Spectral accuracy in fast Ewald-based methods for particle simulations

Lindbo, Dag; Tornberg, Anna‐Karin

doi:10.1016/j.jcp.2011.08.022

Cited by 57 publications

(149 citation statements)

References 29 publications

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“…This support P is tightly related to the free parameter η, which controls the width of the Gaussians. As detailed by Lindbo & Tornberg [33], writing…”

Section: Ifftmentioning

confidence: 99%

“…With this alternative definition of Q we still use the estimates (32) and (33). To illustrate that this works in practice, we set up a system of 4000 quadrature points distributed on 5 particles randomly distributed in a 1 3 box, with maximum radius R = 0.1 and aspect ratios between 3:4 and 4:3.…”

Section: Estimates For Quadrature Points On Surfacesmentioning

confidence: 99%

“…We apply a unit downward force to the particles and solve for q. We then compute the Ewald truncation errors (compared to a fully converged reference result) when evaluating the double layer potential W j [Γ α , q] at the quadrature points for a range of ξ,r c ,K. The results, shown in Figure 4, indicate that the estimates (32) and (33) together with the new definition of Q (34) actually seem to give a rather close upper bound on the error in max norm, while they overestimate the error in 2-norm by roughly one order of magnitude. This indicates that our estimates, though derived as approximations of a statistical measure, are of large practical importance.…”

Section: Estimates For Quadrature Points On Surfacesmentioning

confidence: 99%

See 2 more Smart Citations

Fast Ewald summation for Stokesian particle suspensions

Klinteberg

Tornberg

2014

Numerical Methods in Fluids

118

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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 GMRES accelerated by the spectral Ewald (SE) method, which reduces the computational complexity to O(N log N ), 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.

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“…This support P is tightly related to the free parameter η, which controls the width of the Gaussians. As detailed by Lindbo & Tornberg [33], writing…”

Section: Ifftmentioning

confidence: 99%

Section: Estimates For Quadrature Points On Surfacesmentioning

confidence: 99%

Section: Estimates For Quadrature Points On Surfacesmentioning

confidence: 99%

See 1 more Smart Citation

Fast Ewald summation for Stokesian particle suspensions

Klinteberg

Tornberg

2014

Numerical Methods in Fluids

118

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“…Most of them, as for instance [1][2][3][4][5], are based on the so called Ewald summation approach [6], which splits the slowly converging sum into two rapidly converging parts, where the underlying order of summation takes a central role. The one part is a sum in spatial domain and can be evaluated efficiently.…”

Section: Introductionmentioning

confidence: 99%

“…Most methods use B-splines in order to perform this grid based approximation step, as for example the well-established particle-particle particle-mesh (P 3 M) method [1,4]. Also approximations via a Gaussian have already been considered, see Lindbo and Tornberg [5]. The particle-particle NFFT (P 2 NFFT) approach, which was suggested in Hedman and Laaksonen [9] and Pippig and Potts [7], is based on the FFT for nonequispaced data (NFFT) and allows the usage of various types of approximating window functions, as for example also (Kaiser-)Bessel functions besides B-splines and Gaussian.…”

Section: Introductionmentioning

confidence: 99%

Parameter Tuning for the NFFT Based Fast Ewald Summation

Nestler

2016

Front. Phys.

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The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P 2 NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. Typically B-splines are applied in the scope of particle mesh methods, as for instance within the well-known particle-particle particle-mesh (P 3 M) algorithm. The publicly available P 2 NFFT algorithm allows the application of an oversampled FFT as well as the usage of different window functions. We consider for the first time also an approximation by Bessel functions and show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that, if the parameters are tuned appropriately, the Bessel window function is in many cases even the better choice in terms of computational costs. Moreover, the results indicate that it is often advantageous in terms of efficiency to spend some oversampling within the NFFT while using a window function with a smaller support.

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A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

Barnett

Marple

Veerapaneni

et al. 2018

Comm Pure Appl Math

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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 Green's functions. We describe fast solvers for the two-dimensional (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 "near" part plus a small number of auxiliary sources that represent the (smooth) remaining "far" contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank-1 or rank-3 correction and a Schur complement, leaves a well-conditioned square system that can be solved iteratively using fast multipole acceleration plus a low-rank 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 dimension-and PDE-independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close-to-touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against highaccuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 10 4 randomly located inclusions per unit cell.

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Spectral accuracy in fast Ewald-based methods for particle simulations

Cited by 57 publications

References 29 publications

Fast Ewald summation for Stokesian particle suspensions

Fast Ewald summation for Stokesian particle suspensions

Parameter Tuning for the NFFT Based Fast Ewald Summation

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

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