Non-polynomial Galerkin projection on deforming meshes

Abstract

This paper extends Galerkin projection to a large class of non-polynomial functions typically encountered in graphics. We demonstrate the broad applicability of our approach by applying it to two strikingly different problems: fluid simulation and radiosity rendering, both using deforming meshes. Standard Galerkin projection cannot efficiently approximate these phenomena. Our approach, by contrast, enables the compact representation and approximation of these complex non-polynomial systems, including quotients and roots of polynomials. We rely on representing each function to be model-reduced as a composition of tensor products, matrix inversions, and matrix roots. Once a function has been represented in this form, it can be easily model-reduced, and its reduced form can be evaluated with time and memory costs dependent only on the dimension of the reduced space.

References

1. An, S. S., Kim, T., and James, D. L. 2008. Optimizing cubature for efficient integration of subspace deformations. ACM Transactions on Graphics 27, 5 (Dec.), 165:1--165:10.
2. Ausseur, J., Pinier, J., Glauser, M., and Higuchi, H. 2004. Predicting the dynamics of the flow over a NACA 4412 using POD. APS Meeting Abstracts, D8.
3. Barbič, J., and James, D. 2005. Real-time subspace integration for St. Venant-Kirchhoff deformable models. In Proc. SIGGRAPH '05.
4. Barbič, J., and Popović, J. 2008. Real-time control of physically based simulations using gentle forces. ACM Transactions on Graphics 27, 5.
5. Benzi, M., Golub, G. H., and Liesen, J. 2005. Numerical solution of saddle point problems. Acta Numerica 14, 1--137.
6. Carlberg, K., Bou-Mosleh, C., and Farhat, C. 2011. Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. International Journal for Numerical Methods in Engineering 86, 2, 155--181.
7. Cgal, Computational Geometry Algorithms Library. http://www.cgal.org.
8. Chadwick, J., An, S. S., and James, D. L. 2009. Harmonic shells: A practical nonlinear sound model for near-rigid thin shells. ACM Transactions on Graphics 28, 5 (Dec.), 119:1--119:10.
9. Chahlaoui, Y., and van Dooren, P. 2005. Model reduction of time-varying systems. In Dimension Reduction of Large-Scale Systems. Springer, 131--148.
10. de Witt, T., Lessig, C., and Fiume, E. 2012. Fluid simulation using Laplacian eigenfunctions. ACM Transactions on Graphics 31, 1 (Jan.).
11. Debusschere, B. J., Najm, H. N., Pebay, P. P., Knio, O. M., Ghanem, R. G., and Maitre, O. P. L. 2004. Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comp. 26, 2, 698--719.
12. Deconinck, H., and Ricchiuto, M. 2007. Residual distribution schemes: foundation and analysis. In Encyclopedia of Computational Mechanics, E. Stein, E. de Borst, and T. Hughes, Eds., vol. 3. John Wiley and Sons, Ltd.
13. Dobes, J., and Deconinck, H. 2006. An ALE formulation of the multidimensional residual distribution scheme for computations on moving meshes. In Proc. Int. Conf. CFD.
14. Dorsey, J., Sillion, F., and Greenberg, D. 1991. Design and simulation of opera lighting and projection effects. In Computer Graphics (Proceedings of SIGGRAPH 91), 41--50.
15. Drettakis, G., and Sillion, F. 1997. Interactive update of global illumination using a line-space hierarchy. In Proceedings of SIGGRAPH 97, Computer Graphics Proceedings, Annual Conference Series, 57--64.
16. Ebert, F., and Stykel, T. 2007. Rational interpolation, minimal realization and model reduction. Tech. rep., DFG Research Center Matheon.
17. Elcott, S., Tong, Y., Kanso, E., Schröder, P., and Desbrun, M. 2007. Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics 26, 1 (Jan.).
18. Farhood, M., and Dullerud, G. E. 2007. Model reduction of nonstationary LPV systems. IEEE Transactions on Automatic Control 52, 2, 181--196.
19. Feldman, B. E., O'Brien, J. F., and Klingner, B. M. 2005. Animating gases with hybrid meshes. In Proc. SIGGRAPH '05.
20. Feldman, B. E., O'Brien, J. F., Klingner, B. M., and Goktekin, T. G. 2005. Fluids in deforming meshes. In 2005 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 255--260.
21. Fogleman, M., Lumley, J., Rempfer, D., and Haworth, D. 2004. Application of the proper orthogonal decomposition to datasets of internal combustion engine flows. Journal of Turbulence 5, 23 (June).
22. Foster, N., and Metaxas, D. 1996. Realistic animation of liquids. Graphical Models and Image Processing 58, 5.
23. Gallivan, K., Grimme, E., and Dooren, P. V. 1996. A rational Lanczos algorithm for model reduction. Numerical Algorithms 12, 33--63.
24. Goral, C. M., Torrance, K. E., Greenberg, D. P., and Battaile, B. 1984. Modeling the interaction of light between diffuse surfaces. Computer Graphics 18, 3 (July), 213--222.
25. Grimme, E. J. 1997. Krylov Projection Methods For Model Reduction. PhD thesis, Ohio State University.
26. Gugercin, S., and Antoulas, A. 2004. A survey of model reduction by balanced truncation and some new results. International Journal of Control 77, 8, 748--766.
27. Gugercin, S., Antoulas, A., and Beattie, C. A. 2006. A rational Krylov iteration for optimal H2 model reduction. In Intl. Symposium on Mathematical Theory of Networks and Systems.
28. Gupta, M., and Narasimhan, S. G. 2007. Legendre fluids: A unified framework for analytic reduced space modeling and rendering of participating media. In 2007 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 17--26.
29. Hanrahan, P., Salzman, D., and Aupperle, L. 1991. A rapid hierarchical radiosity algorithm. In Proc of SIGGRAPH.
30. Hauser, K. K., Shen, C., and O'Brien, J. F. 2003. Interactive deformation using modal analysis with constraints. In Graphics Interface, CIPS, Canadian Human-Computer Commnication Society, 247--256.
31. Hossain, M.-S., and Benner, P. 2008. Projection-based model reduction for time-varying descriptor systems using recycled Krylov subspaces. Proceedings in Applied Mathematics and Mechanics 8, 1, 10081--10084.
32. James, D. L., and Fatahalian, K. 2003. Precomputing interactive dynamic deformable scenes. In Proc. SIGGRAPH '03.
33. James, D. L., and Pai, D. K. 2002. DyRT: Dynamic response textures for real time deformation simulation with graphics hardware. ACM Transactions on Graphics 21, 3 (July), 582--585.
34. James, D. L., Barbic, J., and Pai, D. K. 2006. Precomputed acoustic transfer: output-sensitive, accurate sound generation for geometrically complex vibration sources. ACM Transactions on Graphics 25, 3 (July), 987--995.
35. Kim, T., and Delaney, J. 2013. Subspace fluid re-simulation. ACM Transactions on Graphics 32, 4.
36. Kim, T., and James, D. L. 2009. Skipping steps in deformable simulation with online model reduction. ACM Transactions on Graphics 28, 5 (Dec.), 123:1--123:9.
37. Kim, T., and James, D. L. 2011. Physics-based character skinning using multi-domain subspace deformations. In Proc. SCA '11.
38. Klingner, B. M., and Shewchuk, J. R. 2007. Agressive tetrahedral mesh improvement. In Proceedings of the 16th International Meshing Roundtable, 3--23.
39. Klingner, B. M., Feldman, B. E., Chentanez, N., and O'Brien, J. F. 2006. Fluid animation with dynamic meshes. ACM Transactions on Graphics 25, 3 (July), 820--825.
40. Li, J.-R. 2000. Model Reduction of Large Linear Systems. PhD thesis, Massachusetts Institute of Techology.
41. Liua, C., Yuan, X., Mullaguru, A., and Fan, J. 2008. Model order reduction via rational transfer function fitting and eigenmode analysis. In International Conference on Modeling, Identification and Control.
42. Loos, B. J., Antani, L., Mitchell, K., Nowrouzezahrai, D., Jarosz, W., and Sloan, P.-P. 2011. Modular radiance transfer. ACM Transactions on Graphics 30, 6 (Dec.).
43. Loos, B. J., Nowrouzezahrai, D., Jarosz, W., and Sloan, P.-P. 2012. Delta radiance transfer. In Proceedings of the 2012 ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games, ACM, New York, NY, USA, I3D 2012.
44. Mullen, P., Crane, K., Pavlov, D., Tong, Y., and Desbrun, M. 2009. Energy-preserving integrators for fluid animation. ACM Transactions on Graphics 28, 3 (July), 38:1--38:8.
45. Olssen, K. H. A. 2005. Model Order Reduction with Rational Krylov Methods. PhD thesis, KTH Stockholm.
46. Pavlov, D., Mullen, P., Tong, Y., Kanso, E., Marsden, J., and Desbrun, M. 2011. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena 240, 6, 443--458.
47. Pentland, A., and Williams, J. 1989. Good vibrations: Modal dynamics for graphics and animation. Computer Graphics (SIGGRAPH 89) 23, 3 (July), 215--222. Held in Boston, Massachusetts.
48. Pharr, M., and Humphreys, G. 2004. Physically Based Rendering: From Theory to Implementation. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA.
49. Phillips, J. R. 1998. Model reduction of time-varying linear systems using approximate multipoint Krylov-subspace projectors. Computer Aided Design, 96--102.
50. Rowley, C. W., and Marsden, J. E. 2000. Reconstruction equations and the Karhunen-Loéve expansion for systems with symmetry. Phys. D 142, 1-2, 1--19.
51. Rowley, C. W., Kevrekidis, I. G., Marsden, J. E., and Lust, K. 2003. Reduction and reconstruction for self-similar dynamical systems. Nonlinearity 16 (July), 1257--1275.
52. Sandberg, H., and Rantzer, A. 2004. Balanced truncation of linear time-varying systems. IEEE Transactions on Automatic Control 49, 2, 217--229.
53. Savas, B., and Eldén, L. 2009. Krylov subspace methods for tensor computations. Tech. Rep. LITH-MAT-R-2009-02-SE, Department of Mathematics, Linköpings Universitet.
54. Schmit, R., and Glasuer, M. 2002. Low dimensional tools for flow-structure interaction problems: Application to micro air vehicles. APS Meeting Abstracts (Nov.), D1+.
55. Sewall, J., Mecklenburg, P., Mitran, S., and Lin, M. 2007. Fast fluid simulation using residual distribution schemes. In Proc. Eurographics Workshop on Natural Phenomena.
56. Si, H. 2007. A quality tetrahedral mesh generator and 3-dimensional Delaunay triangulator. http://tetgen.berlios.de/.
57. Sloan, P.-P., Kautz, J., and Snyder, J. 2002. Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments. In Proc. SIGGRAPH '02.
58. Sloan, P.-P., Luna, B., and Snyder, J. 2005. Local, deformable precomputed radiance transfer. ACM Transactions on Graphics 24, 3 (Aug.), 1216--1224.
59. Sorkine, O. 2006. Differential representations for mesh processing. Computer Graphics Forum 25, 4, 789--807.
60. Stam, J. 1999. Stable fluids. In Computer Graphics (SIGGRAPH 99).
61. Treuille, A., Lewis, A., and Popović, Z. 2006. Model reduction for real-time fluids. In Proc. SIGGRAPH '06.
62. Wicke, M., Stanton, M., and Treuille, A. 2009. Modular bases for fluid dynamics. ACM Transactions on Graphics 28, 3.
63. Zatz, H. R. 1993. Galerkin radiosity: a higher order solution method for global illumination. In Proc. SIGGRAPH '93, ACM.
64. Zhou, Y. 2002. Numerical Methods for Large Scale Matrix Equations with Applications in LTI System Model Reduction. PhD thesis, Rice University.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

    • Rendering

      • Reflectance modeling

  • Modeling and simulation

    • Simulation types and techniques

      • Simulation by animation