High-volume fraction simulations of two-dimensional vesicle suspensions

Abstract

We consider numerical algorithms for the simulation of the rheology of two-dimensional vesicles suspended in a viscous Stokesian fluid. The vesicle evolution dynamics is governed by hydrodynamic and elastic forces. The elastic forces are due to local inextensibility of the vesicle membrane and resistance to bending. Numerically resolving vesicle flows poses several challenges. For example, we need to resolve moving interfaces, address stiffness due to bending, enforce the inextensibility constraint, and efficiently compute the (non-negligible) long-range hydrodynamic interactions. Our method is based on the work of {\em Rahimian, Veerapaneni, and Biros, "Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain, a boundary integral method", Journal of Computational Physics, 229 (18), 2010}. It is a boundary integral formulation of the Stokes equations coupled to the interface mass continuity and force balance. We extend the algorithms presented in that paper to increase the robustness of the method and enable simulations with concentrated suspensions. In particular, we propose a scheme in which both intra-vesicle and inter-vesicle interactions are treated semi-implicitly. In addition we use special integration for near-singular integrals and we introduce a spectrally accurate collision detection scheme. We test the proposed methodologies on both unconfined and confined flows for vesicles whose internal fluid may have a viscosity contrast with the bulk medium. Our experiments demonstrate the importance of treating both intra-vesicle and inter-vesicle interactions accurately

Stable and contact‐free time stepping for dense rigid particle suspensions

“…Therefore, preconditioners are often applied. There are a variety of preconditioners available for integral equations [13,16,18,31,64,66], we apply a simple block-diagonal preconditioner that was successfully used for vesicle suspensions [62].…”

“…However, integrands with large derivatives must be computed when bodies are in near-contact, and this is a certainty in dense suspensions. We apply an interpolation-based quadrature method [62,82] since it is efficient and extends to three dimensions, but other near-singular integration schemes are possible [7,11,30,37,39,52,73]. The same interpolation-based near-singular integration scheme is used to compute the pressure and stress, but a combination of singularity subtraction and odd-even integration [62,72] is also used to resolve high-order singularities in the integrands.…”

“…The two main sources of error are the quadrature error and time stepping error. We address the quadrature error using a combination of upsampling and interpolation, and details of the method are described in [62]. Time stepping methods have recently received a lot of attention, and some recent works address stiffness [62] and adaptive time stepping [39,63,74].…”

Machine learning acceleration of simulations of Stokesian suspensions

“…Nevertheless, the high computational cost makes simulations of vesicles at realistic concentrations extremely expensive (sometimes they can take several days on large clusters) and limits the ability to perform design optimization and parameter sweeps. Although a numerical scheme based on low resolution discretization in space and time along with a set of correction algorithms enables simulating dense vesicle flows for long time horizons [15,16] and solving a design optimization problem [17], such simulations still demand even faster algorithms [10,18,19]. Let us note that although numerous works have used machine learning to tackle computational physics problems there has been (a)A vesicle in a free-space flow (b)Vesicles in a confined flow FIG.…”

“…Boundaries do not evolve with time so the related matrices are precomputed and their application can be accelerated with fast multipole methods (FMM) [25]. [15,18,[26][27][28]. These methods can be quite expensive to be implemented in MLARM.…”