Two-dimensional Case Research Articles

Emergence of a Hierarchy of Singularities in Zero-Pressure Media. Two-Dimensional Case

Emergence of a Hierarchy of Singularities in Zero-Pressure Media. Two-Dimensional Case

Techniques of sampling the energy characteristics of two-dimensional random signals

The article is devoted to methods of discretization of energy characteristics of two-dimensional random signals when simulating random signals using the original harmonic method, which is a generalization of the well-known algorithm proposed by V. S. Pugachev for the two-dimensional case. Requirements imposed on the sampling method are aimed at reducing the computational complexity of the simulation method and increasing its flexibility thanks to removing restrictions on the form of autocorrelation functions and spectral energy density functions. The use of the simulation error as a criterion for quality assessment is proposed. The discretization method is considered for signals given both on unlimited definition intervals and on limited ones. The article demonstrates results of the software system implementation in which the original simulation method is realized using the described sampling methods in both cases. The proposed technique is shown to be robust and efficient, with the results obtained being of independent scientific and technical value and showing promise for developing new effective spectral techniques of simulating signals for the use in intelligent decision support systems.

Simulating laser-fluid coupling and laser-induced cavitation using embedded boundary and level set methods

A computational method for simulating thermal cavitation induced by long-pulsed laser is presented. This method accounts for the absorption of laser light by a liquid, the formation of vapor bubbles due to localized heating, and the dynamics of the bubbles and the surrounding liquid. The physical model combines the Euler equations for a compressible inviscid two-phase fluid flow, a reduced form of the radiative transfer equation for laser radiation, and a local thermodynamic model of vaporization. The Euler equations are solved using the FInite Volume method with Exact two-phase Riemann solvers (FIVER). Following this method, numerical fluxes across phase boundaries are computed by constructing and solving one-dimensional bimaterial Riemann problems. The paper focuses on numerical methods for coupling the laser and fluid governing equations and tracking the vapor bubbles. An embedded boundary finite volume method is proposed to solve the laser radiation equation on the same mesh created for the Euler equations, which usually does not resolve the boundary and propagation directions of the laser beam. To impose boundary conditions, ghost nodes outside the laser domain are populated by mirroring and interpolation techniques. The existence and uniqueness of solution are proved for the two-dimensional case, leveraging the special geometry of the laser domain. The order of accuracy of the method is also proved, and verified using numerical tests. A method of latent heat reservoir is proposed to predict the onset of vaporization, which accounts for the accumulation and release of latent heat. A unique challenge associated with long-pulsed laser is that the dynamics of vapor bubbles is driven not only by the inertia of the bubble nuclei, but also by the continuation of vaporization. In this work, the localized level set method is employed to track the bubble surface, and a method of local correction and reinitialization is proposed to account for continuous phase transitions. Several numerical tests are presented to verify the convergence of these methods. Two realistic simulations of laser-induced cavitation are presented at the end, showing that the computational method is able to capture the key phenomena in these events, including non-spherical bubble expansion, shock waves, and the “Moses effect”.

AN ANALYTIC INVERSION FORMULA FOR A RADON TRANSFORM ON A CLASS OF CONES

Since the works of Radon and Cormack on the classical Radon transform on straight lines, several generalizations involving integrations on conical surfaces have been studied. In this article, we introduce a new family of conical surfaces with arbitrary vertices of the space and axes through the origin and study the corresponding Radon transform. We derive its analytical inversion in two and three dimensions. Numerical simulations are carried out for the two-dimensional case.

Generalized Weingarten hypersurfaces of radial support type

In this paper we introduce a class of Weingarten hypersurfaces called generalized Weingarten hypersurfaces of radial support type (in short, RSGW hypersurfaces). These hypersurfaces are parameterized using a new technique consisting in obtain locally any hypersurface in Euclidean space as an envelope of a sphere congruence wherein the other envelope is contained in a sphere. We extend the definition of the classical Appell surfaces to hypersurfaces and we characterize both Appell and RSGW hypersurfaces in terms of a same harmonic function in the sphere. For two-dimensional case, we provide a Weierstrass-type representation for RSGW surfaces and from this we get a Weierstrass-type representation for the classical Appell surfaces. These representations depend on two holomorphic functions in such a way that a same pair of functions provides examples in each class. Using it, we give a classification for the rotational cases for both classes. Furthermore is provided a necessary and sufficient condition on the holomorphic data of these classes so that they are parameterized by lines of curvature. We also prove that a compact, complete RSGW surface is a sphere.

The use of Taguchi, ANOVA, and GRA methods to optimize CFD analyses of ventilation performance in buildings

We present a method that can be used to optimize CFD analyses of ventilation performance in buildings; the method was successfully applied in other disciplines. In the method, three statistical methods are used - the Taguchi method, ANOVA, and Grey Relational Analysis (GRA) to achieve optimization. We then model the risk of draft and the mean age of air in a two-dimensional benchmark test case to exemplify the application of the method. We use five independent variables for modeling: temperature, air velocity, and three room dimensions, each with five levels. In short, the method is implemented as follows: At first, using an L25 orthogonal array in the Taguchi method, we decreased the total number of cases to be modeled from 3125 (55) to 25. Then, we modeled the 25 cases with CFD and used the Taguchi method and ANOVA analyses to find the best solutions for draught risk or the age of air, while with the GRA method we found the best solution for both of them considered jointly. Finally, new CFD models for the variables describing the best solutions were run to confirm their superiority within the 25 cases selected for modeling within the orthogonal L25 array. They were not compared against all possible 3125 cases because the computational time would have been prohibitive. Nevertheless, we believe, that the results are credible in identifying the best solutions. This is based on experience in other disciplines. We, therefore, propose that the presented method is considered when complex problems are analyzed related to ventilation performance requiring large computational power.

Tensor product scheme for computing bound states of the quantum mechanical three-body problem

We develop a computationally and numerically efficient method to calculate binding energies and corresponding wave functions of quantum mechanical three-body problems in low dimensions. Our approach exploits the tensor structure of the multidimensional stationary Schrödinger equation, being expressed as a discretized linear eigenvalue problem. In one spatial dimension, we solve the three-body problem with the help of iterative methods. Here the application of the Hamiltonian operator is represented by dense matrix–matrix products. In combination with a newly-designed preconditioner for the Jacobi–Davidson QR, our highly accurate tensor method offers a significantly faster computation of three-body energies and bound states than other existing approaches. For the two-dimensional case, we additionally make use of a hybrid distributed/shared memory parallel implementation to calculate the corresponding three-body energies. Our novel method is of high relevance for the analysis of few-body systems and their universal behavior, which is only governed by the particle masses, overall symmetries, and the spatial dimensionality. Our results have straightforward applications for ultracold atomic gases that are widespread and nowadays utilized in quantum sensors.

Global large solutions and incompressible limit for the compressible Navier–Stokes system with capillarity

Consider the Cauchy problem for the barotropic compressible Navier–Stokes–Korteweg equations in the whole space Rd (d≥2), supplemented with large initial velocity v0 and almost constant initial density ϱ0. In the two-dimensional case, the global solutions are shown in the critical Besov spaces framework without any restrictions on the size of the initial velocity, provided that the pressure admits a stability condition and the volume viscosity is sufficiently large. The result still holds for the higher dimensional case d≥3 under the additional assumption that the classical incompressible Navier-Stokes equations, supplemented with the initial velocity as the Helmholtz projection of v0, admits a global strong solution.

High-order harmonic generation in three-dimensional Weyl semimetals with broken time-reversal symmetry

In this paper the nonlinear interaction of a Weyl semimetal (WSM) with a strong driving electromagnetic wave field is investigated. In the scope of the structure-gauge-invariant low-energy nonlinear electrodynamic theory, the polarization-resolved high-order harmonic generation spectra in the WSM with broken time-reversal symmetry are analyzed. The results obtained show that the spectra in the WSM are completely different compared to the two-dimensional graphene case. In particular, at the noncollinear arrangement of the electric and Weyl node momentum separation vectors, anomalous harmonics are generated which are polarized perpendicular to the pump wave electric field. The intensities of anomalous harmonics are quadratically dependent on the momentum space separation of the Weyl nodes. If the right and the left Weyl fermions are merged, we have a four-component trivial massless Dirac fermion and, as a consequence, the anomalous harmonics vanish. In contrast to the anomalous harmonics, the intensities of normal harmonics do not depend on the Weyl nodes' momentum separation vector and the harmonics spectra resemble the picture for a massless three-dimensional Dirac fermion.

The missing (A,D,r) diagram

In this paper we are interested in “optimal” universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term “optimal” is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two-dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the n-dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke–Santaló diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted (A,D,r). Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears to be quite elementary, it had not been answered until now. This is probably related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.

Conical diffraction modulation in fractional dimensions with a [formula omitted]-symmetric potential

This study analyzes conical diffraction in space-fractional parity-time (PT) symmetry, which arises in a fractional Schrödinger equation with a periodic PT-symmetric potential. The peculiar characteristics of the corresponding Floquet Bloch modes are also examined in detail. We find that the band structure in such a model is different from that in regular PT symmetry, especially the critical point, which becomes symmetric linear for the one-dimensional case, and cone-like for the two-dimensional case. During propagation, the linear band structure has peculiar properties, including splitting or diffraction-free propagation, preferential propagation, and unidirectional propagation. Whether the cone-like band structure leads to cone diffraction or nondiffracting propagation depends on which Floquet Bloch modes are excited by the input. Our results may have potential application in light modulation in PT symmetry and provide a new platform to study other exciting topics related to conical diffraction.

Numerical Solution of the Two-Dimensional Elasticity Problem in Strains

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.

Three-dimensional effects induced by depth variation in a differentially heated cavity

This work investigates a natural convection flow occurring in a differentially heated cavity. The main purpose of this paper is to analyze the influence of the cavity depth variation on heat transfers and flow dynamics. Three-dimensional numerical simulations are conducted. The working fluid is air, the vertical aspect ratio (cavity height over width) is equal to 4, and the Rayleigh Number is equal to 108. The impact of the rear and front boundary conditions on the flow topology is highlighted. When the cavity depth increases, three-dimensional effects are encountered that enhance local heat transfer at the isothermal walls. In particular, for horizontal aspect ratio (cavity depth over width) greater than 1, an alternation of local maximum and minimum Nusselt numbers can be observed along the y-direction toward the center of the wall in the lower half part of the isothermal wall, which are similar to a wave-like behavior. It is shown that they are due to vortex structures generated by a Görtler instability. The depth variation increases the emergence of those structures, but paradoxically several characteristics of both flow and heat transfer, such as the depth-averaged shear stress or the Nusselt numbers, tend to the two-dimensional case value as the cavity gets deeper.

A Study of Contact Detection between Noncircular Particles in Discrete Element Method

One of the key points of modeling noncircular particles in the discrete element method is the contact detection of particles. In this study, a general contact detection algorithm of two-dimensional particles with analytic shape functions is provided. The contact detection of particles with strictly convex shape function, such as ellipses and superellipses, is solved by Newton–Raphson method, and a grid method is provided to deal with heart-shaped particles. The grid method can be generalized into a particle system, in which the shape function is not convex. The accuracy and stability of the algorithm are verified by a series of tests. For the collision of a pair of ellipses with an aspect ratio of α = a / b = 1000, the efficiency is not worse than the Newton–Raphson method. For random packing under gravity, no residual kinetic energy is observed, and the force that acts on the bottom is equal to the gravity, that is, the system reaches a mechanical equilibrium state. After equilibrium, the process of hopper discharge is also simulated. The present method is suitable for arbitrarily shaped particles with analytic shape functions in two-dimensional cases. In addition to superellipses, the method can also be applied to other particle systems as long as the shape functions are given.

A Novel EPANET Integration for the Diffusive–Dispersive Transport of Contaminants

The EPANET model is commonly used to model hydraulic behaviour and water quality within water distribution networks. The standard version of the model solves the advective transport equation by solving a mass balance of the fundamental plug flow substance that considers the advective transport and kinetic reaction processes. Over the years, several versions of the model have been developed, which have made it possible to improve the modelling of water quality through the introduction of additional terms within the transport equation to solve the problem of dispersive transport (EPANET-AZRED) and to consider multiple interacting species in the mass flow and on the pipe walls (EPANET multi-species extension). The present study proposes a novel integration of the EPANET-DD (dynamic-dispersion) model, which enables the advective–diffusive–dispersive transport equation in dynamic flow conditions to be solved in the two-dimensional case, through the classical random walk method, implementing the diffusion and dispersion equations proposed by Romero-Gomez and Choi (2011). The model was applied to the University of Enna “KORE” laboratory network to verify its effectiveness in modelling diffusive–dispersive transport mechanisms in the presence of variable flow regimes. The results showed that the EPANET-DD model could better represent the actual data than previously developed versions of the EPANET model.

Fast prediction of blood flow in stenosed arteries using machine learning and immersed boundary-lattice Boltzmann method.

A fast prediction of blood flow in stenosed arteries with a hybrid framework of machine learning and immersed boundary-lattice Boltzmann method (IB–LBM) is presented. The integrated framework incorporates the immersed boundary method for its excellent capability in handling complex boundaries, the multi-relaxation-time LBM for its efficient modelling for unsteady flows and the deep neural network (DNN) for its high efficiency in artificial learning. Specifically, the stenosed artery is modelled by a channel for two-dimensional (2D) cases or a tube for three-dimensional (3D) cases with a stenosis approximated by a fifth-order polynomial. An IB–LBM is adopted to obtain the training data for the DNN which is constructed to generate an approximate model for the fast flow prediction. In the DNN, the inputs are the characteristic parameters of the stenosis and fluid node coordinates, and the outputs are the mean velocity and pressure at each node. To characterise complex stenosis, a convolutional neural network (CNN) is built to extract the stenosis properties by using the data generated by the aforementioned polynomial. Both 2D and 3D cases (including 3D asymmetrical case) are constructed and examined to demonstrate the effectiveness of the proposed method. Once the DNN model is trained, the prediction efficiency of blood flow in stenosed arteries is much higher compared with the direct computational fluid dynamics simulations. The proposed method has a potential for applications in clinical diagnosis and treatment where the real-time modelling results are desired.

Thermoelastic Damping in Vibrations of Small-Scaled Rings with Rectangular Cross-Section by Considering Size Effect on Both Structural and Thermal Domains

In this paper, thermoelastic dissipation or thermoelastic damping (TED) in micro/nanorings with rectangular cross-section is examined by accounting for small-scale effect on both structural and thermal areas. The modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat conduction model are exploited for incorporating size effect within constitutive relations and heat conduction equation. By employing simple harmonic form for asymmetric vibrations of the miniaturized ring and solving the heat conduction equation, for one-dimensional (1D) and two-dimensional (2D) cases of heat transfer, the solution of temperature distribution in the ring is extracted in the form of infinite series. By employing the definition of TED on the basis of entropy generation approach, an analytical relation in the series form containing structural and thermal scale parameters is established to estimate TED value. To appraise the precision and validity of the developed solution, a comparison study is performed by utilizing the outcomes of researches published in the literature. An exhaustive parametric study is then conducted to ascertain the role of structural and thermal scale parameters in the magnitude of TED. The influence of some key parameters such as vibration mode, geometrical properties, directions of heat conduction (1D and 2D model) and material on TED is also addressed.

Acoustic Mode Attenuation in Ducts (Using CFD) with Time-Domain Impedance Boundary Condition

In the framework of improving aircraft nacelle design to comply international regulation on aircraft noise, a time-domain impedance boundary condition (TDIBC) based on the oscillo-diffusive representation is implemented in an industrial computational fluid dynamics (CFD) solver under a characteristic formalism, in order to take into account acoustic liner attenuation. A validation of such TDIBC is carried out, first on a two-dimensional benchmark case, then on a more realistic three-dimensional cylindrical configuration, with focus on the acoustic modal content and its attenuation with and without flow. Unsteady Reynolds-averaged Navier–Stokes computations are performed, with the grazing flow being considered laminar. Good agreement is found between the CFD results and the references (computational aeroacoustics results and experiments), with differences in the order of for most cases. The formulation chosen for the TDIBC proved to have no significant impact on numerical stability or computational time.

2D Digital Reconstruction of Asphalt Concrete Microstructure for Numerical Modeling Purposes.

In this paper, we deal with the issue of asphalt concrete microstructure recognition for further numerical analysis. An efficient reconstruction of the underlying microstructure makes the composite analysis more reliable. We propose for this purpose a methodology based on the image processing and focus on a two-dimensional case (it can be easily used as a part of the 3D geometry reconstruction, however). Initially obtained geometry of the inclusions is further simplified to reduce the cost of the finite element mesh generation. Three straightforward geometry simplification algorithms are used to perform this process in a controlled way. Subsequently, we present the solutions of two problems, i.e., heat flow and elasticity (plane strain), in order to illustrate the effectiveness of the whole elaborated methodology. The numerical results were obtained using the finite element method. Consequently, an error analysis is demonstrated in order to refer the overkill mesh solutions to the ones presented in this study. The main finding of this paper is the efficient methodology dedicated to a digital reconstruction of the asphalt concrete microstructure by the image processing. It can be also extended to other materials exhibiting similar microstructure.

Lagrange and H({text {curl}},{{mathcal {B}}}) based finite element formulations for the relaxed micromorphic model

Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present significant aspects related to the relaxed micromorphic model realization with the finite element method (FEM). The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation H^1({{mathcal {B}}}) times H^1({{mathcal {B}}}) and a nodal-edge formulation H^1({{mathcal {B}}}) times H({text {curl}}, {{mathcal {B}}}), where the latter employs the Nédélec space. In this framework, the implementation of higher-order Nédélec elements is not trivial and requires some technicalities which are demonstrated. We discuss the computational convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length parameter.