Accelerating the calculation of dipolar interactions in particle based simulations with open boundary conditions by means of the P2NFFT method

Abstract

Magnetic gels are soft elastic materials consisting of magnetic particles embedded in a polymer network. Their shape and elasticity can be controlled by an external magnetic field, which gives rise to both, engineering and biomedical applications. Computer simulations are a commonly used tool to study these materials. A well-known bottleneck of these simulations is the demanding calculation of dipolar interactions. Under periodic boundary conditions established algorithms are available for doing this, however, at the expense of restricting the way in which the gels can deform in an external magnetic field. Moreover, the magnetic properties depend on the sample shape, ruling out periodic boundary conditions entirely for some research questions. In this article we will employ the recently developed dipolar variant of the P 2 NFFT method that is able to calculate dipolar interactions under open boundary conditions with an N log N scaling in the number of particles, rather than the expensive N 2 scaling of a direct summation of pair forces. The dipolar P 2 NFFT method has been implemented within the ScaFaCoS library. The molecular dynamics software ESPResSo has been extended to make use of the library.After a short summary of the method, we will discuss its value for studying magnetic soft matter systems. A particular focus is put on developing a tuning strategy to reach the best performance of the method at a predefined accuracy, and lastly applying the method to a magnetic gel model. Here, adapting to the gel's change in shape during the course of a simulation is of particular interest.

TheENUFmethod—Ewald summation based onnonuniformfast Fourier transform: Implementation, parallelization, and application

“…For the ENUF method, besides the parameters used in the standard Ewald summation method (Equation (31) and Equation (32)), there are additional two parameters from NFFT controlling the approximation errors: the oversampling factor σ s and the number of terms m in spatial domain approximation. [20,21,23] These two parameters are regarded as "knobs"…”

“…The implementation of the ENUF-DPD method uses similar parameters as the ENUF method does, and the correlations between these physical parameters are described by Equations (31) and (32) with predetermined oversampling factor σ s and parameter m controlling the number of terms in spatial domain approximation. [20,21,23] However, due to the fact that n c should be an integer and r c , which is the cutoff for the short-ranged part of electrostatic interactions between charged dissipative particles, should be a suitable value for the link-cell list update scheme in DPD simulations, we adopted another procedure to determine these parameters for the ENUF-DPD method.…”

“…[21] For representative window functions mentioned above, it was found that for a fixed oversampling factor, σ s > 1, the truncation error decays exponentially with m . [20,23,[48][49][50] For a finite number of Fourier coefficientsf k C with k I M , we wish to evaluate the trigonometric polynomial f x ð Þ=…”

“…With given Fourier coefficientsf , the Fourier samples f can be transformed with suitable FFT algorithms in both directions. More in-depth discussion and technical details can be found in publications [20,23,24,26,[48][49][50] and references therein. [20,24,26] which is similar to the other particle-mesh based schemes in handling the long-ranged part of electrostatic interactions.…”

“…The ENUF method is easy-to-implement and efficient for calculating long range electrostatic interactions, and additionally, both electrostatic energy and momentum are conserved to floating point accuracy. Indeed, the ENUF method is the starting point of the subsequent development of particle-particle NFFT with periodic boundary condition [23]. By choosing a set of optimal physical parameters, the ENUF method gives a good precision as desired and bears a computational complexity of O NlogN ð Þ .…”