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Abstract: Vesicles are locally-inextensible fluid membranes that can sustain bending. In this paper, we extend the study of Veerapaneni et al. [S.K. Veerapaneni, D. Gueyffier, G. Biros, D. Zorin, A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows, Journal of Computational Physics 228 (19) (2009) 7233–7249] to general non-axisymmetric vesicle flows in three dimensions. Although the main components of the algorithm are similar in spirit to the axisymmetric case (spectral approximation in space, semi-implicit time-stepping scheme), important new elements need to be introduced for a full 3D method. In particular, spatial quantities are discretized using spherical harmonics, and quadrature rules for singular surface integrals need to be adapted to this case; an algorithm for surface reparameterization is needed to ensure stability of the time-stepping scheme, and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs. To characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi-implicit time-stepping scheme, we perform a spectral analysis of the evolution operator on the unit sphere. By introducing these algorithmic components, we obtain a time-stepping scheme that circumvents the stability constraint on the time-step and achieves spectral accuracy in space. We present results to analyze the cost and convergence rates of the overall scheme. To illustrate the applicability of the new method, we consider a few vesicle-flow interaction problems: a single vesicle in relaxation, sedimentation, shear flows, and many-vesicle flows. [Copyright &y& Elsevier]

• Collision resolution by convex quadratic programming. • Scalable and efficient parallel algorithm and implementation. • Accurate second-kind boundary integral formulation. We describe a computational framework for simulating suspensions of rigid particles in Newtonian Stokes flow. One central building block is a collision-resolution algorithm that overcomes the numerical constraints arising from particle collisions. This algorithm extends the well-known complementarity method for non-smooth multi-body dynamics to resolve collisions in dense rigid body suspensions. This approach formulates the collision resolution problem as a linear complementarity problem with geometric 'non-overlapping' constraints imposed at each time-step. It is then reformulated as a constrained quadratic programming problem and the Barzilai-Borwein projected gradient descent method is applied for its solution. This framework is designed to be applicable for any convex particle shape, e.g., spheres and spherocylinders, and applicable to any Stokes mobility solver, including the Rotne-Prager-Yamakawa approximation, Stokesian Dynamics, and PDE solvers (e.g., boundary integral and immersed boundary methods). In particular, this method imposes Newton's Third Law and records the entire contact network. Further, we describe a fast, parallel, and spectrally-accurate boundary integral method tailored for spherical particles, capable of resolving lubrication effects. We show weak and strong parallel scalings up to 8 × 10 4 particles with approximately 4 × 10 7 degrees of freedom on 1792 cores. We demonstrate the versatility of this framework with several examples, including sedimentation of particle clusters, and active matter systems composed of ensembles of particles driven to rotate. [ABSTRACT FROM AUTHOR]

• A spectrally-accurate close evaluation scheme for Stokes boundary integral operators. • Adaptive panel refinement for arbitrarily shaped boundaries. • Graded meshes to treat corners. • Example problems from microfluidic chip design and vascular network flows. We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex—possibly nonsmooth—geometries in two dimensions. We apply the panel-based quadratures of Helsing and coworkers to evaluate to high accuracy the weakly-singular, hyper-singular, and super-singular integrals arising in the Nyström discretization, and also the near-singular integrals needed for flow and traction evaluation close to boundaries. The resulting linear system is solved iteratively via calls to a Stokes fast multipole method. We include an automatic algorithm to "panelize" a given geometry, and choose a panel order, which will efficiently approximate the density (and hence solution) to a user-prescribed tolerance. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners, or close-to-touching smooth curves. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9-digit solution accuracy. [ABSTRACT FROM AUTHOR]