Periodic Boundary Research Articles

Topological sectors, dimer correlations, and monomers from the transfer-matrix solution of the dimer model

We solve the classical square-lattice dimer model with periodic boundaries and in the presence of a field t that couples to the (vector) flux, by diagonalizing a modified version of Lieb's transfer matrix. After deriving the torus partition function in the thermodynamic limit, we show how the configuration space divides into topological sectors corresponding to distinct values of the flux. Additionally, we demonstrate in general that expectation values are t independent at leading order, and obtain explicit expressions for dimer occupation numbers, dimer-dimer correlation functions, and the monomer distribution function. The last of these is expressed as a Toeplitz determinant, whose asymptotic behavior for large monomer separation is tractable using the Fisher-Hartwig conjecture. Our results reproduce those previously obtained using Pfaffian techniques.

On the efficient enforcement of uniform traction and mortar periodic boundary conditions in computational homogenisation

In this contribution, the enforcement of uniform traction (UT) and periodic boundary (PB) conditions on arbitrarily generated meshes is analysed in detail, and the derivation of the associated macroscopic consistent tangent operators is described. Two different approaches to impose these boundary conditions are examined: the saddle point solution (SPS) resulting from the Lagrange multiplier method and the condensation method (CM). The numerical treatment required to implement the UT and the PB conditions with both methods is presented for rectangular and cuboidal representative volume elements (RVEs). It is shown that the UT condition only needs rigid body motion to be prescribed since it intrinsically restricts rigid body rotations in the finite strain setting. The application of the SPS to the UT condition leads to a formulation where the Lagrange multipliers are the homogenised stresses, and the macroscopic consistent tangent operator for FE2 simulations can be directly derived. The enforcement of PB conditions on non-conforming meshes with the mortar approach is also described for the CM and SPS. The need to modify the Lagrange multiplier dual interpolation functions is emphasised to avoid over-constraints. The proposed implementation recovers the conventional PB condition solution when conforming meshes are used. The computational performance of both methods is compared in terms of time and memory requirements. As anticipated, the results obtained for each boundary condition do not depend on the enforcement method employed. With the direct solvers employed, the SPS is more efficient despite increasing the number of unknowns in the linear system of equations.

Computational and experimental investigation of ignition in a Mach 2 cavity-based flameholder

The ignition and flame propagation in an ethylene fueled Mach 2 supersonic cavity are investigated experimentally and computationally. A laser spark is placed in the base of the cavity and the combustion process is tracked using broadband chemiluminescence. Computationally, hybrid Reynolds-averaged Navier-Stokes/large eddy simulation combined with a 22 species chemistry mechanism is used to simulate the ignition process. The simulation is used to explain the underlying physics and flow features leading to the chemiluminescence intensity profile observed experimentally. Initially, the flame kernel spreads in the spanwise direction to fill the region near the cavity's aft-facing step before moving axially downstream to ignite the entire cavity. The flame causes an increase in pressure, restricting fuel flow from the injectors, subsequently leading to leaner conditions that reduce overall heat release. Heat release reduction lowers the pressure and allows the fuel to flow back into the cavity as the flame adjusts to its steady state configuration. At steady state, the elemental fuel concentration within the cavity is significantly higher compared to the non-reacting condition. Examining the ignition time-scales, it is found that the increase in elemental fuel concentration lags approximately 2 milliseconds behind the increase in pressure as the recirculation and diffusion in the cavity transports the additional fuel into the domain. In addition to the full-width simulation, a partial-width simulation that is 1/6 the size of the experimental domain with periodic boundaries is evaluated as a potential cost-saving option. Due to the decreased domain size, the time scales related to ignition were shortened versus the full-width simulation which showed consistent time scales with the experimental findings. The partial-width simulation, however, was able to capture the major stages of combustion during the ignition process.

Theoretical analysis of transient solution phase concentration field in a porous composite electrode with time-dependent flux boundary condition

Energy conversion and storage in a Li-ion cell involves multiple closely coupled transport processes, such as species diffusion through solid and solution phases in the electrode. Mathematical modeling of these processes is critical for fully understanding and optimizing the performance of a Li-ion cell. While a number of analytical and numerical models have been presented for solution phase diffusion, most of such work is based on the assumption of a constant current density. This paper presents analytical modeling of solution phase diffusion in a separate–electrode composite for a generalized, time-dependent current density. An analytical solution for the concentration field in a separator–electrode composite in such conditions is derived using the method of eigenfunction expansion. Good agreement with past work as well as numerical simulations is shown. Results for linear, periodic and step-function boundary conditions are discussed. The theoretical analysis presented here may help accurately model realistic processes where the applied current changes over time, for example, cyclic charge and discharge in an electric vehicle, or sudden changes in the battery load. Results presented here contribute towards the fundamental understanding of solution phase diffusion in Li-ion cells, and provide a basis for improving electrochemical energy conversion and storage processes.

Subsystem complexity after a global quantum quench

We study the temporal evolution of the circuit complexity for a subsystem in harmonic lattices after a global quantum quench of the mass parameter, choosing the initial reduced density matrix as the reference state. Upper and lower bounds are derived for the temporal evolution of the complexity for the entire system. The subsystem complexity is evaluated by employing the Fisher information geometry for the covariance matrices. We discuss numerical results for the temporal evolutions of the subsystem complexity for a block of consecutive sites in harmonic chains with either periodic or Dirichlet boundary conditions, comparing them with the temporal evolutions of the entanglement entropy. For infinite harmonic chains, the asymptotic value of the subsystem complexity is studied through the generalised Gibbs ensemble.

Boundary Value Problems for Differential Inclusions with ϕ −Laplacian

Combining the method of upper and lower solutions with the topological degree, we show the existence of solutions satisfying periodic or Dirichlet boundary conditions, for the differential inclusion $$(\phi(x^{\prime}(t)))^{\prime} \in F(t,x(t)),$$ where F(⋅,⋅) is a compact lower semi-continuous multi-valued map and ϕ is an homeomorphism.

Modeling Small Structural and Environmental Differences in Solids with 15 N NMR Chemical Shift Tensors.

The ability to theoretically predict accurate NMR chemical shifts in solids is increasingly important due to the role such shifts play in selecting among proposed model structures. Herein, two theoretical methods are evaluated for their ability to assign 15 N shifts from guanosine dihydrate to one of the two independent molecules present in the lattice. The NMR data consist of 15 N shift tensors from 10 resonances. Analysis using periodic boundary or fragment methods consider a benchmark dataset to estimate errors and predict uncertainties of 5.6 and 6.2 ppm, respectively. Despite this high accuracy, only one of the five sites were confidently assigned to a specific molecule of the asymmetric unit. This limitation is not due to negligible differences in experimental data, as most sites exhibit differences of >6.0 ppm between pairs of resonances representing a given position. Instead, the theoretical methods are insufficiently accurate to make assignments at most positions.

Superconvergence of the High-Degree FE Method for Second-Degree Elliptic Problem with Periodic Boundary

It is the first time for us to investigate the superconvergence of finite element method for second-degree elliptic problem with periodic boundary. Using the interpolation postprocessing technique, we obtain $$O(h^{k+1}|\ln h|^{2})$$ convergence for the derivative of approximations with a $$P_{k}$$ finite element method at vertex. Numerical results are used to demonstrate the effectiveness of the proposed method.

On the distributions of fuel droplets and in situ vapor in rotating detonation combustion with prevaporized n-heptane sprays

Rotating detonation combustion fueled with partially prevaporized n-heptane sprays is studied with the Eulerian–Lagrangian method. A flattened two-dimensional domain with periodic boundaries is considered to mimic the annular rotating detonation combustor. This work focuses on the effects of prevaporized gas temperature and equivalence ratio on two-phase rotating detonation wave propagation and n-heptane droplet vaporization characteristics in the refill zone. The results show that gas temperature has a great impact on n-heptane sprays vaporization in the refill zone. The droplet evaporation rate increases with the gas temperature, especially when they are close to the deflagration surface. High evaporation rate can be observed for those droplets that are freshly injected into the chamber because they closely interact with the hot product gas from the previous cycle of the rotating detonation. A vapor layer between the droplet-laden area and deflagration surface exists and high concentrations of n-heptane can be found along the deflagration surface. A conceptual model for the droplet and vapor distribution in the refill zone is proposed. The results also show that the blast waves can encroach the refill zone and therefore influence the droplet thermodynamic properties inside the refill zone. The blast waves influence the droplet evaporation rate but have limited effects on droplet temperature, diameter, and spatial distributions. Also, the detonation propagation speed increases with increased prevaporized gas temperature and/or equivalence ratio. The detonation cell size decreases and becomes more uniform as the reactant temperature increases. Moreover, the size and irregularity of rotating detonation cells increase when the prevaporized gas equivalence ratio decreases.

Topologically induced metastability in a periodic XY chain

Non-trivial topological behavior appears in many different contexts in statistical physics, perhaps the most known one being the Kosterlitz–Thouless phase transition in the two-dimensional XY model. We study the behavior of a simpler, one dimensional, XY chain with periodic boundary and strong interactions, but rather than concentrating on the equilibrium measure, we try to understand its dynamics. The equivalent of the Kosterlitz–Thouless transition in this one-dimensional case happens when the interaction strength scales like the size of system N, yet we show that a sharp transition for the dynamics occurs at the scale of log N. When the interactions are weaker than a certain threshold, topological phases could not be observed over long times, while for interactions that are stronger than that threshold, topological phases become metastable, surviving for diverging time scales.

Torus spectroscopy of the Gross-Neveu-Yukawa quantum field theory: Free Dirac versus chiral Ising fixed point

We establish the universal torus low-energy spectra at the free Dirac fixed point and at the strongly coupled chiral Ising fixed point and their subtle crossover behaviour in the Gross-Neuveu-Yukawa field theory with ${n_\text{D}=4}$ component Dirac spinors in $D=(2+1)$ dimensions. These fixed points and the field theories are directly relevant for the long-wavelength physics of certain interacting Dirac systems, such as repulsive spinless fermions on the honeycomb lattice or $\pi$-flux square lattice. The torus energy spectrum has been shown previously to serve as a characteristic fingerprint of relativistic fixed points and is a powerful tool to discriminate quantum critical behaviour in numerical simulations. Here, we use a combination of exact diagonalization and quantum Monte Carlo simulations of strongly interacting fermionic lattice models, to compute the critical torus energy spectrum on finite-size clusters with periodic boundaries and extrapolate them to the thermodynamic limit. Additionally, we compute the torus energy spectrum analytically using the perturbative expansion in ${\epsilon = 4 - D}$, which is in good agreement with the numerical results, thereby validating the presence of the chiral Ising fixed point in the lattice models at hand. We show that the strong interaction between the spinor field and the scalar order-parameter field strongly influences the critical torus energy spectrum and we observe prominent multiplicity features related to an emergent symmetry predicted from the quantum field theory. Building on these results we are able to address the subtle crossover physics of the low-energy spectrum flowing from the chiral Ising fixed point to the Dirac fixed point, and analyze earlier flawed attempts to extract Fermi velocity renormalizations from the low-energy spectrum.

Analysis for two-dimensional inverse quasilinear parabolic problem by Fourier method

In this work, two-dimensional inverse quasi-linear parabolic problem with periodic boundary and integral overdetermination conditions is investigated. The formal solution is obtained by the Fourier approximation. Under some natural regularity and consistency conditions on the input data,the existence, uniqueness and continuously dependence upon the data of the solution are proved by iteration method. The inverse problem is first examined by linearization and then used implicit finite difference scheme for the numerical solution. Also predictor corrector method is considered in the numerical approach. Some results on the numerical solution with two examples are presented with figures and tables. The sensitivity of the scheme with respect to noisy overdetermination data is illustrated.

Numerical Simulation of Dynamic Electromagnetic Characteristics of Superconducting Electrodynamic Suspension (EDS) Train

Dynamic electromagnetic characteristics is significant for the design and operation of superconducting (SC) electrodynamic suspension (EDS) train. This paper established a three-dimensional transient numerical model in COMSOL. The current density excitation comes from the homogenized model of the superconducting magnet. In order to decrease the computing time, periodic boundary is introduced taking the place of finitely long route. Mapping operator is adopted to ensure the continuity of physical quantities in the case of geometric discontinuity in the process of motion. Induced current and magnetic force at different operation velocity and different structure are simulated. In general, this model could be applied to simulate long term operation. It is also suited for the simulation of propulsion systems for EDS systems. In addition, this model provides electromagnetic environment intuitively that would be helpful for the stability analysis and design of onboard SC magnets during long term operation.

How to quantify and avoid finite size effects in computational studies of crystal nucleation: The case of heterogeneous ice nucleation.

Computational studies of crystal nucleation can be impacted by finite size effects, primarily due to unphysical interactions between crystalline nuclei and their periodic images. It is, however, not always feasible to systematically investigate the sensitivity of nucleation kinetics and mechanism to system size due to large computational costs of nucleation studies. Here, we use jumpy forward flux sampling to accurately compute the rates of heterogeneous ice nucleation in the vicinity of square-shaped model structureless ice nucleating particles (INPs) of different sizes and identify three distinct regimes for the dependence of rate on the INP dimension, L. For small INPs, the rate is a strong function of L due to the artificial spanning of critical nuclei across the periodic boundary. Intermediate-sized INPs, however, give rise to the emergence of non-spanning "proximal" nuclei that are close enough to their periodic images to fully structure the intermediary liquid. While such proximity can facilitate nucleation, its effect is offset by the higher density of the intermediary liquid, leading to artificially small nucleation rates overall. The critical nuclei formed at large INPs are neither spanning nor proximal. Yet, the rate is a weak function of L, with its logarithm scaling linearly with 1/L. The key heuristic emerging from these observations is that finite size effects will be minimal if critical nuclei are neither spanning nor proximal and if the intermediary liquid has a region that is structurally indistinguishable from the supercooled liquid under the same conditions.

Existence results on impulsive stochastic semilinear differential inclusions

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive stochastic semilinear differential inclusions driven by Poisson jumps with periodic boundary conditions.We consider the cases in which the right hand side can be either convex . The results are obtained by using fixed point theorems for multivalued mappings, more precisely, the technique is based on fixed point theorem a nonlinear alternative of Leray-Schauder's fixed point theorem in generalised metric and Banach spaces.

Spontaneous Wave Formation in Stochastic Self-Driven Particle Systems

Waves and oscillations are commonly observed in the dynamics of self-driven agents such as pedestrians or vehicles. Interestingly, many factors may perturb the stability of space homogeneous streaming, leading to the spontaneous formation of collective oscillations of the agents related to stop-and-go waves, jamiton, or phantom jam in the literature. In this article, we demonstrate that even a minimal additive stochastic noise in stable first-order dynamics can initiate stop-and-go phenomena. The noise is not a classic white one, but a colored noise described by a Gaussian Ornstein-Uhlenbeck process. It turns out that the joint dynamics of particles and noises forms again a (Gaussian) Ornstein-Uhlenbeck process whose characteristics can be explicitly expressed in terms of parameters of the model. We analyze its stability and characterize the presence of waves through oscillation patterns in the correlation and autocorrelation of the distance spacing between the particles. We determine exact solutions for the correlation functions for the finite system with periodic boundaries and in the continuum limit when the system size is infinite. Finally, we compare experimental trajectories of single-file pedestrian motions to simulation results.

Двусторонние оценки решений краевых задач для неявных дифференциальных уравнений

We consider a two-point (including periodic) boundary value problem for the following system of differential equations that are not resolved with respect to the derivative of the desired function: f_i (t,x,x ̇,(x_i ) ̇ )=0,i= (1,n) ̅. Here, for any i= (1,n) ̅ the function f_i:[0,1]×R^n×R^n×R→R is measurable in the first argument, continuous in the last argument, right-continuous, and satisfies the special condition of monotonicity in each component of the second and third arguments. Assertions about the existence and two-sided estimates of solutions (of the type of Chaplygin’s theorem on differential inequality) are obtained. Conditions for the existence of the largest and the smallest (with respect to a special order) solution are also obtained. The study is based on results on abstract equations with mappings acting from a partially ordered space to an arbitrary set (see [S. Benarab, Z.T. Zhukovskaya, E.S. Zhukovskiy, S.E. Zhukovskiy. On functional and differential inequalities and their applications to control problems // Differential Equations, 2020, 56:11, 1440–1451]).

An estimation of the eigenfunctions of periodic Sturm-Liouville problems

In this paper, the numerical estimations for the eigenfunctions corresponding to the eigenvalues of Sturm-Liouville problem with periodic and semi-periodic boundary conditions are considered. Eigenfunctions are obtained by using the finite-difference method and shown to be matched with previous asymptotic studies.

Hybrid composite meta-porous structure for improving and broadening sound absorption

In order to improve and broaden the sound absorption performance of porous materials with low flow resistance, this study proposes the composite meta-porous structure embedded multiple lateral plates of different lengths, and investigates the corresponding sound absorption coefficients with the fixed and periodic boundary. Characteristics of spatial fluctuation and slow wave effect are embodied in acoustic pressure distributions. Further, the hybrid composite meta-porous structure with periodic boundary could achieve high sound absorption in range of 0–6.4 kHz. The Theoretical model using transfer matrix method suitable for these models is proposed, and verify the FEM results. Normalized surface impedance, phase and trajectory of the complex reflective coefficient illustrates the loss of acoustic energy inside the structure and leakage to the outside. By detailed discussion on the relationship between the geometric parameters and the corresponding sound absorption coefficients, the three boundaries would have different effects on sound absorption performance. Sound absorption test results by acoustic impedance tube confirm the rationality of the composite meta-porous structure with fixed boundary in range of 0.2–5.716 kHz and its practical application in the noise reduction.

Reconceptualization of status liminality in the sociological theory

This article aims at filling some theoretical gaps in understanding status liminality as a transition state in the processes of social mobility. Based on the ideas of A. van Gennep and V. Turner on the nature of rites de passage, the author reconstructs the types of status liminality - ascending, descending, recursive, permanent liminality and liminoidity. The article identified some features that distinguish liminality from marginality and deviance: transitivity - the altered preliminal position and identity combined with the incomplete metamorphosis; temporality - normative temporal and (possibly) spatial boundaries of the transition period; consequentiality - social significance of the postliminal status transformation for both its bearer and society or social groups involved. The phenomenon of communitas discovered by Turner is of particular importance for understanding the state of liminality for it represents a tendency of liminal people to depart from the mundane domain into the anti- and non-structural social-psychological state in which social ties are vividly affective and social experience has a profound existential effect. For each type of status liminality, the author provides examples from the traditional and modern societies using research in sociology of death, medical sociology, criminal sociology, sociology of tourism, social psychology, etc. To conclude, the author considers such phenomena as precariat, morphological freedom and edgework in the liminality perspective. Thus, the heuristic potential of the concept liminality can make a significant contribution to the study of social changes and understanding mechanisms of reproducing social order at the individual, group and societal levels.