318 research outputs found
Fast Ewald summation for electrostatic potentials with arbitrary periodicity
A unified treatment for fast and spectrally accurate evaluation of
electrostatic potentials subject to periodic boundary conditions in any or none
of the three space dimensions is presented. Ewald decomposition is used to
split the problem into a real space and a Fourier space part, and the FFT based
Spectral Ewald (SE) method is used to accelerate the computation of the latter.
A key component in the unified treatment is an FFT based solution technique for
the free-space Poisson problem in three, two or one dimensions, depending on
the number of non-periodic directions. The cost of calculations is furthermore
reduced by employing an adaptive FFT for the doubly and singly periodic cases,
allowing for different local upsampling rates. The SE method will always be
most efficient for the triply periodic case as the cost for computing FFTs will
be the smallest, whereas the computational cost for the rest of the algorithm
is essentially independent of the periodicity. We show that the cost of
removing periodic boundary conditions from one or two directions out of three
will only marginally increase the total run time. Our comparisons also show
that the computational cost of the SE method for the free-space case is
typically about four times more expensive as compared to the triply periodic
case. The Gaussian window function previously used in the SE method, is here
compared to an approximation of the Kaiser-Bessel window function, recently
introduced. With a carefully tuned shape parameter that is selected based on an
error estimate for this new window function, runtimes for the SE method can be
further reduced. Keywords: Fast Ewald summation, Fast Fourier transform,
Arbitrary periodicity, Coulomb potentials, Adaptive FFT, Fourier integral,
Spectral accuracy.Comment: 21 pages, 11 figure
Analytical solutions for two-dimensional singly periodic Stokes flow singularity arrays near walls
New analytical representations of the Stokes flows due to periodic arrays of point singularities in a two-dimensional no-slip channel and in the half-plane near a no-slip wall are derived. The analysis makes use of a conformal mapping from a concentric annulus (or a disc) to a rectangle and a complex variable formulation of Stokes flow to derive the solutions. The form of the solutions is amenable to fast and accurate numerical computation without the need for Ewald summation or other fast summation techniques
Fast Ewald summation for Stokes flow with arbitrary periodicity
A fast and spectrally accurate Ewald summation method for the evaluation of
stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is
presented. This work extends the previously developed Spectral Ewald method for
Stokes flow to periodic boundary conditions in any number (three, two, one, or
none) of the spatial directions, in a unified framework. The periodic potential
is split into a short-range and a long-range part, where the latter is treated
in Fourier space using the fast Fourier transform. A crucial component of the
method is the modified kernels used to treat singular integration. We derive
new modified kernels, and new improved truncation error estimates for the
stokeslet and stresslet. An automated procedure for selecting parameters based
on a given error tolerance is designed and tested. Analytical formulas for
validation in the doubly and singly periodic cases are presented. We show that
the computational time of the method scales like O(N log N) for N sources and
targets, and investigate how the time depends on the error tolerance and window
function, i.e. the function used to smoothly spread irregular point data to a
uniform grid. The method is fastest in the fully periodic case, while the run
time in the free-space case is around three times as large. Furthermore, the
highest efficiency is reached when applying the method to a uniform source
distribution in a primary cell with low aspect ratio. The work presented in
this paper enables efficient and accurate simulations of three-dimensional
Stokes flow with arbitrary periodicity using e.g. boundary integral and
potential methods.Comment: 54 pages, 15 figure
Fast, High-order Algorithms for Simulating Vesicle Flows Through Periodic Geometries.
This dissertation presents a new boundary integral equation (BIE) method for simulating vesicle flows through periodic geometries. We begin by describing the periodization scheme, in the absence of vesicles, for singly and doubly periodic geometries in 2 dimensions and triply periodic geometries in three dimensions. Later, the periodization scheme will be expanded to include multiple vesicles confined by singly periodic channels of arbitrary shape. Rather than relying on the periodic Greenâs function as classical BIE methods do, the method combines the free-space Greenâs function with a small auxiliary basis and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms to handle a large number of vesicles in a geometrically complex domain. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to O(N) per time step where N is the spatial discretization size. We include two example applications that utilize BIE methods with periodic boundary conditions. The first seeks to determine the equilibrium shapes of periodic planar elastic membranes. The second models the opening and closing of mechanosensitive (MS) channels on the membrane of a vesicle when exposed to shear stress while passing through a constricting channel.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135778/1/gmarple_1.pd
A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions
We present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Greenâs functions. We describe fast solvers for the twoâdimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed ânearâ part plus a small number of auxiliary sources that represent the (smooth) remaining âfarâ contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rankâ1 or rankâ3 correction and a Schur complement, leaves a wellâconditioned square system that can be solved iteratively using fast multipole acceleration plus a lowârank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimensionâ and PDEâindependent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle closeâtoâtouching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against highâaccuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 104 randomly located inclusions per unit cell. © 2018 Wiley Periodicals, Inc.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/146333/1/cpa21759.pd
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