A Prior Reduced Model of Dynamical Systems

  • Haoran Xie
  • Zhiqiang Wang
  • Kazunori Miyata
  • Ye Zhao
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 18)

Abstract

A reduced model technique for simulating dynamical systems in computer graphics is proposed. Most procedural models of physics-based simulations consist of control parameters in a high-dimensional domain in which the real-time controllability of simulations is an ongoing issue. Therefore, we adopt a separated representation of the model solutions that can be preprocessed offline without relying on the knowledge of the complete solutions. To achieve the functional products in this representation, we utilize an iterative method involving enrichment and projection steps in a tensor formulation. The proposed approaches are successfully applied to different parametric and coupled models.

Keywords

Model reduction Dynamical system Separated representation  Fixed-point method Tensor product Enrichment step Projection step 

References

  1. 1.
    T. Adrien, L. Andrew, P. Zoran, Model reduction for real-time fluids. ACM Trans. Graph. 25(3), 826–834 (2006)CrossRefGoogle Scholar
  2. 2.
    G. Beylkin, M. Mohlenkamp, Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26(6), 2133–2159 (2005)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Chinesta, A. Ammar, E. Cueto, Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch. Comput. Methods Eng. 17(4), 327–350 (2010)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    F. Chinesta, A. Leygue, M. Beringhier, L. Nguyen, J. Grandidier, B. Schrefler, F. Pesavento, Towards a framework for non-linear thermal models in shell domains. Int. J. Numer. Methods Heat Fluid Flow 23(1), 55–73 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Chinesta, A. Leygue, F. Bordeu, J.V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar, A. Huerta, Pgd-based computational vademecum for efficient design, optimization and control. Arch. Comput. Methods Eng. 20(1), 31–59 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Chinesta, R. Keunings, A. Leygue, The proper generalized decomposition for advanced numerical simulations: a primer (Springer International Publishing, New York, 2014)Google Scholar
  7. 7.
    A. Dumon, C. Allery, A. Ammar, Proper general decomposition for the resolution of navier stokes equations. J. Comput. Phys. 230(4), 1387–1407 (2011)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Dumon, C. Allery, A. Ammar, Proper generalized decomposition method for incompressible navier-stokes equations with a spectral discretization. Appl. Math. Comput. 219(15), 8145–8162 (2013)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Gonzalez, F. Masson, F. Poulhaon, A. Leygue, E. Cueto, F. Chinesta, Proper generalized decomposition based dynamic data driven inverse identification. Math. Comput. Simul. 82(9), 1677–1695 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Jame, F. Kayvon, Precomputing interactive dynamic deformable scenes. ACM Trans. Graph. 22(3), 879–887 (2003)CrossRefGoogle Scholar
  11. 11.
    E. Ju, J. Won, J. Lee, B. Choi, J. Noh, M. Choi, Data-driven control of flapping flight. ACM Trans. Graph. 32(5), 151:1–151:12 (2013)Google Scholar
  12. 12.
    E. Miguel, R. Tamstorf, D. Bradley, Sara C. Schvartzman, B. Thomaszewski, B. Bickel, W. Matusik, S. Marschner, Miguel A. Otaduy. Modeling and estimation of internal friction in cloth. ACM Trans. Graph. 32(6), 212:1–212:10 (2013)Google Scholar
  13. 13.
    G. Mougin, J. Magnaudet, The generalized kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Int. J. Multiph. Flow 28(11), 1837–1851 (2002)MATHCrossRefGoogle Scholar
  14. 14.
    S. Niroomandi, D. Gonzalez, I. Alfaro, F. Bordeu, A. Leygue, E. Cueto, F. Chinesta, Real-time simulation of biological soft tissues: a pgd approach. Int. J. Numer. Methods Biomed. Eng. 29(5), 586–600 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. Popović, M. Steven, M. Seitz, Erdmann. Motion sketching for control of rigid-body simulations. ACM Trans. Graph. 22(4), 1034–1054 (2003)Google Scholar
  16. 16.
    E. Pruliere, F. Chinesta, A. Ammar, On the deterministic solution of multidimensional parametric models using the proper generalized decomposition. Math. Comput. Simul. 81(4), 791–810 (2010)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Xie, K. Miyata, Stochastic modeling of immersed rigid-body dynamics. SIGGRAPH Asia 2013 Technical Briefs (2013), pp 12:1–12:4Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Haoran Xie
    • 1
  • Zhiqiang Wang
    • 2
  • Kazunori Miyata
    • 1
  • Ye Zhao
    • 2
  1. 1.School of Knowledge ScienceJapan Advanced Institute of Science and Technology/JSPS Research FellowNomiJapan
  2. 2.Department of Computer ScienceKent State UniversityKentUSA

Personalised recommendations