A robust computational framework for simulating the dynamics of large assemblies of highly-flexible fibers immersed in viscous flow. (English) Zbl 07640540
Summary: The dynamic response of flexible filaments immersed in viscous fluids is important in cell mechanics, as well as other biological and industrial processes. In this paper, we propose a parallel computational framework to simulate the fluid-structure interactions in large assemblies of highly-flexible filaments immersed in a viscous fluid. We model the deformation of each filament in 3D with a \(C^1\) geometrically-exact large-deformation finite-element beam formulation and we describe the hydrodynamic interactions by a boundary element discretization of the Stokeslet model. We incorporate a contact algorithm that prevents fiber interpenetration and avoids previously reported numerical instabilities in the flow, thus providing the ability to describe the complex evolution of large clouds of fibers over long time spans. In order to support the required long-term integration, we use implicit integration of the solid-fluid-contact coupling. We address the challenges associated with the solution of the large and dense linear system for the hydrodynamic interactions by taking advantage of the massive parallelization offered by Graphic Processing Units (GPUs), which we test up to 1000 fibers and 45000 degrees of freedom.
We validate the framework against the well-established response of the sedimentation of a single fiber under gravity in the low to moderate flexibility range. We then reproduce previous results and provide additional insights in the large to extreme flexibility range. Finally, we apply the framework to the analysis of the sedimentation of large clouds of filaments under gravity, as a function of fiber flexibility. Owing to the long time spans afforded by our computational framework, our simulations reproduce the breakup response observed experimentally in the lower flexibility range and provide new insights into the breakup of the initial clouds in the higher flexibility range.
We validate the framework against the well-established response of the sedimentation of a single fiber under gravity in the low to moderate flexibility range. We then reproduce previous results and provide additional insights in the large to extreme flexibility range. Finally, we apply the framework to the analysis of the sedimentation of large clouds of filaments under gravity, as a function of fiber flexibility. Owing to the long time spans afforded by our computational framework, our simulations reproduce the breakup response observed experimentally in the lower flexibility range and provide new insights into the breakup of the initial clouds in the higher flexibility range.
MSC:
| 76Dxx | Incompressible viscous fluids |
| 76Mxx | Basic methods in fluid mechanics |
| 76Txx | Multiphase and multicomponent flows |
Keywords:
flexible filaments in Stokes flow; large-deformation beam elements; beam contact; fluid-structure interaction; boundary element method; graphic processing units (GPU)References:
| [1] | Head, D. A.; Levine, A. J.; MacKintosh, F. C., Deformation of cross-linked semiflexible polymer networks, Phys. Rev. Lett., 91, 10 (2003) |
| [2] | Pritchard, R. H.; Shery Huang, Y. Y.; Terentjev, E. M., Mechanics of biological networks: from the cell cytoskeleton to connective tissue, Soft Matter, 10, 12, 1864-1884 (2014) |
| [3] | Gardel, M. L.; Nakamura, F.; Hartwig, J. H.; Crocker, J. C.; Stossel, T. P.; Weitz, D. A., Prestressed F-actin networks cross-linked by hinged filamins replicate mechanical properties of cells, Proc. Natl. Acad. Sci. USA, 103, 6, 1762-1767 (2006) |
| [4] | Shinar, T.; Mana, M.; Piano, F.; Shelley, M. J., A model of cytoplasmically driven microtubule-based motion in the single-celled Caenorhabditis elegans embryo, Proc. Natl. Acad. Sci. USA (2011) |
| [5] | Wong, E. W.; Sheehan, P. E.; Lieber, C. M., Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes, Science, 277, 5334, 1971-1975 (1997) |
| [6] | Falvo, M. R.; Clary, G. J.; Taylor, R. M.; Chi, V.; Brooks, F. P.; Washburn, S.; Superfine, R., Bending and buckling of carbon nanotubes under large strain, Nature, 389, 6651, 582-584 (1997) |
| [7] | Jang, K. I.; Chung, H. U.; Xu, S.; Lee, C. H.; Luan, H.; Jeong, J.; Cheng, H.; Kim, G. T.; Han, S. Y.; Lee, J. W.; Kim, J.; Cho, M.; Miao, F.; Yang, Y.; Jung, H. N.; Flavin, M.; Liu, H.; Kong, G. W.; Yu, K. J.; Rhee, S. I.; Chung, J.; Kim, B.; Kwak, J. W.; Yun, M. H.; Kim, J. Y.; Song, Y. M.; Paik, U.; Zhang, Y.; Huang, Y.; Rogers, J. A., Soft network composite materials with deterministic and bio-inspired designs, Nat. Commun., 6 (2015) |
| [8] | Bharti, B.; Fameau, A. L.; Rubinstein, M.; Velev, O. D., Nanocapillarity-mediated magnetic assembly of nanoparticles into ultraflexible filaments and reconfigurable networks, Nat. Mater., 14, 11, 1104-1109 (2015) |
| [9] | Mhanna, R.; Qiu, F.; Zhang, L.; Ding, Y.; Sugihara, K.; Zenobi-Wong, M.; Nelson, B. J., Artificial bacterial flagella for remote-controlled targeted single-cell drug delivery, Small, 10, 10, 1953-1957 (2014) |
| [10] | Dreyfus, R.; Baudry, J.; Roper, M. L.; Fermigier, M.; Stone, H. A.; Bibette, J., Microscopic artificial swimmers, Nature (2005) |
| [11] | Su, T.; Purohit, P. K., Semiflexible filament networks viewed as fluctuating beam-frames, Soft Matter, 8, 17, 4664-4674 (2012) |
| [12] | Jawed, M. K.; Khouri, N. K.; Da, F.; Grinspun, E.; Reis, P. M., Propulsion and instability of a flexible helical rod rotating in a viscous fluid, Phys. Rev. Lett., 115, Article 168101 pp. (2015) |
| [13] | Batchelor, G. K., Sedimentation in a dilute dispersion of spheres, J. Fluid Mech., 52, 2, 245-268 (1972) · Zbl 0252.76069 |
| [14] | Adachi, K.; Kiriyama, S.; Yoshioka, N., The behavior of a swarm of particles moving in a viscous fluid, Chem. Eng. Sci., 33, 1, 115-121 (1978) |
| [15] | Nitsche, J. M.; Batchelor, G. K., Break-up of a falling drop containing dispersed particles, J. Fluid Mech., 340, 161-175 (1997) · Zbl 0892.76020 |
| [16] | Davis, R. H.; Acrivos, A., Sedimentation of noncolloidal particles at low Reynolds numbers, Annu. Rev. Fluid Mech., 17, 91-118 (1985) |
| [17] | Koch, D. L.; Shaqfeh, E. S., The instability of a dispersion of sedimenting spheroids, J. Fluid Mech., 209, 521-542 (1989) · Zbl 0681.76047 |
| [18] | Herzhaft, B.; Guazzelli, É.; Mackaplow, M. B.; Shaqfeh, E. S., Taylor experimental investigation of the sedimentation of a dilute fiber suspension, Phys. Rev. Lett., 77, 2, 290-293 (1996) |
| [19] | Mackaplow, M. B.; Shaqfeh, E. S., A numerical study of the sedimentation of fibre suspensions, J. Fluid Mech., 376, 149-182 (1998) · Zbl 0937.76068 |
| [20] | Herzhaft, B.; Guazzelli, É., Experimental study of the sedimentation of dilute and semi-dilute suspensions of fibres, J. Fluid Mech., 384, 133-158 (1999) · Zbl 0939.76504 |
| [21] | Butler, J. E.; Shaqfeh, E. S., Dynamic simulations of the inhomogeneous sedimentation of rigid fibres, J. Fluid Mech., 468, 205-237 (2002) · Zbl 1064.76113 |
| [22] | Saintillan, D.; Darve, E.; Shaqfeh, E. S., A smooth particle-mesh Ewald algorithm for Strokes suspension simulations: the sedimentation of fibres, Phys. Fluids, 17, 3 (2005) · Zbl 1187.76458 |
| [23] | Metzger, B.; Guazzelli, É.; Butler, J. E., Large-scale streamers in the sedimentation of a dilute fiber suspension, Phys. Rev. Lett., 95, 16, Article 2 pp. (2005) |
| [24] | Metzger, B.; Nicolas, M.; Guazzelli, É., Falling clouds of particles in viscous fluids, J. Fluid Mech., 580, 283-301 (2007) · Zbl 1113.76005 |
| [25] | Park, J.; Metzger, B.; Guazzelli, É.; Butler, J. E., A cloud of rigid fibres sedimenting in a viscous fluid, J. Fluid Mech., 648, 351-362 (2010) · Zbl 1189.76675 |
| [26] | Alben, S.; Shelley, M.; Zhang, J., Drag reduction through self-similar bending of a flexible body, Nature, 420, 479-481 (2002) |
| [27] | Gosselin, F.; De Langre, E.; Machado-Almeida, B. A., Drag reduction of flexible plates by reconfiguration, J. Fluid Mech., 650, 319-341 (2010) · Zbl 1189.76026 |
| [28] | Manghi, M.; Schlagberger, X.; Kim, Y.; Netz, R., Hydrodynamic effects in driven soft matter, Soft Matter, 2, 653-668 (2006) |
| [29] | Witten, T.; Diamant, H., A review of shaped colloidal particles in fluids: anisotropy and chirality, Rep. Prog. Phys., 83, Article 116601 pp. (2020) |
| [30] | Bukowicki, M.; Ekiel-Jeżewska, M. L., Different bending models predict different dynamics of sedimenting elastic trumbbells, Soft Matter, 14, 5786-5799 (2018) |
| [31] | du Roure, O.; Lindner, A.; Nazockdast, E. N.; Shelley, M. J., Dynamics of flexible fibers in viscous flows and fluids, Annu. Rev. Fluid Mech. (2019) · Zbl 1412.76030 |
| [32] | Tornberg, A. K.; Gustavsson, K., A numerical method for simulations of rigid fiber suspensions, J. Comput. Phys., 215, 1, 172-196 (2006) · Zbl 1140.76486 |
| [33] | Gustavsson, K.; Tornberg, A. K., Gravity induced sedimentation of slender fibers, Phys. Fluids, 21, 12, 1-15 (2009) · Zbl 1183.76230 |
| [34] | Bosse, T.; Kleiser, L.; Härtel, C.; Meiburg, E., Numerical simulation of finite Reynolds number suspension drops settling under gravity, Phys. Fluids, 17, 3, Article 037101 pp. (2005) · Zbl 1187.76064 |
| [35] | Machu, G.; Meile, W.; Nitsche, L. C.; Schaflinger, U., Coalescence, torus formation and breakup of sedimenting drops: experiments and computer simulations, J. Fluid Mech., 447, 299-336 (2001) · Zbl 0991.76501 |
| [36] | Abade, G. C.; Cunha, F. R., Computer simulation of particle aggregates during sedimentation, Comput. Methods Appl. Mech. Eng., 196, 45, 4597-4612 (2007) · Zbl 1173.76419 |
| [37] | Bülow, F.; Nirschl, H.; Dörfler, W., On the settling behaviour of polydisperse particle clouds in viscous fluids, Eur. J. Mech. B, Fluids, 50, 19-26 (2015) · Zbl 1408.76176 |
| [38] | Ayeni, O.; Tiwari, S. S.; Wu, C.; Joshi, J. B.; Nandakumar, K., Behavior of particle swarms at low and moderate Reynolds numbers using computational fluid dynamics—discrete element model, Phys. Fluids, 32, 7, Article 073304 pp. (2020) |
| [39] | Lin, Y.; Tan, J. H.; Phan-Thien, N.; Khoo, B. C., Settling of particle-suspension drops at low to moderate Reynolds numbers, Eur. J. Mech. B, Fluids, 61, 72-76 (2017) · Zbl 1408.76534 |
| [40] | Nazockdast, E.; Rahimian, A.; Zorin, D.; Shelley, M., A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics, J. Comput. Phys., 329, 173-209 (2017) · Zbl 1406.92032 |
| [41] | Schoeller, S. F.; Townsend, A. K.; Westwood, T. A.; Keaveny, E. E., Methods for suspensions of passive and active filaments, J. Comput. Phys., 424 (2021) · Zbl 07508451 |
| [42] | Xu, X.; Nadim, A., Deformation and orientation of an elastic slender body sedimenting in a viscous liquid, Phys. Fluids, 6, 9, 2889-2893 (1994) · Zbl 0829.76029 |
| [43] | Lagomarsino, M. C.; Pagonabarraga, I.; Lowe, C. P., Hydrodynamic induced deformation and orientation of a microscopic elastic filament, Phys. Rev. Lett. (2005) |
| [44] | Li, L.; Manikantan, H.; Saintillan, D.; Spagnolie, S. E., The sedimentation of flexible filaments, J. Fluid Mech. (2013) · Zbl 1294.76090 |
| [45] | Saggiorato, G.; Elgeti, J.; Winkler, R. G.; Gompper, G., Conformations, hydrodynamic interactions, and instabilities of sedimenting semiflexible filaments, Soft Matter (2015) |
| [46] | Tornberg, A.-K.; Shelley, M. J., Simulating the dynamics and interactions of flexible fibers in Stokes flows, J. Comput. Phys., 196, 1, 8-40 (2004) · Zbl 1115.76413 |
| [47] | Manikantan, H.; Saintillan, D., Effect of flexibility on the growth of concentration fluctuations in a suspension of sedimenting fibers: particle simulations, Phys. Fluids, 28, 1 (2016) |
| [48] | Maxian, O.; Mogilner, A.; Donev, A., Integral-based spectral method for inextensible slender fibers in Stokes flow, Phys. Rev. Fluids, 6, 14102-14157 (2021) |
| [49] | Schlagberger, X.; Netz, R. R., Orientation of elastic rods in homogeneous Stokes flow, Europhys. Lett., 70, 129-135 (2005) |
| [50] | Llopis, I.; Pagonabarraga, I.; Lagomarsino, M. C.; Lowe, C. P., Sedimentation of pairs of hydrodynamically interacting semiflexible filaments, Phys. Rev. E, Stat. Nonlinear Soft Matter Phys., 76, Article 061901 pp. (2007) |
| [51] | Delmotte, B.; Climent, E.; Plouraboué, F., A general formulation of Bead Models applied to flexible fibers and active filaments at low Reynolds number, J. Comput. Phys., 286, 14-37 (2015) · Zbl 1351.76294 |
| [52] | Marchetti, B.; Raspa, V.; Lindner, A.; Du Roure, O.; Bergougnoux, L.; Guazzelli, É.; Duprat, C., Deformation of a flexible fiber settling in a quiescent viscous fluid, Phys. Rev. Fluids (2018) |
| [53] | Słowicka, A. M.; Wajnryb, E.; Ekiel-Jeewska, M. L., Dynamics of flexible fibers in shear flow, J. Chem. Phys., 143, 12, Article 124904 pp. (2015) |
| [54] | Sasayama, T.; Inagaki, M., Simplified bead-chain model for direct fiber simulation in viscous flow, J. Non-Newton. Fluid Mech., 250, 52-58 (2017) |
| [55] | Sasayama, T.; Inagaki, M., Efficient bead-chain model for predicting fiber motion during molding of fiber-reinforced thermoplastics, J. Non-Newton. Fluid Mech., 264, 135-143 (2019) |
| [56] | Cyron, C. J.; Wall, W. A., Finite-element approach to Brownian dynamics of polymers, Phys. Rev. E, Stat. Nonlinear Soft Matter Phys., 80, 6 (2009) |
| [57] | Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 135, 2, 280-292 (1997) · Zbl 0898.70002 |
| [58] | Meier, C.; Popp, A.; Wall, W. A., A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods, Comput. Methods Appl. Mech. Eng., 290, 314-341 (2015) · Zbl 1423.74502 |
| [59] | Meier, C.; Popp, A.; Wall, W. A., A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation, Comput. Methods Appl. Mech. Eng., 308, 377-413 (2016) · Zbl 1439.74228 |
| [60] | Smith, D. J., A boundary element regularized Stokeslet method applied to cilia- and flagella-driven flow, Proc. R. Soc. A, Math. Phys. Eng. Sci. (2009) · Zbl 1195.76452 |
| [61] | Wriggers, P.; Zavarise, G., On contact between three-dimensional beams undergoing large deflections, Commun. Numer. Methods Eng., 13, 6, 429-438 (1997) · Zbl 0878.73063 |
| [62] | Batchelor, G. K., Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech. (1970) · Zbl 0216.52401 |
| [63] | Meier, C.; Wall, W. A.; Popp, A., A unified approach for beam-to-beam contact, Comput. Methods Appl. Mech. Eng., 315, 972-1010 (2017) · Zbl 1439.74160 |
| [64] | Stockie, J. M.; Green, S. I., Simulating the motion of flexible pulp fibres using the immersed boundary method, J. Comput. Phys., 147, 1, 147-165 (1998) · Zbl 0935.76065 |
| [65] | Lim, S.; Peskin, C. S., Simulations of the whirling instability by the immersed boundary method, SIAM J. Sci. Comput., 25, 6, 2066-2083 (2004) · Zbl 1133.65300 |
| [66] | Wiens, J. K.; Stockie, J. M., Simulating flexible fiber suspensions using a scalable immersed boundary algorithm, Comput. Methods Appl. Mech. Eng., 290, 1-18 (2015) · Zbl 1423.74290 |
| [67] | Zhu, L.; Peskin, C. S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. Comput. Phys., 179, 2, 452-468 (2002) · Zbl 1130.76406 |
| [68] | Zhu, L.; Chin, R. C.Y., Simulation of elastic filaments interacting with a viscous pulsatile flow, Comput. Methods Appl. Mech. Eng., 197, 25, 2265-2274 (2008) · Zbl 1158.76458 |
| [69] | Hua, R.-N.; Zhu, L.; Lu, X.-Y., Dynamics of fluid flow over a circular flexible plate, J. Fluid Mech., 759 (2014) |
| [70] | Kim, Y.; Peskin, C. S., Penalty immersed boundary method for an elastic boundary with mass, Phys. Fluids, 19 (2007) · Zbl 1146.76441 |
| [71] | Kim, Y.; Peskin, C. S., A penalty immersed boundary method for a rigid body in fluid, Phys. Fluids, 28, Article 033603 pp. (2016) |
| [72] | Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow (1992), Cambridge University Press · Zbl 0772.76005 |
| [73] | Cox, R. G., The motion of long slender bodies in a viscous fluid. Part 1. General theory, J. Fluid Mech., 44, 04, 791 (1970) · Zbl 0267.76015 |
| [74] | Lighthill, J., Flagellar hydrodynamics, SIAM Rev., 18, 2, 161-226 (1976) · Zbl 0366.76099 |
| [75] | Johnson, R. E., An improved slender-body theory, J. Fluid Mech., 99, 2, 411-431 (1980) · Zbl 0447.76037 |
| [76] | Cortez, R., The method of regularized stokeslets, SIAM J. Sci. Comput., 23, 4, 1204-1225 (2002) · Zbl 1064.76080 |
| [77] | Cortez, R.; Fauci, L.; Medovikov, A., The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming, Phys. Fluids (2005) · Zbl 1187.76105 |
| [78] | Cortez, R., Regularized Stokeslet segments, J. Comput. Phys., 375, 1, 783-796 (2018) · Zbl 1416.76208 |
| [79] | Smith, D. J., A nearest-neighbour discretisation of the regularized stokeslet boundary integral equation, J. Comput. Phys., 358, 88-102 (2018) · Zbl 1381.76052 |
| [80] | ΣMIT, a scalable computational framework for large-scale simulation of complex mechanical response of materials (2021) |
| [81] | Balay, S.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Eijkhout, V.; Gropp, W.; Karpeyev, D.; Kaushik, D.; Knepley, M.; McInnes, L. C.; Smith, B.; Zampini, S.; Zhang, H., Petsc users manual (2020), Argonne National Laboratory, Tech. Rep. ANL-95/11 - Revision 3.13 |
| [82] | Nvidia; Vingelmann, P.; Fitzek, F. H., Cuda, release: 10.2.89 (2020) |
| [83] | Inc, W. R., Mathematica, version 12.0.0 (2020), Champaign, IL |
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