Let p(ξ) be a nonnegative polynomial in ξ∈ℝ^d . The differential operator P(Dx)—the Friedrichs extension of the operator p(Dx) defined on C_0^∞(Ω ) —is considered. The Newton polyhedron of the polynomial p is introduced. The asymptotics of eigenvalues is obtained for the case when the polynomial is nondegenerate with respect to its Newton polyhedron. The geometry of the polyhedron shows whether the asymptotics is given by Weyl’s law or by another formula.
A new model of a piezoelectric transducer is described, which makes it possible to measure an electric signal proportional to either the shock-wave stress propagating through the rod or the deformation of the rod’s end. For the case of one-dimensional axisymmetric loading, for particle displacement, a consistent derivation of the wave equation with a source in which electrical energy arises is presented. The solution of the problem of describing the motion of an electro-elastic medium is based on the postulate of the continuity of elastic mechanical and electrodynamic properties of the medium. It is shown that the source on the right-hand side of the resulting wave equation is an expression proportional to electric current or electrical voltage of the piezoelectric sensor. The operation of a piezoelectric sensor mounted on the free end of the rod is considered when the duration of the shock wave pulse is longer than the time of passage of the elastic wave through the sensor material. In this case, the electrical signal taken from the piezoelectric sensor is proportional to the acceleration of the rod’s end, and the integral of this signal is proportional to the rod’s deformation. It is also shown that the electrical voltage measured at the piezoelectric transducer placed in the middle of the rod is proportional to the mechanical stress. The operation of the piezoelectric sensor under short pulse loads, the duration of which is shorter than the time of wave passage through the thickness of the piezoelectric sensor material, is analyzed.
Bernoulli sequences of independent random variables X1, X2, … taking the value 1 with some probability 0 < p < 1 and the value 0 with the probability q = 1 – p are considered. The events Xn = 1 and Xn = 0 are treated, respectively, as “success” and “failure” in the nth Bernoulli trial. Works regularly appear that explore various situations related to the occurrence of such successes, failures, and series of them in Bernoulli sequences. In their earlier studies, the authors of this work obtained a number of new results for series of successes. Note that if the probabilities p and q are interchanged in the relations for the number of series of successes, similar results can be stated for series of failures. These earlier works studied properties of distributions of random variables such as the number of distinct success streaks among n variables, the number of distinct success streaks observed at the occurrence of the nth success, the number of success streaks at the occurrence of the mth failure, and the number of distinct success streaks observed at the occurrence of the first kth success streak. This work studies the properties of new random variables associated with the occurrence of various combinations of success and failure streaks in Bernoulli sequences.
In this paper, we present a construction of polynomials approximating a function that is analytic in a domain and continuous on a fixed arc of the boundary, and besides, to construct the polynomials, we use only the values of the function on this arc. More precisely, we consider a bounded, simply connected domain in the complex plane with a chord-arc boundary. Furthermore, we assume that the function grows no faster than a power of the distance to the boundary as it approaches points on the boundary that do not lie on the fixed arc. At one of the stages of constructing the approximating polynomials, we use an analogue of the famous Carleman’s formula, which allows us to reconstruct the value of an analytic function from its values on a boundary arc.
This article investigates the computational properties of a functional‑type a posteriori error estimate for analyzing the accuracy of solutions to plane linear elasticity problems obtained by neural networks. Problems with boundary conditions in displacements and boundary tractions are considered. Special attention is paid to a comparison of solutions calculated using the classical finite element method (FEM) and the physics-informed neural networks (PINN). The functional a posteriori error estimate obtained by S.I. Repin is used as the error analysis tool. In order to ensure exact satisfaction of the Dirichlet boundary condition, the neural network solution is constructed based on bubble functions. In addition to analytical specification of the bubble functions, a method for constructing them by solving Poisson’s equation on the same computational domain is employed. Numerical experiments conducted using adaptive algorithms based on the first-order Raviart–Thomas element showed that the efficiency index of the estimate tends to an optimal value with local mesh refinement in areas of maximum error. The obtained results confirm that functional a posteriori estimates provide reliable and efficient error control for solutions obtained by neural networks.
The problems of design and operation of various pipelines, supports, tanks, elements of offshore structures, which are modeled by heterogeneous cylindrical shells, remain relevant for modern industry. The study analyzes a shell whose inhomogeneous elastic parameters depend on the radial coordinate and can be described by piecewise continuous functions. The application of various methods for averaging the elastic parameters of shells of the Kirchhoff–Love and Timoshenko type with inhomogeneity of the Poisson’s ratio over its thickness is considered. Calculations are carried out using the example of harmonic vibrations of a two-layer circular cylindrical shell of the Kirchhoff–Love type. Its shell modes are considered. The potential energy of the shell element is analyzed and the dependence of the shell’s natural frequencies and its reduced stiffness values as functions of the Poisson’s ratio are examined. The study is a continuation of testing these averaging methods in the case of inhomogeneity of the Young’s modulus over the shell thickness. The results obtained are compared with calculations using the ANSYS finite element package. The range of applicability of various models for averaging inhomogeneous shell parameters is analyzed. A method is proposed that makes it possible to increase the accuracy of calculations and reduce their complexity when considering various profiles of Poisson’s ratio. Together with previously obtained results, the methods considered make it possible to study the effects due to the inhomogeneity of shells; for example, to assess the change in shell frequencies due to its hydrogen embrittlement.
The paper presents conditions under which the probability of a linear combination of independent identically distributed random vectors hitting an n -dimensional rectangular parallelepiped is a Schur-concave function of the vector corresponding to this linear combination. It is required that the parallelepiped contain the point 0, its edges be parallel to the coordinate axes, and the vector distribution density be a logarithmically concave sign-invariant function. In addition, a new pre-order within the majorization is proposed, and a characterization of functions preserving this pre-order is obtained in differential form.
Transonic flow past the Dsma523b airfoil equipped with an aileron is studied in the free-stream Mach number range from 0.81 to 0.85. The airfoil angle of attack varies from –0.5° to 2°. The numerical solution of the Reynolds-averaged Navier–Stokes equations is found using the finite volume method using the Ansys CFX software package. The k−ω SST and BSL Reynolds Stress turbulence models are used. They yield similar results. At small angles of attack from –0.5° to 0.5°, the airfoil lift coefficient decreases sharply with increasing free-stream Mach number if the aileron is not deflected. Upward deflection of the aileron by 4° results in a lift coefficient that depends weakly on the Mach number. In the lower part of the studied range of Mach numbers, an upward aileron deflection of several degrees causes anomalously large changes in lift. Conversely, in the upper part of the Mach number range, these changes are anomalously small, and at an aileron deflection of 1° to 3°, they are completely absent. Increasing the angle of attack of the airfoil to 2° leads to the disappearance of this anomaly. The two anomalous regimes correspond to different sizes and locations of the supersonic zones adjacent to the airfoil.
Accumulation of impurities in material affects the stress-strain state of the host medium. The modeling of coupled mass transfer and deformation processes is typically performed under the assumption of small relative changes in the concentration of the diffusing species and in the deformation of the matrix. In this approach, the concentration and the small deformation tensor are considered as independent quantities, implying a direct influence of concentration on the stress field. Such models demonstrate good agreement with experimental data when the change in diffusant concentration only weakly affects the material behavior. However, they can be inadequate from a physical standpoint in the general case. The work proposes a more general approach, according to which changes in concentration within the material’s elementary fragments first alter their unloaded configuration, and the change in this unloaded configuration, in turn, leads to the development of stresses in the solid. The unloaded configuration is understood as the one that the fragment would assume in the absence of constraints from the surrounding medium. Numerical modeling has been performed based on this approach, and a comparison between the classical and the proposed methods has been made using an initial-boundary value problem for a long cylinder with a free surface. The ranges of parameters of the medium for which the models are equivalent, as well as those where significant discrepancies occur, have been identified. It is shown that the proposed approach can be applied to describe the behavior of a wide class of materials, including geometrically nonlinear ones.
Time series of spherical expansion coefficients of global maps of the total electron content of the Earth’s ionosphere are studied. Zonal harmonics are shown to correlate well with the 11-year cycle of solar activity, and tesseral harmonics of the first order feature a pronounced diurnal periodicity. The obtained series of spherical expansion coefficients are used to construct an ionosphere model.
In this paper, a new extension of Tychonoff and Darbo theorems via the family of measures of noncompactness (FMN) is given. Moreover, several results on coupled fixed points and multi-variable condensing operators in a Fréchet space are presented. To support main results, the existence of solutions for a class of systems of n -variable Fredholm integral equations on unbounded domains is also discussed, which can include some previous systems in the existing literature.
The possibility of using the Reynolds-averaged Navier–Stokes equations to describe asymmetric flow regimes past rows of flat plates on a screen is studied. The k–ω and shear stress transport (SST) turbulence models are used. Two-dimensional calculation methods predict the occurrence of two asymmetric flow regimes past a cascade of plates for sufficiently dense cascades. However, the boundary value of the cascade density, which separates symmetric and asymmetric flow regimes, takes different values for different turbulence models. Three-dimensional calculations of the flow past four plates mounted on a screen showed that the effective density can serve as the parameter determining the flow regime (symmetric or asymmetric). The sizes and shapes of the recirculation zones were determined using the k–ω turbulence model. They are close to the sizes and shapes of the recirculation zones observed in the experiment. Another turbulence model, SST, does not predict an asymmetric flow regime. Calculations showed an unsteady flow regime exists at a large plate height-to-width ratio. This regime is characterized by periodic oscillations of the forces acting on the plates.
As demonstrated by both experimental results and numerical modeling, elastic carbon nanotubes have proven to be remarkably strong materials, both in tension and bending. The discovered and other remarkable mechanical properties of single-walled carbon nanotubes open up promising opportunities for various important applications. The analysis of a single-walled carbon nanotube presents a complex problem, the solution of which should be grounded in continuum theories that take into account its discrete (crystalline) structure. The previously developed moment-membrane theory of elastic shells, particularly cylindrical shells, fully meets the above requirements. This theory offers extensive possibilities for investigating a wide range of mechanical problems related to single-walled carbon nanotubes. In this work, various equilibrium problems of a single-walled carbon nanotube are studied based on the moment-membrane theory of cylindrical shells. Problems concerning the static stability of the initial momentless state of the nanotube are also investigated (with respect to both critical force and distributed critical surface load), as well as problems related to the free vibrations of the nanotube (revealing that the lowest vibration frequencies lie in the terahertz range). All the problems considered are carried through to specific numerical results.
One of the important issues of modern nonequilibrium gas dynamics is the correct modeling of all possible molecular distributions over internal energy levels, which is necessary for the accurate calculation of physical properties, transport coefficients, and flux terms. In particular, it is of importance for modeling of the thermal protection of re-entry spacecrafts, as well as for hypersonic flight regimes. This study is devoted to the modeling of various non-equilibrium vibrational distributions of molecular species that differ significantly from the well-known Boltzmann and Treanor distributions. The obtained distributions are implemented for the problem of calculation of state-specific transport coefficients, and, in particular, for the calculation of thermal conductivity and shear viscosity coefficients. In addition, the study evaluates the effect of an increasing collision diameter to the transport coefficients calculation. It is shown that the effect of an account for increasing collision diameter is of importance under conditions of strong vibrational non-equilibrium. It is shown that for N2 the contribution of the variable collision diameter to the state-specific thermal conductivity coefficient is significant at temperatures above 8000 K, while for shear viscosity the effect is negligible up to 15 000 K.
Applying the integral Laplace transform to a wide class of problems leads to a simpler equation with respect to the image of the sought original. The next step is the inversion problem, i.e., finding the original from its image. This step is typically not analytically feasible, necessitating the use of approximate inversion methods. In this case, the approximate solution is represented as a linear combination of the image and its derivatives at a number of points in the complex half-plane in which the image is analytic. However, the original, unlike its image, may even have discontinuity points. Finding frameworks and approximate solutions is reduced to solving systems of linear algebraic equations (SLAE) constructed using classical orthogonal Laguerre, Legendre, and Chebyshev polynomials and their generalizations. SLAE matrices have various properties, which, when taken into account, allow decreasing their condition number as compared to known regularization methods. The results of numerical experiments confirming the effectiveness of the proposed inversion algorithms are presented.
Multilayer cylindrical structures are widely used in pipelines, chemical reactors, etc. The main objective of this work is to investigate dynamic nonstationary processes in a two-layer cylindrical body. A procedure for the numerical solution of the dynamic elasticity problem in terms of displacements using the finite difference method is considered. In this case, the numerical algorithm combines an explicit three-layer scheme for the system of equations of motion inside the domain with an implicit single-layer approximation of the boundary conditions. The process of elastic wave propagation under a sudden change in internal pressure caused by a longitudinal impact is described. The results of dynamic calculations for a two-layer cylindrical body for specific initial data are presented. Results are also given that shed light on certain features of the behavior of the two-layer cylinder during the first milliseconds. A methodology and an algorithm based on the numerical solution of the problem by the finite difference method have been developed. An example of the calculation of a two-layer pipe is presented.
The surface of a body under impingement of solid particles of two-phase flow is subject to deformation and destruction and, thus, changes its roughness. The depth of a crater, which originates from the impact of a solid particle, is determined on the base of semi-empirical theory of penetration of a spherical pellet into the surface of a target. The inertial penetration of an absolutely rigid sphere into a metallic half-space of the target is considered. The expression for maximum depth of the penetration of a spherical striker into a half-space in a wide range of dimensionless parameters changing is obtained. The obtained data of calculation of penetration depth and experimental data are compared. The evolution of a surface roughness during the action of solid particles of two-phase flow is reviewed. The value of probability of a particle fall into a crater on the surface left by previous particles is obtained. The estimation is conducted on how fast the surface of the model is covered by the craters under impingement of solid particles. It is revealed, that some seconds of presence of the experimental model in two-phase flow are enough for the solid particles to begin to drop on the surface that has been deformed by the previous particles. Therefore, when defining experimentally the coefficients of restitutions under interaction of solid particles of two-phase flow with the model it is necessary to allow for a surface roughness changing, which originates from impacts of particles.
The stability of axisymmetric equilibrium states of an isotropic non-homogeneous circular plate under uniform pressure is considered. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinates. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. It is shown that if the elasticity modulus decreases away from the center of a plate, the critical pressure for unsymmetric buckling is sufficiently lower than for a plate with constant mechanical properties. The folds in the narrow zone at the periphery of the lamina cibrosa (LC) of the human eye could be explained by the bucking of the axisymmetric state of the LC in the nonaxisymmetric state.
The article investigates the influence of particle shape on its rebound parameters. A comparison is presented of the results of calculations of collision of particles in the shape of a sphere and an ellipsoid of revolution with a flat surface. It is shown that particle shape significantly affects the rebound parameters. It is found that the particle may interact with the surface more than once during impact, and a significant portion of the energy of the reflected particle is concentrated in the rotational degree of freedom ([9], Fig. 7).
An algorithm for finding the eigenvalues and eigenvectors of a perturbation matrix is considered. The case when the original problem is reduced to the eigenvalue problem for operator pencils (non-self-adjoint perturbation) is analyzed.
