1,384 research outputs found
Sharp quadrature error bounds for the nearest-neighbor discretization of the regularized stokeslet boundary integral equation
The method of regularized stokeslets is a powerful numerical method to solve
the Stokes flow equations for problems in biological fluid mechanics. A recent
variation of this method incorporates a nearest-neighbor discretization to
improve accuracy and efficiency while maintaining the ease-of-implementation of
the original meshless method. This method contains three sources of numerical
error, the regularization error associated from using the regularized form of
the boundary integral equations (with parameter ), and two sources
of discretization error associated with the force and quadrature
discretizations (with lengthscales and ). A key issue to address is
the quadrature error: initial work has not fully explained observed numerical
convergence phenomena. In the present manuscript we construct sharp quadrature
error bounds for the nearest-neighbor discretisation, noting that the error for
a single evaluation of the kernel depends on the smallest distance ()
between these discretization sets. The quadrature error bounds are described
for two cases: with disjoint sets () being close to linear in
and insensitive to , and contained sets () being
quadratic in with inverse dependence on . The practical
implications of these error bounds are discussed with reference to the
condition number of the matrix system for the nearest-neighbor method, with the
analysis revealing that the condition number is insensitive to
for disjoint sets, and grows linearly with for contained sets.
Error bounds for the general case () are revealed to be
proportional to the sum of the errors for each case.Comment: 12 pages, 6 figure
Passively parallel regularized stokeslets
Stokes flow, discussed by G.G. Stokes in 1851, describes many microscopic
biological flow phenomena, including cilia-driven transport and flagellar
motility; the need to quantify and understand these flows has motivated decades
of mathematical and computational research. Regularized stokeslet methods,
which have been used and refined over the past twenty years, offer significant
advantages in simplicity of implementation, with a recent modification based on
nearest-neighbour interpolation providing significant improvements in
efficiency and accuracy. Moreover this method can be implemented with the
majority of the computation taking place through built-in linear algebra,
entailing that state-of-the-art hardware and software developments in the
latter, in particular multicore and GPU computing, can be exploited through
minimal modifications ('passive parallelism') to existing MATLAB computer code.
Hence, and with widely-available GPU hardware, significant improvements in the
efficiency of the regularized stokeslet method can be obtained. The approach is
demonstrated through computational experiments on three model biological flows:
undulatory propulsion of multiple C. Elegans, simulation of progression and
transport by multiple sperm in a geometrically confined region, and left-right
symmetry breaking particle transport in the ventral node of the mouse embryo.
In general an order-of-magnitude improvement in efficiency is observed. This
development further widens the complexity of biological flow systems that are
accessible without the need for extensive code development or specialist
facilities.Comment: 21 pages, 7 figures, submitte
Quantum information processing with single photons and atomic ensembles in microwave coplanar waveguide resonators
We show that pairs of atoms optically excited to the Rydberg states can
strongly interact with each other via effective long-range dipole-dipole or van
der Waals interactions mediated by their non-resonant coupling to a common
microwave field mode of a superconducting coplanar waveguide cavity. These
cavity mediated interactions can be employed to generate single photons and to
realize in a scalable configuration a universal phase gate between pairs of
single photon pulses propagating or stored in atomic ensembles in the regime of
electromagnetically induced transparency
Steady-state crystallization of Rydberg excitations in an optically driven lattice gas
We study resonant optical excitations of atoms in a one-dimensional lattice
to the Rydberg states interacting via the van der Waals potential which
suppresses simultaneous excitation of neighboring atoms. Considering two- and
three-level excitation schemes, we analyze the dynamics and stationary state of
the continuously-driven, dissipative many-body system employing time-dependent
density-matrix renormalization group (t-DMRG) simulations. We show that
two-level atoms can exhibit only nearest neighbor correlations, while
three-level atoms under dark-state resonant driving can develop finite-range
crystalline order of Rydberg excitations. We present an approximate rate
equation model whose analytic solution yields qualitative understanding of the
numerical results.Comment: 5 pages,3 figure
Efficient Implementation of Elastohydrodynamics via Integral Operators
The dynamics of geometrically non-linear flexible filaments play an important
role in a host of biological processes, from flagella-driven cell transport to
the polymeric structure of complex fluids. Such problems have historically been
computationally expensive due to numerical stiffness associated with the
inextensibility constraint, as well as the often non-trivial boundary
conditions on the governing high-order PDEs. Formulating the problem for the
evolving shape of a filament via an integral equation in the tangent angle has
recently been found to greatly alleviate this numerical stiffness. The
contribution of the present manuscript is to enable the simulation of non-local
interactions of multiple filaments in a computationally efficient manner using
the method of regularized stokeslets within this framework. The proposed method
is benchmarked against a non-local bead and link model, and recent code
utilizing a local drag velocity law. Systems of multiple filaments (1) in a
background fluid flow, (2) under a constant body force, and (3) undergoing
active self-motility are modeled efficiently. Buckling instabilities are
analyzed by examining the evolving filament curvature, as well as by
coarse-graining the body frame tangent angles using a Chebyshev approximation
for various choices of the relevant non-dimensional parameters. From these
experiments, insight is gained into how filament-filament interactions can
promote buckling, and further reveal the complex fluid dynamics resulting from
arrays of these interacting fibers. By examining active moment-driven
filaments, we investigate the speed of worm- and sperm-like swimmers for
different governing parameters. The MATLAB(R) implementation is made available
as an open-source library, enabling flexible extension for alternate
discretizations and different surrounding flows.Comment: 37 pages, 17 figure
Binary Oscillatory Crossflow Electrophoresis
Electrophoresis has long been recognized as an effective analytic technique for the separation of proteins and other charged species, however attempts at scaling up to accommodate commercial volumes have met with limited success. In this report we describe a novel electrophoretic separation technique - Binary Oscillatory Crossflow Electrophoresis (BOCE). Numerical simulations indicate that the technique has the potential for preparative scale throughputs with high resolution, while simultaneously avoiding many problems common to conventional electrophoresis. The technique utilizes the interaction of an oscillatory electric field and a transverse oscillatory shear flow to create an active binary filter for the separation of charged protein species. An oscillatory electric field is applied across the narrow gap of a rectangular channel inducing a periodic motion of charged protein species. The amplitude of this motion depends on the dimensionless electrophoretic mobility, alpha = E(sub o)mu/(omega)d, where E(sub o) is the amplitude of the electric field oscillations, mu is the dimensional mobility, omega is the angular frequency of oscillation and d is the channel gap width. An oscillatory shear flow is induced along the length of the channel resulting in the separation of species with different mobilities. We present a model that predicts the oscillatory behavior of charged species and allows estimation of both the magnitude of the induced convective velocity and the effective diffusivity as a function of a in infinitely long channels. Numerical results indicate that in addition to the mobility dependence, the steady state behavior of solute species may be strongly affected by oscillating fluid into and out of the active electric field region at the ends of the cell. The effect is most pronounced using time dependent shear flows of the same frequency (cos((omega)t)) flow mode) as the electric field oscillations. Under such conditions, experiments indicate that solute is drawn into the cell from reservoirs at both ends of the cell leading to a large mass build up. As a consequence, any initially induced mass flux will vanish after short times. This effect was not captured by the infinite channel model and hence numerical and experimental results deviated significantly. The revised model including finite cell lengths and reservoir volumes allowed quantitative predictions of the time history of the concentration profile throughout the system. This latter model accurately describes the fluxes observed for both oscillatory flow modes in experiments using single protein species. Based on the results obtained from research funded under NASA grant NAG-8-1080.S, we conclude that binary separations are not possible using purely oscillatory flow modes because of end effects associated with the cos((omega)t) mode. Our research shows, however, that a combination of cos(2(omega)t) and steady flow should lead to efficient separation free of end effects. This possibility is currently under investigation
Clustering of solutions in the random satisfiability problem
Using elementary rigorous methods we prove the existence of a clustered phase
in the random -SAT problem, for . In this phase the solutions are
grouped into clusters which are far away from each other. The results are in
agreement with previous predictions of the cavity method and give a rigorous
confirmation to one of its main building blocks. It can be generalized to other
systems of both physical and computational interest.Comment: 4 pages, 1 figur
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