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Low-Mach-number and slenderness limit for elastic Cosserat rods and its numerical investigation. (English) Zbl 1465.35372

In this paper the authors investigate the relation between the dynamic elastic Cosserat rod model and the Kirchhoff beam equation. They prove that the Kirchhoff beam equation (without angular inertia) can be obtained as the asymptotic limit of the Cosserat rod when the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and the speed of sound) approach to zero. The asymptotic framework is exact up to fourth order in the small parameter. It also reveals that the strucuture allows a uniform handling of the transition regime for the models. This is also done numerically applying a scheme based on an \(\alpha\)-method in time and a Gauss-Legendre collocation in space (finite differences).

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74A35 Polar materials
74A60 Micromechanical theories
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S20 Finite difference methods applied to problems in solid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs

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[1] T. Alazard, Low Mach number limit for the full Navier-Stokes equations,Arch. Rat. Mech. Anal.180(2006), 1-73. doi:10.1007/s00205-005-0393-2. · Zbl 1108.76061
[2] S.S. Antman,Nonlinear Problems of Elasticity, Springer, New York, 2006.
[3] B. Audoly and Y. Pomeau,Elasticity and Geometry, Oxford University Press, Oxford, 2010. · Zbl 1223.74001
[4] M. Bergou, M. Wardetzky, S. Robinson, B. Audoly and E. Grinspun, Discrete elastic rods,ACM Transaction Graphics27 (2008), 63:1-63:12.
[5] F. Bertails, B. Audoly, M. Cani, B. Querleux, F. Leroy and J. Lévéque, Super-helices for predicting the dynamics of natural hair,ACM Transaction Graphics25(2006), 1180-1187. doi:10.1145/1141911.1142012.
[6] P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams,Comp. Mech.31(2003), 49-59. doi:10. 1007/s00466-002-0392-1. · Zbl 1038.74580
[7] R.E. Caflisch and J.H. Maddocks, Nonlinear dynamical theory of the elastica,Proc. Roy. Soc. Edinburgh: Sec. A Math. 99(1984), 1-23. doi:10.1017/S0308210500025920. · Zbl 0589.73057
[8] N. Chouaieb and J.H. Maddocks, Kirchhoff’s problem of helical equilibria of uniform rods,J. Elast.77(2004), 221-247. doi:10.1007/s10659-005-0931-z. · Zbl 1071.74031
[9] J. Chung and G.M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-alpha method,J. Appl. Mech.60(1993), 371-375. doi:10.1115/1.2900803. · Zbl 0775.73337
[10] B.D. Coleman, E.H. Dill, M. Lembo, Z. Lu and I. Tobias, On the dynamics of rods in the theory of Kirchhoff and Clebsch, Arch. Rat. Mech. Anal.121(1993), 339-359. doi:10.1007/BF00375625. · Zbl 0784.73044
[11] B.D. Coleman, E.H. Dill and D. Swigon, On the dynamics of flexure and stretch in the theory of elastic rods,Arch. Rat. Mech. Anal.129(1995), 147-174. doi:10.1007/BF00379919. · Zbl 0872.73020
[12] E. Cosserat and F. Cosserat,Théorie des Corps Déformables, Hermann, Paris, 1909. · JFM 40.0862.02
[13] D.J. Dichmann, Y. Li and J.H. Maddocks, Hamiltonian formulations and symmetries in rod mechanics, in:Mathematical Approaches to Biomolecular Structure and Dynamics, J.P. Mesirov, K. Schulten and D.W. Sumners, eds, Springer, New York, 1996, pp. 71-113. doi:10.1007/978-1-4612-4066-2_6. · Zbl 0864.92004
[14] D.J. Dichmann and J.H. Maddocks, An impetus-striction simulation of the dynamics of an elastica,J. Nonlinear Sci.6 (1996), 271-292. doi:10.1007/BF02439312. · Zbl 0854.73032
[15] T. Fütterer, A. Klar and R. Wegener, An energy conserving numerical scheme for the dynamics of hyperelastic rods,Int. J. Diff. Eqs.2012(2012), 718308. · Zbl 1402.74062
[16] H. Guillard and C. Viozat, On the behavior of upwind schemes in the low Mach number limit,Computers & Fluids28 (1999), 63-86. doi:10.1016/S0045-7930(98)00017-6. · Zbl 0963.76062
[17] E. Hairer, C. Lubich and M. Roche,The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Springer, Berlin, 1989. · Zbl 0683.65050
[18] J. Hämäläinen, S.B. Lindström, T. Hämäläinen and H. Niskanen, Papermaking fibre-suspension flow simulations at multiple scales,J. Eng. Math.71(2011), 55-79. doi:10.1007/s10665-010-9433-5. · Zbl 1254.76156
[19] T.Y. Hou, I. Klapper and H. Si, Removing the stiffness of curvature in computing 3-d filaments,J. Comp. Phys.143 (1998), 628-664. doi:10.1006/jcph.1998.5977. · Zbl 0917.76063
[20] G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes,Journal für die reine und angewandte Mathematik56(1859), 285-316. · ERAM 056.1494cj
[21] A. Klar, N. Marheineke and R. Wegener, Hierarchy of mathematical models for production processes of technical textiles, ZAMM - J. Appl. Math. Mech.89(2009), 941-961. · Zbl 1230.74076
[22] R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow,J. Comp. Phys.121(1995), 213-237. doi:10.1016/S0021-9991(95)90034-9. · Zbl 0842.76053
[23] L.D. Landau and E.M. Lifschitz,Elastizitätstheorie, Lehrbuch der Theoretischen Physik, Vol. VII, Akademie-Verlag, Berlin, 1970. · Zbl 0201.30501
[24] H. Lang, J. Linn and M. Arnold, Multi-body dynamics simulation of geometrically exact Cosserat rods,Multiboldy System Dyn.25(2011), 285-312. doi:10.1007/s11044-010-9223-x. · Zbl 1271.74264
[25] J. Langer and D.A. Singer, Lagrangian aspects of the Kirchhoff elastic rod,SIAM Rev.38(1996), 605-618. doi:10.1137/ S0036144593253290. · Zbl 0859.73040
[26] S. Leyendecker, P. Betsch and P. Steinmann, Objective energy-momentum conserving integration for the constrained dynamics of geometrically exact beams,Comput. Meth. Appl. Mech. Eng.195(2006), 2313-2333. doi:10.1016/j.cma. 2005.05.002. · Zbl 1142.74045
[27] P.L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,J. Math. Pures. Appl.77(1998), 585-627. doi:10.1016/S0021-7824(98)80139-6. · Zbl 0909.35101
[28] A.E.H. Love,A Treatise on the Mathematical Theory of Elasticity, 4th edn, Cambridge University Press, Cambridge, 1927. · JFM 53.0752.01
[29] J.H. Maddocks, Stability of nonlinearly elastic rods,Arch. Rat. Mech. Anal.85(1984), 311-354. doi:10.1007/ BF00275737. · Zbl 0545.73039
[30] J.H. Maddocks and D.J. Dichmann, Conservation laws in the dynamics of rods,J. Elast.34(1994), 83-96. doi:10.1007/ BF00042427. · Zbl 0808.73042
[31] N. Marheineke and R. Wegener, Fiber dynamics in turbulent flows: General modeling framework,SIAM J. Appl. Math. 66(2006), 1703-1726. doi:10.1137/050637182. · Zbl 1108.76036
[32] N. Marheineke and R. Wegener, Modeling and application of a stochastic drag for fiber dynamics in turbulent flows,Int. J. Multiphase Flow37(2011), 136-148. doi:10.1016/j.ijmultiphaseflow.2010.10.001.
[33] A. Meister, Asymptotic single and multiple scale expansions in the low Mach number limit,SIAM J. Appl. Math.60 (1999), 256-271. doi:10.1137/S0036139998343198. · Zbl 0941.35052
[34] A.H. Nayfeh and P.F. Pai, Non-linear non-planar parametric responses of an inextensional beam,Int. J. Non-Lin. Mech. 24(1998), 139-158. doi:10.1016/0020-7462(89)90005-X. · Zbl 0673.73043
[35] I. Romero and F. Armero, An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics,Int. J. Numer. Meth. Engng.54(2002), 1683-1716. doi:10.1002/nme.486. · Zbl 1098.74713
[36] M.B. Rubin,Cosserat Theories, Kluwer, Dordrecht, 2000. · Zbl 0984.74003
[37] J.C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem - Part I,Comput. Meth. Appl. Mech. Eng.49(1985), 55-70. doi:10.1016/0045-7825(85)90050-7. · Zbl 0583.73037
[38] J.C. Simo, J.E. Marsden and P.S. Krishnaprasad, The Hamiltonian structure of nonlinear elasticity: The material, spatial and convective representations of solids, rods and plates,Archive Rat. Mech. Analysis104(1988), 125-183. doi:10.1007/ BF00251673. · Zbl 0668.73014
[39] J.C. Simo, N. Tarnow and M. Doblare, Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms,Int. J. Numer. Meth. Engng.38(1995), 1431-1473. doi:10.1002/nme.1620380903. · Zbl 0860.73025
[40] J.C. Simo and L. Vu-Quoc, Three-dimensional finite strain rod model. Part II: Computational aspects,Comput. Meth. Appl. Mech. Eng.58(1986), 79-116. doi:10.1016/0045-7825(86)90079-4. · Zbl 0608.73070
[41] J.C. Simo and L. Vu-Quoc, On the dynamics in space of rods undergoing large motions - A geometrically exact approach, Comput. Meth. Appl. Mech. Eng.66(1988), 125-161. doi:10.1016/0045-7825(88)90073-4. · Zbl 0618.73100
[42] G. Sobottka, T. Lay and A. Weber, Stable integration of the dynamic Cosserat equations with application to hair modeling, J. WSCG16(2008), 73-80.
[43] J. Spillmann and M. Teschner, CoRDE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects, in:Proc. of the 2007 ACM SIGGRAPH / Eurographics Symposium on Computer Animation, Aire-la-Ville, Eurographics Association , Switzerland, 2007, pp. 63-72.
[44] W.
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