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Issue No.12 - Dec. (2012 vol.18)

pp: 2325-2334

Blake Nelson , University of Utah

Eric Liu , MIT

Robert M. Kirby , University of Utah

Robert Haimes , MIT

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2012.218

ABSTRACT

This paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with highorder finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through runtime plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA&#8217;s OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis&#8217; pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers&#8217; debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of uncertainty for engineers using ElVis.

INDEX TERMS

Finite element methods, Isosurfaces, Polynomials, Geometry, Rendering (computer graphics), Data models, isosurfaces, High-order finite elements, spectral/hp elements, discontinuous Galerkin, fluid flow simulation, cut surface extraction, contours

CITATION

Blake Nelson, Eric Liu, Robert M. Kirby, Robert Haimes, "ElVis: A System for the Accurate and Interactive Visualization of High-Order Finite Element Solutions",

*IEEE Transactions on Visualization & Computer Graphics*, vol.18, no. 12, pp. 2325-2334, Dec. 2012, doi:10.1109/TVCG.2012.218REFERENCES

- [1] Cuda. developer.nvidia.com/category/zone/cudazone, 2012.
- [2] Nektar++. http:/www.nektar.info. 2012.
- [3] J. Akin, W. Gray, and Q. Zhang, Colouring isoparametric contours
Engineering Computations, 1: 36-41, 1984.- [4] M. Brasher and R. Haimes., Rendering planar cuts through quadratic and cubic finite elements. In
Proceedings of the conference on Visualization ‘04, VIS ‘04, pages 409-416, Washington, DC, USA, 2004. IEEE Computer Society. - [5] G. Coppola,S. J. Sherwin,, and J. Peiró., Nonlinear particle tracking for high-order elements.
J. Comput. Phys., 172: 356-386, September 2001.- [6] L. T. Diosady and D. L. Darmofal., Massively parallel solution techniques for higher-order finite-element discretizations in CFD. In
Adaptive High-Order Methods in Computational Fluid Dynamics. World Scientific, 2011.- [7] K. J. Fidkowski and D. L. Darmofal., A triangular cut-cell adaptive method for higher-order discretizations of the compressible Navier-Stokes equations
J. Comput. Phys., 225: 1653-1672, 2007.- [8] B. Haasdonk, M. Ohlberger, M. Rumpf., A. Schmidt, and K. G. Siebert., Multiresolution visualization of higher order adaptive finite element simulations
Computing, 70: 181-204, July 2003.- [9] R. Haimes and D. Darmofal., Visualization in computational fluid dynamics: A case study. In
IEEE Computer Society, Visualization, pages 392-397. IEEE Computer Society Press, 1991.- [10] R. Haimes and M. Giles., Visual3: Interactive unsteady unstructured 3d visualization. AIAA 91–0794, 1991.
- [11] G. Karniadakis, E. Bullister, and A. Patera., A spectral element method for solution of two- and three-dimensional time dependent Navier-Stokes equations. In
Finite Element Methods for Nonlinear Problems, Springer-Verlag, page 803, 1985.- [12] G. E. Karniadakis and S. J., Sherwin
Spectral/hp element methods for CFD. Oxford University Press, New-York, NY, USA, 1999.- [13] G. D. Kontopidis and D. E. Limbert., A predictor-corrector contouring algorithm for isoparametric 3d elements
International Journal for Numerical Methods in Engineering, 19(7): 995-1004, 1983.- [14] T. Leicht and R. Hartmann, Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations
J. Comput. Phys., 229: 7344-7360, 2010.- [15] J. L. Meek and G. Beer., Contour plotting of data using isoparametric element representation
International Journal for Numerical Methods in Engineering, 10(4): 954-957, 1976.- [16] M. Meyer, B. Nelson, R. Kirby,, and R. Whitaker., Particle systems for efficient and accurate high-order finite element visualization.
IEEE Transactions on Visualization and Computer Graphics, 13: 1015-1026, 2007.- [17] T. Michal and J. Krakos., Anisotropic mesh adaptation through edge primitive operations. AIAA 2012-159, 2012.
- [18] B. Nelson.,
Accurate and Interactive Visualization of High-Order Finite Element Fields. PhD thesis, University of Utah, 2012.- [19] B. Nelson and R. M. Kirby., Ray-tracing polymorphic multidomain spectral/hp elements for isosurface rendering
IEEE Transactions on Visualization and Computer Graphics, 12: 114-125, January 2006.- [20] B. Nelson,R. M. Kirby,, and R. Haimes., GPU-based interactive cut-surface extraction from high-order finite element fields.
IEEE Transactions on Visualization and Computer Graphics, 17(12): 1803-1811, Dec. 2011.- [21] T. Oliver and D. Darmofal., Impact of turbulence model irregularity on high-order discretizations. AIAA 2009–953, 2009.
- [22] C. Pagot, D. Osmari, F. Sadlo., D. Weiskopf, T. Ertl,, and J. Comba., Efficient parallel vectors feature extraction from higher-order data.
Computer Graphics Forum, 30(3): 751-760, 2011.- [23] C. Pagot, J. Vollrath, F. Sadlo., D. Weiskopf, T. Ertl,, and J. ao Luiz Dihl Comba., Interactive Isocontouring of High-Order Surfaces. In H. Hagen, editor,
Scientific Visualization: Interactions, Features, Metaphors, 2 of Dagstuhl Follow-Ups, pages 276-291. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2011.- [24] S. G. Parker, J. Bigler, A. Dietrich., H. Friedrich, J. Hoberock., D. Luebke, D. McAllister., M. McGuire, K. Morley., A. Robison, and M. Stich., Optix: A general purpose ray tracing engine
ACM Transactions on Graphics, August 2010.- [25] A. Patera, A spectral method for fluid dynamics: Laminar flow in a chan-nel expansion
J. Compo Phys., 54: 468, 1984.- [26] V. Schmitt and F. Charpin., Pressure distributions on the onera-m6-wing at transonic mach numbers. AGARD AR 138, 1979.
- [27] W. Schroeder, F. Bertel, M. Malaterre., D. Thompson, P. Pebay,R. O’Barall,, and S. Tendulkar., Framework for visualizing higher-order basis functions. In
Visualization, 2005. VIS 05. IEEE, pages 43-50, 2005.- [28] C. Singh and D. Sarkar, A simple and fast algorithm for the plotting of contours using quadrilateral meshes
Finite Elem. Anal. Des., 7: 217-228, December 1990.- [29] C. Singh and J. Singh, Accurate contour plotting using 6-node triangular elements in 2d
Finite Elem. Anal. Des., 45: 81-93, January 2009.- [30] M. Uffinger, S. Frey, and T. Ertl, Interactive high-quality visualization of higher-order finite elements
Computer Graphics Forum, 29(2): 337-346, 2010.- [31] D. Walfisch,J. K. Ryan,R. M. Kirby,, and R. Haimes., One-sided smoothness-increasing accuracy-conserving filtering for enhanced streamline integration through discontinuous fields.
J. Sci. Comput., 38(2): 164-184, Feb. 2009.- [32] D. F. Wiley, H. Childs, B. Hamann,, and K. Joy., Ray casting curved-quadratic elements. In O. Deussen,C. D. Hansen, D. Keirn, and D. Saupe, editors,
Data Visualization 2004, pages 201–209. Eurographics/IEEE TCVG, ACM Siggraph, 2004.- [33] D. F. Wiley,H. R. Childs,B. F. Gregorski, B. Hamann, and K. I. Joy., Contouring curved quadratic elements. In
Proceedings of the symposium on Data visualisation 2003, VISSYM ‘03, pages 167-176, Aire-Ia-Ville, Switzerland, Switzerland, 2003. Eurographics Association. - [34] P. L. Williams,N. L. Max,, and C. M. Stein., A high accuracy volume renderer for unstructured data
IEEE Transactions on Visualization and Computer Graphics, 4: 37-54, January 1998.- [35] M. Yano and D. Darmofal., An optimization framework for anisotropic simplex mesh adaptation: application to aerodynamic flows. AIAA 2012-0079, Jan. 2012.
- [36] M. Yano,J. M. Modisette,, and D. Darmofal., The importance of mesh adaptation for higher-order discretizations of aerodynamic flows. AIAA 2011-3852, June 2011.
- [37] Y. Zhou and M. Garland, Interactive point-based rendering of higher-order tetrahedral data
IEEE Transactions on Visualization and Computer Graphics, 12(5): 2006, 2006. |