Fast Stokesian dynamics.

*(English)*Zbl 1430.76397Summary: We present a new method for large scale dynamic simulation of colloidal particles with hydrodynamic interactions and Brownian forces, which we call fast Stokesian dynamics (FSD). The approach for modelling the hydrodynamic interactions between particles is based on the Stokesian dynamics (SD) algorithm [A. Sierou and J. F. Brady, ibid. 448, 115-146 (2001; Zbl 1045.76034)], which decomposes the interactions into near-field (short-ranged, pairwise additive and diverging) and far-field (long-ranged many-body) contributions. In FSD, the standard system of linear equations for SD is reformulated using a single saddle point matrix. We show that this reformulation is generalizable to a host of particular simulation methods enabling the self-consistent inclusion of a wide range of constraints, geometries and physics in the SD simulation scheme. Importantly for fast, large scale simulations, we show that the saddle point equation is solved very efficiently by iterative methods for which novel preconditioners are derived. In contrast to existing approaches to accelerating SD algorithms, the FSD algorithm avoids explicit inversion of ill-conditioned hydrodynamic operators without adequate preconditioning, which drastically reduces computation time. Furthermore, the FSD formulation is combined with advanced sampling techniques in order to rapidly generate the stochastic forces required for Brownian motion. Specifically, we adopt the standard approach of decomposing the stochastic forces into near-field and far-field parts. The near-field Brownian force is readily computed using an iterative Krylov subspace method, for which a novel preconditioner is developed, while the far-field Brownian force is efficiently computed by linearly transforming those forces into a fluctuating velocity field, computed easily using the positively split Ewald approach [A. M. Fiore et al., “Rapid sampling of stochastic displacements in Brownian dynamics simulations”, J. Chem. Phys. 146, No. 12, Article ID 124116 (2017; doi:10.1063/1.4978242)]. The resultant effect of this field on the particle motion is determined through solution of a system of linear equations using the same saddle point matrix used for deterministic calculations. Thus, this calculation is also very efficient. Additionally, application of the saddle point formulation to develop high-resolution hydrodynamic models from constrained collections of particles (similar to the immersed boundary method) is demonstrated and the convergence of such models is discussed in detail. Finally, an optimized graphics processing unit implementation of FSD for mono-disperse spherical particles is used to demonstrated performance and accuracy of dynamic simulations of \(O(10^5)\) particles, and an open source plugin for the HOOMD-blue suite of molecular dynamics software is included in the supplementary material.

##### MSC:

76M28 | Particle methods and lattice-gas methods |

76T20 | Suspensions |

76D07 | Stokes and related (Oseen, etc.) flows |

##### Keywords:

Stokesian dynamics
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\textit{A. M. Fiore} and \textit{J. W. Swan}, J. Fluid Mech. 878, 544--597 (2019; Zbl 1430.76397)

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##### References:

[1] | ,; ,, Handbook of Mathematical Function: with Formulas, Graphs and Mathematical Tables, (1965), Dover |

[2] | ,; ,, Computer Simulation of Liquids, (1989), Oxford University Press · Zbl 0703.68099 |

[3] | ,; ,; ,, General purpose molecular dynamics simulations fully implemented on graphics processing units, J. Comput. Phys., 227, 10, 5342-5359, (2008) · Zbl 1148.81301 |

[4] | ,; ,; ,; ,; ,; ,, Hydrodynamics of suspensions of passive and active rigid particles: a rigid multiblob approach, Commun. Appl. Maths Comput. Sci., 11, 2, 217-296, (2016) |

[5] | ,; ,, Accelerated Stokesian dynamics: Brownian motion, J. Chem. Phys., 118, 22, 10323-10332, (2003) |

[6] | ,; ,; ,, Numerical solution of saddle point problems, Acta Numerica, 14, 1-137, (2005) · Zbl 1115.65034 |

[7] | ,, Brownian motion, hydrodynamics, and the osmotic pressure, J. Chem. Phys., 98, 4, 3335-3341, (1993) |

[8] | ,; ,, Stokesian dynamics, Annu. Rev. Fluid Mech., 20, 111-157, (1988) |

[9] | ,; ,; ,; ,, Dynamic simulation of hydrodynamically interacting suspensions, J. Fluid Mech., 195, 257-280, (1988) · Zbl 0653.76024 |

[10] | , 2011 RCM Reverse Cuthill McKee Ordering C++ Library. https://people.sc.fsu.edu/∼jburkardt/f_src/rcm/rcm.html. |

[11] | ,; ,, Preconditioned Krylov subspace methods for sampling multivariate Gaussian distributions, SIAM J. Sci. Comput., 36, 2, A588-A608, (2014) |

[12] | ,; ,; ,, Lubrication corrections for three-particle contribution to short-time self-diffusion coefficients in colloidal dispersions, J. Chem. Phys., 111, 7, 3265-3273, (1999) |

[13] | ,; ,; ,; ,; ,, Friction and mobility of many spheres in Stokes flow, J. Chem. Phys., 100, 5, 3780-3790, (1994) |

[14] | , & , 1969Reducing the bandwidth of sparse symmetric matrices. In Proceedings of the 1969 24th National Conference, pp. 157-172. ACM. |

[15] | , , , , , & , 2014 Cusp: generic parallel algorithms for sparse matrix and graph computations. Version 0.5.0. |

[16] | ,; ,; ,, Particle mesh Ewald: an Nlog(N) method for Ewald sums in large systems, J. Chem. Phys., 98, 12, 10089-10092, (1993) |

[17] | ,; ,, Simulating Brownian suspensions with fluctuating hydrodynamics, J. Chem. Phys., 143, 24, (2015) |

[18] | ,; ,; ,, Brownian dynamics of confined rigid bodies, J. Chem. Phys., 143, 14, (2015) |

[19] | ,; ,; ,; ,; ,, Brownian dynamics without Green’s functions, J. Chem. Phys., 140, 13, (2014) |

[20] | ,; ,, Brownian dynamics with hydrodynamic interactions, J. Chem. Phys., 69, 4, 1352-1360, (1978) |

[21] | ,, Die berechnung optischer und elektrostatischer gitterpotentiale, Ann. Phys., 369, 3, 253-287, (1921) · JFM 48.0566.02 |

[22] | ,; ,, Rapid sampling of stochastic displacements in Brownian dynamics simulations with stresslet constraints, J. Chem. Phys., 148, 4, (2018) |

[23] | ,; ,; ,; ,, Rapid sampling of stochastic displacements in Brownian dynamics simulations, J. Chem. Phys., 146, 12, (2017) |

[24] | ,; ,; ,, From hindered to promoted settling in dispersions of attractive colloids: simulation, modeling, and application to macromolecular characterization, Phys. Rev. Fluids, 3, 6, (2018) |

[25] | ,, Construction of Langevin forces in the simulation of hydrodynamic interaction, Macromolecules, 19, 4, 1204-1207, (1986) |

[26] | ,; ,, Computer Solution of Large Sparse Positive Definite Systems, (1981), Prentice-Hall · Zbl 0516.65010 |

[27] | ,; ,; ,; ,; ,; ,; ,; ,, Strong scaling of general-purpose molecular dynamics simulations on GPUs, Comput. Phys. Commun., 192, 97-107, (2015) |

[28] | ,, A class of multi-symmetric polyhedra, Tohoku Math. J. (1), 43, 104-108, (1937) · JFM 63.0585.04 |

[29] | , , , , , & , 2009Multi-particle collision dynamics: a particle-based mesoscale simulation approach to the hydrodynamics of complex fluids. In Advanced Computer Simulation Approaches for Soft Matter Sciences III, pp. 1-87. Springer. |

[30] | ,, An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner, J. Comput. Phys., 228, 20, 7565-7595, (2009) · Zbl 1391.76474 |

[31] | ,; ,; ,; ,; ,, Efficient particle-mesh spreading on GPUs, Proc. Comput. Sci., 51, 120-129, (2015) |

[32] | ,, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, J. Fluid Mech., 5, 317-328, (1959) · Zbl 0086.19901 |

[33] | ,, Application of the Langevin equation to fluid suspensions, J. Fluid Mech., 72, 3, 499-511, (1975) · Zbl 0327.76044 |

[34] | ,; ,, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhys. Lett., 19, 3, 155-160, (1992) |

[35] | ,, Improvement of the Stokesian dynamics method for systems with a finite number of particles, J. Fluid Mech., 452, 231-262, (2002) · Zbl 1059.76059 |

[36] | , 1992a Programs for Stokes resistance functions. https://www.uwo.ca/apmaths/faculty/jeffrey/research/Resistance.html. |

[37] | ,, The calculation of the low Reynolds number resistance functions for two unequal spheres, Phys. Fluids A, 4, 1, 16-29, (1992) |

[38] | ,; ,, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow, J. Fluid Mech., 139, 261-290, (1984) · Zbl 0545.76037 |

[39] | ,, Fluctuating force-coupling method for simulations of colloidal suspensions, J. Comput. Phys., 269, 61-79, (2014) · Zbl 1349.76862 |

[40] | ,; ,, Microhydrodynamics: Principles and Selected Applications, (2005), Dover |

[41] | ,, The fluctuation – dissipation theorem, Rep. Prog. Phys., 29, 1, 255-284, (1966) · Zbl 0163.23102 |

[42] | ,, Hydrodynamic transport coefficients of random dispersions of hard spheres, J. Chem. Phys., 93, 5, 3484-3494, (1990) |

[43] | ,; ,; ,, Application of lattice-gas cellular automata to the Brownian motion of solids in suspension, Phys. Rev. Lett., 60, 11, 975-978, (1988) |

[44] | , & , 1969The Mathematical Theory of Viscous Incompressible Flow, vol. 12. Gordon & Breach. · Zbl 0184.52603 |

[45] | ,; ,; ,, An accurate method to include lubrication forces in numerical simulations of dense Stokesian suspensions, J. Fluid Mech., 769, 369-386, (2015) |

[46] | ,; ,, Spectrally accurate fast summation for periodic Stokes potentials, J. Comput. Phys., 229, 23, 8994-9010, (2010) · Zbl 1282.76151 |

[47] | ,; ,, Force-coupling method for particulate two-phase flow: Stokes flow, J. Comput. Phys., 184, 2, 381-405, (2003) · Zbl 1047.76100 |

[48] | ,; ,, Large scale dynamic simulation of plate-like particle suspensions. Part I. Non-Brownian simulation, J. Rheol., 52, 1, 1-36, (2008) |

[49] | ,; ,, Large scale dynamic simulation of plate-like particle suspensions. Part II. Brownian simulation, J. Rheol., 52, 1, 37-65, (2008) |

[50] | ,; ,; ,; ,, The Rotne-Prager-Yamakawa approximation for periodic systems in a shear flow, J. Chem. Phys., 140, 18, (2014) · Zbl 1325.76189 |

[51] | NVIDIA2018 CUDA Toolkit Documentation. |

[52] | ,; ,, Application of the adhesive sphere model to the structure of colloidal suspensions, J. Chem. Phys., 91, 2, 1211-1221, (1989) |

[53] | ,; ,, Erratum: application of the adhesive sphere model to the structure of colloidal suspensions [J. Chem. Phys. 9 1, 1211 (1989)], J. Chem. Phys., 92, 5, 3250-3250, (1990) |

[54] | ,; ,, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys., 50, 11, 4831-4837, (1969) |

[55] | ,, Brownian particles at different times scales: a new derivation of the Smoluchowski equation, Physica A, 188, 4, 526-552, (1992) |

[56] | ,; ,, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869, (1986) · Zbl 0599.65018 |

[57] | ,; ,; ,, A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: the sedimentation of fibers, Phys. Fluids, 17, 3, (2005) · Zbl 1187.76458 |

[58] | ,; ,, Accelerated Stokesian dynamics simulations, J. Fluid Mech., 448, 115-146, (2001) · Zbl 1045.76034 |

[59] | ,; ,; ,; ,, Large scale Brownian dynamics of confined suspensions of rigid particles, J. Chem. Phys., 147, 24, (2017) |

[60] | ,; ,, Rapid calculation of hydrodynamic and transport properties in concentrated solutions of colloidal particles and macromolecules, Phys. Fluids, 28, 1, (2016) |

[61] | ,; ,, A fast multipole method for the three-dimensional Stokes equations, J. Comput. Phys., 227, 3, 1613-1619, (2008) · Zbl 1290.76116 |

[62] | , 2002 Large scale simulations of Brownian suspensions. PhD thesis, University of Illinois at Urbana-Champaign. |

[63] | ,; ,; ,, On the viscosity of adhesive hard sphere dispersions: critical scaling and the role of rigid contacts, J. Rheol., 63, 2, 229-245, (2019) |

[64] | ,; ,, Spectral Ewald acceleration of Stokesian dynamics for polydisperse suspensions, J. Comput. Phys., 306, 443-477, (2016) · Zbl 1351.76299 |

[65] | ,; ,, Simulation of concentrated suspensions using the force-coupling method, J. Comput. Phys., 229, 6, 2401-2421, (2010) · Zbl 1303.76012 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.