Non-periodic Boundary Conditions Research Articles

NFFT based Ewald summation for electrostatic systems with charges and dipoles

The efficient computation of Coulomb interactions in charged particle systems is of great importance in the field of molecular dynamics simulations. It is widely known that an approximation can be realized based on the Ewald summation approach and the fast Fourier transform (FFT). In the present paper we consider particle systems containing a mixture of N point charges as well as point dipoles. New cutoff errors in the Ewald summation formulas concerning charge–dipole interactions are derived and, moreover, validated by numerical examples. Furthermore, we present for the first time an O(Nlog⁡N) particle mesh algorithm for computing mixed charge–dipole interactions based on the FFT for nonequispaced data (NFFT). We present first numerical results for charge–dipole systems, showing that the introduced method can be tuned to a high precision and verifying the O(Nlog⁡N) scaling. In order to calculate the interactions with dipoles efficiently, two new variants of the NFFT, namely the Hessian NFFT as well as the adjoint gradient NFFT, are derived and implemented. In the context of NFFT, these new variants are of great importance on their own. The presented particle mesh method is an extension of the particle–particle NFFT (P2NFFT) framework. Therefore, all the formerly derived P2NFFT features, which cover for instance the treatment of arbitrary combinations of periodic and non-periodic boundary conditions, the handling of triclinic box shapes and a massively parallel implementation, are now also supported for mixed charge–dipole as well as pure dipole systems. The algorithms are publicly available as a part of the ScaFaCoS library.

Measuring nanometre-scale electric fields in scanning transmission electron microscopy using segmented detectors

Electric field mapping using segmented detectors in the scanning transmission electron microscope has recently been achieved at the nanometre scale. However, converting these results to quantitative field measurements involves assumptions whose validity is unclear for thick specimens. We consider three approaches to quantitative reconstruction of the projected electric potential using segmented detectors: a segmented detector approximation to differential phase contrast and two variants on ptychographical reconstruction. Limitations to these approaches are also studied, particularly errors arising from detector segment size, inelastic scattering, and non-periodic boundary conditions. A simple calibration experiment is described which corrects the differential phase contrast reconstruction to give reliable quantitative results despite the finite detector segment size and the effects of plasmon scattering in thick specimens. A plasmon scattering correction to the segmented detector ptychography approaches is also given. Avoiding the imposition of periodic boundary conditions on the reconstructed projected electric potential leads to more realistic reconstructions.

Вычисление ВАХ для систем джозефсоновских переходов.О построении асимптотики вблизи точки излома

Solving the system of essentially nonlinear differential equations for different we obtain thecurrent-voltage characteristics (IVC) for a system of Josephson junctions (JJ) as a hysteresisloop. When the current approaches on the back way the breakpoint the voltage ()falls sharply to zero. In addition, in numerical modelling (non-periodic boundary conditions(NPBC)) IVC multiple branching is observed near . It is interesting to study this phenomenonanalytically developing asymptotic methods. There had been developed simple “asymptotic”formulas suitable for calculation of all IVC points except near to . A numerical-analyticalmethod allowing to shorten IVC calculation time essentially was proposed. This method showedgood results in IVC multiple branching calculation in particular. All calculations were performedusing the REDUCE system. We succeeded first to calculate analytically all points of IVC. Anapproximate solution at the breakpoint region (periodic boundary conditions (PBC)) has beendeveloped using the Bogolyubov-Krylov method.

The Annealed Entropy of Wiener Number on Random Double Hexagonal Chains

We study a random planar honeycomb lattice model, namely the random double hexagonal chains. This is a lattice system with nonperiodic boundary condition. The Wiener number is an important molecular descriptor based on the distances, which was introduced by the chemist Harold Wiener in 1947. By applying probabilistic method and combinatorial techniques we obtained an explicit analytical expression for the expected value of Wiener number of a random double hexagonal chain, and the limiting behaviors on the annealed entropy of Wiener number when the random double hexagonal chain becomes infinite in length are analyzed.

EXPLICIT LEAST-DEGREE BOUNDARY FILTERS FOR DISCONTINUOUS GALERKIN.

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation. The paper introduces a new class of PSIAC filters NP0 that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP0 filters outperform the more complex known boundary filters. NP0 filters typically reduce the L∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP0 filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute.

Efficient, Convergent SENSE MRI Reconstruction for Nonperiodic Boundary Conditions via Tridiagonal Solvers.

Undersampling is an effective method for reducing scan acquisition time for MRI. Strategies for accelerated MRI such as parallel MRI and Compressed Sensing MRI present challenging image reconstruction problems with non-differentiable cost functions and computationally demanding operations. Variable splitting (VS) can simplify implementation of difficult image reconstruction problems, such as the combination of parallel MRI and Compressed Sensing, CS-SENSE-MRI. Combined with augmented Lagrangian (AL) and alternating minimization strategies, variable splitting can yield iterative minimization algorithms with simpler auxiliary variable updates. However, arbitrary variable splitting schemes are not guaranteed to converge. Many variable splitting strategies are combined with periodic boundary conditions. The resultant circulant Hessians enable 𝒪(n log n) computation but may compromise image accuracy at the spatial boundaries. We propose two methods for CS-SENSE-MRI that use regularization with non-periodic boundary conditions to prevent wrap-around artifacts. Each algorithm computes one of the resulting variable updates efficiently in 𝒪(n) time using a parallelizable tridiagonal solver. AL-tridiag is a VS method designed to enable efficient computation for non-periodic boundary conditions. Another proposed algorithm, ADMM-tridiag, uses a similar VS scheme but also ensures convergence to a minimizer of the proposed cost function using the Alternating Direction Method of Multipliers (ADMM). AL-tridiag and ADMM-tridiag show speeds competitive with previous VS CS-SENSE-MRI reconstruction algorithm AL-P2. We also apply the tridiagonal VS approach to a simple image inpainting problem.

Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary

Given a compact manifold M with boundary ∂M, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to (M,∂M). The symbols of operators with boundary conditions on ∂M are defined in terms of the biorthogonal expansions in eigenfunctions of a fixed operator L with the same boundary conditions on ∂M. The boundary ∂M is allowed to have (arbitrary) singularities. As an application, several criteria for the membership in Schatten classes on L2(M) and r-nuclearity on Lp(M) are obtained. We also describe a new addition to the Grothendieck–Lidskii formula in this setting. Examples and applications are given to operators on M=[0,1]n with non-periodic boundary conditions, and of operators with non-local boundary conditions.

Shapiro step at nonequilibrium conditions

Detailed numerical simulations of intrinsic Josephson junctions of high-temperature superconductors under external electromagnetic radiation are performed taking into account a charge imbalance effect. We demonstrate that the charge imbalance is responsible for a slope in the Shapiro step in the IV-characteristic. The value of slope increases with a nonequilibrium parameter. Coupling between junctions leads to the distribution of the slope's values along the stack. The nonperiodic boundary conditions shift the Shapiro step from the canonical position determined by , where f is a frequency of external radiation. This fact makes the interpretation of the experimentally found Shapiro step shift by the charge imbalance effect ambiguous.

Erratum to: A study on boundary force model used in multiscale simulations with non-periodic boundary condition

Erratum to: A study on boundary force model used in multiscale simulations with non-periodic boundary condition

Ab initio calculations of optical properties of silver clusters: cross-over from molecular to nanoscale behavior

Electronic and optical properties of silver clusters were calculated using two different \textit{ab initio} approaches: 1) based on all-electron full-potential linearized-augmented plane-wave method and 2) local basis function pseudopotential approach. Agreement is found between the two methods for small and intermediate sized clusters for which the former method is limited due to its all-electron formulation. The latter, due to non-periodic boundary conditions, is the more natural approach to simulate small clusters. The effect of cluster size is then explored using the local basis function approach. We find that as the cluster size increases, the electronic structure undergoes a transition from molecular behavior to nanoparticle behavior at a cluster size of 140 atoms (diameter $\sim 1.7$\,nm). Above this cluster size the step-like electronic structure, evident as several features in the imaginary part of the polarizability of all clusters smaller than Ag$_\mathrm{147}$, gives way to a dominant plasmon peak localized at wavelengths 350\,nm$\le\lambda\le$ 600\,nm. It is, thus, at this length-scale that the conduction electrons' collective oscillations that are responsible for plasmonic resonances begin to dominate the opto-electronic properties of silver nanoclusters.

Multilevel summation with B-spline interpolation for pairwise interactions in molecular dynamics simulations.

The multilevel summation method for calculating electrostatic interactions in molecular dynamics simulations constructs an approximation to a pairwise interaction kernel and its gradient, which can be evaluated at a cost that scales linearly with the number of atoms. The method smoothly splits the kernel into a sum of partial kernels of increasing range and decreasing variability with the longer-range parts interpolated from grids of increasing coarseness. Multilevel summation is especially appropriate in the context of dynamics and minimization, because it can produce continuous gradients. This article explores the use of B-splines to increase the accuracy of the multilevel summation method (for nonperiodic boundaries) without incurring additional computation other than a preprocessing step (whose cost also scales linearly). To obtain accurate results efficiently involves technical difficulties, which are overcome by a novel preprocessing algorithm. Numerical experiments demonstrate that the resulting method offers substantial improvements in accuracy and that its performance is competitive with an implementation of the fast multipole method in general and markedly better for Hamiltonian formulations of molecular dynamics. The improvement is great enough to establish multilevel summation as a serious contender for calculating pairwise interactions in molecular dynamics simulations. In particular, the method appears to be uniquely capable for molecular dynamics in two situations, nonperiodic boundary conditions and massively parallel computation, where the fast Fourier transform employed in the particle-mesh Ewald method falls short.

Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries

Most reacting and two-phase flows of practical interest are turbulent but take place at low Mach numbers or under incompressible conditions. In order to investigate the properties of such complex flows with high accuracy but acceptable computing times, a suitable tool for Direct Numerical Simulations (DNS), called DINOSOARS, has been developed. The present article describes the numerical components and methods implemented in this code, together with a detailed verification and validation phase, and finishes with two examples of full-scale simulations. We hope it might be useful as a “verification and validation guideline” for other researchers working on DNS of reacting flows. Since applications of growing complexity are considered by DNS, a Direct Boundary Immersed Boundary Method (DB-IBM) has been implemented, allowing a description of arbitrary geometries on a fixed, but possibly refined, Cartesian mesh. A direct force IBM is implemented as well in DINOSOARS in order to resolve large moving spherical particles (much larger than the Kolmogorov scale) on the grid. Particles below the Kolmogorov scale are treated as point particles, taking into account additionally heat and mass transfer with the continuous flow. The efficient parallelization of the code relies on the open-source library 2DECOMP&FFT. The underlying Poisson equation is solved in a fast and accurate manner by FFT, even for non-periodic boundary conditions. The flexibility of DINOSOARS makes it a very promising tool for analyzing a variety of problems and applications involving turbulent reacting and/or two-phase flows.

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

Searching for new ferroelectrics and multiferroics: A user’s point of view

AbstractA perspective on computational studies of ferroelectrics and multiferroics is given that emphasises what has yet to be done, along with some subtleties in previously studied systems. Beginning with the extensive data-mining studies of Abrahams and more recently, Rabe, a survey is given of magnetostrictive effects in antiferromagnetic antiferroelectrics (after Toledano and Toledano), which has an nonmagnetic analogy in the antiferroelectric phase of tris-sarcosine calcium chloride and a reminder of the unusual spin–phonon coupling of Holden et al. in systems such as KCoF3 and EuTiO3. Attention is also paid to field-temperature phase diagrams, finite non-periodic boundary conditions, and processing-dependent structures.

Phase field approach to dislocation evolution at large strains: Computational aspects

Computational aspects of the phase field simulations of dislocation nucleation and evolution are addressed. The complete system of equations for the coupled phase field approach to dislocation nucleation and evolution and nonlinear mechanics for large strains is formulated. Analytical solutions for a stationary and propagating single dislocation, dislocation velocity, core energy, and core width are found. Dislocation parameters for nickel are identified based on existing molecular dynamics simulations. In contrast to all previous efforts that are based on the spectral approach, finite element method (FEM) is utilized, which allowed us to treat large strain problems and non-periodic boundary conditions. The single dislocation order parameter profile and the stationary distance between two neighboring dislocations at a semicoherent sharp austenite–martensite interface are in perfect agreement with analytical expressions. The main focus is on proving that the new points of the developed theory can be confirmed in simulations, including possibility of obtaining the desired dislocation height for aligned and inclined dislocations, eliminating spurious stresses, resolving dislocation cores and interaction between cores of different dislocations. Mesh independence of the solutions is demonstrated and the effect of approximating finite element polynomials is analyzed, exhibiting possibility of significant numerical errors when special care is not taken of. Problems of nucleation and evolution of multiple dislocations along the single and multiple slip systems near martensitic lath, and along the sharp austenite –martensite interface, the activity of dislocations with two different orientations in a nanograined material under shear and pressure, and the interaction between two intersecting dislocation systems are studied. Surface-modified partial dislocation was revealed. These problems represent the first step in the future study of interaction of phase transformation and dislocations.

Thermodynamically consistent phase field approach to dislocation evolution at small and large strains

A thermodynamically consistent, large strain phase field approach to dislocation nucleation and evolution at the nanoscale is developed. Each dislocation is defined by an order parameter, which determines the magnitude of the Burgers vector for the given slip planes and directions. The kinematics is based on the multiplicative decomposition of the deformation gradient into elastic and plastic contributions. The relationship between the rates of the plastic deformation gradient and the order parameters is consistent with phenomenological crystal plasticity. Thermodynamic and stability conditions for homogeneous states are formulated and satisfied by the proper choice of the Helmholtz free energy and the order parameter dependence on the Burgers vector. They allow us to reproduce desired lattice instability conditions and a stress-order parameter curve, as well as to obtain a stress-independent equilibrium Burgers vector and to avoid artificial dissipation during elastic deformation. The Ginzburg–Landau equations are obtained as the linear kinetic relations between the rate of change of the order parameters and the conjugate thermodynamic driving forces. A crystalline energy coefficient for dislocations is defined as a periodic step-wise function of the coordinate along the normal to the slip plane, which provides an energy barrier normal to the slip plane and determines the desired, mesh-independent height of the dislocation bands for any slip system orientation. Gradient energy contains an additional term, which excludes the localization of a dislocation within a height smaller than the prescribed height, but it does not produce artificial interface energy. An additional energy term is introduced that penalizes the interaction of different dislocations at the same point. Non-periodic boundary conditions for dislocations are introduced which include the change of the surface energy due to the exit of dislocations from the crystal. Obtained kinematics, thermodynamics, and kinetics of dislocations at large strains are simplified for small strains and rotations, as well.

Quantum energy teleportation across a three-spin Ising chain in a Gibbs state

In general, it is important to identify what is the informational resource for quantum tasks. Quantum energy teleportation (QET) is a quantum task, which attains energy transfer in an operational sense by local operations and classical communication, and is expected to play a role in future development of nano-scale smart grids. We consider QET protocols in a three-element Ising spin system with non-periodic boundary conditions coupled to a thermal bath. The open chain is the minimal model of QET between two edge spins that allows the measurement and operation steps of the QET protocol to be optimized without restriction. It is possible to analyze how two-body correlations of the system, such as mutual information, entanglement and quantum discord, can be resources of this QET at each temperature. In particular, we stress that quantum discord is not the QET resource in some cases, even if arbitrary measurements and operations are available.

Soft-spring wall based non-periodic boundary conditions for non-equilibrium molecular dynamics of dense fluids.

Non-equilibrium molecular dynamics (MD) simulations require imposition of non-periodic boundary conditions (NPBCs) that seamlessly account for the effect of the truncated bulk region on the simulated MD region. Standard implementation of specular boundary conditions in such simulations results in spurious density and force fluctuations near the domain boundary and is therefore inappropriate for coupled atomistic-continuum calculations. In this work, we present a novel NPBC model that relies on boundary atoms attached to a simple cubic lattice with soft springs to account for interactions from particles which would have been present in an untruncated full domain treatment. We show that the proposed model suppresses the unphysical fluctuations in the density to less than 1% of the mean while simultaneously eliminating spurious oscillations in both mean and boundary forces. The model allows for an effective coupling of atomistic and continuum solvers as demonstrated through multiscale simulation of boundary driven singular flow in a cavity. The geometric flexibility of the model enables straightforward extension to nonplanar complex domains without any adverse effects on dynamic properties such as the diffusion coefficient.

Organic solvent simulations under non-periodic boundary conditions: A library of effective potentials for the GLOB model

We extend the library of solvents that can be treated using the GLOB (general liquid optimized boundary) method, that allows to perform MD simulations under non-periodic boundary conditions (NPBC) optimizing effective potentials between explicit molecules and the boundary for four organic solvents: CHCl3, CCl4, CH3OH and CH3CN. We show that GLOB allows reducing the number of explicit solvent shells to be included, while yielding results comparable with PBC and significant advantages over simulations without explicit boundaries. Finally, we provide polynomial fittings for all available GLOB effective potentials (including SPC water) to simplify their implementation in NPBC MD simulations.

Toward the correction of effective electrostatic forces in explicit-solvent molecular dynamics simulations: restraints on solvent-generated electrostatic potential and solvent polarization.

Despite considerable advances in computing power, atomistic simulations under nonperiodic boundary conditions, with Coulombic electrostatic interactions and in systems large enough to reduce finite-size associated errors in thermodynamic quantities to within the thermal energy, are still not affordable. As a result, periodic boundary conditions, systems of microscopic size and effective electrostatic interaction functions are frequently resorted to. Ensuing artifacts in thermodynamic quantities are nowadays routinely corrected a posteriori, but the underlying configurational sampling still descends from spurious forces. The present study addresses this problem through the introduction of on-the-fly corrections to the physical forces during an atomistic molecular dynamics simulation. Two different approaches are suggested, where the force corrections are derived from special potential energy terms. In the first approach, the solvent-generated electrostatic potential sampled at a given atom site is restrained to a target value involving corrections for electrostatic artifacts. In the second approach, the long-range regime of the solvent polarization around a given atom site is restrained to the Born polarization, i.e., the solvent polarization corresponding to the ideal situation of a macroscopic system under nonperiodic boundary conditions and governed by Coulombic electrostatic interactions. The restraints are applied to the explicit-water simulation of a hydrated sodium ion, and the effect of the restraints on the structural and energetic properties of the solvent is illustrated. Furthermore, by means of the calculation of the charging free energy of a hydrated sodium ion, it is shown how the electrostatic potential restraint translates into the on-the-fly consideration of the corresponding free-energy correction terms. It is discussed how the restraints can be generalized to situations involving several solute particles. Although the present study considers a very simple system only, it is an important step toward the on-the-fly elimination of finite-size and approximate-electrostatic artifacts during atomistic molecular dynamics simulations.Electronic supplementary materialThe online version of this article (doi:10.1007/s00214-014-1600-8) contains supplementary material, which is available to authorized users.