Two-dimensional Case Research Articles

Feedback semiglobal stabilization to trajectories for the Kuramoto–Sivashinsky equation

AbstractIt is shown that an oblique projection-based feedback control is able to stabilize the state of the Kuramoto–Sivashinsky equation, evolving in rectangular domains, to a given time-dependent trajectory. The actuators consist of a finite number of indicator functions supported in small subdomains. Simulations are presented, in the one-dimensional case under periodic boundary conditions and in the two-dimensional case under Neumann boundary conditions, showing the stabilizing performance of the feedback control.

On the effects of suitably designed space microstructures in the propagation of waves in time modulated composites

In the one-dimensional case, the amplitude of a pulse that propagates in a homogeneous material whose properties are instantaneously changed in time will undergo an exponential increase due to the interference between the reflected and transmitted pulses generated at each sudden switch. Here, we resolve the issue by designing suitable reciprocal PT-symmetric space-time microstructures so that the interference between the scattered waves is such that the overall amplitude of the wave will be constant in time in each constituent material. Remarkably, for the geometries proposed here, a pulse will propagate with constant amplitude regardless of the impedance between the constituent materials, and for some, regardless of the wave speed mismatch. We extend, then, these results to the two-dimensional case, by proposing suitable geometries that avoid the blow up of the wave amplitude at the source point due to the scattering associated with time modulation. Given that the energy associated with the wave will increase exponentially in time, this creates the possibility to exploit the stable propagation of the pulse to accumulate energy for harvesting.

Morphology evolution of a melting solid layer above its melt heated from below

We numerically study the melting process of a solid layer heated from below such that a liquid melt layer develops underneath. The objective is to quantitatively describe and understand the emerging topography of the structures (characterized by the amplitude and wavelength), which evolve out of an initially smooth surface. By performing both two-dimensional (achieving Rayleigh number up to$Ra=10^{11}$) and three-dimensional (achieving Rayleigh number up to$Ra=10^9$) direct numerical simulations with an advanced finite difference solver coupled to the phase-field method, we show how the interface roughness is spontaneously generated by thermal convection. With increasing height of the melt the convective flow intensifies and eventually even becomes turbulent. As a consequence, the interface becomes rougher but is still coupled to the flow topology. The emerging structure of the interface coincides with the regions of rising hot plumes and descending cold plumes. We find that the roughness amplitude scales with the mean height of the liquid layer. We derive this scaling relation from the Stefan boundary condition and relate it to the non-uniform distribution of heat flux at the interface. For two-dimensional cases, we further quantify the horizontal length scale of the morphology, based on the theoretical upper and lower bounds given for the size of convective cells known from Wanget al.(Phys. Rev. Lett., vol. 125, 2020, 074501). These bounds agree with our simulation results. Our findings highlight the key connection between the morphology of the melting solid and the convective flow structures in the melt beneath.

A non-stationary iterative Tikhonov regularization method for simultaneous inversion in a time-fractional diffusion equation

The present paper is devoted to recovering the source term and the initial data simultaneously for a time-fractional diffusion equation from additional temperature data at two fixed times t=T1 and t=T2. Firstly, we prove the uniqueness result for the direct problem. And then we apply a non-stationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm. Three numerical examples in both one-dimensional and two-dimensional cases are provided to demonstrate the effectiveness and feasibility of the proposed method.

Two-Measurement Tomography of High-Dimensional Orbital Angular Momentum Entanglement

High-dimensional (HD) entanglement enables an encoding of more bits than in the two-dimensional case and promises to increase communication capacity over quantum channels and to improve robustness to noise. In practice, however, one of the central challenges is to devise efficient methods to quantify the HD entanglement explicitly. Full quantum state tomography is a standard technology to obtain all the information about the quantum state, but it becomes impractical because the required measurements increase exponentially with the dimension in HD systems. Hence, it is highly anticipated that a new method will be found for characterizing the HD entanglement with as few measurements as possible and without introducing unwarranted assumptions. Here, we present and demonstrate a scan-free tomography method independent of dimension, which only requires two measurements for the characterization of two-photon HD orbital angular momentum (OAM) entanglement. Taking Laguerre-Gaussian modes of photons as an example, the density matrices of OAM entangled states are experimentally reconstructed with very high fidelity. Our method is also generalized to the mixed HD OAM entanglement. Our results provide realistic approaches for quantifying more complex OAM entanglement in many scientific and engineering fields such as multiphoton HD quantum systems and quantum process tomography.

Wake vortex structures and hydrodynamics performance of a power-extraction flapping hydrofoil

Numerical simulations are used to investigate wake vortex structures with different aspect ratios (ARs) of flapping hydrofoil on power extraction. Simulations employ a finite-volume method with an overset grid technique allowing relatively large heaving and pitching motions of the flapping hydrofoil. In a two-dimensional case, the wake is dominated by a Karman-like vortex street composed of spanwise vortices. In contrast, the wake in a three-dimensional condition is dominated by two sets of vortex structures that are oblique to the streamwise, which is significantly different from the two-dimensional condition. The two sets of vortex structures interact with each other as they convect downstream. For a large AR flapping hydrofoil, the two sets of vortex loops stay intertwined with each other. The vortex loops evolve into vortex rings for small AR. In addition, the leading edge vortex on the foil surface is compressed by the tip vortices, affecting the synchronization of vertical force and foil motion. This effect of compression is weakened as the AR increases. The power-extraction efficiency reaches an optimal value of 32.4% at AR = 8 in a three-dimensional condition.

Gravito-turbulence in local disc simulations with an adaptive moving mesh

ABSTRACTSelf-gravity plays an important role in the evolution of rotationally supported systems such as protoplanetary discs, accretion discs around black holes, or galactic discs, as it can both feed turbulence and lead to gravitational fragmentation. While such systems can be studied in the shearing box approximation with high local resolution, the large density contrasts that are possible in the case of fragmentation still limit the utility of Eulerian codes with constant spatial resolution. In this paper, we present a novel self-gravity solver for the shearing box based on the TreePM method of the moving-mesh code arepo. The spatial gravitational resolution is adaptive, which is important to make full use of the quasi-Lagrangian hydrodynamical resolution of the code. We apply our new implementation to two- and three-dimensional, self-gravitating discs combined with a simple β-cooling prescription. For weak cooling we find a steady, gravito-turbulent state, while for strong cooling the formation of fragments is inevitable. To reach convergence for the critical cooling efficiency above which fragmentation occurs, we require a smoothing of the gravitational force in the two-dimensional case that mimics the stratification of the three-dimensional simulations. The critical cooling efficiency we find, β ≈ 3, as well as the box-averaged quantities characterizing the gravito-turbulent state, agrees well with various previous results in the literature. Interestingly, we observe stochastic fragmentation for β > 3, which slightly decreases the cooling efficiency required to observe fragmentation over the lifetime of a protoplanetary disc. The numerical method outlined here appears well suited to study the problem of galactic discs as well as the magnetized, self-gravitating discs.

Multimetric Finsler geometry

Motivated in part by the bi-gravity approach to massive gravity, we introduce and study the multimetric Finsler geometry. For the case of an arbitrary number of dimensions, we study some general properties of the geometry in terms of its Riemannian ingredients, while in the two-dimensional case, we derive all the Cartan equations as well as explicitly find the Holmes–Thompson measure.

Inhomogeneous Fokker–Planck equation from framework of Kaniadakis statistics

In this work we study an inhomogeneous Fokker–Planck equation (FPE) emerging in the framework of Kaniadakis statistics. The resultant FPE presents the features: (a) the solution is an special case of the Johnson’s SU-distribution as the response of the system to a delta form solicitation, (b) the mean standard deviation increases exponentially with a characteristic time depending on the deformation parameter κ; (c) the associated κ-deformed entropy functional is obtained assuming the validity of H-Theorem in κ-deformed space with the entropy contribution of the medium in terms of the deformation; and (d) the deformed derivatives carry the information about the inhomogeneities. Homogeneous diffusion is recovered in the limit of null deformation, and the results are generalized to the two-dimensional case with the presence of two deformation parameters κ1,κ2 controlling inhomogeneities in the directions x and y.

Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher

Building on the work of Crouseilles and Faou on the 2D case, we construct C∞ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer N⩾1 we prove that any Lq initial stream function can be approximated in Lq (strongly when 1⩽q<∞ and weak-⁎ when q=∞) by smooth initial data whose solutions are dense on N-dimensional tori.

Convexification Numerical Method for a Coefficient Inverse Problem for the Radiative Transport Equation

An -D coefficient inverse problem for the stationary radiative transport equation is considered for the first time. A globally convergent so-called convexification numerical method is developed and its convergence analysis is provided. The analysis is based on a Carleman estimate. Extensive numerical studies in the two-dimensional case are presented.

Basic aspects of topological technology of automated on-board equipment layout in space vehicle compartments using the example of “Yantar-2k” Earth remote sensing space vehicle

The article presents the theoretical foundations of automated design of on-board equipment layout in spacecraft compartments in terms of topology. The purpose of the work is to create a flexible arrangement model that would allow solving the problem for any configuration of the structure, taking into account the engineering and operational restrictions imposed on the objects located in a limited space. The technology is formalized according to the methods of the foundations of topology and in this work is demonstrated for a two-dimensional case by comparing it with a reference design-layout scheme. The functional, installation, dimensional, thermal, vibroacoustic requirements are taken into account with the mass-centering restrictions of the layout. In the course of the study, a new approach to describing the design process of assembling on-board equipment was obtained, which can be fully digitized and then integrated into known computer-aided design systems. It has been verified using the example of the power supply system of the Yantar-2k spacecraft.

Partial differential equations to determine elasto-plastic stress–strain behavior from measured kinematic fields

A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed equations that assume pure plastic behavior without elasticity. A method to numerically solve these equations is also presented. In addition to force balance, the equations are derived from the elastic–plastic decomposition of the deformation gradient, the assumption of isotropy, and the assumption that the function mapping the elastic strain to stress is known. The system of equations can be directly applied to complex geometries, finite deformation, non-linear elasticity and plasticity, compressible materials, rate dependent materials, and a variety of hardening laws. This system of PDEs is non-linear and time dependent. Furthermore, it overcomes an important prior limitation: it can be directly applied to cases where some regions of a body are elastically deforming while others are elasto-plastically deforming. A two-dimensional case study of necking in a uniaxial tensile specimen is investigated to illustrate and validate the method. The governing equations are numerically solved using strain fields output from a finite element simulation and validated against this same simulation showing accurate results.

Buckling in a rotationally invariant spin-elastic model.

Scanning tunneling microscopy experiments have revealed a spontaneous rippled-to-buckled transition in heated graphene sheets, in absence of any mechanical load. Several models relying on a simplified picture of the interaction between elastic and internal, electronic, degrees of freedom have been proposed to understand this phenomenon. Nevertheless, these models are not fully consistent with the classical theory of elasticity, since they do not preserve rotational invariance. Herein, we develop and analyze an alternative classical spin-elastic model that preserves rotational invariance while giving a qualitative account of the rippled-to-buckled transition. By integrating over the internal degrees of freedom, an effective free energy for the elastic modes is derived, which only depends on the curvature. Minimization of this free energy gives rise to the emergence of different mechanical phases, whose thermodynamic stability is thoroughly analyzed, both analytically and numerically. All phases are characterized by a spatially homogeneous curvature, which plays the role of the order parameter for the rippled-to-buckled transition, in both the one- and two-dimensional cases. In the latter, our focus is put on the honeycomb lattice, which is representative of actual graphene.

Investigations on a high order SPH scheme using WENO reconstruction

Theoretically, a 2nd order convergence can be reached with the Smoothed Particle Hydrodynamics (SPH) method. However, depending on the spatial disorder of particles, the order of convergence observed in practice can be lower than one. In this paper, a methodology is proposed for the reconstruction of high order numerical fluxes in Riemann-SPH formulations for weakly-compressible flows, so as to increase the global order of convergence of the scheme. This methodology is based on the use of one-dimensional Weighted Essentially Non-Oscillatory (WENO) reconstructions applied at each pair of interacting particles, and each 1D WENO stencil is completed using Moving-Least-Squares (MLS) reconstructions. It is shown that a 6th order convergence can be reached with the proposed SPH-WENO scheme. The gain in accuracy and convergence properties of this scheme is shown and discussed through its application to various one-dimensional and two-dimensional test cases. In particular, the influence of the number of neighbor particles and of the spatial particle disorder is studied. Finally, it is shown that the proposed high-order SPH scheme provides a better accuracy to CPU time ratio than usual Riemann-SPH schemes. The treatment of boundary conditions on rigid walls with the proposed scheme is also discussed.

Scale-invariant random geometry from mating of trees: A numerical study

The search for scale-invariant random geometries is central to the asymptotic safety hypothesis for the Euclidean path integral in quantum gravity. In an attempt to uncover new universality classes of scale-invariant random geometries that go beyond surface topology, we explore a generalization of the mating of trees approach introduced by Duplantier, Miller, and Sheffield. The latter provides an encoding of Liouville quantum gravity on the 2-sphere decorated by a certain random space-filling curve in terms of a two-dimensional correlated Brownian motion, that can be viewed as describing a pair of random trees. The random geometry of Liouville quantum gravity can be conveniently studied and simulated numerically by discretizing the mating of trees using the mated-CRT maps of Gwynne, Miller, and Sheffield. Considering higher-dimensional correlated Brownian motions, one is naturally led to a sequence of nonplanar random graphs generalizing the mated-CRT maps that may belong to new universality classes of scale-invariant random geometries. We develop a numerical method to efficiently simulate these random graphs and explore their possible scaling limits through distance measurements, allowing us in particular to estimate Hausdorff dimensions in the two- and three-dimensional setting. In the two-dimensional case these estimates accurately reproduce previous known analytic and numerical results, while in the three-dimensional case they provide a first window on a potential three-parameter family of new scale-invariant random geometries.

Reconstruction of pointwise sources in a time-fractional diffusion equation

This paper is concerned with an inverse pointwise source problem for the time-fractional diffusion equation in the two-dimensional case. The source term to be identified models the action of a finite number of small particles. Each particle is assumed to be no larger than a single point, characterized by its location and intensity. Both theoretical and numerical aspects are discussed. In the theoretical part, we analyse the well-posedness of the Dirac time-fractional diffusion problem. For the inverse problem, we establish that the unknown point sources can be uniquely identified from local measured data and we derive a local Lipschitz stability result. In the numerical part, we develop a fast and accurate reconstruction approach. The unknown pointwise sources are characterized as solution to an optimization problem minimizing a tracking-type functional. A noniterative reconstruction algorithm is devised, allowing us to determine the number, locations and intensities of the pointwise sources. The efficiency of the proposed approach is confirmed by some numerical examples.

The obstacle problem in no-tension structures and cable nets

We consider the problem of finding a net that supports prescribed point forces, yet avoids certain obstacles, with all the elements of the net being under compression (or all being under tension), and being confined within a suitable bounding box. In the case of masonry structures, when described through the simple, no-tension constitutive model, this consists, for instance, in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region Γ available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region Γ ′ available to the obstacle with Γ ⊂ Γ ′ . The case where some of the forces are reactive is also treated.

Ефективний чисельний метод длярозв’язання рiвняння Рiчардса-Клюта звiдстеженням зони повного насичення

The article presents modifications for numerical methods for modeling of mass transfer process in porous medium with full saturated zone tracking. The goal of the article is to increase computational efficiency of finding an approximate solution process using division of the area into the two non-intersecting parts: unsaturated zone and zone with full saturation. Numerical methods for solving the one-dimensional Richards-Klute equation with tracking of the full saturated zone have been developed. The cases of monotonic solution and solution with general properties of Richards-Klute equation were considered. A modification of the full saturated zone tracking process using a doubly connected edge list structure have been developed for two-dimensional case. Efficiency increase estimation is proven for one- and two-dimensional cases using probability distibution for a measure of the full saturated zone. A comparative analysis of the proposed modifications was carried out. The results of numerical experiments coincide with the estimates predicted by theory.

Fractured inclusion localization and characterization based on deep convolutional neural networks

Seismic exploration is a standard method of prospecting for hydrocarbon deposits. The major goal is to correctly localize the spatial position of the fractured inclusion and estimate its parameters. In this work we investigate the applicability of machine-learning algorithms for this problem. To generate synthetic seismograms the model of a deformable solid medium containing slip planes with nonlinear slip conditions on them is used. The explicit-implicit scheme is applied for obtaining the numerical solution of a constitutive system of equations. The problem of the seismic wave propagation in an inhomogeneous fractured geological model based on the well-known Marmousi2 model in a two-dimensional case is considered. Deep convolutional neural networks are used to provide a fast solution for the inverse problem of restoring the parameters of the fractured inclusion based on the surface measurements. The neural network architecture follows a segmentation approach and targets primarily a spatial localization of the fractured inclusion. However, the mechanical parameters of the inclusion are also estimated using a single run of the same network.