The Community for Technology Leaders
RSS Icon
Issue No.03 - May/June (2008 vol.14)
pp: 680-692
Streamline integration of fields produced by computational fluid mechanics simulations is a commonly used tool for the investigation and analysis of fluid flow phenomena. Integration is often accomplished through the application of ordinary differential equation (ODE) integrators -- integrators whose error characteristics are predicated on the smoothness of the field through which the streamline is being integrated -- smoothness which is not available at the inter-element level of finite volume and finite element data. Adaptive error control techniques are often used to ameliorate the challenge posed by inter-element discontinuities. As the root of the difficulties is the discontinuous nature of the data, we present a complementary approach of applying smoothness-enhancing accuracy-conserving filters to the data prior to streamline integration. We investigate whether such an approach applied to uniform quadrilateral discontinuous Galerkin (high-order finite volume) data can be used to augment current adaptive error control approaches. We discuss and demonstrate through numerical example the computational trade-offs exhibited when one applies such a strategy.
streamline integration, finite element
Sean Curtis, Robert M. Kirby, Jennifer K. Ryan, "Investigation of Smoothness-Increasing Accuracy-Conserving Filters for Improving Streamline Integration through Discontinuous Fields", IEEE Transactions on Visualization & Computer Graphics, vol.14, no. 3, pp. 680-692, May/June 2008, doi:10.1109/TVCG.2008.9
[1] D.H. Laidlaw, R.M. Kirby, C.D. Jackson, J.S. Davidson, T.S. Miller, M. da Silva, W.H. Warren, and M.J. Tarr, “Comparing 2D Vector Field Visualization Methods: A User Study,” IEEE Trans. Visualization and Computer Graphics, vol. 11, no. 1, pp. 59-70, Jan./Feb. 2005.
[2] D. Weiskopf and G. Erlebacher, “Overview of Flow Visualization,” The Visualization Handbook, C.D. Hansen and C.R. Johnson, eds., Elsevier, 2005.
[3] C.W. Gear, “Solving Ordinary Differential Equations with Discontinuities,” ACM Trans. Math. Software, vol. 10, no. 1, pp.23-44, 1984.
[4] J. Bramble and A. Schatz, “Higher Order Local Accuracy by Averaging in the Finite Element Method,” Math. Computation, vol. 31, pp. 94-111, 1977.
[5] J. Ryan, C.-W. Shu, and H. Atkins, “Extension of a Post-Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem,” SIAM J.Scientific Computing, vol. 26, pp. 821-843, 2005.
[6] S. Curtis, R.M. Kirby, J.K. Ryan, and C.-W. Shu, “Post-Processing for the Discontinuous Galerkin Method over Non-Uniform Meshes,” SIAM J. Scientific Computing, to be published.
[7] J. Ryan and C.-W. Shu, “On a One-Sided Post-Processing Technique for the Discontinuous Galerkin Methods,” Methods and Applications of Analysis, vol. 10, pp. 295-307, 2003.
[8] E. Tufte, The Visual Display of Quantitative Information. Graphics Press, 1983.
[9] J.V. Wijk, “Spot Noise Texture Synthesis for Data Visualization,” Computer Graphics, vol. 25, no. 4, pp. 309-318, 1991.
[10] B. Cabral and L. Leedom, “Imaging Vector Fields Using Line Integral Convolution,” Computer Graphics, vol. 27, pp. 263-272, 1993.
[11] G. Turk, “Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion Textures,” Computer Graphics, vol. 25, no. 4, pp. 289-298, 1991.
[12] A. Witkin and M. Kass, “Reaction-Diffusion Textures,” Computer Graphics, vol. 25, no. 4, pp. 299-308, 1991.
[13] D. Kenwright and G. Mallinson, “A 3-D Streamline Tracking Algorithm Using Dual Stream Functions,” Proc. IEEE Conf. Visualization (VIS '92), pp. 62-68, 1992.
[14] D. Watanabe, X. Mao, K. Ono, and A. Imamiya, “Gaze-Directed Streamline Seeding,” ACM Int'l Conf. Proc. Series, vol. 73, p. 170, 2004.
[15] G. Turk and D. Banks, “Image-Guided Streamline Placement,” Proc. ACM SIGGRAPH '96, pp. 453-460, 1996.
[16] B. Jobard and W. Lefer, “Creating Evenly-Spaced Streamlines of Arbitrary Density,” Proc. Eighth Eurographics Workshop Visualization in Scientific Computing, 1997.
[17] X. Ye, D. Kao, and A. Pang, “Strategy for Seeding 3D Streamlines,” Proc. IEEE Conf. Visualization (VIS), 2005.
[18] A. Mebarki, P. Alliez, and O. Devillers, “Farthest Point Seeding for Efficient Placement of Streamlines,” Proc. IEEE Conf. Visualization (VIS), 2005.
[19] D. Stalling and H.-C. Hege, “Fast and Resolution Independent Line Integral Convolution,” Proc. ACM SIGGRAPH '95, pp.249-256, 1995.
[20] C. Teitzel, R. Grosso, and T. Ertl, “Efficient and Reliable Integration Methods for Particle Tracing in Unsteady Flows on Discrete Meshes,” Proc. Eighth Eurographics Workshop Visualization in Scientific Computing, pp. 49-56, , 1997.
[21] Z. Liu and R.J. Moorhead, “Accelerated Unsteady Flow Line Integral Convolution,” IEEE Trans. Visualization and Computer Graphics, vol. 11, no. 2, pp. 113-125, Mar./Apr. 2005.
[22] J. Parker, R. Kenyon, and D. Troxel, “Comparison of Interpolating Methods for Image Resampling,” IEEE Trans. Medical Imaging, vol. 2, no. 1, pp. 31-39, 1983.
[23] H. Hou and H. Andrews, “Cubic Splines for Image Interpolation and Digital Filtering,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 26, pp. 508-517, 1978.
[24] A. Entezari, R. Dyer, and T. Möller, “Linear and Cubic Box Splines for the Body Centered Cubic Lattice,” Proc. IEEE Conf. Visualization (VIS '04), pp. 11-18, 2004.
[25] D. Mitchell and A. Netravali, “Reconstruction Filters in Computer-Graphics,” Proc. ACM SIGGRAPH '88, pp. 221-228, 1988.
[26] G. Nürnberger, L. Slatexchumaker, and F. Zeilfelder, Local Lagrange Interpolation by Bivariate C1 Cubic Splines. Vanderbilt Univ., 2001.
[27] P. Sablonniere, “Positive Spline Operators and Orthogonal Splines,” J. Approximation Theory, vol. 52, no. 1, pp. 28-42, 1988.
[28] 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.
[29] I. Ihm, D. Cha, and B. Kang, “Controllable Local Monotonic Cubic Interpolation in Fluid Animations: Natural Phenomena and Special Effects,” Computer Animation and Virtual Worlds, vol. 16, no. 3-4, pp. 365-375, 2005.
[30] T. Möller, R. Machiraju, K. Mueller, and R. Yagel, “Evaluation and Design of Filters Using a Taylor Series Expansion,” IEEE Trans. Visualization and Computer Graphics, vol. 3, no. 2, pp. 184-199, June 1997.
[31] T. Möller, K. Mueller, Y. Kurzion, R. Machiraju, and R. Yagel, “Design of Accurate and Smooth Filters for Function and Derivative Reconstruction,” Proc. IEEE Symp. Volume Visualization (VVS '98), pp. 143-151, 1998.
[32] D. Stalling, “Fast Texture-Based Algorithms for Vector Field Visualization,” PhD dissertation, Konrad-Zuse-Zentrum für Informationstechnik, 1998.
[33] D. Stalling and H.-C. Hege, “Fast and Resolution Independent Line Integral Convolution,” Proc. ACM SIGGRAPH '95, pp. 249-256, 1995.
[34] V. Thomée, “High Order Local Approximations to Derivatives inthe Finite Element Method,” Math. Computation, vol. 31, pp.652-660, 1977.
[35] I. Shoenberg, “Contributions to the Problem of Approximation ofEquidistant Data by Analytic Functions, Parts A, B,” Quarterly Applied Math., vol. 4, pp. 45-99, 112-141, 1946.
[36] C.-W.S.B. Cockburn, M. Luskin, and E. Suli, “Enhanced Accuracy by Post-Processing for Finite Element Methods for Hyperbolic Equations,” Math. Computation, vol. 72, pp. 577-606, 2003.
[37] M. Mock and P. Lax, “The Computation of Discontinuous Solutions of Linear Hyperbolic Equations,” Comm. Pure and Applied Math., vol. 18, pp. 423-430, 1978.
[38] D.G.W. Cai and C.-W. Shu, “On One-Sided Filters for Spectral Fourier Approximations of Discontinuous Functions,” SIAM J. Numerical Analysis, vol. 29, pp. 905-916, 1992.
[39] P.G. Ciarlet, “The Finite Element Method for Elliptic Problems,” SIAM Classics in Applied Math., Soc. of Industrial and Applied Math., 2002.
[40] P.G. Ciarlet, “Interpolation Error Estimates for the Reduced Hsieh-Clough-Tocher Triangle,” Math. Computation, vol. 32, no. 142, pp. 335-344, 1978.
[41] J.H. Bramble and M. Zlámal, “Triangular Elements in the FiniteElement Method,” Math. Computation, vol. 24, no. 112, pp. 809-820, 1970.
[42] P. Alfeld and L. Schumaker, “Smooth Finite Elements Based on Clough-Tocher Triangle Splits,” Numerishe Mathematik, vol. 90, pp.597-616, 2002.
[43] M. jun Lai and L. Schumaker, “Macro-Elements and Stable Local Bases for Splines on Clough-Tocher Triangulations,” Numerishe Mathematik, vol. 88, pp. 105-119, 2001.
[44] O. Davydov and L.L. Schumaker, “On Stable Local Bases for Bivariate Polynomial Spline Spaces,” Constructive Approximations, vol. 18, pp. 87-116, 2004.
[45] T.J.R. Hughes, The Finite Element Method. Prentice Hall, 1987.
[46] G.E. Karniadakis and S.J. Sherwin, Spectral/HP Element Methods for CFD. Oxford Univ. Press, 1999.
[47] E. Hairer, S.P. Norrsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, second revised ed. Springer, 1993.
[48] R. Burden and J. Faires, Numerical Analysis. PWS, 1993.
[49] C. Canuto and A. Quarteroni, “Approximation Results for Orthogonal Polynomials in Sobolev Spaces,” Math. Computation, vol. 38, no. 157, pp. 67-86, 1982.
[50] E. Murman and K. Powell, “Trajectory Integration in Vertical Flows,” AIAA J., vol. 27, no. 7, pp. 982-984, 1988.
[51] G. Scheuermann, X. Tricoche, and H. Hagen, “C1-Interpolation forVector Field Topology Visualization,” Proc. IEEE Conf. Visualization (VIS '99), pp. 271-278, 1999.
[52] R.L. Burden and J.D. Faires, Numerical Analysis, fifth ed. PWS, 1993.
[53] I. Babuska and J.T. Oden, “Verification and Validation in Computational Engineering and Science: Basic Concepts,” Computer Methods in Applied Mechanics and Eng., vol. 193, no. 36-38, pp. 4057-4066, 2004.
15 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool