×

Numerical solution of a bending-torsion model for elastic rods. (English) Zbl 1456.65153

From the mathematical point of view the authors carry out a finite element discretization, based on linear and cubic elements, of a minimization problem representing the total of the energy functional attached to a general elastic rod. Actually, the discretization is based on a reformulation of the energy functional that provides coercivity properties even when the frame constraints are only approximately satisfied. For the iterative minimization scheme applied, the convergence (Gamma) and stability is proved. A lot of interesting numerical experiments are reported.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G65 Energy minimization in equilibrium problems in solid mechanics
74B20 Nonlinear elasticity
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Antman, SS, Nonlinear Problems of Elasticity, Volume 107 of Applied Mathematical Sciences (2005), New York: Springer, New York · Zbl 1098.74001
[2] Arunakirinathar, K.; Reddy, BD, Mixed finite element methods for elastic rods of arbitrary geometry, Numer. Math., 64, 1, 13-43 (1993) · Zbl 0794.73070
[3] Barrett, JW; Garcke, H.; Nürnberg, R., Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves, Numer. Math., 120, 3, 489-542 (2012) · Zbl 1242.65188
[4] Bartels, S., A simple scheme for the approximation of the elastic flow of inextensible curves, IMA J. Numer. Anal., 33, 4, 1115-1125 (2013) · Zbl 1298.65121
[5] Bartels, S., Finite Element Simulation of Nonlinear Bending Models for Thin Elastic Rods and Plates (2019), Amsterdam: Elsevier, Amsterdam · Zbl 1455.35255
[6] Bartels, S., Reiter, Ph.: Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves. arXiv e-prints (2018)
[7] Bartels, S., Reiter, Ph., Riege, J.: A simple scheme for the approximation of self-avoiding inextensible curves. IMA J. Numer. Anal. drx021 (2017) · Zbl 1403.65128
[8] Bergou, M.; Wardetzky, M.; Robinson, S.; Audoly, B.; Grinspun, E., Discrete elastic rods, ACM Trans. Graph., 27, 3, 63:1-63:12 (2008)
[9] Blatt, S., The energy spaces of the tangent point energies, J. Topol. Anal., 5, 3, 261-270 (2013) · Zbl 1277.28005
[10] Blatt, S.; Reiter, Ph, Regularity theory for tangent-point energies: the non-degenerate sub-critical case, Adv. Calc. Var., 8, 2, 93-116 (2015) · Zbl 1322.49060
[11] Buck, G.; Orloff, J., A simple energy function for knots, Topol. Appl., 61, 3, 205-214 (1995) · Zbl 0829.57005
[12] Călugăreanu, G., L’intégrale de Gauss et l’analyse des nœuds tridimensionnels, Rev. Math. Pures Appl., 4, 5-20 (1959) · Zbl 0134.43005
[13] Călugăreanu, G., Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants, Czechoslovak Math. J., 11, 86, 588-625 (1961) · Zbl 0118.16005
[14] Clauvelin, N.; Audoly, B.; Neukirch, S., Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non-trivial contact topology, J. Mech. Phys. Solids, 57, 9, 1623-1656 (2009) · Zbl 1371.74165
[15] Coleman, BD; Dill, EH; Lembo, M.; Lu, Z.; Tobias, I., On the dynamics of rods in the theory of Kirchhoff and Clebsch, Arch. Ration. Mech. Anal., 121, 4, 339-359 (1992) · Zbl 0784.73044
[16] Coleman, BD; Swigon, D., Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids, J. Elast., 60, 3, 173-221 (2001) · Zbl 1014.74038
[17] Coleman, B.D., Swigon, D.: Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362(1820), 1281-1299 (2004) · Zbl 1091.92002
[18] Coleman, BD; Swigon, D.; Tobias, I., Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact, Phys. Rev. E (3), 61, 1, 759-770 (2000)
[19] da Costa e Silva, C., Maassen, S.F., Pimenta, P.M., Schröder, J.: A simple finite element for the geometrically exact analysis of Bernoulli-Euler rods. Comput. Mech. 65(4), 905-923 (2020) · Zbl 1468.74061
[20] Dall’Acqua, A.; Lin, C-C; Pozzi, P., Evolution of open elastic curves in \(\mathbb{R}^n\) subject to fixed length and natural boundary conditions, Analysis (Berlin), 34, 2, 209-222 (2014) · Zbl 1293.35136
[21] Dall’Acqua, A.; Pluda, A., Some minimization problems for planar networks of elastic curves, Geom. Flows, 2, 1, 105-124 (2017) · Zbl 1383.49017
[22] Deckelnick, K.; Dziuk, G., Error analysis for the elastic flow of parametrized curves, Math. Comp., 78, 266, 645-671 (2009) · Zbl 1198.65183
[23] Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mathematical approaches to biomolecular structure and dynamics (Minneapolis, MN, 1994), volume 82 of IMA Vol. Mathematical Applications, pp. 71-113. Springer, New York (1996) · Zbl 0864.92004
[24] Djondjorov, P.A., Hadzhilazova, M.T., Mladenov, I.M., Vassilev, V.M.: Explicit Parameterization of Euler’s Elastica. Geometry, Integrability and Quantization, pp. 175-186. Softex, Sofia (2008) · Zbl 1196.53004
[25] Dziuk, G.; Kuwert, E.; Schätzle, R., Evolution of elastic curves in \(\mathbb{R}^n\): existence and computation, SIAM J. Math. Anal., 33, 5, 1228-1245 (2002) · Zbl 1031.53092
[26] Gerlach, H.; Reiter, Ph; von der Mosel, H., The elastic trefoil is the doubly covered circle, Arch. Ration. Mech. Anal., 225, 1, 89-139 (2017) · Zbl 1385.53002
[27] Gonzalez, O.; Maddocks, JH, Global curvature, thickness, and the ideal shapes of knots, Proc. Natl. Acad. Sci. USA, 96, 9, 4769-4773 (1999) · Zbl 1057.57500
[28] Gonzalez, O.; Maddocks, JH; Schuricht, F.; von der Mosel, H., Global curvature and self-contact of nonlinearly elastic curves and rods, Calc. Var. Partial Differ. Equ., 14, 1, 29-68 (2002) · Zbl 1006.49001
[29] Goriely, A., Twisted elastic rings and the rediscoveries of Michell’s instability, J. Elast., 84, 3, 281-299 (2006) · Zbl 1098.74034
[30] Goriely, A.; Tabor, M., Nonlinear dynamics of filaments. I. Dynamical instabilities, Phys. D, 105, 1-3, 20-44 (1997) · Zbl 0962.74513
[31] Goriely, A.; Tabor, M., Nonlinear dynamics of filaments. II. Nonlinear analysis, Phys. D, 105, 1-3, 45-61 (1997) · Zbl 0962.74514
[32] Goyal, S.; Perkins, N.; Lee, C., Non-linear dynamic intertwining of rods with self-contact, Int. J. Non-Linear Mech., 43, 1, 65-73 (2008) · Zbl 1203.74081
[33] Goyal, S.; Perkins, NC; Lee, CL, Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables, J. Comput. Phys., 209, 1, 371-389 (2005) · Zbl 1329.74154
[34] Hoffman, KA; Seidman, TI, A variational characterization of a hyperelastic rod with hard self-contact, Nonlinear Anal., 74, 16, 5388-5401 (2011) · Zbl 1402.74064
[35] Hoffman, KA; Seidman, TI, A variational rod model with a singular nonlocal potential, Arch. Ration. Mech. Anal., 200, 1, 255-284 (2011) · Zbl 1294.74040
[36] Hu, K., Buckling of some isotropic, intrinsically curved elastics induced by a terminal twist, Appl. Math. Lett., 16, 2, 193-197 (2003) · Zbl 1135.74313
[37] Ivey, TA; Singer, DA, Knot types, homotopies and stability of closed elastic rods, Proc. Lond. Math. Soc. (3), 79, 2, 429-450 (1999) · Zbl 1036.53001
[38] Kehrbaum, S., Maddocks, J.H.: Elastic rods, rigid bodies, quaternions and the last quadrature. Philos. Trans. R. Soc. Lond. Ser. A 355(1732), 2117-2136 (1997) · Zbl 0895.73029
[39] Krömer, S.; Valdman, J., Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms, Math. Mech. Solids, 24, 11, 3644-3673 (2019) · Zbl 07273387
[40] Langer, J.; Singer, DA, Curve straightening and a minimax argument for closed elastic curves, Topology, 24, 1, 75-88 (1985) · Zbl 0561.53004
[41] Langer, J.; Singer, DA, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev., 38, 4, 605-618 (1996) · Zbl 0859.73040
[42] Le Tallec, P.; Mani, S.; Rochinha, FA, Finite element computation of hyperelastic rods in large displacements, RAIRO Modél. Math. Anal. Numér., 26, 5, 595-625 (1992) · Zbl 0758.73048
[43] Levien, R.: The elastica: a mathematical history. Technical Report UCB/EECS-2008-103, EECS Department, University of California, Berkeley (2008)
[44] Lin, C-C; Schwetlick, HR, On the geometric flow of Kirchhoff elastic rods, SIAM J. Appl. Math., 65, 2, 720-736 (2004) · Zbl 1074.74039
[45] Maddocks, J.H.: Bifurcation theory, symmetry breaking and homogenization in continuum mechanics descriptions of DNA. Mathematical modelling of the physics of the double helix. In: A Celebration of Mathematical Modeling, pp. 113-136. Kluwer, Dordrecht (2004)
[46] Manhart, A.; Oelz, D.; Schmeiser, C.; Sfakianakis, N., An extended filament based lamellipodium model produces various moving cell shapes in the presence of chemotactic signals, J. Theor. Biol., 382, 244-258 (2015) · Zbl 1343.92075
[47] Manning, RS; Maddocks, JH, Symmetry breaking and the twisted elastic ring, Comput. Methods Appl. Mech. Eng., 170, 3-4, 313-330 (1999) · Zbl 0949.74032
[48] Manning, R.S., Rogers, K.A., Maddocks, J.H.: Isoperimetric conjugate points with application to the stability of DNA minicircles. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1980), 3047-3074 (1998) · Zbl 1002.92507
[49] Meier, C.; Popp, A.; Wall, WA, An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods, Comput. Methods Appl. Mech. Eng., 278, 445-478 (2014) · Zbl 1423.74501
[50] Michell, J.H.: On the stability of a bent and twisted wire. Messenger Math. 11, 181-184; 1889-1890. Reprinted in [29] · JFM 22.1014.01
[51] Moffatt, HK; Ricca, RL, Helicity and the Călugăreanu invariant, Proc. R. Soc. Lond. Ser. A, 439, 1906, 411-429 (1992) · Zbl 0771.57013
[52] Mora, MG; Müller, S., Derivation of the nonlinear bending-torsion theory for inextensible rods by \(\Gamma \)-convergence, Calc. Var. Partial Differ. Equ., 18, 3, 287-305 (2003) · Zbl 1053.74027
[53] Needham, T., Kähler structures on spaces of framed curves, Ann. Glob. Anal. Geom., 54, 1, 123-153 (2018) · Zbl 1396.58012
[54] Neukirch, S.; Henderson, ME, Classification of the spatial equilibria of the clamped elastica: symmetries and zoology of solutions, J. Elast., 68, 1-3, 95-121 (2003) · Zbl 1073.74037
[55] O’Hara, J., Family of energy functionals of knots, Topol. Appl., 48, 2, 147-161 (1992) · Zbl 0769.57006
[56] O’Hara, J.: Energy of Knots and Conformal Geometry. Series on Knots and Everything, Vol. 33. World Scientific, River Edge (2003) · Zbl 1034.57008
[57] Pozzi, P.; Stinner, B., Curve shortening flow coupled to lateral diffusion, Numer. Math., 135, 4, 1171-1205 (2017) · Zbl 1369.65111
[58] Ranner, T.: A stable finite element method for low inertia undulatory locomotion in three dimensions. arXiv e-prints (2019) · Zbl 1443.74266
[59] Reiter, Ph, Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family \(E^{(\alpha )},\alpha \in [2,3)\), Math. Nachr., 285, 7, 889-913 (2012) · Zbl 1248.42009
[60] Romero, I., The interpolation of rotations and its application to finite element models of geometrically exact rods, Comput. Mech., 34, 2, 121-133 (2004) · Zbl 1138.74406
[61] Sachkov, YL, Closed Euler elasticae, Proc. Steklov Inst. Math., 278, 1, 218-232 (2012) · Zbl 1334.70030
[62] Sander, O., Geodesic finite elements for Cosserat rods, Int. J. Numer. Methods Eng., 82, 13, 1645-1670 (2010) · Zbl 1193.74157
[63] Scholtes, S., Schumacher, H., Wardetzky, M.: Variational Convergence of Discrete Elasticae. arXiv e-prints (2019)
[64] Schuricht, F.; von der Mosel, H., Euler-Lagrange equations for nonlinearly elastic rods with self-contact, Arch. Ration. Mech. Anal., 168, 1, 35-82 (2003) · Zbl 1030.74029
[65] Spillmann, J.; Teschner, M., An adaptive contact model for the robust simulation of knots, Comput. Graph. Forum, 27, 2, 497-506 (2008)
[66] Starostin, E.L.: Symmetric equilibria of a thin elastic rod with self-contacts. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362, 1317-1334 (1820), 2004 · Zbl 1091.92007
[67] Starostin, EL; van der Heijden, GHM, Theory of equilibria of elastic 2-braids with interstrand interaction, J. Mech. Phys. Solids, 64, 83-132 (2014)
[68] Starostin, EL; van der Heijden, GHM, Tightening elastic \((n, 2)\)-torus knots, J. Phys. Conf. Ser., 544, 1, 012007 (2014)
[69] Starostin, EL; van der Heijden, GHM; Blatt, S.; Reiter, Ph; Schikorra, A., Equilibria of elastic cable knots and links, New Directions in Geometric and Applied Knot Theory, 258-275 (2018), Berlin: De Gruyter, Berlin · Zbl 1421.57014
[70] Strzelecki, P.; von der Mosel, H., Tangent-point self-avoidance energies for curves, J. Knot Theory Ramif., 21, 5, 1250044-28 (2012) · Zbl 1245.57012
[71] Tobias, I.; Coleman, BD; Lembo, M., A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids, J. Chem. Phys., 105, 6, 2517-2526 (1996)
[72] Tobias, I.; Olson, WK, The effect of intrinsic curvature on supercoiling: predictions of elasticity theory, Biopolymers, 33, 4, 639-646 (1993)
[73] Tobias, I.; Swigon, D.; Coleman, BD, Elastic stability of DNA configurations. I. General theory, Phys. Rev. E (3), 61, 1, 747-758 (2000)
[74] van der Heijden, G.; Neukirch, S.; Goss, V.; Thompson, J., Instability and self-contact phenomena in the writhing of clamped rods, Int. J. Mech. Sci., 45, 1, 161-196 (2003) · Zbl 1051.74571
[75] von der Mosel, H., Minimizing the elastic energy of knots, Asymptot. Anal., 18, 1-2, 49-65 (1998) · Zbl 0935.49024
[76] Zajac, EE, Stability of two planar loop elasticas, Trans. ASME Ser. E. J. Appl. Mech., 29, 136-142 (1962) · Zbl 0106.38005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.