Dmitry Karpeev

Mesh algorithms for PDE with Sieve I: Mesh distribution

We have developed a new programming framework, called Sieve, to support parallel numerical partial differential equation(s) (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 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.

Effective viscosity of dilute bacterial suspensions: a two-dimensional model

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 Einsteins 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.

A Model of Hydrodynamic Interaction Between Swimming Bacteria

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.

Polymer Piezoelectric Energy Harvesters for Low Wind Speed

We fabricated polymer piezoelectric energy harvesters (PEHs) that can generate electric power at wind speed of less than 4.7 m/s due to their high sensitivity to wind. In order to optimize their operating conditions, we evaluated three distinct PEH operation modes under the boundary conditions of single-side clamping. We found that a PEH connected to an external load of 120 kΩ shows the largest output power of 0.98 μW at 3.9 m/s, with wind incident on its side (mode I). We attribute this result to large bending and torsion involved in this operation mode.

A Kinetic Model for Semidilute Bacterial Suspensions

Suspensions of self-propelled microscopic particles, such as swimming bacteria, exhibit collective motion leading to remarkable experimentally observable macroscopic properties. Rigorous mathematical analysis of this emergent behavior can provide significant insight into the mechanisms behind these experimental observations; however, there are many theoretical questions remaining unanswered. In this paper, we study a coupled PDE/ODE system first introduced in the physics literature and used to investigate numerically the effective viscosity of a bacterial suspension. We then examine the kinetic theory associated with the coupled system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles (infinitesimal spheroids representing self-propelled particles) with Lennard-Jones--type repulsion. A planar shear background flow is imposed on the suspension through the novel use of Lees--Edwards quasi-periodic boundary conditions applied to a representative volume. We s...

Effective viscosity of puller-like microswimmers: a renormalization approach

Effective viscosity (EV) of suspensions of puller-like microswimmers (pullers), for example Chlamydamonas algae, is difficult to measure or simulate for all swimmer concentrations. Although there are good reasons to expect that the EV of pullers is similar to that of passive suspensions, analytical determination of the passive EV for all concentrations remains unsatisfactory. At the same time, the EV of bacterial suspensions is closely linked to collective motion in these systems and is biologically significant. We develop an approach for determining analytical EV estimates at all concentrations for suspensions of pullers as well as for passive suspensions. The proposed methods are based on the ideas of renormalization group (RG) theory and construct the EV formula based on the known asymptotics for small concentrations and near the critical point (i.e. approaching dense packing). For passive suspensions, the method is verified by comparison against known theoretical results. We find that the method performs much better than an earlier RG-based technique. For pullers, the validation is done by comparing them to experiments conducted on Chlamydamonas suspensions.

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

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 Greens 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.

Patterns and intrinsic fluctuations in semi-dilute motor-filament systems

We perform Brownian dynamics simulations of molecular motor-induced ordering and structure formations in semi-dilute cytoskeletal filament solutions. In contrast to the previously studied dilute case where binary filament interactions prevail, the semi-dilute regime is characterized by multiple motor-mediated interactions. Moreover, the forces and torques exerted by motors on filaments are intrinsically fluctuating quantities. We incorporate the influences of thermal and motor fluctuations into our model as additive and multiplicative noises, respectively. Numerical simulations reveal that filament bundles and vortices emerge from a disordered initial state. Subsequent analysis of motor noise effects reveals: i) Pattern formation is very robust against fluctuations in motor force; ii) bundle formation is associated with a significant reduction of the motor fluctuation contributions; iii) the time scale of vortex formation and coalescence decreases with increases in motor noise amplitude.