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An asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. (English) Zbl 1440.74480
Summary: We present an efficient numerical scheme based on asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. Using quaternions to represent rotation, the equations of static equilibria of special Cosserat rods are posed as a system of thirteen first order ordinary differential equations having cubic nonlinearity. The derivatives in these equations are further discretized to yield a system of cubic polynomial equations. As asymptotic-numerical methods are typically applied to polynomial systems having quadratic nonlinearity, a modified version of this method is presented in order to apply it directly to our cubic nonlinear system. We then use our method for continuation of equilibria of the follower load problem and demonstrate our method to be highly efficient when compared to conventional solvers based on the finite element method. Finally, we demonstrate how our method can be used for computing the buckling load as well as for continuation of postbuckled equilibria of hemitropic rods.
MSC:
74S99 Numerical and other methods in solid mechanics
65N99 Numerical methods for partial differential equations, boundary value problems
74S05 Finite element methods applied to problems in solid mechanics
74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Software:
AUTO
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[1] Antman, S. S., Nonlinear Problems of Elasticity (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0820.73002
[2] Manning, R. S.; Maddocks, J. H.; Kahn, J. D., A continuum rod model of sequence-dependent DNA structure, J. Chem. Phys., 105, 13, 5626-5646 (1996)
[3] Swigon, D.; Coleman, B. D.; Tobias, I., The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes, Biophys. J., 74, 5, 2515-2530 (1998)
[4] Kosikov, K. M.; Gorin, A. A.; Zhurkin, V. B.; Olson, W. K., DNA stretching and compression: large-scale simulations of double helical structures, J. Mol. Biol., 289, 5, 1301-1326 (1999)
[5] Thompson, J. T.; van der Heijden, G. M.; Neukirch, S., Supercoiling of DNA plasmids: mechanics of the generalized ply, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 458 (2002) · Zbl 1065.74048
[6] Bozec, L.; van der Heijden, G.; Horton, M., Collagen fibrils: ranoscale ropes, Biophys. J., 92, 70-75 (2007)
[7] Marko, J. F.; Neukirch, S., Competition between curls and plectonemes near the buckling transition of stretched supercoiled DNA, Phys. Rev. E, 85, 1, 011908 (2012)
[8] Kumar, A.; Kumar, S.; Gupta, P., A helical cauchy-born rule for special cosserat rod modeling of nano and continuum rods, J. Elasticity, 124, 81-106 (2016) · Zbl 1338.74012
[9] Gupta, P.; Kumar, A., Effect of material nonlinearity on spatial buckling of nanorods and nanotubes, J. Elasticity, 126, 2, 155-171 (2017) · Zbl 1354.74023
[10] Nuti, S.; Ruimi, A.; Reddy, J. N., Modeling the dynamics of filaments for medical applications, Internat. J. Non-Linear Mech., 66, 139-148 (2014)
[11] Sobottka, G.; Lay, T.; Weber, A., Stable integration of the dynamic Cosserat equations with application to hair modeling, J. WSCG, 16, 73-80 (2008)
[12] Audoly, B.; Pomeau, Y., Elasticity and Geometry: From Hair Curls to the Non-Linear Response of Shells (2010), Oxford Press · Zbl 1223.74001
[13] Miller, J. T.; Lazarus, A.; Audoly, B.; Reis, P. M., Shapes of a suspended curly hair, Phys. Rev. Lett., 112, 068103 (2014)
[14] Costello, G. A., Theory of Wire Rope (1997), Springer-Verlag: Springer-Verlag New York
[15] Goriely, A.; Tabor, M., Nonlinear dynamics of filaments I. Dynamical instabilities, Phys. D, 105, 1-3, 20-44 (1997) · Zbl 0962.74513
[16] Bergou, M.; Wardetzky, M.; Robinson, S.; Audoly, B.; Grinspun, E., Discrete elastic rods, (ACM Transactions on Graphics, Vol. 27. ACM Transactions on Graphics, Vol. 27, (TOG) (2008), ACM), 63
[17] Lang, H.; Linn, J.; Arnold, M., Multi-body dynamics simulation of geometrically exact Cosserat rods, Multibody Syst. Dyn., 25, 3, 285-312 (2011) · Zbl 1271.74264
[18] Simo, J. C.; Vu-Quoc, L., A three-dimensional finite-strain rod model. Part II: Computational aspects, Comput. Methods Appl. Mech. Engrg., 58, 1, 79-116 (1986) · Zbl 0608.73070
[19] Dichmann, D. J.; Li, Y.; Maddocks, J. H., Hamiltonian formulations and symmetries in rod mechanics, IMA Vol. Math. Appl., 82, 71-114 (1996) · Zbl 0864.92004
[20] Furrer, P. B.; Manning, R. S.; Maddocks, J. H., DNA rings with multiple energy minima, Biophys. J., 79, 1, 116-136 (2000)
[21] Domokos, G.; Szeberényi, I., A hybrid parallel approach to one-parameter nonlinear boundary value problems, Comput. Assist. Mech. Engrg. Sci., 11, 1, 15-34 (2004)
[22] Healey, T. J.; Mehta, P. G., Straightforward computation of spatial equilibria of geometrically exact Cosserat rods, Internat. J. Bifur. Chaos, 15, 03, 949-965 (2005) · Zbl 1081.74026
[23] Marino, E., Locking-free isogeometric collocation formulation for three-dimensional geometrically exact shear-deformable beams with arbitrary initial curvature, Comput. Methods Appl. Mech. Engrg., 324, 546-572 (2017)
[24] Weeger, O.; Yeung, S. K.; Dunn, M. L., Isogeometric collocation methods for Cosserat rods and rod structures, Comput. Methods Appl. Mech. Engrg., 316, 100-122 (2017)
[25] Kumar, A.; Healey, T. J., A generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints, Comput. Methods Appl. Mech. Engrg., 199, 1805-1815 (2010) · Zbl 1231.74484
[26] Vu-Quoc, L., Dynamics of Flexible Structures Performing Large Overall Motions: A Geometrically- Nonlinear Approach (1986), EECS Department, University of California: EECS Department, University of California Berkeley, UCB/ERL M86/36
[27] Lazarus, A.; Miller, J. T.; Reis, P. M., Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method, J. Mech. Phys. Solids, 61, 8, 1712-1736 (2013) · Zbl 1294.74041
[28] Zhong, H.; Zhang, R.; Xiao, N., A quaternion-based weak form quadrature element formulation for spatial geometrically exact beams, Arch. Appl. Mech., 84, 12, 1825-1840 (2014)
[29] McRobie, F. A.; Lasenby, J., Simo-Vu Quoc rods using Clifford algebra, Internat. J. Numer. Methods Engrg., 45, 4, 377-398 (1999) · Zbl 0940.74081
[30] Simo, J. C.; Vu-Quoc, L., On the dynamics in space of rods undergoing large motionsa geometrically exact approach, Comput. Methods Appl. Mech. Engrg., 66, 2, 125-161 (1988) · Zbl 0618.73100
[31] Riks, E., An incremental approach to the solution of snapping and buckling problems, Int. J. Solids Struct., 15, 7, 529-551 (1979) · Zbl 0408.73040
[32] Doedel, E. J., AUTO: a program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30, 265-284 (1981)
[33] Crisfield, M. A., Nonlinear finite element analysis of solids and structures, (Essentials, Vol. 1 (1991), Wiley: Wiley New York) · Zbl 0809.73005
[34] Davies, M. A.; Moon, F. C., 3D spatial chaos in the elastica and the spinning top: Kirchhoff analogy, Chaos, 3, 1, 93-99 (1993) · Zbl 1055.37587
[35] Damil, N.; Potier-Ferry, M., A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures, Internat. J. Engrg. Sci., 28, 9, 943-957 (1990) · Zbl 0721.73018
[36] Cochelin, B.; Damil, N.; PotierFerry, M., Asymptotic-numerical methods and Pade approximants for nonlinear elastic structures, Internat. J. Numer. Methods Engrg., 37, 7, 1187-1213 (1994) · Zbl 0805.73076
[37] Vannucci, P.; Cochelin, B.; Damil, N.; Potier-Ferry, M., An asymptotic-numerical method to compute bifurcating branches, Internat. J. Numer. Methods Engrg., 41, 8, 1365-1389 (1998) · Zbl 0911.73078
[38] Zahrouni, H.; Cochelin, B.; Potier-Ferry, M., Computing finite rotations of shells by an asymptotic-numerical method, Comput. Methods Appl. Mech. Engrg., 175, 1-2, 71-85 (1999) · Zbl 0963.74035
[39] Chen, X.; Zheng, C.; Xu, W.; Zhou, K., An asymptotic numerical method for inverse elastic shape design, Proc. SIGGRAPH, 33, 4, 95 (2014) · Zbl 1396.65037
[40] Karkar, S.; Cochelin, B.; Vergez, C., A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities, J. Sound Vib., 332, 4, 968-977 (2013)
[41] Ed-Dinari, A.; Mottaqui, H.; Braikat, B.; Jamal, M.; Mohri, F.; Damil, N., Large torsion analysis of thin-walled open sections beams by the Asymptotic Numerical Method, Eng. Struct., 81, 240-255 (2014)
[42] Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity (2000), Dover Books
[43] Healey, T. J.; Papadopoulos, C. M., Bifurcation of hemitropic elastic rods under axial thrust, Appl. Math., 71, 729-753 (2013) · Zbl 1282.74047
[44] Healey, T. J., Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids, 7, 405-420 (2002) · Zbl 1090.74610
[45] Cochelin, B., A path-following technique via an asymptotic-numerical method, Comput. Struct., 53, 5, 1181-1192 (1994) · Zbl 0918.73337
[46] Singh, R.; Kumar, S.; Kumar, A., Effect of intrinsic twist and orthotropy on extension-twist-inflation coupling in compressible circular tubes, J. Elasticity, 128, 2, 175-201 (2017) · Zbl 1374.74017
[47] Kumar, A.; Mukherjee, S., A Geometrically Exact Rod Model including in-plane cross-sectional deformation, J. Appl. Mech., 78, 011010 (2011)
[48] Cochelin, B.; Medale, M., Power series analysis as a major breakthrough to improve the efficiency of Asymptotic Numerical Method in the vicinity of bifurcations, J. Comput. Phys., 236, 594-607 (2013)
[49] Medale, M.; Cochelin, B., High performance computations of steady-state bifurcations in 3D incompressible fluid flows by Asymptotic Numerical Method, J. Comput. Phys., 299, 581-596 (2015)
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