the properties of flow representing physical phenomena such as incompressibility and vorticity. The HHD has proven to be an important tool in the analysis of fluids, making it one of the fundamental theorems in fluid dynamics. The recent advances in the area of flow analysis have led to the application of the HHD in a number of research communities such as flow visualization, topological analysis, imaging, and robotics. However, because the initial body of work, primarily in the physics communities, research on the topic has become fragmented with different communities working largely in isolation often repeating and sometimes contradicting each others results. Additionally, different nomenclature has evolved which further obscures the fundamental connections between fields making the transfer of knowledge difficult. This survey attempts to address these problems by collecting a comprehensive list of relevant references and examining them using a common terminology. A particular focus is the discussion of boundary conditions when computing the HHD. The goal is to promote further research in the field by creating a common repository of techniques to compute the HHD as well as a large collection of example applications in a broad range of areas.
Vectors, Communities, Boundary conditions, Visualization, Physics, Conferences, Helmholtz-Hodge decomposition, Vector fields, incompressibility, boundary conditions
H. Bhatia, G. Norgard, V. Pascucci, Peer-Timo Bremer, "The Helmholtz-Hodge Decomposition—A Survey", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 8, pp. 1386-1404, Aug. 2013, doi:10.1109/TVCG.2012.316
[1] R. Abraham, J.E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis and Applications, vol. 75, Applied Math. Science, second ed. Springer-Verlag, 1988.
[2] E. Ahusborde, M. Azaiez, and J.-P. Caltagirone, "A Primal Formulation for the Helmholtz Decomposition," J. Computational Physics, vol. 225, no. 1, pp. 13-19, 2007.
[3] K. Aki and P.G. Richards, Quantitative Seismology. Univ. Science Books, 2002.
[4] M. Akram and V. Michel, "Regularisation of the Helmholtz Decomposition and its Application to Geomagnetic Field Modelling," Int'l J. Geomath., vol. 1, no. 1, pp. 101-120, 2010.
[5] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, "Vector Potentials in Three-Dimensional Nonsmooth Domains," Math. Methods in the Applied Sciences, vol. 21, pp. 823-864, 1998.
[6] H. Aref, N. Rott, and H. Thomann, "Gröbli's Solution of the Three-Vortex Problem," Ann. Rev. Fluid Mechanics, vol. 24, pp. 1-21, 1992.
[7] G. Arfken, Mathematical Methods for Physicists, second ed. Academic Press, 1970.
[8] G. Auchmuty, "Potential Representation of Incompressible Vector Fields," Nonlinear Problems in Applied Mathematics, pp. 43-49, SIAM, 1995.
[9] J.B. Bell, P. Colella, and H.M. Glaz, "A Second Order Projection Method for the Incompressible Navier-Stokes Equations," J. Computational Physics, vol. 85, pp. 257-283, 1989.
[10] J.B. Bell and D.L. Marcus, "A Second-Order Projection Method for Variable-Density Flows," J. Computational Physics, vol. 101, pp. 334-348, 1992.
[11] H. Bhatia, G. Norgard, V. Pascucci, and P.-T. Bremer, "Comments on the 'Meshless Helmholtz-Hodge Decomposition'," IEEE Trans. Visualization and Computer Graphics, vol. 19, no. 3, pp. 527-528, Mar. 2013.
[12] O. Blumenthal, "Über Die Zerlegung Unendlicher Vektorfelder," Mathematische Annalen, vol. 61, no. 2, 235-250, 1905.
[13] H. Braun and A. Hauck, "Tomographic Reconstruction of Vector Fields," IEEE Trans. Signal Processing, vol. 39, no. 2, pp. 464-471, Feb. 1991.
[14] R. Bridson, Fluid Simulation For Computer Graphics. A.K. Peters, 2008.
[15] D.L. Brown, R. Cortez, and M.L. Minion, "Accurate Projection Methods for the Incompressible Navier-Stokes Equations," J. Computational Physics, vol. 168, pp. 464-499, 2001.
[16] E.B. Bykhovskiy and N.V. Smirnov, On Orthogonal Expansions of the Space of Vector Functions Which Are Square-Summable over a Given Domain and the Vector Analysis Operators. Academy of Sciences USSR Press, 1960.
[17] J. Caltagirone and J. Breil, "A Vector Projection Method for Solving the Navier-Stokes Equations," Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy, vol. 327, no. 11, pp. 1179-1184, 1999.
[18] J. Cantarella, D. DeTurck, and H. Gluck, "Vector Calculus and the Topology of Domains in 3-Space," Am. Math. Monthly, vol. 109, no. 5, pp. 409-442, 2002.
[19] A.J. Chorin, "A Numerical Method for Solving Incompressible Viscous Flow Problems," J. Computational Physics, vol. 2, pp. 12-26, 1967.
[20] A.J. Chorin, "Numerical Solution of the Navier-Stokes Equations," Math. Computations, vol. 22, pp. 745-762, 1968.
[21] A.J. Chorin, "On the Convergence of Discrete Approximations to the Navier-Stokes equations," Math. Computations, vol. 23, pp. 341-353, 1969.
[22] A.J. Chorin and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer, 1993.
[23] F. Colin, R. Egli, and F. Lin, "Computing a Null Divergence Velocity Field Using Smoothed Particle Hydrodynamics," J. Computational Physics, vol. 217, no. 2, pp. 680-692, 2006.
[24] R. Courant and D. Hilbert, Methods of Mathematical Physics. New York Univ., 1953.
[25] R. Courant, E. Isaacson, and M. Rees, "On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences," Comm. Pure and Applied Math., vol. 5, pp. 243-255, 1952.
[26] S.J. Cummins and M. Rudman, "An SPH Projection Method," J. Computational Physics, vol. 152, pp. 584-607, 1999.
[27] 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.
[28] E. Deriaz and V. Perrier, "Divergence-Free and Curl-Free Wavelets in 2D and 3D, Application to Turbuent Flows," J. Turbulence, vol. 7, no. 3, pp. 1-37, 2006.
[29] E. Deriaz and V. Perrier, "Direct Numerical Simulation of Turbulence Using Divergence-Free Wavelets," Multiscale Modeling and Simulation, vol. 7, no. 3, pp. 1101-1129, 2008.
[30] E. Deriaz and V. Perrier, "Orthogonal Helmholtz Decomposition in Arbitrary Dimension using Divergence-Free and Curl-Free Wavelets," Applied and Computational Harmonic Analysis, vol. 26, no. 2, pp. 249-269, 2009.
[31] W. Weinan and J.G. Liu, "Projection Method I: Convergence and Numerical Boundary Layers," SIAM J. Numerical Analysis, vol. 32, no. 4, pp. 1017-1057, 1995.
[32] R. Fedkiw, J. Stam, and H.W. Jensen, "Visual Simulation of Smoke," Proc. 28th Ann. Conf. Computer Graphics and Interactive Techniques (SIGGRAPH '01), pp. 15-22, 2001.
[33] M. Fisher, P. Schröder, M. Desburn, and H. Hoppe, "Design of Tangent Vector Fields," ACM Trans. Graphics, vol. 26, no. 3,article 56, 2007.
[34] N. Foster and D. Metaxas, "Realistic Animation of Liquids," Graphical Models and Image Processing, vol. 58, no. 5, pp. 471-483, 1996.
[35] N. Foster and D. Metaxas, "Modeling the Motion of a Hot, Turbulent Gas," Proc. 24th Ann. Conf. Computer Graphics and Interactive Techniques (SIGGRAPH '97), pp. 181-188, 1997.
[36] D. Fujiwara and H. Morimoto, "An $L_r$ Theorem of the Helmholtz Decomposition of Vector Fields," J. Faculty of Science, vol. 44, pp. 685-700, 1977.
[37] E.J. Fuselier, "Refined Error Estimates for Matrix-Valued Radial Basis Functions," PhD thesis, Texas A & M Univ., 2006.
[38] H. Gao, M. Mandal, G. Guo, and J. Wan, "Singular Point Detection using Discrete Hodge Helmholtz Decomposition in Fingerprint Images," Proc. IEEE Int'l Conf. Acoustics Speech and Signal Processing (ICASSP), pp. 1094-1097, 2010.
[39] J. Geng and Z. Shen, "The Neumann Problem and Helmholtz Decomposition in Convex Domains," J. Functional Analysis, vol. 259, pp. 2147-2164, 2010.
[40] D. Georgobiani, N. Mansour, A. Kosovichev, R. Stein, and Å. Nordlund, "Velocity Field Decomposition in 3D Numerical Simulations of Solar Turbulent Convection," NASA Center for Turbulence Research - Ann. Research Briefs, pp. 355-340, 2004.
[41] D.J. Griffiths, Introduction to Electrodynamics, second ed. Addison Wesley, 1999.
[42] Q. Guo, "Cardiac Video Analysis using the Hodge Helmholtz Field Decomposition," master's thesis, Dept. of Electrical and Computer Eng., Univ. of Alberta, 2004.
[43] Q. Guo, M.K. Mandal, and M.Y. Li, "Efficient Hodge-Helmholtz Decomposition of Motion Fields," Pattern Recognition Letters, vol. 26, no. 4, pp. 493-501, 2005.
[44] Q. Guo, M.K. Mandal, G. Liu, and K.M. Kavanagh, "Cardiac Video Analysis Using Hodge-Helmholtz Field Decomposition," Computers in Biology and Medicine, vol. 36, no. 1, pp. 1-20, 2006.
[45] R. Guy and A. Fogelson, "Stability of Approximate Projection Methods on Cell-Centered Grids," J. Computational Physics, vol. 203, pp. 517-538, 2005.
[46] F.H. Harlow and J.E. Welch, "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface," Physics of Fluids, vol. 8, no. 12, pp. 2182-2189, 1965.
[47] R. Hatton and H. Choset, "Optimizing Coordinate Choice for Locomoting Systems," Proc. IEEE Int'l conf. Robotics and Automation (ICRA), pp. 4493-4498, 2010.
[48] W. Hauser, "On the Fundamental Equations of Electromagnetism," Am. J. Physics, vol. 38, no. 1, pp. 80-85, 1970.
[49] W. Hauser, Introduction to the Principles of Electromagnetism. Addison-Wesley Educational Publishers Inc, 1971.
[50] H. Helmholtz, "Über Integrale der Hydrodynamischen Gleichungen, Welche den Wirbelbewegungen Entsprechen," J. für die reine und angewandte Mathematik, vol. 1858, no. 55, pp. 25-55, Jan. 1858.
[51] H. Helmholtz, "On Integrals of the Hydrodynamical Equations, which Express Vortex-Motion," Philosophical Magazine and J. Science, vol. 33, no. 226, pp. 485-512, 1867.
[52] J. Hinkle, P.T. Fletcher, B. Wang, B. Salter, and S. Joshi, "4D MAP Image Reconstruction Incorporating Organ Motion," Proc. 21st Int'l Conf. Information Processing in Medical Imaging, (IPMI), 2009.
[53] W. Hodge, The Theory and Applications of Harmonic Integrals. Cambridge Univ. Press, 1952.
[54] M.S. Ingber and S.N. Kempka, "A Galerkin Implementation of the Generalized Helmholtz Decomposition for Vorticity Formulations," J. Computational Physics, vol. 169, no. 1, pp. 215-237, 2001.
[55] T. Jie and Y. Xubo, "Physically-Based Fluid Animation: A Survey," Science in China Series F: Information Sciences, vol. 52, no. 1, pp. 1-17, 2007.
[56] S.A. Johnson, J.F. Greenleaf, M. Tanaka, and G. Flandro, Acoustical Holography, vol. 7. Plenum Press, 1977.
[57] D.D. Joseph, "Helmholtz Decomposition Coupling Rotational to Irrotational Flow of a Viscous Fluid," Proc. Nat'l Academy of Sciences of USA, vol. 103, no. 39, pp. 14272-14277, 2006.
[58] J. Kim and P. Moin, "Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations," J. Computational Physics, vol. 59, no. 2, pp. 308-323, 1985.
[59] D.H. Kobe, "Helmholtz Theorem for Antisymmetric Second-Rank Tensor Fields and Electromagnetism with Magnetic Monopoles," Am. J. Physics, vol. 52, no. 4, pp. 354-358, 1984.
[60] D.H. Kobe, "Helmholtz's Theorem Revisited," Am. J. Physics, vol. 54, no. 6, pp. 552-554, 1986.
[61] K. Kodaira, "Harmonic Fields in Riemannian Manifolds," Annals of Math., vol. 50, pp. 587-665, 1949.
[62] O.A. Ladyzhenskaja, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 1963.
[63] H. Lamb, Hydrodynamics, sixth ed. Cambridge Univ. Press, 1932.
[64] F. Losasso, F. Gibou, and R. Fedkiw, "Simulating Water and Smoke with an Octree Data Structure," ACM Trans. Graphics, vol. 23, pp. 457-462, 2004.
[65] L.B. Lucy, "A Numerical Approach to the Testing of the Fission Hypothesis," The Astronomical J., vol. 82, pp. 1013-1024, 1977.
[66] I. Mac$\hat{\rm e}$ do and R. Castro, "Learning Divergence-Free and Curl-Free Vector Fields with Matrix-Valued Kernels," technical report, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, 2010.
[67] N.N. Mansour, A. Kosovichev, D. Georgobiani, A. Wray, and M. Miesch, "Turbulence Convection and Oscillations in the Sun," Proc. SOHO14/GONG Workshop, "Helio- and Astero-Seismology: Towards a Golden Future," 2004.
[68] S.F. McCormick, "The Finite Volume Method," Multilevel Adaptive Methods for Partial Differential Equations, chapter 3, vol. 6. SIAM, 1989.
[69] C.A. Micchelli and M. Pontil, "On Learning Vector-Valued Functions," Neural Computation, vol. 17, pp. 177-204, 2005.
[70] B.P. Miller, "Interpretations from Helmholtz Theorem in Classical Electromagnetism," Am. J. Physics, vol. 52, p. 948, 1984.
[71] C. Min and F. Gibou, "A Second Order Accurate Projection Method for the Incompressible Navier-Stokes Equations on Non-Graded Adaptive Grids," J. Computational Physics, vol. 219, pp. 912-929, 2006.
[72] C. Min, F. Gibou, and H.D. Ceniceros, "A Supra-Convergent Finite Difference Scheme for the Variable Coefficient Poisson Equation on Non-Graded Grids," J. Computational Physics, vol. 218, no. 1, pp. 123-140, 2006.
[73] Y. Mochizuki and A. Imiya, "Spatial Reasoning for Robot Navigation Using the Helmholtz-Hodge Decomposition of Omnidirectional Optical Flow," Proc. 24th Int'l Conf. Image and Vision Computing (IVCNZ), pp. 1-6, 2009.
[74] J.J. Monaghan, "Smoothed Particle Hydrodynamics," Ann. Rev. of Astronomy and Astrophysics, vol. 30, pp. 543-574, 1992.
[75] L. Morino, "Helmholtz Decomposition Revisited: Vorticity Generation and Trailing Edge Condition," Computational Mechanics, vol. 1, pp. 65-90, 1986.
[76] F.J. Narcowich and J.D. Ward, "Generalized Hermite Interpolation Via Matrix-Valued Conditionally Positive Definite Functions," Math. of Computations, vol. 63, no. 208, pp. 661-687, 1994.
[77] D.Q. Nguyen, R. Fedkiw, and H.W. Jensen, "Physically based Modeling and Animation of Fire," ACM Trans. Graphics, vol. 21, pp. 721-728, 2002.
[78] S.J. Norton, "Tomographic Reconstruction of 2-D Vector Fields: Applications to Flow Imaging," Geophysics J. Int'l, vol. 97, no. 1, pp. 161-168, 1988.
[79] S.J. Norton, "Unique Tomographic Reconstruction of Vector Fields Using Boundary Data," IEEE Trans. Image Processing, vol. 1, no. 3, pp. 406-412, July 1992.
[80] B. Palit, A. Basu, and M. Mandal, "Applications of the Discrete Hodge-Helmholtz Decomposition to Image and Video Processing," Proc. First Int'l Conf. Pattern Recognition and Machine Intelligence, S. Pal, S. Bandyopadhyay, and S. Biswas, eds., pp. 497-502, 2005.
[81] 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. 338-342, Mar./Apr. 2010.
[82] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation. Morgan Kaufmann Publishers, Inc., 2004.
[83] K. Polthier and E. Preuß, "Variational Approach to Vector Field Decomposition," Proc. Eurographics Workshop Scientific Visualization, 2000.
[84] K. Polthier and E. Preuß, "Identifying Vector Fields Singularities Using a Discrete Hodge Decomposition," Mathematical Visualization III, H.C. Hege, K. Polthier, eds., pp. 112-134, Springer, 2003.
[85] S. Popinet, "A Tree-Based Adaptive Solver for the Incompressible Euler Equations in Complex Geometries," J. Computational Physics, vol. 190, pp. 572-600, 2003.
[86] J.L. Prince, "Convolution Backprojection Formulas for 3-D Vector Tomography with Application to MRI," IEEE Trans. Image Processing, vol. 5, no. 10, pp. 1462-1472, Oct. 1996.
[87] R. Scharstein, "Helmholtz Decomposition of Surface Electric Current in Electromagnetic Scattering Problems," Proc. 23rd Southeastern Symp. System Theory, pp. 424-426, 1991.
[88] N. Schleifer, "Differential Forms as a Basis for Vector Analysis - with Applications to Electrodynamics," Am. J. Physics, vol. 51, no. 45, p. 1983, 1983.
[89] G. Schwarz, Hodge Decomposition - A Method for Solving Boundary Value Problems. Springer, 1995.
[90] J.M. Solomon and W.G. Szymczak, "Finite Difference Solutions for the Incompressible Navier-Stokes Equations Using Galerkin Techniques," Proc. Fifth IMACS Int'l Symp. Computer Methods for Partial Differential Equations, June 1984.
[91] W. Sprössig, "On Helmholtz Decompositions and their Generalizations - An Overview," Math. Methods in the Applied Sciences, vol. 33, pp. 374-383, 2010.
[92] J. Stam, "Stable Fluids," Proc. ACM SIGGRAPH '99, pp. 121-128, 1999.
[93] J. Stam, "A Simple Fluid Solver Based on the FFT," J. Graphics Tools, vol. 6, pp. 43-52, 2002.
[94] J. Stam and E. Fiume, "Depicting Fire and Other Gaseous Phenomena Using Diffusion Processes," Proc. ACM SIGGRAPH '95, pp. 129-136, 1995.
[95] R. Stein and Å. Nordlund, "Realistic Solar Convection Simulations," Solar Physics, vol. 192, pp. 91-108, 2000.
[96] A.B. Stephens, J.B. Bell, J.M. Solomon, and L.B. Hackerman, "A Finite Difference Galerkin Formulation for the Incompressible Navier-Stokes Equations," J. Computational Physics, vol. 53, pp. 152-172, 1984.
[97] 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.
[98] J. van Kan, "A Second-Order Accurate Pressure-Correction Scheme for Viscous Incompressible Flow," SIAM J. Scientific and Statistical Computing, vol. 7, no. 3, pp. 870-891, 1986.
[99] K. Wang Weiwei, Y. Tong, M. Desburn, and P. Schröder, "Edge Subdivision Schemes and the Construction of Smooth Vector Fields," ACM Trans. Graphics, vol. 25, no. 3, pp. 1041-1048, 2006.
[100] H. Weyl, "The Method of Orthogonal Projection in Potential Theory," Duke Math. J., vol. 7, no. 7, pp. 411-444, 1940.
[101] A. Wiebel, "Feature Detection in Vector Fields Using the Helmholtz-Hodge Decomposition," Diploma thesis, Diplomarbeit, Univ. Kaiserslautern, 2004.
[102] A. Wiebel, C. Garth, and G. Scheuermann, "Computation of Localized Flow for Steady and Unsteady Vector Fields and its Applications," IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 4, pp. 641-651, July/Aug. 2007.
45 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool