Semi-implicit methods for the dynamics of elastic sheets. (English) Zbl 1453.74081
Summary: Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints that arise in explicit methods while avoiding much of the complexity of fully-implicit approaches. For a triangular lattice discretization with stretching and bending springs, our semi-implicit approach involves discrete Laplacian and biharmonic operators, and is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced.
MSC:
| 74S20 | Finite difference methods applied to problems in solid mechanics |
| 74K35 | Thin films |
| 74B05 | Classical linear elasticity |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Software:
RODASReferences:
| [1] | Calladine, C. R., Theory of Shell Structures (1983), Cambridge University Press · Zbl 0507.73079 |
| [2] | Freund, L. B.; Suresh, S., Thin Film Materials: Stress, Defect Formation and Surface Evolution (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1152.74301 |
| [3] | Armon, Shahaf; Efrati, Efi; Kupferman, Raz; Sharon, Eran, Geometry and mechanics in the opening of chiral seed pods, Science, 333, 6050, 1726-1730 (2011) |
| [4] | Audoly, B.; Boudaoud, A., Self-similar structures near boundaries in strained systems, Phys. Rev. Lett., 91, 8 (2003) |
| [5] | Santangelo, C. D., Buckling thin disks and ribbons with non-Euclidean metrics, Europhys. Lett., 86, 3 (2009) |
| [6] | Gemmer, John A.; Venkataramani, Shankar C., Shape selection in non-Euclidean plates, Physica D, 240, 19, 1536-1552 (2011) · Zbl 1419.74159 |
| [7] | Sharon, Eran, Swell approaches for changing polymer shapes, Science, 335, 6073, 1179-1180 (2012) |
| [8] | Sharon, Eran; Aharoni, Hillel, Frustrated shapes, Nat. Mater., 15, 7, 707-709 (2016) |
| [9] | Efrati, E.; Sharon, E.; Kupferman, R., Elastic theory of unconstrained non-Euclidean plates, J. Mech. Phys. Solids, 57, 4, 762-775 (2009) · Zbl 1306.74031 |
| [10] | Efrati, Efi; Sharon, Eran; Kupferman, Raz, The metric description of elasticity in residually stressed soft materials, Soft Matter, 9, 34, 8187-8197 (2013) |
| [11] | Na, Jun-Hee; Evans, Arthur A.; Bae, Jinhye; Chiappelli, Maria C.; Santangelo, Christian D.; Lang, Robert J.; Hull, Thomas C.; Hayward, Ryan C., Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers, Adv. Mater., 27, 1, 79-85 (2015) |
| [12] | Klein, Yael; Efrati, Efi; Sharon, Eran, Shaping of elastic sheets by prescription of non-Euclidean metrics, Science, 315, 5815, 1116-1120 (2007) · Zbl 1226.74013 |
| [13] | Kim, Jungwook; Hanna, James A.; Byun, Myunghwan; Santangelo, Christian D.; Hayward, Ryan C., Designing responsive buckled surfaces by halftone gel lithography, Science, 335, 6073, 1201-1205 (2012) · Zbl 1355.74027 |
| [14] | Liang Wu, Zi; Moshe, Michael; Greener, Jesse; Therien-Aubin, Heloise; Nie, Zhihong; Sharon, Eran; Kumacheva, Eugenia, Three-dimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal stresses, Nat. Commun., 4, 1586 (2013) |
| [15] | Ware, T. H.; McConney, M. E.; Wie, J. J.; Tondiglia, V. P.; White, T. J., Voxelated liquid crystal elastomers, Science, 347, 6225, 982-984 (2015) |
| [16] | White, Timothy J.; Broer, Dirk J., Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers, Nat. Mater., 14, 11, 1087-1098 (2015) |
| [17] | Sydney Gladman, A.; Matsumoto, Elisabetta A.; Nuzzo, Ralph G.; Mahadevan, L.; Lewis, Jennifer A., Biomimetic 4D printing, Nat. Mater., 15, 4, 413 (2016) |
| [18] | Lin, Yen-Chih; Sun, Ji-Ming; Hsiao, Jen-Hao; Hwu, Yeukuang; Wang, C. L.; Hong, Tzay-Ming, Spontaneous emergence of ordered phases in crumpled sheets, Phys. Rev. Lett., 103, Article 263902 pp. (2009) |
| [19] | Roman, B.; Bico, J., Elasto-capillarity: deforming an elastic structure with a liquid droplet, J. Phys. Condens. Matter, 22, 49, Article 493101 pp. (2010) |
| [20] | Aharoni, Hillel; Sharon, Eran, Direct observation of the temporal and spatial dynamics during crumpling, Nat. Mater., 9, 12, 993-997 (2010) |
| [21] | Paulsen, Joseph D.; Demery, Vincent; Bugra Toga, K.; Qiu, Zhanlong; Russell, Thomas P.; Davidovitch, Benny; Menon, Narayanan, Geometry-driven folding of a floating annular sheet, Phys. Rev. Lett., 118, 4, Article 27 pp. (2017) |
| [22] | Yoshida, R.; Takahashi, T.; Yamaguchi, T.; Ichijo, H., Self-oscillating gel, J. Am. Chem. Soc., 118, 21, 5134-5135 (1996) |
| [23] | Maeda, Shingo; Hara, Yusuke; Sakai, Takamasa; Yoshida, Ryo; Hashimoto, Shuji, Self-walking gel, Adv. Mater., 19, 21, 3480 (2007) |
| [24] | Tabata, O.; Hirasawa, H.; Aoki, S.; Yoshida, R.; Kokufuta, E., Ciliary motion actuator using self-oscillating gel, Sens. Actuators A, Phys., 95, 2-3, 234-238 (2002) |
| [25] | Tabata, O.; Kojima, H.; Kasatani, T.; Isono, Y.; Yoshida, R., Chemo-mechanical actuator using self-oscillating gel for artificial cilia, (MEMS-03: IEEE the Sixteenth Annual International Conference on Micro Electro Mechanical Systems, Proceedings: IEEE Micro Electro Mechanical Systems (2003), IEEE), 12-15 |
| [26] | Maeda, Shingo; Hara, Yusuke; Yoshida, Ryo; Hashimoto, Shuji, Peristaltic motion of polymer gels, Angew. Chem., Int. Ed. Engl., 47, 35, 6690-6693 (2008) |
| [27] | Shiraki, Yusuke; Yoshida, Ryo, Autonomous intestine-like motion of tubular self-oscillating gel, Angew. Chem., Int. Ed. Engl., 51, 25, 6112-6116 (2012) |
| [28] | Yashin, V. V.; Balazs, A. C., Modeling polymer gels exhibiting self-oscillations due to the Belousov-Zhabotinsky reaction, Macromolecules, 39, 6, 2024-2026 (2006) |
| [29] | Kuksenok, Olga; Yashin, Victor V.; Balazs, Anna C., Mechanically induced chemical oscillations and motion in responsive gels, Soft Matter, 3, 9, 1138-1144 (2007) |
| [30] | Yashin, Victor V.; Balazs, Anna C., Theoretical and computational modeling of self-oscillating polymer gels, J. Chem. Phys., 126, 12, Article 124707 pp. (2007) |
| [31] | Kuksenok, Olga; Deb, Debabrata; Dayal, Pratyush; Balazs, Anna C., Modeling chemoresponsive polymer gels, Annu. Rev. Chem. Biomol. Eng., 5, 1, 35-54 (2014), PMID: 2449895424498954 |
| [32] | Boncheva, Mila; Andreev, Stefan A.; Mahadevan, L.; Winkleman, Adam; Reichman, David R.; Prentiss, Mara G.; Whitesides, Sue; Whitesides, George M., Magnetic self-assembly of three-dimensional surfaces from planar sheets, Proc. Natl. Acad. Sci. USA, 102, 11, 3924-3929 (2005) |
| [33] | Alben, Silas; Brenner, Michael P., Self-assembly of flat sheets into closed surfaces, Phys. Rev. E, 75, 5, Article 056113 pp. (2007) |
| [34] | Alben, Silas; Balakrisnan, Bavani; Smela, Elisabeth, Edge effects determine the direction of bilayer bending, Nano Lett., 11, 6, 2280-2285 (2011) |
| [35] | Alben, Silas, Bending of bilayers with general initial shapes, Adv. Comput. Math., 41, 1, 1-22 (2015) · Zbl 1431.74083 |
| [36] | Veerapaneni, Shravan K.; Rahimian, Abtin; Biros, George; Zorin, Denis, A fast algorithm for simulating vesicle flows in three dimensions, J. Comput. Phys., 230, 14, 5610-5634 (2011) · Zbl 1419.76475 |
| [37] | Rosenbrock, H. H., Some general implicit processes for the numerical solution of differential equations, Comput. J., 5, 4, 329-330 (1963) · Zbl 0112.07805 |
| [38] | Gottlieb, David; Orszag, Steven A., Numerical Analysis of Spectral Methods: Theory and Applications, vol. 26 (1977), SIAM · Zbl 0412.65058 |
| [39] | Hou, Thomas Y.; Lowengrub, John S.; Shelley, Michael J., Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114, 2, 312-338 (1994) · Zbl 0810.76095 |
| [40] | Desbrun, Mathieu; Schröder, Peter; Barr, Alan, Interactive animation of structured deformable objects, (Graphics Interface, vol. 99 (1999)), 10 |
| [41] | Eberhardt, Bernhard; Etzmuß, Olaf; Hauth, Michael, Implicit-explicit schemes for fast animation with particle systems, (Computer Animation and Simulation 2000 (2000), Springer), 137-151 |
| [42] | Hou, Thomas Y.; Lowengrub, John S.; Shelley, Michael J., Boundary integral methods for multicomponent fluids and multiphase materials, J. Comput. Phys., 169, 2, 302-362 (2001) · Zbl 1046.76029 |
| [43] | Selle, Andrew; Lentine, Michael; Fedkiw, Ronald, A mass spring model for hair simulation, ACM Trans. Graph., 27, 64 (2008) |
| [44] | Alben, Silas, An implicit method for coupled flow-body dynamics, J. Comput. Phys., 227, 10, 4912-4933 (2008) · Zbl 1388.76194 |
| [45] | Alben, Silas, Simulating the dynamics of flexible bodies and vortex sheets, J. Comput. Phys., 228, 7, 2587-2603 (2009) · Zbl 1158.74015 |
| [46] | Chen, Hsiao-Yu; Sastry, Arnav; van Rees, Wim M.; Vouga, Etienne, Physical simulation of environmentally induced thin shell deformation, ACM Trans. Graph., 37, 4, 146 (2018) |
| [47] | Seung, H. S.; Nelson, D. R., Defects in flexible membranes with crystalline order, Phys. Rev. A, 38, 2, 1005-1018 (1988) |
| [48] | Koiter, Warner Tjardus, On the nonlinear theory of thin elastic shells, Proc. K. Ned. Akad. Wet., Ser. B, 69, 1-54 (1966) |
| [49] | Ciarlet, Philippe G., Un modèle bi-dimensionnel non linéaire de coque analogue à celui de wt Koiter, C. R. Acad. Sci., Sér. 1 Math., 331, 5, 405-410 (2000) · Zbl 0994.74044 |
| [50] | Vetter, Roman; Stoop, Norbert; Jenni, Thomas; Wittel, Falk K.; Herrmann, Hans J., Subdivision shell elements with anisotropic growth, Int. J. Numer. Methods Eng., 95, 9, 791-810 (2013) · Zbl 1352.74179 |
| [51] | Efrati, Efi; Klein, Yael; Aharoni, Hillel; Sharon, Eran, Spontaneous buckling of elastic sheets with a prescribed non-Euclidean metric, Physica D, 235, 1-2, 29-32 (2007) |
| [52] | Nocedal, Jorge; Wright, Stephen, Numerical Optimization (2006), Springer Science & Business Media · Zbl 1104.65059 |
| [53] | Hairer, Ernst; Norsett, Syvert P.; Wanner, Gerhard, Solving Ordinary Differential Equations II: Stiff Problems (1987), Springer · Zbl 0638.65058 |
| [54] | Gompper, Gerhard; Kroll, Daniel M., Fluctuations of polymerized, fluid and hexatic membranes: continuum models and simulations, Curr. Opin. Colloid Interface Sci., 2, 4, 373-381 (1997) |
| [55] | Lobkovsky, Alexander E.; Witten, T. A., Properties of ridges in elastic membranes, Phys. Rev. E, 55, 2, 1577 (1997) |
| [56] | Lidmar, Jack; Mirny, Leonid; Nelson, David R., Virus shapes and buckling transitions in spherical shells, Phys. Rev. E, 68, 5, Article 051910 pp. (2003) |
| [57] | Vliegenthart, G. A.; Gompper, G., Forced crumpling of self-avoiding elastic sheets, Nat. Mater., 5, 3, 216 (2006) |
| [58] | Katifori, Eleni; Alben, Silas; Cerda, Enrique; Nelson, David R.; Dumais, Jacques, Foldable structures and the natural design of pollen grains, Proc. Natl. Acad. Sci. USA, 107, 17, 7635-7639 (2010) |
| [59] | Couturier, Etienne; Dumais, J.; Cerda, E.; Katifori, Elenei, Folding of an opened spherical shell, Soft Matter, 9, 34, 8359-8367 (2013) |
| [60] | Funkhouser, Chloe M.; Sknepnek, Rastko; Olvera, Monica; Cruz, De La, Topological defects in the buckling of elastic membranes, Soft Matter, 9, 1, 60-68 (2013) |
| [61] | Wan, Duanduan; Nelson, David R.; Bowick, Mark J., Thermal stiffening of clamped elastic ribbons, Phys. Rev. B, 96, 1, Article 014106 pp. (2017) |
| [62] | Nelson, David R.; Piran, Tsvi; Weinberg, Steven, Statistical Mechanics of Membranes and Surfaces (2004), World Scientific · Zbl 1059.82002 |
| [63] | Schmidt, Bernd; Fraternali, Fernando, Universal formulae for the limiting elastic energy of membrane networks, J. Mech. Phys. Solids, 60, 1, 172-180 (2012) · Zbl 1244.74080 |
| [64] | DiDonna, Brian Anthony, Scaling of the buckling transition of ridges in thin sheets, Phys. Rev. E, 66, 1, Article 016601 pp. (2002) |
| [65] | Weiner, Joel L., On a problem of Chen, Willmore et al., Indiana Univ. Math. J., 27, 1, 19-35 (1978) · Zbl 0343.53038 |
| [66] | Efrati, Efi; Sharon, Eran; Kupferman, Raz, Buckling transition and boundary layer in non-Euclidean plates, Phys. Rev. E, 80, 1 (2009) |
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.
