Boundary Integral Methods for Stokes Flow: Quadrature Techniques and Fast Ewald Methods

Fluid phenomena dominated by viscous effects can be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the three-dimensional problem to two-dimensional integral equations over the boundaries of the domain. Hence for the study of objects immersed in a fluid, such as drops or elastic/solid particles, integral equations are to be discretized over the surfaces of these objects only. However, as outer boundaries or confinements are added these must also be included in the formulation.

An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations, e.g., the so-called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type \(1/|x|\). To assess the convergence properties of this quadrature rule a theoretical analysis has been performed. The slightly more complicated singularity of the Stokeslet required certain modifications of the integration rule developed for \(1/|x|\).

Another difficulty associated with boundary integral problems is introduced by periodic boundary conditions. For a set of particles in a periodic domain, periodicity is imposed by requiring that the motion of each particle has an added contribution from all periodic images of all particles all the way up to infinity. This leads to an infinite sum which is not absolutely convergent, and an additional physical constraint which removes the divergence needs to be imposed. The sum is decomposed into two fast converging sums, one that handles the short-range interactions in real space and the other that sums up the long-range interactions in Fourier space. Such decompositions are already available in the literature for kernels that are commonly used in boundary integral formulations. Here a decomposition in faster decaying sums than the ones present in the literature is derived for the periodic kernel of the stress tensor.

The computational complexity of the sums, regardless of the decomposition they stem from, is \(O(N^{2})\). This complexity can be lowered using a fast summation method which we introduce for simulating a sedimenting fiber suspension. The fast summation method was initially designed for point particles, which could be used for fibers discretized numerically almost without any changes. However, when two fibers are very close to each other, analytical integration is used to eliminate numerical inaccuracies due to the nearly singular behavior of the kernel and the real space part in the fast summation method is modified to allow for this analytical treatment.