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A novel quaternion integration approach for describing the behaviour of non-spherical particles. (English) Zbl 1356.70009
Summary: 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.

##### MSC:
 70E15 Free motion of a rigid body 76T25 Granular flows
ITK
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##### References:
 [1] Allen M.P., Tildesley D.J.: The Computer Simulation of Liquids, Vol. 42. Oxford University Press, Oxford (1989) · Zbl 1372.82005 [2] Altmann S.L.: Rotations, Quaternions, and Double Groups, Vol. 3. Clarendon Press Oxford, England (1986) · Zbl 0683.20037 [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 · Zbl 1116.70013 [4] Baraff, D.: An Introduction to Physically Based Modeling: Rigid Body Simulation I–Unconstrained Rigid Body Dynamics. SIGGRAPH Course Notes (1997) [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) · Zbl 1171.70300 [6] Betsch P., Steinmann P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191(3–5), 467–488 (2001) · Zbl 1004.70005 [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) · Zbl 1166.70005 [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) · Zbl 1158.70309 [9] Chou J.: Quaternion kinematic and dynamic differential equations. IEEE Trans. Robot. Autom. 8(1), 53–64 (1992) [10] Cundall P., Strack O.D.L.: A discrete numerical model for granular assemblies. Géotechnique 29(1), 47–65 (1979) [11] Delaney G.W., Cleary P.: The packing properties of superellipsoids. EPL (Europhys. Lett.) 89(3), 34,002 (2010) [12] Diebel, J.: Representing attitude: Euler angles, unit Quaternions, and rotation vectors. Tech. rep., Stanford University, California, USA (2006) [13] Eberly, D.: Quaternion algebra and calculus. Magic Software, Inc. (2002) · Zbl 1219.81038 [14] Eberly, D.: Rotation representations and performance issues. Magic Software, Inc. (2002) · Zbl 1219.81038 [15] Evans D., Murad S.: Singularity free algorithm for molecular dynamics simulation of rigid polyatomics. Mol. Phys. 34(2), 327–331 (1977) [16] Grassia F.S.: Practical parameterization of rotations using the exponential map. J. Graphics Tools 3, 1–13 (1998) · Zbl 05467624 [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) [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) · Zbl 1127.70002 [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) [20] Hoffmann, G.: Application of Quaternions. Tech. Rep. February 1978, Technische Universität Braunschweig (1978) [21] Hoover W.G.: Lecture Notes in Physics–Molecular Dynamics. Springer, US (1986) [22] Ibanez, L.: Tutorial on Quaternions Part I. Insight Segmentation and Registration Toolkit (ITK) (2001) [23] Karney C.F.F.: Quaternions in molecular modeling. J. Mol. Graph. Model. 25(5), 595–604 (2007) [24] Kleppmann, M.: Simulation of colliding constrained rigid bodies. Tech. Rep. 683, University of Cambridge, UCAM-CL-TR-683, ISSN 1476-2986 (2007) [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) · Zbl 1050.70504 [26] Kuipers J.B.: Quaternions and rotation sequences. Mater. Sci. Eng. A 271(d), 322–333 (1999) [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) [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) · Zbl 1205.74025 [29] Mclachlan, R.I., Zanna, A.: The discrete Moser-Veselov algorithm for the free rigid body, revisited. Tech. rep., University of Bergen (2003) · Zbl 1123.70012 [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) · Zbl 1182.76539 [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) · Zbl 0754.58017 [32] Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes in C, 2nd ed. Cambridge University Press, Cambridge (1992) · Zbl 0778.65003 [33] Qi D.: Direct simulations of flexible cylindrical fiber suspensions in finite Reynolds number flows. J. Chem. Phys. 125(11), 114,901 (2006) [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) [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) [36] Whitmore, S.A., Hughes, L.: Calif: Closed-form Integrator for the Quaternion (Euler Angle) Kinematics Equations, US patent (2000)
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