Two-dimensional Static Problems Research Articles

СТІЙКІСТЬ ҐРУНТОВИХ СПОРУД З УРАХУВАННЯМ ПРОСТОРОВОГО ЕФЕКТУ

Problems of stability of soil structures and the development of landslides, including the Black Sea coast, pose a significant danger to the population and existing buildings and structures. The development of landslides and landslide-prone slopes requires careful and detailed study, the results of which should be the basis for the implementation of measures for engineering protection of territories, taken into account in the design and construction of new facilities. When calculating the stability of slopes, there is a difference in the calculated values of the coefficients of stability in two-dimensional and three-dimensional problem statements. Analysis of the research indicates the need to take into account the influence of topographic conditions of the entire slope when calculating it in two-dimensional schemes within the designated areas. Determination of the coefficient of stability can be carried out by both analytical and graph-analytical methods. In most cases, the calculation of the stability of slopes and escarpment and the determination of the coefficient of stability is performed for two-dimensional sections constructed on lines that coincide with the most probable direction of shift of the shear body. The most common methods used in calculating the stability of slopes in a two-dimensional problem statement are the most used because of their simplicity, but these methods are based on the acceptance of some assumptions described above. Based on the analysis of a number of works on two-dimensional and three-dimensional modeling of soil slopes, it is shown that the assessment of the stability of landslides is a complex, responsible and poorly studied task. Performing calculations of soil slopes in the three-dimensional formulation of the problem gives results close to reality, in contrast to the results obtained by conventional and common engineering methods for two-dimensional schemes. This is due to the fact that the three-dimensional formulation of the problem takes into account the combination of engineering and geological conditions of the entire slope. Three-dimensional modeling of soil slopes also allows taking into account factors that have a significant impact on stability, which cannot be taken into account in two-dimensional modeling (features of topography, the presence of trimming slopes, ravines, point structures, etc.).

ГИДРОДИНАМИЧЕСКИЙ АНАЛИЗ ПРОЦЕССОВ ТОПЛИВНЫХ СИСТЕМ ТИПА «COMMON RAIL» В СРЕДЕ ИМИТАЦИОННОГО МОДЕЛИРОВАНИЯ AVL BOOST HYDSIM

The study objective is modeling hydro-dynamic processes occurring in an electro-magnetic diesel fuel injector using AVL Boost Hydsim software package (SP).
 The task to which the paper is devoted is the design of equipment for Common Rail battery injection and fuel atomization systems; description of methods for developing battery fuel systems using Hydsim SP.
 Research methods: numerical simulation of fuel supply based on hydrodynamic dependencies describing the compressible viscous fluid motion in Hydsim software environment.
 Novelty of the work: due to constant tightening of environmental, technical and economic requirements for diesel internal combustion engines (ICE), which directly depend on the perfection of the fuel supply system, there is a need for software that allows engineers to design the appropriate fuel equipment. It is also important to have an effective and comprehensive methodology for working with such software solutions.
 Research results: a model of an electromagnetic fuel injector of a diesel engine is developed, a brief description is given of the methodology for constructing a hydraulic model of the nozzle (in a two-dimensional problem statement) and an analysis of the simulation results obtained.
 Conclusions: AVL Boost Hydsim SP has a number of advantages over its competitors, which can significantly reduce the time spent on the design and implementation of modern fuel equipment, opportunities for dynamic analysis of hydraulic and hydromechanical engine systems appear.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ЗАДАЧ ТЕОРИИ УПРУГОСТИ МЕТОДОМ ГРАНИЧНЫХ УРАВНЕНИЙ

The article describes the features of mathematical modeling of a two-dimensional static problem of elasticity theory by the method of boundary integral equations. The resulting equations make it possible to investigate the stress-strain state of a plane problem under various external influences. As a result of applying the spline approximation of the boundary parameters, the system of integral equations is transformed into a system of linear algebraic equations with unknown displacement and stress components.

Assessment of radiation heat transfer influence on parameters of temperature fields of various design fuel rods

VVER-1000 reactors use cylindrical smooth-core fuel rods. Previously, a model to determe the fuel rod temperature field in a two-dimensional problem statement has been developed and verified. However, modelling assumptions do not consider the influence of variable thermophysical properties, radiation heat transfer, and the opening in the fuel rod on the final parameters of the temperature fields. The impact assessment is an urgent task to improve the economic efficiency of the fuel cycle and the capacity of power units. To develop models and study the features of energy release in nuclear reactors, a numerical package of thermophysical modeling COMSOL Multiphysics software is used. The simulation of temperature fields is performed based on the heat equation with an internal heat source, under the boundary conditions of the second kind at the ends of the fuel rod and the boundary conditions of the third kind on the side surface of the rod. Аn axisymmetric model in two-dimensional problem statement and a three-dimensional model of the fuel rod are developed. The temperature distribution fields are determined by the finite element method. The results of calculations of various design fuel rods are presented. The results have showen that the radiation heat transfer significantly affects the maximum fuel temperature (UO2). The impact degree of variability of thermophysical properties and radiation heat transfer is determined. It was found that the temperature characteristics under different specified conditions have a difference in the range of 15,5–282,0 K (0,8–14,4 %). The developed models are reliable and confirmed by the previously verified model, the characteristics of the fuel assembly used on the VVER-1000 units. The results presented can be used for mathematical modeling of heat transfer processes, both during the modernization of the equipment in operation, and during the development, design, and operation, which will increase the efficiency of electric energy generation at the power unit of a nuclear power plant.

Development of the model to determine the fuel temperature field in a two-dimensional problem statement

Water-cooled water-moderated reactors (VVER) are widely used at Russian nuclear power plants. The VVER reactor core is formed by fuel assemblies consisting of fuel rods. The fuel in fuel rods is uranium dioxide. The safety of the reactor operation is ensured through stringent requirements for the maximum nuclear fuel temperature. Calculation of temperature fields within the reactor core requires associated problems to be solved to determine the internal energy release in fuel based on neutronic characteristics. Dedicated software for such calculations is not available to a broad range of users. At the present time, there are numerical thermophysical modeling packages available for training or noncommercial applications which are used extensively, including Elcut, Flow Vision, Ansys Fluent, and Comsol Multiphysics. Verification of the obtained results is becoming an important issue in building models using these calculation packages. An analytical solution was obtained as part of the study for the fuel temperature field determination. A program was developed in MathCAD based on this solution. A model was developed in Comsol Multiphysics to determine the fuel temperature field with constant thermophysical properties in a two-dimensional problem statement. The numerical model was verified using the analytical solution. The influence of the number of the grid nodes on the solution accuracy was established. The analytical solution can be used to determine the fuel temperature field at any radial coordinate of the reactor. The temperature field determination model developed in MathCAD can be used to verify numerical models of the fuel temperature field determination developed in dedicated packages.

Разработка модели по определению температурного поля твэла в двумерной постановке задачи

Разработка модели по определению температурного поля твэла в двумерной постановке задачи

The singular boundary method for two-dimensional static thermoelasticity analysis

Based on the singular boundary method (SBM) coupled with the radial basis functions (RBF), a meshless algorithm is proposed to solve two-dimensional static thermoelasticity problems. The displacements and stresses can be written as the sum of a homogeneous solution and a particular solution. The RBF and SBM are used to construct the corresponding approximate particular and homogeneous parts, respectively. All unknowns are determined by satisfying the governing equations and the boundary conditions. Numerical experiments are performed for different cases, and the proposed meshless method is effectively validated by comparing the results with the available analytical solutions.

Analysis of Magneto-Piezoelastic Anisotropic Materials

The paper is concerned with the analysis of magneto-piezoelastic anistropic materials. Analytical modeling of magneto-piezoelastic materials is essential for the design and applications in the smart composite structures incorporating them as actuating and sensing constituents. It is shown that Green’s function method is applicable to time harmonic magneto-elastic-piezoelectricity problems using the boundary integral technique, and the exact analytical solutions are obtained. As an application, a two-dimensional static plane-strain problem is considered to investigate the effect of magnetic field on piezoelectric materials. The closed-form analytical solutions are obtained for a number of boundary conditions for all components of the magneto-piezoelectric field. As a special case, numerical results are presented for two-dimensional static magneto-electroelastic field of a piezoelectric solid subjected to a concentrated line load and an electric charge. The numerical solutions are obtained for three different piezoelectric materials and they demonstrate a substantial dependence of the stress and electric field distribution on the constitutive properties and magnetic flux.

Axisymmetric deformation of cylinders made of homogeneous and fiber-reinforced elastic materials in butt-end torsion

The equations of axisymmetric deformation of a circular cylinder made of homogeneous and weakly fiber-reinforced materials in a macroscopically two-dimensional problem statement are presented. The fibers are placed in the axial, circumferential, and radial directions and along spirals in the cylinder wall. They also can lie along opposite spirals in adjacent cylindrical layers, forming an angle-ply reinforcement scheme. At a low content of fibers, it is assumed that they are in the uniaxial stress state, determined by the axial tension or constraint compression in the deformed cylinder. Based on this mathematical model, the deformation of homogeneous and fiber-reinforced elastic cylinders of various length, with free outer surfaces, in butt-end torsion is investigated. The effect of length of the cylinders on their deformation character in torsion is evaluated. The torsion of the cylinders under the conditions of sliding fit along the inner surface and under the action of pressure on the surface, when the initial and deformed configurations are practically congruent, is considered.

Canonical moduli and general solution of equations of a two-dimensional static problem of anisotropic elasticity

Equations of a two-dimensional static problem of anisotropic elasticity are brought to a simple form with the use of orthogonal and affine transformations of coordinates and corresponding transformations of mechanical quantities. It is proved that an arbitrary matrix of elasticity moduli containing six independent components can be always converted by a congruent transformation to a matrix with two independent components, which are called the canonical moduli. Depending on the relations between the canonical moduli, the determinant of the matrix of operators of equations in displacements is presented as a product of various quadratic terms. A general presentation of the solution of equations in displacements in the form of a linear combination of the first derivatives of two quasi-harmonic functions satisfying two independent equations is given. A symmetry operator (i.e., a formula of production of new solutions) is found to correspond to each presentation. In a three-dimensional case, the matrix of elasticity moduli with 21 independent components is congruent to a matrix with 12 independent canonical moduli.

A finite element finite-strain formulation for modeling colliding blocks of Gent materials

The present paper is devoted to the analysis of the contact/impact problems with Coulomb friction and large deformation between two hyperelastic bodies of Gent model. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the incremental law of Gent model is derived. A first order algorithm is applied for the numerical integration of the time-discretized equation of motion. Efficiency and accuracy of the resulting method is illustrated on a two-dimensional static contact problem and a three-dimensional dynamic contact problem as compared with ANSYS simulations.

Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

Results on stress concentration in thin shells with curvilinear holes subject to plastic deformation and finite deflections are reviewed. The holes (circular, elliptical) are reinforced with thin-walled elements (rings, rods) of different stiffness. A numerical method of solving doubly nonlinear problems of statics for shells of complex geometry is outlined. The stress distribution near curvilinear holes in spherical, cylindrical, and conical shells under statical loading is studied. The numerical results are analyzed

Redistribution of finite elastic strains after the formation of inclusions. Approximate analytical solution

A class of two-dimensional static problems of the stress-strain state of non-linearly elastic bodies, in which domains with different elastic properties (inclusions) arise after preloading, is considered. Problems are formulated and solved using the theory of the repeated superposition of finite strains. The mechanical properties of the initial material and the material of the inclusions are described by Murnaghan-type or Mooney-type constitutive relations. Two ways of specifying the constitutive relations for the material of an inclusion are considered: when there are inherent strains in this material and when there are not. Approximate analytical methods are used for the solution.

Stress-strain analysis of closed nonthin orthotropic conical shells of varying thickness

The approach developed to solve two-dimensional static problems for nonthin conical shells of varying thickness is used to examine the effect of the geometrical parameters on the stress-strain state of shells. The approach is based on spline-approximation and a stable numerical method of solving one-dimensional problems

Two-dimensional fractal-like finite element method for thermoelastic crack analysis

The fractal-like finite element method has been proved to be very efficient and accurate in two-dimensional static and dynamic crack problems. In this paper, we extend our previous study to include the thermal effect for two-dimensional isotropic thermal crack problems. Both the temperature intensity factor and thermal stress intensity factor can be calculated directly. The temperature distribution is first found, which is imposed thereafter as a thermal load in the elastic problem. The transformation function used in the study has been found analytically. The effects of different thermal loading on the thermal stress intensity factor are presented. The numerical examples are compared with the results from other methods and find to be in good agreement.

A Finite Element Method Code to Analyse Waveguide Dispersion

The aim of our current work is to construct an algorithm for solving problems of radiation and detection of transient electromagnetic waves in the analysis of permeable bodies. The Time-Domain Finite Element Method (TDFEM) will be used in combination with other numerical techniques. TDFEM is necessary to design — with good precision — the broadband antennas to be used. Here we report the first step achieved in this goal. We designed a Finite Element Method (FEM) code to solve two-dimensional static problems in homogeneous waveguides. In order to test the algorithm constructed, it was applied to calculate dispersion features and to obtain the electromagnetic field in the transversal section of the waveguides for the propagation modes. The results obtained are in good agreement with the analytical results and those of other authors. Currently, we are developing a more complete code that includes time (TDFEM) and that focuses on three-dimensional problems.

Analytical Two-dimensional Solutions For HeatTransfer In A System With Rectangular Fin

A two-dimensional heat transfer in the element of a periodic system with a rectangular fin by steady-state conditions has been studied analytically. Firstly we formulate the two-dimensional statement of steady-state heat transfer through the element of the periodical system with the rectangular fin. Then the well known one-dimensional statement of the problem is examined. Further we solve the system of ordinary differential equations, which is obtained from the twodimensional problem statement by the method of conservative averaging, and we receive an approximate solution. Finally we receive the exact solution of the two-dimensional problem by using Green functions for two rectangles (the wall and the fin). This solution is obtained in the form of Fredholm’s integral equation of the second kind.

Studying the Effect of the Spatial Frequency and Amplitude of Corrugation on the Stress–Strain State of Cylindrical Shells

An approach developed earlier to solve two-dimensional static problems for noncircular cylindrical shells is used to analyze the effect of the spatial frequency and amplitude of corrugation on the stress–strain state of shells. The results of the analysis are presented in the form of plots and tables

SOME APPROACHES TO THE SOLUTION OF PROBLEMS ON THIN SHELLS WITH VARIABLE GEOMETRICAL AND MECHANICAL PARAMETERS

A number of approaches to the solution of stress problems for anisotropic inhomogeneous shells in the classical formulation are discussed. A review is made of approaches to the solution of one- and two-dimensional static problems for thin shells with variable parameters and to the solution of stress–strain problems for anisotropic shells of revolution under axisymmetric and non-axisymmetric loading, shallow convexo-convex shells, noncircular cylindrical shells, plates of various shapes, and shells of complex geometry

Moving least square reproducing kernel method for electromagnetic field computation

This paper presents the meshless moving least square reproducing kernel method, originating from mechanics, which is applied for the first time to the solution of electromagnetic problems. Two-dimensional static problems are studied and simulation results show good agreement with analytical and other numerical solutions.