Periodic Domain Research Articles

Pressure Control of Nonferroelastic Ferroelectric Domains in ErMnO3.

Mechanical pressure controls the structural, electric, and magnetic order in solid-state systems, allowing tailoring of their physical properties. A well-established example is ferroelastic ferroelectrics, where the coupling between pressure and the primary symmetry-breaking order parameter enables hysteretic switching of the strain state and ferroelectric domain engineering. Here, we study the pressure-driven response in a nonferroelastic ferroelectric, ErMnO3, where the classical stress-strain coupling is absent and the domain formation is governed by creation-annihilation processes of topological defects. By annealing ErMnO3 polycrystals under variable pressures in the MPa regime, we transform nonferroelastic vortex-like domains into stripe-like domains. The width of the stripe-like domains is determined by the applied pressure as we confirm by three-dimensional phase field simulations, showing that pressure leads to oriented layer-like periodic domains. Our work demonstrates the possibility to utilize mechanical pressure for domain engineering in nonferroelastic ferroelectrics, providing a lever to control their dielectric and piezoelectric responses.

An explicit dynamic FFT method for homogenizing heterogeneous solids under large deformations

We propose an explicit displacement-based FFT method for computational homogenization and multi-scale modeling of heterogeneous solids. Unlike existing FFT methods that solve the static equilibrium equation using implicit iterative techniques, our proposed explicit method solves dynamic equations with the Central Difference method. The voxel-like discretizations of the periodic domain and the direct calculation of gradient and divergence operations using Discrete Fourier Transform and its inverse result in a straightforward algorithm that avoids convergence issues and does not require material consistent tangents. The stability condition of this explicit method is also addressed. Four numerical examples (2D particle-reinforced composites of hyperelasticity and viscoelasticity, a 3D fiber-reinforced composite, and a 3D polycrystalline metal) illustrate that the proposed method calculates as accurately and efficiently as implicit FFT methods for quasi-static problems. Furthermore, the proposed explicit displacement-based FFT method outperforms implicit FFT methods when dealing with microstructures comprising phases with very high contrast. Moreover, since the method does not require material consistent tangents, it extends the applicability of FFT methods to the homogenization of materials with complex or black-boxed constitutive models, where consistent tangents are challenging to compute.

Energy conservation of weak solutions for the incompressible Euler equations via vorticity

Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the Lp control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion ω=curlv∈L3(0,T;L3nn+2(Ω))(n=2,3), that generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatio-temporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and ∇ρ belongs to L∞(0,T;Ln(Tn))(n=2,3), which gives an affirmative answer to a problem proposed by Chen and Yu in [5].

A combined ensemble-volume average homogenization method for lattice structures with defects under dynamic and static loading

In the study of lattices structures, both experiments and numerical simulations are often conducted with small samples. Using combined ensemble and volume averaging, this work introduces a method to extract a macroscopic constitutive response of a lattice material from numerical simulations performed in periodic domains. The domain size needed to obtain statistically accurate results is investigated. Similar to molecular dynamics, the concept of the virial stress is introduced after homogenized equations are derived using the ensemble averaging method. Under static conditions, the virial stress is shown to agree with the volume averaged solid stress.Using the homogenization method, constitutive relations for this stress can be obtained from systems with uniform strains. Application of such obtained constitutive relations to more general cases results in an error proportional to the square of the ratio between the lattice length scale and the macroscopic length scale. Taking advantage of this property, numerical simulations are performed in systems with a uniform gradient of the average velocity. The volume average method is then used to accelerate convergence when studying lattices with defects. To avoid the artificial numerical time scale from the size of a representative volume element divided by the wave speed, a numerical scheme is developed to enforce a spatially uniform velocity gradient within the computational domain while allowing fluctuations of the velocity or displacement to develop naturally. To account for probability distribution of lattice defects, the stress is calculated as the ensemble-volume averaged value. For dynamic systems, energy dissipation properties are also studied.

Dynamics and steady state of squirmer motion in liquid crystal

AbstractWe analyse a nonlinear partial differential equation system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer’s motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer’s surface. The liquid crystal is described using the well-established Beris–Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem in a periodic domain. Finally, we will use a two-scale asymptotic expansion to derive a homogenised model for the collective swimming of squirmers as they reach their steady-state orientation and speed.

FLUPS - A Flexible and Performant Massively Parallel Fourier Transform Library

Massively parallel Fourier transforms are widely used in computational sciences, and specifically in computational fluid dynamics which involves unbounded Poisson problems. In practice the latter is usually the most time-consuming operation due to its inescapable all-to-all communication pattern. The original <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flups</monospace> library tackles that issue with an implementation of the distributed Fourier transform tailor-made for successive resolutions of unbounded Poisson problems. However the proposed implementation lacks of flexibility as it only supports cell-centered data layout and features a plain communication strategy. This work extends the library along two directions. First, <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flups</monospace> ' implementation is generalized to support a node-centered data layout. Second, three distinct approaches are provided to handle the communications: one all-to-all, and two non-blocking implementations relying on manual packing and <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MPI_Datatype</monospace> to communicate over the network. The proposed software is validated against analytical solutions for unbounded, semi-unbounded, and periodic domains. The performance of the approaches is then compared against <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">accFFT</monospace> , another distributed FFT implementation, using a periodic case. Finally the performance metrics of each implementation are analyzed and detailed on various top-tier European facilities up to 49,152 cores. This work brings <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flups</monospace> up to a fully production-ready and performant distributed FFT library, featuring all the possible types of FFTs and with flexibility in the data-layout. The code is available under a BSD-3 license at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">github.com/vortexlab-uclouvain/flups</uri> .

Exponential mixing for random dynamical systems and an example of Pierrehumbert

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions.

Modeling the evolution of representative dislocation structures under high thermo-mechanical conditions during Additive Manufacturing of Alloys

Mesoscale simulations of discrete defects in metals provide an ideal framework to investigate the micro-scale mechanisms governing the plastic deformation under high thermal and mechanical loading conditions. To bridge size and time-scale while limiting computational effort, typically the concept of representative volume elements (RVEs) is employed. This approach considers the microstructure evolution in a volume that is representative of the overall material behavior. However, in settings with complex thermal and mechanical loading histories careful consideration of the impact of modeling constraints in terms of time scale and simulation domain on predicted results is required. We address the representation of heterogeneous dislocation structure formation in simulation volumes using the example of residual stress formation during cool-down of laser powder-bed fusion (LPBF) of AISI 316L stainless steel. This is achieved by a series of large-scale three-dimensional discrete dislocation dynamics (DDD) simulations assisted by thermo-mechanical finite element modeling of the LPBF process. Our results show that insufficient size of periodic simulation domains can result in dislocation patterns that reflect the boundaries of the primary cell. More pronounced dislocation interaction observed for larger domains highlight the significance of simulation domain constraints for predicting mechanical properties. We formulate criteria that characterize representative volume elements by capturing the conformity of the dislocation structure to the bulk material. This work provides a basis for future investigations of heterogeneous microstructure formation in mesoscale simulations of bulk material behavior.

Multiscale Asymptotic Computations for the Elastic Quadratic Eigenvalue Problem in Periodically Composite Structure

A multiscale analysis and computational method based on the Second-Order Two-Scale (SOTS) approach are proposed for the elastic quadratic eigenvalue problems in the periodic composite domain. Two typical quadratic eigenvalue problems with different damping effects are considered, and by the asymptotic expansions of both the eigenfunctions and eigenvalues, the first- and second-order cell functions, the microscale features of this heterogeneous materials are defined successively. Then, the homogenized quadratic eigenvalue problems are derived and the second-order expansions of the eigenfunctions are formed. The eigenvalues are also broadened to the second-order terms by introducing proper auxiliary elastic functions defined in the composite structure, and the nonlinear expressions of the correctors of the eigenvalues are derived. The finite element procedures are established, linearized methods are discussed for solving the quadratic eigenvalue problems and the second-order asymptotic computations are performed. Effectiveness of the asymptotic model is demonstrated by both the qualitative and quantitative comparisons between the computed SOTS approximations and the reference solutions, and the converging behavior of the eigenfunctions are numerically verified. It is also indicated that the second-order correctors are of importance to reconstruct the detailed information of the original eigenfunctions within the micro cells.

Effect of boundary conditions on structure formation in fuzzy dark matter

We illustrate the effect of boundary conditions on the evolution of structure in fuzzy dark matter. Scenarios explored include the evolution of single, ground-state equilibrium solutions of the Schr\"odinger-Poisson system, relaxation of a Gaussian density fluctuation, mergers of two equilibrium configurations, and the random merger of many solitons. For comparison, each scenario is evolved twice, with isolation vs periodic boundary conditions, the two commonly used to simulate isolated systems and structure formation, respectively. Replacing isolation boundary conditions (implemented by an absorbing sponge at large radius) by periodic boundary conditions changes the domain topology and dynamics of each scenario by affecting the outcome of gravitational cooling. With periodic boundary conditions, the initial ground-state equilibrium solution and Gaussian fluctuation each evolve toward the single equilibrium solitonic core of the isolated case, but surrounded by an envelope, or tail, in which additional mass is distributed nearly uniformly, unlike the isolated versions. The case of head-on, binary merger introduces additional effects caused by the pull on the system due to the infinite network of periodic images along each axis of the domain. Adding angular momentum to this binary merger results in a tail with polynomial profile when using a periodic domain. Finally, the 3D merger of many, randomly placed solitonic cores of different mass makes a solitonic core surrounded by a tail with power-law-like profile, for periodic boundary conditions, while producing a core with a much sharper falloff in the isolated case. This suggests the conclusion of earlier work, that the ground-state equilibrium solution is an attractor for the asymptotic state, is true even in 3D and for more general circumstances than previously considered, but only if gravitational cooling is able to carry mass and energy off to infinity, which isolation boundary conditions allow, but periodic ones do not.

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

We prove exponential expression rates of deep operator networks (deep ONets) between infinite-dimensional spaces that emulate the coefficient-to-solution map of linear, elliptic, second-order, divergence-form partial differential equations (PDEs). In particular, we consider problems set in -dimensional periodic domains, , with analytic right-hand sides and coefficients. Our analysis covers diffusion-reaction problems, parametric diffusion equations, and certain elliptic systems such as linear isotropic elastostatics in heterogeneous materials. In the constructive proofs, we leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the branch and trunk ONet architecture of Chen and Chen [IEEE. Trans. Nural Netw., 4 (1993), pp. 910–918] and of Lu et al. [Nat. Mach. Intell., 3 (2021), pp. 218–229], we construct deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the norm, uniformly over the PDE coefficient set. The ONets have size , where is the approximation accuracy, for some depending on the physical space dimension.

The burden of COVID‐19 on all‐cause mortality in older adults living in Wales

AbstractBackgroundThe COVID‐19 pandemic has profoundly impacted the lives of people living with dementia (PLWD). Mitigating measures have led to extreme social isolation and reduced or limited capacity within routine health and social care services. Using anonymised individual‐level, population‐scale linked electronic healthcare, demographic and mortality records in Wales, we have examined the impact of the COVID‐19 pandemic on all‐cause mortality in PLWD and in older adults living with other comorbid conditions.MethodUsing the Secure Anonymised Information Linkage (SAIL) Databank (Lyons et al., 2009), we identified six cohorts of individuals aged 50‐101 and living in Wales on or before 1st February 2020, with conditions in the following domains: dementia, cardiovascular, respiratory, metabolic, renal, and cerebrovascular. For each, we identified paired samples matched by age and sex using propensity sampling. Evidence for comorbidities were extracted using ICD‐10 and READ codes. We quantified excess mortality in PLWD during the COVID‐19 period (July 2020 – May 2021 at the point of the analysis) by comparing weekly mortality to a baseline rate (2015‐2019 average) and to the population paired sample. We then extracted retrospective datasets across sixteen‐month periods from 2010‐2020. Finally, we estimated all‐cause mortality odds‐ratios for each period and comorbid domain using logistic regression, controlling for age and sex.ResultPLWD had higher risk of all‐cause mortality compared to a matched sample of individuals without dementia (Figure 1a), especially during the first wave of the pandemic (April 2020, Figure 1b), and in older PLWD (Figure 1c). Examining evidence across other comorbid conditions (Figure 2), logistic regression modelling showed the greatest increase in mortality odds among PLWD (Figure 2f) compared to people without dementia, increasing from 4.47 in 2018 to 5.61 in 2020. The magnitude of the increase in odds among people with cardiovascular, metabolic, renal, and cerebrovascular conditions from 2020‐2021 compared to previous years was smaller compared to the increase in PLWD (Figures 2a, 2c‐e).ConclusionPLWD experienced increased mortality risk compared to those living with other comorbid conditions during 2020. Further work will focus on the composition of death causes and the contribution of other comorbidities to all‐cause mortality among PLWD.

Global Weak Solutions for Compressible Navier-Stokes-Vlasov-Fokker-Planck System

The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper. The existence, uniqueness, and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain. Moreover, the long time behavior of the weak solution is analyzed. It is shown that as the time grows, the distribution function of the particles converges to the global Maxwellian, and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.

Large Enhancements in Optical and Piezoelectric Properties in Ferroelectric Zn1‐xMgxO Thin Films through Engineering Electronic and Ionic Anharmonicities

AbstractMultifunctionality as a paradigm requires materials exhibiting multiple superior properties. Integrating second‐order optical nonlinearity and large bandgap with piezoelectricity can, for example, enable broadband, strain‐tunable photonics. Though very different phenomena at distinct frequencies, both second‐order optical nonlinearity and piezoelectricity are third‐rank polar tensors present only in acentric crystal structures. However, simultaneously enhancing both phenomena is highly challenging since it involves competing effects with tradeoffs. Recently, a large switchable ferroelectric polarization of ≈80 μC cm−2 was reported in Zn1‐xMgxO films. Here, ferroelectric Zn1‐xMgxO is demonstrated to be a platform that hosts simultaneously a 30% increase in the electronic bandgap, a 50% enhancement in the second harmonic generation (SHG) coefficients, and a near 200% improvement in the piezoelectric coefficients over pure ZnO. These enhancements are shown to be due to a 400% increase in the electronic anharmonicity and a ≈200% decrease in the ionic anharmonicity with Mg substitution. Precisely controllable periodic ferroelectric domain gratings are demonstrated down to 800 nm domain width, enabling ultraviolet quasi‐phase‐matched optical harmonic generation as well as domain‐engineered piezoelectric devices.

Uniqueness of global weak solutions for the general Ericksen–Leslie system with Ginzburg–Landau penalization in {mathbb {T}}^2

The Ericksen–Leslie system is a fundamental hydrodynamic model that describes the evolution of incompressible liquid crystal flows of nematic type. In this paper, we prove the uniqueness of global weak solutions to the general Ericksen–Leslie system with a Ginzburg–Landau type approximation in a two dimensional periodic domain. The proof is based on some delicate energy estimates for the difference of two weak solutions within a suitable functional framework that is less regular than the usual one at the natural energy level, combined with the Osgood lemma involving a specific double-logarithmic type modulus of continuity. We overcome the essential mathematical difficulties arising from those highly nonlinear terms in the Leslie stress tensor and in particular, the lack of maximum principle for the director equation due to the stretching effect of the fluid on the director field. Our argument makes full use of the coupling structure as well as the dissipative nature of the system, and relies on some techniques from harmonic analysis and paradifferential calculus in the periodic setting.

Surface‐Atmosphere Decoupling Prolongs Cloud Lifetime Under Warm Advection Due To Reduced Entrainment Drying

AbstractAn initially well‐mixed stratocumulus deck can remain overcast for several tens of hours under warm‐advection conditions, although moisture supply is cut off from the ocean due to surface‐atmosphere decoupling (stabilization of the surface‐atmosphere interface). In this study, a set of idealized large‐eddy simulations were performed to investigate the physical mechanism of how warm‐air advection impacts the evolution of a pre‐existing stratocumulus deck. To mimic warm‐air advection, we decrease the sea surface temperature linearly over time in a doubly periodic domain. Given the same initial conditions, the stratocumulus deck is more persistent when experiencing warm‐air advection than cold‐air advection. This persistence is caused by reduced cloud‐top entrainment drying due to decoupling, a process more influential than the decoupling‐induced cutoff of moisture supply. This mechanism is more notable when the free troposphere becomes more humid. The relevance of the mechanism to previous observations of less low‐level cloudiness under warm‐advection conditions is discussed.

Period domains for gravitational instantons

Based on the uniformization theorems of gravitation instantons by Chen–Chen [Acta Math. 227 (2021), pp. 263–307], Chen–Viaclovsky [Gravitational instantons with quadratic volume growth, 2021], Collins–Jacob–Lin [Forum Math. Sigma (2021)], and Hein–Sun–Viaclovsky–Zhang [Gravitational instantons and del Pezzo surfaces], we prove that the period maps for the A L H ∗ \mathrm {ALH}^{\ast } , A L G \mathrm {ALG} , and A L G ∗ \mathrm {ALG}^{\ast } gravitational instantons are surjective. In particular, the period domains of these gravitational instantons are exactly their moduli spaces.

Factorization‐splitting algorithm based on system incorporated method for periodic nonuniform domains in open regions

AbstractThe original Factorization‐Splitting (FS) algorithm solves large number of coefficients and components in each iteration cycle which significantly affects the computational resources and calculation efficiency. Meanwhile, it can merely solve problems in uniform domains due to the limitation of discretized mesh sizes. In order alleviate these drawbacks, system incorporated factorization‐splitting algorithm is proposed to efficiently solve problems in nonuniform domains. The finite‐difference time‐domain (FDTD) domain is terminated by the higher order perfectly matched layer (PML) formulation which shows advantages in improving the absorbing performance. As the employment of the original domain decomposition method results in the algorithm unstable, an alternative domain decomposition method is proposed which can split the entire nonuniform domain into several sub‐regions. Metamaterial structure is introduced to demonstrate the performance of the proposed algorithms. Through results, the proposed algorithm can decrease memory consumption and simulation duration due to the calculation prevention of repeated components and coefficients. The absorption can be further improved especially in the low‐frequency band due to the enhanced absorption of propagation waves. In addition, the alternative domain decomposition method shows considerable advantages in the simulation of nonuniform domains.

Planar Viscous Shocks with Periodic Perturbations for Scalar Multidimensional Viscous Conservation Laws

This paper studies a Cauchy problem for a scalar multidimensional (multi-d) viscous convex conservation law, in which the initial data is a planar viscous shock with a multi-d periodic perturbation. We show that if the wave-strength and perturbation are both small, then the viscous shock is stable in the space with an exponential decay rate. One contribution of the paper is to establish a new framework to study the stability of viscous shocks in multiple dimensions, where the elementary energy method with the antiderivative technique can be used. The idea is to decompose the multi-d perturbation into a one-dimensional function and a multi-d remainder, where the former can well define its antiderivative and the latter satisfies the Poincaré inequality over a periodic domain.

Enhancement of second-order optical nonlinearities and nanoscale periodic domain patterning in ferroelectric boron-substituted aluminum nitride thin films

The discovery and development of CMOS-compatible, nonlinear optical materials is essential to produce integrated photonic devices with advanced functionalities. AlN is a strong candidate for on-chip device demonstration due to its intrinsic second-order optical nonlinearities, large bandgap, and well-established fabrication techniques. However, AlN is not easily phase matched for the largest coefficient d33; the coefficients that could potentially be dispersion phase-matched, d31 and d15, have weak nonlinearities. This work investigates ferroelectric Al1-xBxN (x = 0 to 0.11) for viability as a large bandgap nonlinear optical material with unique suitability towards ultraviolet light generation using second harmonic generation. The linear and nonlinear optical properties are characterized accounting for material anisotropy. With increasing B concentration, a large enhancement from near negligible values to d31 = 0.9 ± 0.1 pm/V and d15= 1.2 ± 0.1 pm/V is observed. This compares favorably to other large bandgap materials like β-Ba(BO2)2, where the largest nonlinear coefficient is d22 ∼ 2.3 pm/V at 800 nm. This is accompanied by a change in the bandgap from 6.1 eV to 5.8 eV as B substitution goes from 0 to 11%. A periodically poled, quasi-phase-matched ferroelectric domain pattern with 400 nm domain size and a wall roughness of &lt;16 nm is demonstrated.