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Development Trefftz Method for Problems of Nonhomogeneous Media 非均质介质问题的特雷弗兹法发展
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602534
D. B. Volkov-Bogorodskiy

Abstract

A new scheme for solving problems in the mechanics of structurally nonhomogeneous media is proposed. This scheme is based on dividing of the initial domain into system of subdomain-blocks similar to finite elements and on approximation of the solution in each block by a systems of functions that exactly satisfy the equation and do not assume unlike the finite element method continuity at the block boundaries. This scheme is based on the Papkovich–Neuber analytical representation through the auxiliary potentials, which makes it possible to construct the complete approximation systems in nonhomogeneous media that analytically satisfy the initial equations and contact conditions on the boundaries of inhomogeneities. Also this scheme is based on the generalization of a direct Trefftz method in the system of subdomain-blocks, which approximates the solution in discontinuous energy space. It is shown that generalized Trefftz method has the ability simultaneously with minimizing of the energy functional to stitch together all the necessary quantities at the block boundaries. They are displacements, surface forces and for gradient elasticity models also derivatives and cohesion moments. This ability is achieved solely due to the analytical representation of the used functions. This analytical representation opens up the possibility of construction of finite element approximations for complex nonhomogeneous media on unstructured meshes and inconsistent shape functions, that analytically accurately reproduce the stress state at the vicinity of inclusions and can be considered as a new technology of finite element approximations.

摘要 提出了一种解决结构非均质介质力学问题的新方案。该方案的基础是将初始域划分为类似有限元的子域块系统,并在每个块中通过函数系统近似求解,这些函数系统完全满足方程要求,且不像有限元方法那样假定块边界的连续性。该方案基于帕普科维奇-纽伯(Papkovich-Neuber)分析表示法,通过辅助电势,可以在非均质介质中构建完整的近似系统,通过分析满足初始方程和非均质边界的接触条件。此外,该方案还基于子域块系统中直接特雷弗兹方法的广义化,可逼近非连续能量空间中的解。研究表明,广义特里夫兹法在最小化能量函数的同时,还能拼接块边界的所有必要量。这些量包括位移、表面力,梯度弹性模型还包括导数和内聚力矩。这种能力的实现完全归功于所使用函数的分析表示。这种分析表示法为在非结构网格和不一致的形状函数上构建复杂非均质介质的有限元近似提供了可能性,可以通过分析精确地再现夹杂物附近的应力状态,可视为有限元近似的一项新技术。
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引用次数: 0
Stationary Modes of Compressible Fluid Flow in a Thermodynamically Consistent Coupled Model 热力学一致耦合模型中可压缩流体流动的静态模式
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602492
N. N. Nazarenko, A. G. Knyazeva

Abstract

Processes of fluid flow in porous media are encountered in various spheres of human activity. The structure of porous media is extremely diverse, and the gases, liquids, mixtures, suspensions, suspensions, etc. moving in them are significantly different in terms of transport and rheological properties. The models used by different authors to describe fluid flows in porous media are also different. In this paper, classical models of filtration theory are supplemented with thermodynamically consistent constitutive relations that take into account the phenomenon of barodiffusion and an example of a coupled two-dimensional model that takes into account the pressure change associated with the redistribution of impurities due to different transport phenomena is presented. Different flow regimes in a flat layer with asymmetric inlet and outlet are demonstrated.

摘要 多孔介质中的流体流动过程在人类活动的各个领域都会遇到。多孔介质的结构千差万别,在其中运动的气体、液体、混合物、悬浮物、悬浮液等的输运和流变特性也大不相同。不同学者用来描述多孔介质中流体流动的模型也不尽相同。在本文中,过滤理论的经典模型得到了热力学一致的构成关系的补充,这些构成关系考虑到了巴氏扩散现象,并介绍了一个耦合二维模型的实例,该模型考虑到了与不同输运现象导致的杂质再分布相关的压力变化。演示了具有不对称入口和出口的扁平层中的不同流动状态。
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引用次数: 0
Numerical Analysis of Gas Well Test Results in Naturally Fractured Reservoirs 天然裂缝储层气井测试结果的数值分析
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602352
M. N. Shamsiev, V. R. Gadil’shina

Abstract

In this paper, a mathematical model of real gas filtration to a vertical well in naturally fractured reservoir is constructed. The curves of bottomhole pressure and pressure derivative depending on the filtration parameters of the reservoir are analyzed. A method for well test analysis is proposed based on solving an ill-posed inverse problem.

摘要 本文构建了天然裂缝储层垂直井实际气体过滤的数学模型。分析了取决于储层过滤参数的井底压力和压力导数曲线。提出了一种基于求解拟反问题的试井分析方法。
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引用次数: 0
Oscillations of Nanofilms in a Fluid 流体中纳米薄膜的振荡
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602212
M. A. Ilgamov

Abstract

Oscillations and waves in a nanofilm in contact with a gaseous medium are considered. It is assumed that the excitation frequencies are in the ultrasonic range. The simplest model is constructed, based on Timoshenko theory of plate bending and on the first approximation of the reaction from the gaseous medium. This takes into account the surface effect caused by the difference in elastic characteristics in the near-surface layer and in the main volume of the material. The derived relations within the Timoshenko model are simplified, which makes it possible to obtain visible results. The contribution of surface effects and reactions from the gaseous medium is assessed. Linear dynamics of a semi-infinite film is studied.

摘要 研究考虑了与气体介质接触的纳米薄膜中的振荡和波。假设激励频率在超声波范围内。根据季莫申科板弯曲理论和气体介质反应的第一近似值,建立了最简单的模型。该模型考虑到了近表面层和材料主体积弹性特性不同所造成的表面效应。简化了季莫申科模型中的推导关系,从而可以获得可见的结果。对表面效应和气体介质反应的贡献进行了评估。研究了半无限薄膜的线性动力学。
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引用次数: 0
Effect of a Rigid Cone Inserted in a Tube on Resonant Gas Oscillations 插入管中的刚性锥体对共振气体振荡的影响
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602340
L. R. Shaidullin, S. A. Fadeev

Abstract

Resonant oscillations of gas in a tube with an end in the form of an inverted cone have been investigated. As the height of the inverted cone inserted into the tube increases, a decrease in the resonant frequency and amplitude of gas pressure oscillations is observed. The dispersion of the resonant frequency is due to the viscosity and thermal conductivity of the gas. It is found that tubes with an inverted cone-shaped end are less efficient resonators compared to uniform tubes with a flat closed end. The decrease in the amplitude of gas pressure oscillations is attributed to the increase in the near-wall losses due to the increase in the total sidewall extent.

摘要 研究了末端为倒锥形的管中气体的共振振荡。随着插入管中的倒锥高度的增加,观察到气体压力振荡的共振频率和振幅减小。共振频率的分散是由于气体的粘度和热导率造成的。研究发现,与具有扁平封闭端头的均匀管相比,具有倒锥形端头的管子是效率较低的谐振器。气体压力振荡幅度的减小是由于侧壁总范围的增加导致近壁损失的增加。
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引用次数: 0
Mathematical Modeling of Anisotropic Thermal Protection with a High Degree of Longitudinal Anisotropy 具有高度纵向各向异性的各向异性热保护数学模型
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s199508022460256x
V. F. Formalev, B. A. Garibyan

Abstract

In this work, based on a new analytical solution to the third initial-boundary value problem of thermal conductivity in an anisotropic strip, an effective method for thermal protection of high-speed aircraft is proposed by channeling heat flows from the central part of the strip to its periphery using an anisotropic material with a high degree of longitudinal anisotropy (longitudinal to transverse thermal conductivity coefficient ratio not less than twenty).

摘要 本文基于对各向异性带材导热系数第三初始边界值问题的一种新的解析解,提出了一种有效的高速飞机热防护方法,即利用纵向各向异性程度高(纵向与横向导热系数比不小于 20)的各向异性材料,将热流从带材的中心部分引向外围。
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引用次数: 0
The Asperities Density and Height Distribution Combined Effect on Rough Elastic Bodies Contact Characteristics 尖晶石密度和高度分布对粗糙弹性体接触特性的综合影响
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602595
I. G. Goryacheva, A. A. Yakovenko

Abstract

The combined effect of the asperities density and their height distribution in contact of the rough rigid and smooth elastic half-spaces is investigated, taking into account the mutual influence of contact spots and statistical nature of the height distribution of asperities. The normal and exponential types of the height distribution of asperities with various densities and the dispersions of their height distribution are used in calculations of the contact characteristics at macroscale (the approach of contacting bodies and the relative contact area under given normal pressure) and the microscale (real contact pressure distributions). The obtained results are compared with ones calculated from the simplified models neglecting the mutual influence of individual contact spots. It is shown that neglecting the mutual influence of asperities in contact, which is made in many models of the discrete contact, can lead to significant errors in determining the contact characteristics on both the macro- and microscales. Based on the asymptotic analysis the analytical expressions for the approach of the contacting bodies and the relative contact area were derived and used for analysis of the microgeometry parameters effects on the rigid rough half-space penetration into the elastic half-space at high nominal pressures.

摘要 研究了粗糙刚性半空间和光滑弹性半空间接触中的非主流密度及其高度分布的综合影响,同时考虑了接触点的相互影响和非主流高度分布的统计性质。在计算宏观(给定法向压力下接触体的接近和相对接触面积)和微观(实际接触压力分布)的接触特性时,使用了不同密度的尖面高度分布的正态和指数类型及其高度分布的分散性。计算结果与忽略单个接触点相互影响的简化模型计算结果进行了比较。结果表明,许多离散接触模型都忽略了接触面的相互影响,这会导致在确定宏观和微观尺度的接触特性时出现重大误差。在渐近分析的基础上,推导出了接触体接近和相对接触面积的分析表达式,并用于分析微观几何参数对高名义压力下刚性粗糙半空间穿透弹性半空间的影响。
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引用次数: 0
One Problem for the Bessel Equation with a Spectral Parameter in the Boundary Condition 贝塞尔方程的一个问题,边界条件中有一个频谱参数
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s199508022460242x
N. Kapustin, A. Kholomeeva

Abstract

In this paper, we consider the spectral problem for thesemi-integer Bessel equation with a boundary condition containingthe square of the spectral parameter and a complex physicalparameter. The system of eigenfunctions of the problem and thecharacteristic equation for the eigenvalues are derived. Theequation for multiple roots of the characteristic equation isderived. The results on the basis properties (Riesz basis) of thesystem of eigenfunctions at different values of the parameter areobtained. For each case a biorthogonally conjugate system isconstructed. At the end of the paper there is an example for theorder of Bessel functions equal to (1/2).

摘要 本文考虑了半整数贝塞尔方程的谱问题,其边界条件包含谱参数的平方和复物理参数。导出了问题的特征函数系和特征值的特征方程。得出了特征方程的多根方程。得出了不同参数值下特征函数系的基础性质(Riesz 基础)。针对每种情况,都构建了一个双对映共轭系统。在论文末尾有一个贝塞尔函数阶等于 (1/2)的例子。
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引用次数: 0
Theory of Poisson’s Ratio for a Thermoelastic Micropolar Acentric Isotropic Solid 热弹性微极性同心各向同性固体的泊松比理论
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602480
E. V. Murashkin, Y. N. Radayev

Abstract

The present paper deals with the triple weights pseudotensor formulation of multivariant thermoelasticity for acentric isotropic micropolar solids. The fundamental concepts of pseudoinvariant volume/area/arc elements of odd integer weights in three-dimensional spaces are discussed. The developed theory of acentric isotropic micropolar thermoelasticity is formulated in terms of contravariant pseudovector of spinor displacements having positive odd algebraic weight and covariant absolute vector of translational displacements. Three energetic forms (H), (E), and (A) of thermoelasticity potential are proposed. The latter is derived from the irreducible system of algebraic invarinats/pseudoinvariants being actually their linear span with coefficients thus allowing us to introduce the conventional thermoelasticity moduli (shear modulus of elasticity, Poisson’s ratio, characteristic nano/microlength, etc.). For others energetic forms thermoelasticity anisotropic micropolar (E) moduli are determined via (A) moduli and then (H) moduli are found in terms of (A) moduli. The triple weights formulation of multivariant constitutive equations for acentric isotropic thermoelastic solid are obtained and analyzed. A comparison of proposed multivariant constitutive equations elucidates the absolute invariance of Poisson’s ratio, i.e., it insensibility to mirror reflections and prohibition of assigning any algebraic weight to this constitutive scalar.

摘要 本文论述了等向各向同性微极性固体的多变量热弹性的三权重伪张量公式。讨论了三维空间中奇数整数权的伪不变体积/面积/弧元的基本概念。用具有正奇数代数权重的旋转位移的协变伪向量和平移位移的协变绝对向量来表述所发展的同心各向同性微极热弹性理论。提出了热弹性势的三种能量形式 (H)、(E) 和 (A)。后者是从代数不变式/伪不变式的不可还原系统中推导出来的,实际上是它们与系数的线性跨度,因此允许我们引入传统的热弹性模量(剪切弹性模量、泊松比、特征纳米/微米长度等)。对于其他能量形式的热弹性各向异性微波(E)模量是通过(A)模量确定的,然后根据(A)模量找到(H)模量。得到并分析了中心各向同性热弹性固体的多变量构成方程的三重权值公式。通过对所提出的多变量构成方程进行比较,阐明了泊松比的绝对不变性,即泊松比不受镜面反射的影响,并禁止对这一构成标量赋予任何代数权重。
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引用次数: 0
Dynamics of a Wheel with a Deformable Periphery 外围可变形车轮的动力学特性
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602510
V. G. Vil’ke, I. F. Kozhevnikov

Abstract

We consider a model of a wheel consisting of a disc and a continuous set of rods joined to the disc. The rods are replaced by a continuous set of masses at free ends, joined by springs and dampers (the longitudinal and transverse stiffness of the tread rods) to the wheel disc. The viscous friction acts at the contact points of the rods with the road. The equations of motion of the wheel in the vertical plane are obtained, taking into account the impact phenomena at the boundary points of the contact area. The shape of the deformed periphery, the contact area, the frequencies of rods vibrations in steady-state regime are found. The value of external forces power required to existence of a steady-state regime is determined when wheel translational motion speed and its angular velocity are constant. The wheel vibrations in the vertical plane about the equilibrium position of the loaded wheel are also studied.

摘要 我们考虑的车轮模型由一个轮盘和一组与轮盘相连的连续杆组成。这些杆由一组自由端连续的质量块代替,通过弹簧和阻尼器(花纹杆的纵向和横向刚度)连接到轮盘上。粘性摩擦作用于杆与路面的接触点。考虑到接触区域边界点的冲击现象,得到了车轮在垂直面上的运动方程。求出了变形外围的形状、接触面积、稳定状态下杆的振动频率。在车轮平移速度和角速度恒定的情况下,确定了稳态机制存在所需的外力功率值。此外,还研究了车轮在垂直面上围绕加载车轮平衡位置的振动。
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引用次数: 0
期刊
Lobachevskii Journal of Mathematics
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