Non-periodic Boundary Research Articles

Influence of Structural Configurations, Boundary Conditions, and Atomic Number on the Curie Transition Temperature of Bulk Cobalt: A Monte Carlo Simulation Study

This study investigates the impact of structural configurations, boundary conditions, and atomic number on the Curie transition temperature ([Formula: see text] of bulk cobalt (Co) utilizing Monte Carlo simulations. The analysis reveals that the Curie temperature increases as the structure transitions from a simple cubic (SC) to a body-centered cubic (BCC), hexagonal close-packed (HCP), and face-centered cubic (FCC) configuration. The SC structure, exhibiting the lowest [Formula: see text], serves as the reference for evaluating these influencing factors. The study also demonstrates that nonperiodic boundary conditions yield the lowest [Formula: see text] compared to periodic ones. Additionally, an increase in the number of atoms from 4000 to 108,000 under nonperiodic boundary conditions correlates with an increase in [Formula: see text]. These findings are consistent with experimental data and provide a foundational basis for researchers involved in the synthesis and application of bulk Co in devices, biomedical applications at room temperature, and emerging technologies.

Extended FFT-based micromechanical formulation to consider general non-periodic boundary conditions

Extended FFT-based micromechanical formulation to consider general non-periodic boundary conditions

Dirichlet Boundary Conditions for FFT‐Based Micromechanics of Bicontinuous Stochastic Microstructures

ABSTRACTComputational homogenization may be used to evaluate effective properties of microstructured materials. As digital imaging techniques generally generate microstructure information on a regular grid, a common approach to compute the effective properties of the microstructured materials is to use fast Fourier transform (FFT)‐based methods. Traditionally, this approach involves periodic boundary conditions. However, for certain microstructure types, nonperiodic boundary conditions are of interest. In this work, we show how to efficiently impose Dirichlet boundary conditions for FFT‐based micromechanics on stochastic bicontinuous microstructures in combinations with a linear conjugate gradient (CG) solver. Consequently, we compute the effective properties of bicontinuous stochastic microstructures and compare the results to values reported in the literature computed by a finite element (FE) code.

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

An exponential spectral deferred correction method for multidimensional parabolic problems

We present some efficient algorithms based on an exponential time differencing spectral deferred correction (ETDSDC) method for multidimensional second and fourth-order parabolic problems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. Similar to the Fourier spectral method for periodic problems, the key to the efficiency of our algorithms is to construct diagonal discrete linear operators via Legendre–Galerkin methods with Fourier-like basis functions. In combination with the ETDSDC scheme, the proposed methods are spectrally accurate in space and up to 10th-order accurate in time (as shown in this work). We demonstrate the high-order of convergence and efficiency of our algorithms in solving parabolic equations through a series of two-dimensional and three-dimensional examples including Ginzburg–Landau and Allen–Cahn equations.

FFT‐based computational micromechanics with Dirichlet boundary conditions on the rotated staggered grid

AbstractImposing nonperiodic boundary conditions for unit cell analyses may be necessary for a number of reasons in applications, for example, for validation purposes and specific computational setups. The work at hand discusses a strategy for utilizing the powerful technology behind fast Fourier transform (FFT)‐based computational micromechanics—initially developed with periodic boundary conditions in mind—for essential boundary conditions in mechanics, as well, for the case of the discretization on a rotated staggered grid. Introduced by F. Willot into the community, the rotated staggered grid is presumably the most popular discretization, and was shown to be equivalent to underintegrated trilinear hexahedral elements. We leverage insights from previous work on the Moulinec–Suquet discretization, exploiting a finite‐strain preconditioner for small‐strain problems and utilize specific discrete sine and cosine transforms. We demonstrate the computational performance of the novel scheme by dedicated numerical experiments and compare displacement‐based methods to implementations on the deformation gradient.

The Simplified Approach to the Bose Gas Without Translation Invariance

The simplified approach to the Bose gas was introduced by Lieb in 1963 to study the ground state of systems of interacting Bosons. In a series of recent papers, it has been shown that the simplified approach exceeds earlier expectations, and gives asymptotically accurate predictions at both low and high density. In the intermediate density regime, the qualitative predictions of the simplified approach have also been found to agree very well with quantum Monte Carlo computations. Until now, the simplified approach had only been formulated for translation invariant systems, thus excluding external potentials, and non-periodic boundary conditions. In this paper, we extend the formulation of the simplified approach to a wide class of systems without translation invariance. This also allows us to study observables in translation invariant systems whose computation requires the symmetry to be broken. Such an observable is the momentum distribution, which counts the number of particles in excited states of the Laplacian. In this paper, we show how to compute the momentum distribution in the simplified approach, and show that, for the simple equation, our prediction matches up with Bogolyubov’s prediction at low densities, for momenta extending up to the inverse healing length.

Efficient all-electron hybrid density functionals for atomistic simulations beyond 10 000 atoms.

Hybrid density functional approximations (DFAs) offer compelling accuracy for abinitio electronic-structure simulations of molecules, nanosystems, and bulk materials, addressing some deficiencies of computationally cheaper, frequently used semilocal DFAs. However, the computational bottleneck of hybrid DFAs is the evaluation of the non-local exact exchange contribution, which is the limiting factor for the application of the method for large-scale simulations. In this work, we present a drastically optimized resolution-of-identity-based real-space implementation of the exact exchange evaluation for both non-periodic and periodic boundary conditions in the all-electron code FHI-aims, targeting high-performance central processing unit (CPU) compute clusters. The introduction of several new refined message passing interface (MPI) parallelization layers and shared memory arrays according to the MPI-3 standard were the key components of the optimization. We demonstrate significant improvements of memory and performance efficiency, scalability, and workload distribution, extending the reach of hybrid DFAs to simulation sizes beyond ten thousand atoms. In addition, we also compare the runtime performance of the PBE, HSE06, and PBE0 functionals. As a necessary byproduct of this work, other code parts in FHI-aims have been optimized as well, e.g., the computation of the Hartree potential and the evaluation of the force and stress components. We benchmark the performance and scaling of the hybrid DFA-based simulations for a broad range of chemical systems, including hybrid organic-inorganic perovskites, organic crystals, and ice crystals with up to 30 576 atoms (101 920 electrons described by 244 608 basis functions).

Render unto Numerics: Orthogonal Polynomial Neural Operator for PDEs with Nonperiodic Boundary Conditions

Render unto Numerics: Orthogonal Polynomial Neural Operator for PDEs with Nonperiodic Boundary Conditions

Numerical Simulation of Flows Using the Fourier Pseudospectral Method and the Immersed Boundary Method

The present work proposes the application of a computational methodology based on the coupling of the Fourier Pseudospectral Method (FPSM) and the Immersed Boundary Method (IBM) for conducting flow simulations over slender airfoils. This methodology, termed IMERSPEC, leverages the benefits of both high accuracy and low computational cost inherent in pseudospectral methods, thanks to the utilization of the Fast Fourier Transform algorithm. IBM is employed to impose non-periodic boundary conditions in the Navier–Stokes equations, addressing the requirement of periodicity at boundaries for FPSM convergence and to accurately represent the immersed slender airfoil in the flow. The aerodynamic behavior of the analyzed profiles was assessed by calculating lift and drag coefficients, which were then compared with existing literature results. Consistently favorable outcomes were observed, particularly in flows at lower Reynolds numbers, demonstrating the effectiveness of the IMERSPEC methodology for simulating complex flows computationally. Additionally, weight functions, fundamental to IBM, are employed flexibly for aerodynamic force calculations. Specifically, within the same simulation, a Cubic function is utilized for drag calculation while a Hat function is employed for lift calculation, yielding results more closely aligned with the literature’s findings. This approach offers an alternative to previously proposed methods for IBM implementation.

Error estimates of a space–time Legendre spectral method for solving the Korteweg–de Vries equation

In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in L2-norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some (2m+1)th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation.

Homogenization of 3D laminated micro-structures including bending effects

In this paper, a homogenization method which captures intrinsic size effect associated with fiber diameter is revisited and adapted for three-dimensional laminated micro-structures. Based on a unit-cell composed of matrix and reinforcement layers, enhanced deformation gradients varying through the thickness, are introduced with the aid of an additional kinematic variable reflecting the difference between the homogenized and constituent level deformation gradients. In the current work, as opposed to the original formulation, higher order terms are preserved for both phases and therefore bending stiffness of the matrix phase can be taken into account as well. The formulation is implemented within the commercial finite element solver Abaqus through user element (UEL) subroutine considering a finite strain hyperelastic response for the reinforcement layers and a von Mises type hyper-elastoplastic one for the matrix phase. Explicitly discretized unit-cells with varying reinforcement phase fraction, layer inclination angle and layer thicknesses are used as references to assess the predictive capabilities of the homogenized model and the significance of bending stiffness of the phases. Similarly, explicitly discretized model of a beam type structure with a crossed lamellar micro-structure is used to evaluate the performance of the homogenized model under more general, non-periodic boundary conditions. The findings of both cases support the effectiveness of the homogenized model.

High order numerical methods for Vlasov-Poisson models of plasma sheaths

This article is a report of the CEMRACS 2022 project, called HIVLASHEA, standing for ”High order methods for Vlasov-Poisson models for sheaths”. A two-species Vlasov-Poisson model is described together with some numerical simulations, permitting to exhibit the formation of a plasma sheath. The numerical simulations are performed with two different methods: a first order classical finite difference (FD) scheme and a high order semi-Lagrangian (SL) scheme with Strang splitting; for the latter one, the implementation of (non-periodic) boundary conditions is discussed and different solvers for the electric field are compared. The codes are first evaluated on a one-species case, where an analytical solution is known. For the two-species case, cross-comparisons are performed and the numerical convergence of the SL method is investigated. The use of high order method is emphasized in this context. The (parallel) SL code turns out to be a robust tool to provide reliable numerical approximations of sheath stationary solution with realistic physical parameters.

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional integral equations.

The linear Lugiato–Lefever equation with forcing and nonzero periodic or nonperiodic boundary conditions

The linear Lugiato–Lefever equation with forcing and nonzero periodic or nonperiodic boundary conditions

A new treatment of boundary conditions in PDE solution with Galerkin methods via Partial Integral Equation framework

We present a new mathematical framework for solution of Partial Differential Equations (PDEs), which is based on an exact transformation of the underlying PDE that removes the boundary constraints from the solution state and moves them into the dynamics of the equivalent transformed equation. The framework is based on a Partial Integral Equation (PIE) representation of a PDE or a system of PDEs, where Partial Integral Equation does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized and consistent treatment of boundary conditions in constructing spectrally convergent solution approximations to a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries, among others. The significance of this result is that a solution to almost any linear PDE in a form of a function series approximation can now be systematically constructed, irrespective of the boundary conditions. Furthermore, in many cases, the resulting ODE system for the expansion coefficients can now be integrated analytically in time, which allows us to obtain solution approximations to a broad class of unsteady PDEs with unprecedented accuracy. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. We also present comparison of the PIE methods with some classical direct PDE solution methods, further demonstrating advantages and potential limitations of the PIE approach. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

The longest edge of the one-dimensional soft random geometric graph with boundaries

The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant α, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power α and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger[ 2 ].

Vlasov-Fokker-Planck-Maxwell simulations for plasmas in inertial confinement fusion

This work presents a code to simulate collisions and laser-plasma interactions of multiple-species plasmas by solving 1D3V Vlasov-Fokker-Planck-Maxwell equations. The distribution functions of all species are expanded into spherical harmonic series of arbitrary order. Each electron or ion species possesses its distinct and suitable velocity mesh and the number of harmonics. Collisions between different ion species are considered and treated with the Fokker-Planck collision term. The code employs the total variation diminishing (TVD) scheme to avoid numerical instabilities in the region of steep density, rarefied plasma-vacuum interface, and non-periodic boundary conditions, which makes it possible to simulate cases that are close to actual physical scenes. The number of particles is globally conserved for each species, the energy for each species is conserved for collisions between the same species, and total energy is conserved for collisions between different species.

Comparison between periodic and non-periodic boundary conditions in the multi-particle finite element modelling of ductile powders

In recent years, the multi-particle finite element method (MPFEM) has demonstrated its suitability for the micromechanical modelling of granular systems made up of highly deformable particles. This method combines the advantages of the finite element method to compute particles’ deformations and of the discrete element method to simulate particles’ interactions. Being computationally expensive, its main drawback is the limitation of the number of particles which can be included in a simulation. However, using a multi-scale approach, it is possible to deduce mesoscopic material properties from the simulation of a relatively small number of particles within an elementary volume, through appropriate averaging techniques. Such simulations are significantly influenced by the boundary conditions (BCs) applied. Periodic BCs are commonly admitted to be the most accurate. However, they are technically rather difficult to implement in the MPFEM because of the multiple contacts which need to be handled in the model. As a result, simplified BCs involving for example, rigid walls, remain used. This study analyses the effect of different types of BCs on the simulated mechanical response of a numerical granular sample, under quasi-static loading. Two different techniques for implementing full or simplified periodic BCs, introduced by Schmidt et al. [1] and Loidolt et al. [2], respectively, were compared with simpler, non-periodic BCs. In the comparison, the same initial microstructure was used; it consisted of an assembly of 50 spherical particles with a periodic topology. The results suggest that considering the additional computational cost induced by the implementation of multi-point constraints, the advantage brought by periodic BCs is relatively limited. The study also highlights the differences between BC types and provides interpretations relative to the particularities of the numerical method used.

Towards Digital Twins of Plasmonic Sensors: Constructing the Complex Numerical Model of a Plasmonic Sensor Based on Hexagonally Arranged Gold Nanoparticles.

In this work, we aim to design the digital twin of a plasmonic sensor based on hexagonally arranged ellipsoidal gold nanoparticles fixed to a glass substrate. Based on electron microscopy images of three experimentally realized nanoparticle arrangement types, we constructed numerical models in environments based on finite element method (FEM) and boundary element method (BEM), namely COMSOL Multiphysics for FEM and the MNPBEM Matlab Toolbox for BEM. Models with nonperiodic and periodic boundary conditions with different unit cells were constructed to study the plasmonic behavior of both the single ellipsoidal particles and the hexagonal nanoparticle arrangements. The effect of the geometrical parameters, namely the interparticle distance, the nanoparticle diameter and thickness, on the resulting LSPR peak positions and bulk refractive index sensitivities were studied in detail, also taking into account the effect of the SiO2 substrate (pillars) under the ellipsoidal particles. We have demonstrated that by optimizing the models, the LSPR peak positions (and sensitivities) can match the experimentally measured values within 1 nm (nm/RIU) precision. The comparison of simulation conditions and the detailed discussion of the effect of the geometrical parameters and used gold dielectric functions on the obtained sensitivities can be very beneficial for the optimization of plasmonic sensor constructions through numerical simulations.