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Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires O(N^2) operations, where N is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green's function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method to evaluate the integrals in O(N) operations, with O(N) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the contest of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. The results from these test cases show that our parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications., Comment: 29 pages, 8 figures

Classical density functional theory (DFT) of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses $\mathcal{O}(N\log N)$ operations and $\mathcal{O}(N)$ memory, where $N$ is the number of grid points. This system-size scaling is significant because of the very large $N$ required for three-dimensional systems. The algorithm uses fast Fourier transforms (FFT) to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard (iterative substitution) iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT we use Fundamental Measure Theory. For the electrostatic DFT we present two algorithms. One is for the \textquotedblleft bulk-fluid\textquotedblright functional of Rosenfeld [Y. Rosenfeld. \textit{J. Chem. Phys.} 98, 8126 (1993)] that uses $\mathcal{O}(N\log N)$ operations. The other is for the \textquotedblleft reference fluid density\textquotedblright (RFD) functional [D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 (2002)]. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires $\mathcal{O}(N^{2})$ operations., Comment: 23 pages, 4 figures

We have developed a new programming framework, called Sieve, to support parallel numerical PDE algorithms operating over distributed meshes. We have also developed a reference implementation of Sieve in C++ as a library of generic algorithms operating on distributed containers conforming to the Sieve interface. Sieve makes instances of the incidence relation, or \emph{arrows}, the conceptual first-class objects represented in the containers. Further, generic algorithms acting on this arrow container are systematically used to provide natural geometric operations on the topology and also, through duality, on the data. Finally, coverings and duality are used to encode not only individual meshes, but all types of hierarchies underlying PDE data structures, including multigrid and mesh partitions. In order to demonstrate the usefulness of the framework, we show how the mesh partition data can be represented and manipulated using the same fundamental mechanisms used to represent meshes. We present the complete description of an algorithm to encode a mesh partition and then distribute a mesh, which is independent of the mesh dimension, element shape, or embedding. Moreover, data associated with the mesh can be similarly distributed with exactly the same algorithm. The use of a high level of abstraction within the Sieve leads to several benefits in terms of code reuse, simplicity, and extensibility. We discuss these benefits and compare our approach to other existing mesh libraries., Comment: 36 pages, 22 figures

We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by pushing" or "pulling" both in three- and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which, complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria., Comment: 28 pages, 20 figures

Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions., Comment: 15 pages, 3 figures, submitted to Physical Biology