A fast multipole method for the three-dimensional Stokes equations

Many problems in Stokes flow (and linear elasticity) require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions. In the fluid mechanics setting, these are typically the Stokeslet (the kernel of the single layer potential) or the Stresslet (the kernel of the double layer potential). In this paper, we present a simple and efficient method for the rapid evaluation of such fields, using a decomposition into a small number of Coulombic N-body problems, following an approach similar to that of Fu and Rodin [Y. Fu, G.J. Rodin, Fast solution methods for three-dimensional Stokesian many-particle problems, Commun. Numer. Meth. En. 16 (2000) 145–149]. While any fast summation algorithm for Coulombic interactions can be employed, we present numerical results from a scheme based on the most modern version of the fast multipole method [H. Cheng, L. Greengard, V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys. 155 (1999) 468–498]. This approach should be of value in both the solution of boundary integral equations and multiparticle dynamics.     

A numerical method for soluble surfactants on moving interfaces

We present a numerical method modeling soluble surfactants on deforming interfaces. The method uses an explicit Eulerian discretization of the interface allowing the use of standard finite difference schemes to solve coupled time-dependent differential equations for the concentration of surfactant on the interface and for the concentration of surfactant in the bulk. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive kernel-split quadrature method for parameter-dependent layer potentials

Advances in Computational Mathematics - It is well known in the literature that standard hierarchical matrix ( ${\mathscr{H}}$ -matrix)-based methods, although very efficient for asymptotically...     

Fast Ewald summation for Stokesian particle suspensions

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to , where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Raman spectra of rubidium dihydrogen arsenate in its paraelectric phase1

The Raman spectra of RbH₂AsO₄, are presented as a function of temperature for the paraelectric phase and are used to investigate the molecular structure of this phase and to investigate the coupled proton—phonon mode responsible for the onset of the ferroelectric phase.     

Soft mode behavior in RbH2AsO4

The low frequency xy Raman spectra of RbH₂AsO₄ are reported as a function of temperature in the paraelectric phase and are fitted to a coupled damped harmonic oscillator model to yield the soft mode behavior of the ferroelectric model.     

An integral equation method for closely interacting surfactant-covered droplets in wall-confined Stokes flow

A highly accurate method for simulating surfactant-covered droplets in two-dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for closely interacting droplets. The problem is considered in a periodic setting and Ewald decompositions for the Stokeslet and stresslet are derived. Computations are accelerated using the spectral Ewald method. The time evolution is handled with a fourth-order, adaptive, implicit-explicit time-stepping scheme. The numerical method is tested through several convergence studies and other challenging examples and is shown to handle drops in close proximity both to other drops and solid objects to high accuracy.     

Spectral accuracy in fast Ewald-based methods for particle simulations

A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.