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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
Unsteady Contact Interaction of Liquid and Solid Bodies 液体和固体物体的非稳态接触相互作用
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602558
G. V. Fedotenkov, A. A. Orekhov, L. N. Rabinskiy

Abstract

The processes of unsteady contact interaction of liquids described by different mathematical models with solid deformable bodies are considered. Closed mathematical formulations of unsteady contact problems in the case of various models of liquids and linear-elastic bodies are developed. The analytical solution of the nonstationary problem of interaction between an acoustic fluid and a deformable solid body is obtained. The time integral Laplace transform is used to construct the solution. The distributions of displacements and stresses in the solid body, as well as pressure and velocity fields in the fluid during unsteady contact interaction are analyzed.

摘要 研究了不同数学模型描述的液体与固体可变形体的非稳态接触相互作用过程。建立了各种液体模型和线性弹性体非稳态接触问题的封闭数学公式。获得了声学流体与可变形固体体之间相互作用的非稳态问题的解析解。解法采用时间积分拉普拉斯变换。分析了非稳态接触相互作用过程中固体中的位移和应力分布,以及流体中的压力和速度场。
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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
Mathematical Modeling of Non-isothermal Flow of Two-phase Media in Curved Channels 弯曲通道中两相介质非等温流动的数学建模
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602200
R. I. Ibyatov, F. G. Akhmadiev

Abstract

The mathematical modeling of the non-isothermal flow of two-phase media in curved channels and pipes of complex geometric shapes is considered. Simplified equations of motion of a two-phase medium, taking into account the flow characteristics, written in an orthogonal coordinate system associated with the flow region, are solved by the method of equal flow surfaces. An algorithm for calculating the flow is constructed for the implementation of a computational experiment. This takes into account changes in the physical characteristics of the two-phase medium depending on temperature. Numerical calculations have been performed for channels of parabolic and conical shapes, taking into account changes in the effective viscosity of the medium from temperature, the initial section of the flow, and the influence of the centrifugal force field. Based on the conducted computational experiment, various flow regimes and the influence of various parameters on the hydrodynamic situation in the flow region are studied.

摘要 研究了两相介质在几何形状复杂的弯曲通道和管道中的非等温流动数学模型。考虑到流动特性,在与流动区域相关的正交坐标系中写入了两相介质的简化运动方程,并用等流面法进行了求解。为实施计算实验,构建了计算流动的算法。该算法考虑到了两相介质物理特性随温度的变化。考虑到介质的有效粘度随温度的变化、流动的初始截面以及离心力场的影响,对抛物线形和锥形通道进行了数值计算。根据所进行的计算实验,研究了各种流动状态以及各种参数对流动区域流体力学状况的影响。
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引用次数: 0
Numerical Solution of the Inverse Problem of Non-stationary Filtration of Bingham Non-Newtonian Fluid to a Horizontal Well 宾汉非牛顿流体对水平井非静态过滤反问题的数值求解
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602224
M. Kh. Khairullin, E. R. Badertdinova

Abstract

Unsteady filtration of a Bingham non-Newtonian fluid to a horizontal well is considered. The experimental results show that when such liquids flow in porous media at low pressure gradients, deviations from the linear Darcy law appear. A feature of the movement of Bingham non-Newtonian fluids in a porous medium is the fact that filtration becomes noticeable only after the pressure gradient reaches a certain critical value—the limiting pressure gradient. The formulation of the inverse coefficient problem for determining filtration parameters during the flow of Bingham non-Newtonian fluid to a horizontal well is given. Pressure change curves measured at the well are used as initial information. To numerically solve the inverse coefficient problem, a computational algorithm based on regularization methods is proposed.

摘要 研究了宾汉非牛顿流体对水平井的非稳态过滤。实验结果表明,当这类液体在多孔介质中以低压力梯度流动时,会出现偏离线性达西定律的现象。宾厄姆非牛顿流体在多孔介质中流动的一个特点是,只有当压力梯度达到某个临界值--极限压力梯度--时,才会出现明显的过滤现象。本文给出了宾汉非牛顿流体流向水平井时确定过滤参数的反系数问题的公式。井上测得的压力变化曲线被用作初始信息。为了对反系数问题进行数值求解,提出了一种基于正则化方法的计算算法。
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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
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
Interaction of Two Gas Bubbles Rising One after Another in a Liquid 液体中相继上升的两个气泡的相互作用
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602285
I. V. Morenko

Abstract

The dynamics of two gas bubbles rising in a stagnant viscous liquid is studied. The mathematical model is based on the laws of conservation of mass, momentum and energy, taking into account the compressibility of media. The gas is assumed to be calorically perfect. To trace the gas–liquid interface, the volume of fluid method is used. The solution to the problem is carried out using the finite volume method. The evolution of the bubble shape during the process of ascent and hydrodynamic interaction is shown. The change in the bubble shapes occurs under the influence of buoyancy force, drag force, viscous force, inertia force, and surface tension force. The results of the test calculations are in good agreement with the known data of other authors. The mechanism of coalescence of bubbles is described in the case of their movement one after another, when one bubble falls into the region of the hydrodynamic wake of another. Dependencies of bubble volume and temperature change on time are established.

摘要 研究了停滞粘性液体中两个气泡上升的动力学。数学模型基于质量、动量和能量守恒定律,并考虑了介质的可压缩性。假设气体在热量上是完美的。为了追踪气液界面,采用了流体体积法。问题的求解采用有限体积法。图中显示了气泡在上升和流体动力学相互作用过程中的形状演变。气泡形状的变化是在浮力、阻力、粘性力、惯性力和表面张力的影响下发生的。试验计算结果与其他作者的已知数据非常吻合。在气泡一个接一个运动的情况下,当一个气泡落入另一个气泡的流体动力尾流区域时,气泡的凝聚机制得到了描述。确定了气泡体积和温度变化与时间的关系。
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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
Solution of the Biharmonic Problem with the Steklov-type and Farwig Boundary Conditions 利用斯特克洛夫型和 Farwig 边界条件求解双谐波问题
IF 0.7 Q2 MATHEMATICS Pub Date : 2024-08-28 DOI: 10.1134/s1995080224602479
Giovanni Migliaccio, Hovik A. Matevossian

Abstract

In this paper, we consider a biharmonic problem with Steklov-type boundary conditions on one part of the boundary and with the Farwig condition on the other part. For this problem, questions of uniqueness of solutions are studied, and in the case of non-uniqueness, provided that the weighted Dirichlet integral is bounded, the exact number of linear independent solutions to the problem under consideration is established. Using the variational principle, uniqueness (non-uniqueness) theorems are obtained, as well as exact formulas for calculating the dimension of the space of solutions depending on the value of the parameter included in the weighted Dirichlet integral.

摘要 在本文中,我们考虑了一个双谐波问题,其边界的一部分具有 Steklov 型边界条件,另一部分具有 Farwig 条件。对于这个问题,研究了解的唯一性问题,在非唯一性情况下,只要加权 Dirichlet 积分是有界的,就可以确定所考虑问题的线性独立解的精确数目。利用变分原理,可以得到唯一性(非唯一性)定理,以及计算解空间维度的精确公式,这取决于加权狄利克特积分中包含的参数值。
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引用次数: 0
期刊
Lobachevskii Journal of Mathematics
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