Numerical simulation of an extensible capsule using regularized Stokes kernels and overset finite differences

In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that enables asymptotically faster computations compared to spherical-harmonics based representations. We use a boundary integral equation formulation to represent and discretize hydrodynamic interactions. The boundary integrals are weakly singular. We use the quadrature scheme based on the regularized Stokes kernels. We also use partition-of unity based finite differences that are required for the computational of interfacial forces. Given an N-point surface discretization, our numerical scheme has fourth-order accuracy and O(N) asymptotic complexity, which is an improvement over the O(N^2 log(N)) complexity of a spherical harmonics based spectral scheme that uses product-rule quadratures. We use GPU acceleration and demonstrate the ability of our code to simulate the complex shapes with high resolution. We study capsules that resist shear and tension and their dynamics in shear and Poiseuille flows. We demonstrate the convergence of the scheme and compare with the state of the art

FMM-accelerated solvers for the Laplace-Beltrami problem on complex surfaces in three dimensions

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory,the Laplace-Beltrami operator can be pre-/post-conditioned with integral operators whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard Laplace kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.Comment: 18 pages, 5 tables, 3 figure

Numerical methods for fast simulation of a red blood cell

In this dissertation, we study Stokesian particulate flows. In particular, we are interested in the dynamics of vesicles and red blood cells (RBCs) suspended in Stokes flow. We aim to develop mathematical models and numerical techniques for accurate simulation of their dynamics in microcirculation. Vesicles are closed membranes made of a phospholipid bilayer and are filled with fluid. Red blood cells are highly deformable nucleus-free cells and have rich dynamics when subjected to viscous forcing. Understanding single RBC dynamics is a complex fluid-membrane interaction problem of fundamental importance in expanding our understanding of red blood cell suspensions. For example, one of the fundamental problems is the construction of phase diagrams for the red blood cell shapes as a function of the imposed flow and the mechanical properties of the cell. Accurate knowledge of their shape dynamics has also led to interesting approaches for cell sorting based on mechanical properties in lateral displacement devices. We model an RBC using two different models, namely, “vesicle" and “capsule". We use the term particle to refer to both of them. Vesicles are inextensible surfaces with bending resistance and serve as a good model for RBC in 2D. But in 3D, vesicles miss important features of RBC dynamics because they have zero shear resistance. In contrast, an inextensible capsule resists shear in addition to the bending and is a more accurate model of RBC in 3D. For both the particles, we use a boundary integral formulation to simulate their long time horizon dynamics using spherical harmonics based spectral singular quadratures, differentiation and reparameterization techniques. We demonstrate the full relevance of our simulations using quantitative comparisons with existing experimental results with RBCs and vesicles. Once we have verified and validated our code, we use it to study the bistability (two RBC equilibrium states depending on initial state of RBC) observed under same flow conditions in our simulations. We plot the phase diagrams of equilibrium shapes of vesicles and RBCs in confined and unconfined Poiseuille flow. Finally, we also develop a novel scheme for Stokesian particle simulation using regularized Stokes kernels and overset finite differences based on overlapping patchwise discretization of the surface. Our scheme has lower work complexity than the spherical harmonics based scheme and also exhibits a high order convergence (typically fourth order) than the quadratic convergence of the triangulation based schemes. Furthermore, the patchwise discretization approach allows for more local independent control over resolution of the different parts of the surface than the global spherical harmonics based scheme. We verify this new scheme for extensible capsule simulation by quantitative comparison with the previous results in the literature for extensible capsules. We also demonstrate easy acceleration of singular quadrature using all-pairs evaluation algorithm implemented for the GPU architecture. The GPU acceleration allows us to do long time horizon simulation of capsules of low reduced volume resulting in complex shapes. Our scheme is also easily accessible to further acceleration using the fast multipole methods (FMMs).Computational Science, Engineering, and Mathematic