Advertisement

Acta Mechanica

, Volume 224, Issue 12, pp 3091–3109 | Cite as

A novel Quaternion integration approach for describing the behaviour of non-spherical particles

  • F. Zhao
  • B. G. M. van Wachem
Open Access
Article

Abstract

There are three main frameworks to describe the orientation and rotation of non-spherical particles: Euler angles, rotation matrices and unit Quaternions. Of these methods, the latter seems the most attractive for describing the behaviour of non-spherical particles. However, there are a number of drawbacks when using unit Quaternions: the necessity of applying rotation matrices in conjunction to facilitate the transformation from body space to world space, and the algorithm integrating the Quaternion should inherently conserve the length of the Quaternion. Both drawbacks are addressed in this paper. The present paper derives a new framework to transform vectors and tensors by unit Quaternions, and the requirement of explicitly using rotation matrices is removed altogether. This means that the algorithm derived in this paper can describe the rotation of a non-spherical particle with four parameters only. Moreover, this paper introduces a novel corrector-predictor method to integrate unit Quaternions, which inherently conserves the length of the Quaternion. The novel framework and method are compared to a number of other methods put forward in the literature. All the integration methods are discussed, scrutinised and compared to each other by comparing the results of four test cases, involving a single falling particle, nine falling and interacting particles, a 2D prescribed torque on a sphere and a 3D prescribed torque on a non-spherical particle. Moreover, a convergence study is presented, comparing the rate of convergence of the various methods. All the test cases show a significant improvement of the new framework put forward in this paper over existing algorithms. Moreover, the new method requires less computational memory and fewer operations, due to the complete omission of the rotation matrix in the algorithm.

Keywords

Rotation Matrix Euler Angle Time Level Euler Method Rotation Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Allen M.P., Tildesley D.J.: The Computer Simulation of Liquids, Vol. 42. Oxford University Press, Oxford (1989)Google Scholar
  2. 2.
    Altmann S.L.: Rotations, Quaternions, and Double Groups, Vol. 3. Clarendon Press Oxford, England (1986)Google Scholar
  3. 3.
    Arribas M., Elipe A., Palacios M.: Quaternions and the rotation of a rigid body. Celestial Mech. Dyn. Astr. 96(3-4), 239–251 (2006). doi: 10.1007/s10569-006-9037-6 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baraff, D.: An Introduction to Physically Based Modeling: Rigid Body Simulation I–Unconstrained Rigid Body Dynamics. SIGGRAPH Course Notes (1997)Google Scholar
  5. 5.
    Betsch P., Siebert R.: Rigid body dynamics in terms of Quaternions: Hamiltonian formulation and conserving numerical integration. Int. J. Numer. Methods Eng. 79(4), 444–473 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Betsch P., Steinmann P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191(3–5), 467–488 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Celledoni E., Fassò F., Säfström N., Zanna A.: The exact computation of the free rigid body motion and its use in splitting methods. SIAM J. Sci. Comput. 30, 2084–2112 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Celledoni E., Säfström N.: Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions. J. Phys. A. Math. General 39(19), 5463–5478 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chou J.: Quaternion kinematic and dynamic differential equations. IEEE Trans. Robot. Autom. 8(1), 53–64 (1992)CrossRefGoogle Scholar
  10. 10.
    Cundall P., Strack O.D.L.: A discrete numerical model for granular assemblies. Géotechnique 29(1), 47–65 (1979)CrossRefGoogle Scholar
  11. 11.
    Delaney G.W., Cleary P.: The packing properties of superellipsoids. EPL (Europhys. Lett.) 89(3), 34,002 (2010)CrossRefGoogle Scholar
  12. 12.
    Diebel, J.: Representing attitude: Euler angles, unit Quaternions, and rotation vectors. Tech. rep., Stanford University, California, USA (2006)Google Scholar
  13. 13.
    Eberly, D.: Quaternion algebra and calculus. Magic Software, Inc. (2002)Google Scholar
  14. 14.
    Eberly, D.: Rotation representations and performance issues. Magic Software, Inc. (2002)Google Scholar
  15. 15.
    Evans D., Murad S.: Singularity free algorithm for molecular dynamics simulation of rigid polyatomics. Mol. Phys. 34(2), 327–331 (1977)CrossRefGoogle Scholar
  16. 16.
    Grassia F.S.: Practical parameterization of rotations using the exponential map. J. Graphics Tools 3, 1–13 (1998)CrossRefGoogle Scholar
  17. 17.
    Gsponer, A., Hurni, J.P.: The physical heritage of Sir W.R. Hamilton. In: The Mathematical Heritage of Sir William Rowan Hamilton—commemorating the sesquicentennial of the invention of Quaternions, pp. 1–37. Trinity College, Dublin (1993)Google Scholar
  18. 18.
    Hairer E., Vilmart G.: Preprocessed discrete Moser Veselov algorithm for the full dynamics of a rigid body. J. Phys. A Math. General 39(42), 13,225–13,235 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hamilton W.R.: On Quaternions; or on a new system of imaginaries in Algebra. Lond. Edinburgh Dublin Philos. Mag. J. Sci. (3d Series) 15-36, 1–306 (1844)Google Scholar
  20. 20.
    Hoffmann, G.: Application of Quaternions. Tech. Rep. February 1978, Technische Universität Braunschweig (1978)Google Scholar
  21. 21.
    Hoover W.G.: Lecture Notes in Physics–Molecular Dynamics. Springer, US (1986)Google Scholar
  22. 22.
    Ibanez, L.: Tutorial on Quaternions Part I. Insight Segmentation and Registration Toolkit (ITK) (2001)Google Scholar
  23. 23.
    Karney C.F.F.: Quaternions in molecular modeling. J. Mol. Graph. Model. 25(5), 595–604 (2007)CrossRefGoogle Scholar
  24. 24.
    Kleppmann, M.: Simulation of colliding constrained rigid bodies. Tech. Rep. 683, University of Cambridge, UCAM-CL-TR-683, ISSN 1476-2986 (2007)Google Scholar
  25. 25.
    Kosenko I.: Integration of the equations of a rotational motion of a rigid body in quaternion algebra. The Euler case. J. Appl. Math. Mech. 62(2), 193–200 (1998)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kuipers J.B.: Quaternions and rotation sequences. Mater. Sci. Eng. A 271(d), 322–333 (1999)Google Scholar
  27. 27.
    Langston P.A., Al-Awamleh M.A., Fraige F.Y., Asmar B.N.: Distinct element modelling of non-spherical frictionless particle flow. Chem. Eng. Sci. 59(2), 425–435 (2004)CrossRefGoogle Scholar
  28. 28.
    Latham, J., Munjiza, A.: The modelling of particle systems with real shapes. Philos. Trans. Roy. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 362(1822), 1953 (2004)Google Scholar
  29. 29.
    Mclachlan, R.I., Zanna, A.: The discrete Moser-Veselov algorithm for the free rigid body, revisited. Tech. rep., University of Bergen (2003)Google Scholar
  30. 30.
    Mortensen P., Andersson H., Gillissen J., Boersma B.J.: Dynamics of prolate ellipsoidal particles in a turbulent channel flow. Phys. Fluids 20(9), 093,302 (2008)CrossRefGoogle Scholar
  31. 31.
    Moser J., Veselov A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139(2), 217–243 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes in C, 2nd ed. Cambridge University Press, Cambridge (1992)Google Scholar
  33. 33.
    Qi D.: Direct simulations of flexible cylindrical fiber suspensions in finite Reynolds number flows. J. Chem. Phys. 125(11), 114,901 (2006)CrossRefGoogle Scholar
  34. 34.
    Sabatini A.M.: Quaternion-based strap-down integration method for applications of inertial sensing to gait analysis. Med. Biol. Eng. Comput. 43(1), 94–101 (2005)CrossRefGoogle Scholar
  35. 35.
    Walton, O.R., Braun, R.L.: Simulation of rotary-drum and repose tests for frictional spheres and rigid sphere clusters. DOE/NSF Workshop on Flow of Particulates (1993)Google Scholar
  36. 36.
    Whitmore, S.A., Hughes, L.: Calif: Closed-form Integrator for the Quaternion (Euler Angle) Kinematics Equations, US patent (2000)Google Scholar

Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Division of Thermofluids, Department of Mechanical EngineeringImperial College LondonLondonUK

Personalised recommendations