An integral equation formulation for rigid bodies in Stokes flow in three dimensions

Abstract We present a new derivation of a boundary integral equation (BIE) for simulating the three-dimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in O ( n ) time, where n denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.

[1]  Johan Helsing,et al.  On the evaluation of layer potentials close to their sources , 2008, J. Comput. Phys..

[2]  Ludvig af Klinteberg,et al.  Fast Ewald summation for Stokesian particle suspensions , 2014 .

[3]  George Biros,et al.  A fast algorithm for simulating vesicle flows in three dimensions , 2011, J. Comput. Phys..

[4]  L. Fauci,et al.  The method of regularized Stokeslets in three dimensions : Analysis, validation, and application to helical swimming , 2005 .

[5]  Yijun Liu.  Fast Multipole Boundary Element Method: Theory and Applications in Engineering , 2009 .

[6]  George Biros,et al.  On preconditioners for the Laplace double‐layer in 2D , 2013, Numer. Linear Algebra Appl..

[7]  N. Nishimura.  Fast multipole accelerated boundary integral equation methods , 2002 .

[8]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[9]  Lexing Ying,et al.  A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains , 2006, J. Comput. Phys..

[10]  C. Pozrikidis.  Boundary Integral and Singularity Methods for Linearized Viscous Flow: Index , 1992 .

[11]  Denis Zorin,et al.  A Tensor-Train accelerated solver for integral equations in complex geometries , 2015, J. Comput. Phys..

[12]  Mario Bebendorf,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Hierarchical Lu Decomposition Based Preconditioners for Bem Hierarchical Lu Decomposition Based Preconditioners for Bem , 2022 .

[13]  Lexing Ying,et al.  Hierarchical Interpolative Factorization for Elliptic Operators: Integral Equations , 2013, 1307.2666.

[14]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[15]  Adrianna Gillman,et al.  A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape , 2015, SIAM J. Sci. Comput..

[16]  Leslie Greengard,et al.  Quadrature by expansion: A new method for the evaluation of layer potentials , 2012, J. Comput. Phys..

[17]  Leslie Greengard,et al.  Integral Equation Methods for Elastance and Mobility Problems in Two Dimensions , 2015, SIAM J. Numer. Anal..

[18]  C. Pozrikidis,et al.  Boundary Integral and Singularity Methods for Linearized Viscous Flow: Green's functions , 1992 .

[19]  George Biros,et al.  High-volume fraction simulations of two-dimensional vesicle suspensions , 2013, J. Comput. Phys..

[20]  H. T. Banks,et al.  Simulations of Particle Dynamics in Magnetorheological Fluids , 1999 .

[21]  Jian-Ming Jin,et al.  Fast and Efficient Algorithms in Computational Electromagnetics , 2001 .

[22]  David Saintillan,et al.  Electrohydrodynamic interaction of spherical particles under Quincke rotation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Ivan P. Gavrilyuk.  Book Review: Fast multipole boundary element method , 2011 .

[24]  Kazufumi Ito,et al.  Regular Article: Simulations of Particle Dynamics in Magnetorheological Fluids , 1999 .

[25]  Shravan Veerapaneni,et al.  Spectrally Accurate Quadratures for Evaluation of Layer Potentials Close to the Boundary for the 2D Stokes and Laplace Equations , 2014, SIAM J. Sci. Comput..

[26]  Seppo Karrila,et al.  INTEGRAL EQUATIONS OF THE SECOND KIND FOR STOKES FLOW: DIRECT SOLUTION FOR PHYSICAL VARIABLES AND REMOVAL OF INHERENT ACCURACY LIMITATIONS , 1989 .

[27]  A. Najafi,et al.  Simple swimmer at low Reynolds number: three linked spheres. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Mary Catherine A. Kropinski,et al.  Integral equation methods for particle simulations in creeping flows , 1999 .

[29]  Anna-Karin Tornberg,et al.  A fast integral equation method for solid particles in viscous flow using quadrature by expansion , 2016, J. Comput. Phys..

[30]  Martin J. Mohlenkamp.  A fast transform for spherical harmonics , 1997 .

[31]  Steven A. Orszag,et al.  Fourier Series on Spheres , 1974 .

[32]  J. Thomas Beale,et al.  A Method for Computing Nearly Singular Integrals , 2000, SIAM J. Numer. Anal..

[33]  Martin R. Maxey,et al.  Modeling the magnetic interactions between paramagnetic beads in magnetorheological fluids , 2008, J. Comput. Phys..

[34]  D. Zorin,et al.  A kernel-independent adaptive fast multipole algorithm in two and three dimensions , 2004 .

[35]  Zydrunas Gimbutas,et al.  A Fast Algorithm for Spherical Grid Rotations and Its Application to Singular Quadrature , 2013, SIAM J. Sci. Comput..

[36]  Eric Darve,et al.  The Inverse Fast Multipole Method: Using a Fast Approximate Direct Solver as a Preconditioner for Dense Linear Systems , 2015, SIAM J. Sci. Comput..

[37]  Ramin Golestanian,et al.  Analytic results for the three-sphere swimmer at low Reynolds number. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Manas Rachh,et al.  Integral equation methods for problems in electrostatics, elastostatics and viscous flow , 2015 .

[39]  Rikard Ojala,et al.  An accurate integral equation method for simulating multi-phase Stokes flow , 2014, J. Comput. Phys..

[40]  Stephen A. Vavasis,et al.  Preconditioning for Boundary Integral Equations , 1992, SIAM J. Matrix Anal. Appl..

[41]  R. Kress.  Linear Integral Equations , 1989 .

[42]  Per-Gunnar Martinsson,et al.  An O(N) Direct Solver for Integral Equations on the Plane , 2013, 1303.5466.

[43]  Svetlana Tlupova,et al.  Nearly Singular Integrals in 3D Stokes Flow , 2013 .