Three-dimensional Formulation Research Articles

Numerical simulations of the region of possible sprite inception in the mesosphere above winter thunderstorms under wind shear

Transient luminous events (TLEs) is the collective name given to mesospheric electrical breakdown phenomena occurring in conjunction with strong lightning discharges in tropospheric thunderstorms. They include elves, sprites, halos, and jets, and are characterized by short lived optical emissions, mostly of red (665 nm) and blue (337 nm) wavelengths. Sprites are caused by the brief quasi-electrostatic field induced in the mesosphere, mostly after the removal of the upper positive charge of the thundercloud by a +CG, and they have been recorded above most of the lightning activity centers on Earth. In wintertime, there are just a few areas where lightning occurs, and of those, sprites have been observed over the Sea of Japan, the British Channel, and the Mediterranean Sea. Unlike their summer counterparts, winter thunderstorms tend to have weaker updrafts and as a result, reduced vertical dimensions and compact charge structures, whose positive and negative centers are located at lower altitudes. These storms are often susceptible to significant wind shear and as a result may exhibit a tilted dipole charge structure and a lateral offset of the upper positive charge relative to the main negative charge. We present results of numerical simulations using a three-dimensional explicit formulation of the mesospheric quasi-electrostatic electrical field following a lightning discharge from a typical mid-latitude winter thunderstorm exhibiting tilt due to wind shear and evaluate the regions of possible sprite inception. Our results show, as numerous observations suggest, that sprites can be shifted a large distance from the location of the parent +CG in the direction of the shear and will occur over a larger region compared with non-sheared storms.

CALCULATION OF THE MAIN CHARACTERISTICS OF A HYBRID VACUUM PUMP

Data are presented for calculating the main characteristics of a hybrid turbomolecular vacuum pump operating in the molecular gas flow mode. The pump contains a block of turbomolecular stages, an intermediate stage and a molecular stage with screw channels made on a cylindrical rotor. To determine the probabilities of gas molecules passing through the entire flow part of the pump in the forward and reverse directions, the Monte Carlo method in a three-dimensional formulation was used. The program allows you to evaluate the influence of the dynamic and geometric parameters of the flow path on the main characteristics of the pump in order to optimize them.

On the Schapery nonlinear viscoelastic model: A review

Among the various models presented so far to describe the nonlinear viscoelastic behavior of materials, the model proposed by Schapery in the 1960s stands out for its wide acceptance and significant contributions to the field. Through a detailed inspection of the main articles presented by Schapery and other related studies, the present paper reviews the Schapery nonlinear viscoelastic model. One-dimensional and three-dimensional formulations of the model for both isotropic and anisotropic materials, along with model accuracy, are discussed. The present article highlights the challenges of material characterization methods, such as creep-recovery testing, stress relaxation testing, and dynamic mechanical analysis. The article also addresses the issues of characterizing the material's behavior under non-recoverable strain and varying temperatures. Finally, the future research needs in this field are outlined, emphasizing the potential for further advancements in our understanding of nonlinear viscoelasticity.

Numerical Simulation of Hydrodynamics and Heat Transfer in a Reactor with a Fluidized Bed of Catalyst Particles in a Three-Dimensional Formulation

The hydrodynamics and heat transfer in a reactor with a fluidized bed of catalyst particles and an inert material were simulated. The particle bed (the particle density was 2350 kg/m3, and the particle diameter was 1.5 to 4 mm) was located in a distribution device which was a grid of 90 × 90 × 60 mm vertical baffles. The behavior of the liquefying medium (air) was modeled using a realizable k-ε turbulence model. The behavior of particles was modeled using the discrete element method (DEM). In order to reduce the slugging effect, the particles were divided into four separate horizontal layers. It was determined that with the velocity of the liquefying medium close to the minimum fluidization velocity (1 m/s), slugging fluidization is observed. At a velocity of the liquefying medium of 3 m/s, turbulent fluidization in the lowest particle layer and bubbling fluidization on subsequent particle layers are observed. With an increase in the velocity of the liquefying medium over 3 m/s, entrainment of particles is observed. It was shown that a decrease in the density of the liquefying medium from 1.205 kg/m3 to 0.383 kg/m3 when it is heated from 298 K to 923 K would not significantly affect the hydraulic resistance of the bed. Based on the obtained results, it can be stated that the obtained model is optimal for such problems and is suitable for the further description of experimental data.

Determination of Temperature Stresses during the Construction of Massive Monolithic Foundation Slabs, Taking into Account the Subgrade Compliance

Background The problem of early cracking caused by the heat of concrete hardening is relevant for massive reinforced concrete structures, including foundation slabs. The purpose of this work is to develop the methodology for determining temperature stresses during the construction of foundation slabs, taking into account the interaction with the subgrade. Methods The Pasternak elastic foundation model with two-bed coefficients is used for the soil. The temperature of the foundation slab is considered a function of only one coordinate z (temperature changes only along the thickness of the slab). As a result, to determine the stress-strain state of the slab, a fourth-order differential equation for deflection was obtained. A technique for numerically solving the resulting equation using the finite difference method is proposed. The calculation of the stress-strain state is preceded by the calculation of the temperature field, which is performed by the finite element method in a simplified one-dimensional formulation. Results The solution to the test problem is presented for a constant modulus of elasticity of concrete over time. The results were compared with finite element calculations in a three-dimensional formulation in the ASNYS software. The calculation was also performed taking into account the dependence of the mechanical characteristics of concrete on its degree of maturity. In this case, the picture of the stress-strain state changes significantly. The proposed method was also successfully tested on experimental data. Conclusion The proposed approach can significantly save calculation time compared to the finite element analysis in a three-dimensional setting.

INVESTIGATION OF THE CYLINDRICAL RESONATING CAVITY EFFECT ON A JET’S NEAR ACOUSTIC FIELD IN THE TWO-CHANNEL SYSTEM

Numerical gas flow modeling in the two-channel resonance system path of ITAM SB RAS is carried out. Numerical simulation of a nonstationary acoustic-convective flow was carried out using the Ansys Fluent software package in a three-dimensional formulation. As a result of the simulation, the gas-dynamic characteristics of the flow in the system central part, where a supersonic under-expanded jet is formed, are obtained. The analysis of pressure distributions in the first barrel of the jet near field, as well as inside its structure under the conditions of a two-channel system, is carried out. Six points in the near field of the jet were selected for the study. Taking into account the unsteadiness of the flow in the resonant system path, the studied points record the fluctuations of the formed flow and the change in the jet structure. The spectral composition is estimated near and inside the jet undisturbed by the resonator and a comparison with the spectrum of the resonant system is presented. The possibility of influencing the jet structure and acoustic characteristics near the jet using a resonator is shown.

A simplified model for the wear prediction of plain bearings in the variable stator vane system

The wear of plain bearings in the variable stator vane (VSV) systems in a turbojet engine is a significant problem for compressor efficiency and stability requiring adapted models. The VSV bearings are located in a complex system and are exposed to combined and oscillating loads. This study aims to predict the wear evolution of the VSV bearings, considering the complexity of the system’s geometry and the loads with adapted computational time. This paper presents a three-dimensional formulation of this highly nonlinear wear problem under simplifying hypotheses in the objective to obtain a compromise between the quality of the prediction and the CPU time.

Скінченно-елементна модель вертикального резервуара на жорсткій основі

This study addresses the problem of finite element modeling of a 20,000 m3 vertical steel tank subjected to static loads. The structure includes a cylindrical wall of total height 17,880 mm and diameter 39,900 mm. The shell thicknesses of the cylindrical wall are determined according to strength and buckling design standards. The geometric model is axisymmetric. The analysis involves the calculation of the stress and strain fields of the cylindrical wall and the contact zone between the flat bottom and the rigid foundation under various combinations of external loads, namely, excessive and hydrostatic pressures. The ANSYS Mechanical software is used for finite element analysis. Three-dimensional SOLID186 and SHELL281 elements are used for axisymmetric modeling of the shell structure in a three-dimensional formulation. To simulate the contact zone, CONTA174 and TARGE170 finite elements are used to model the moving contact surface of the bottom and the fixed surface of the rigid foundation, respectively. The model is verified by comparing the radial displacements calculated numerically and analytically. The discrepancy does not exceed 4%, thus evidencing the adequacy of the finite element model. The contact zone is analyzed for non-standard service conditions, such an excessive internal pressure in the tank (2.5 and 3 kPa compared to 2 kPa under normal conditions). The unilaterally constrained "bottom–foundation" contact zone model allows the bottom to detach from the foundation, thus leading to contact opening. A full detachment occurs under a certain combination of the excessive and the hydrostatic pressure. For certain liquid levels in the tank, the gap decreases, which may be due to a reduced effect of the excessive pressure. This is accompanied by the development of internal detachment caused by the increasing moment from the hydrostatic pressure. The internal detachment increases the bending moment at the wall–bottom junction, which, under certain conditions, may cause plastic deformations followed by the development of an emergency state.

Numerical Simulation of Compensation Injection Problems Near Deep Excavations

The stress-strain state (SSS) of the “ground foundation – foundation pit under the protection of a fence – compensation injection zone” system is complex, multifactorial and transforming in space. Taking into account a wide range of phenomena when changing the SSS of the soil base is associated with significant difficulties, especially without the use of numerical modeling using software and computer systems. Numerical modeling in the implementation of the finite element method (FEM) for the problems of complex interaction of a soil basis with underground structures during compensatory grouting makes it possible to assign injection zones in any position in space (horizontally, vertically, at an angle), determine the necessary injection parameters to achieve the building lifting of the soil basis when solving the inverse problem. In this study, using examples of numerical modeling of compensation grouting problems near the pit fence with different locations of existing buildings, the authors identified the determining factors affecting the distribution of internal forces in the enclosing structures of the excavation, as well as on the existing structures of the underground structure. The results of comparison of calculations in two-dimensional and three-dimensional formulations are presented, an increase in the bending moment of the retaining wall when carrying out work on compensatory injection by 187-279% is shown for different parameters of the distance to the grouting horizon. The results of the conducted studies confirm the need to take into account the increase in internal efforts during the work on compensatory injection in the structures of the pit fences when assigning reinforcement. In addition, in some cases it is necessary to provide additional protective measures to prevent a negative impact on the constructed underground structures to reduce emergency situations.

The stress state of a thick-walled shell with allowance for contact with a hydrogen-containing medium

Numerical and experimental methods are used to solve a multidisciplinary problem on determining the stress state of a steel shell of revolution under mechanical loading and thermal effect with allowance for its contact with a hydrogen-containing medium. The study uses a well-developed mathematical tool for solving heat conduction problems in order to solve the problem of hydrogen diffusion into metal. The effective stresses and their invariants are determined by solving the nonlinear boundary value problem of thermoplasticity of a thick-walled shell of revolution in a three-dimensional formulation. The study takes into account the experimentally found effect of changes in the mechanical properties of steel affected by hydrogen. The correctness of the proposed method and the performed calculations is quantitatively estimated by comparison with a well-known problem having an analytical solution. The paper shows that it is possible and necessary to take into account the change in mechanical properties when determining the stress state of steel structures operating in contact with a hydrogen-containing medium.

Comparative analysis of two-dimensional and three-dimensional modeling of heat transfer during operation of a gas infrared heater indoor

Relevance. It is proposed to heat local work areas with systems based on gas infrared heaters, capable of directing radiative heat flow to reduce heating costs in large premises. However, the widespread use of gas infrared heaters is hampered by the existing difficulties with the preliminary assessment of convective-radiative heat flows movement, on which the number and location of heating devices depends. The preliminary assessment is complicated by the need in some cases for 3D modeling of complex physical processes. It is necessary to evaluate the possibility of replacing labor-intensive 3D modeling with a method for calculating a heating system using gas infrared emitters based on a 2D approach to reduce the time spent on calculations. Aim. To prove that the use of a two-dimensional model of the processes under consideration makes it possible to obtain the main characteristics of the thermal regime of the premises, making it possible to replace spatial modeling. Objects. Heating system with a light-type gas infrared heater and an air exchange system. Methods. Two-dimensional and three-dimensional mathematical modeling of conjugate heat transfer processes using the finite element method. Mathematical modeling was carried out in the COMSOL Multiphysics software environment using the modules: “Heat Transfer Interface in Liquids”, “Radiation between Surfaces” and “Turbulent Flow, k-ε Interface”. Results. The article presents the results of mathematical modeling performed in three-dimensional and two-dimensional formulations. The distribution of temperatures in the air and enclosing structures, as well as the flow lines of heated air and air, which was heating, in the volume of the premise are presented. The results of two-dimensional and three-dimensional modeling were compared. Satisfactory similarity of the calculated average air temperatures in the local working area was established based on the results. The difference was less than 2℃ for different spatial modeling approaches.

3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory.

This work develops a three-dimensional (3D) weak formulation, based on the consistent couple stress theory (CCST), for analyzing the size-dependent dynamic instability behavior of simply-supported, functionally graded (FG) cylindrical microshells that are subjected to combinations of periodic axial compression and external pressure. In our formulation, the microshells are artificially divided into nl layers. The displacement components of each individual layer are selected as the primary variables, which are expanded as a double Fourier series in the in-plane domain and are interpolated with Hermitian C2 polynomials in the thickness direction. Incorporating the layer-wise displacement models into our weak formulation, we develop a Hermitian C2 finite layer method (FLM) for addressing the current issue. The accuracy and the convergence rate of our Hermitian C2 FLM are validated by comparing the solutions it produces with the accurate two-dimensional solutions of critical loads and critical pressures of FG cylindrical macroshells and single-walled carbon nanotubes, which were reported in the literature. The numerical results show the effects of the material length-scale parameter, the inhomogeneity index, the radius-to-thickness and length-to-radius ratios, the load magnitude ratio, and the static and dynamic load factors on the first principal and first secondary instability regions of parametric resonance of simply-supported FG cylindrical microshells are significant.

‘The International Style’ multilingual three-dimensional grammar and hybrid designs

The present paper describes a multilingual shape grammar to recreate the work of three designers that preconized the ‘International Style’. This is a top-down three-dimensional formulation. Whilst most grammars are monolingual and univocal, this work creates a multilingual formulation within the same artistic movement. Typically, a grammar describes a family of design solutions. These are helpful mechanisms to describe patterns that can be used for analysis or design exploration. This multilingual three-dimensional parametric grammar allows the independent recreation of original designs from three designers and hybrid solutions. Shape grammar's difficult application can be linked to its evaluation process. The original contribution of the study lies in the use of quantitative and qualitative methods for grammar evaluation, using Principal Components Analysis and user questionnaires. The PCA provides a pragmatic analysis, and the questionnaires an intuitive reading. Together they corroborate the results which are discussed using hybrids that assist in delineating parametric spaces.

Численное исследование высокоскоростного взаимодействия вращающихся ударников с различной формой головной части с преградой конечной толщины

Numerical simulation methods are used to study the effect of rotation of cylindrical strikers made of high-strength steel on the high-velocity interaction with a steel barrier. Three types of striker head shapes are considered: ogival, hemispherical, and flat. The initial velocity of the striker is 1000 m/s, and the rotation frequency varies from 0 to 10000 revolutions per second. The striker-barrier interaction angle varies from 0° to 75°. The modeling is carried out in a three-dimensional formulation using the author’s EFES 2.0 software package. This allows the simulation of the fragmentation of interacting bodies with the formation of new contact and free surfaces, as well as the erosion of materials. The obtained results show that the difference in the penetrating power of rotating and non-rotating strikers is due to the presence of a stress-strain state in the rotating striker caused by rotational motion. The effect of the rotation of the striker on its penetrating power is studied. The conditions for the striker ricochet are determined for various striker-barrier interaction angles, rotation velocities, and head part shapes.

Сравнение методов исследования терморадиационного режима производственных помещений и воздействия теплового излучения на работников

The thermoradiational situation in the steelmaking shop and at the aluminum casting site was investigated. The calculation of the heat radiance of a steelworker and a forklift operator of metallurgical production is carried out in two ways. The first method involves the engineering calculation of heat radiation and the construction of its plot in two-dimensional and three-dimensional space, based on data on the temperatures of radiation sources, the distances from them to the irradiated objects, the angles between the normal to the sources and the directions to them and the degree of blackness of radiation sources. For a workplace located in a volume limited by walls, the degree of blackness of the wall material was additionally considered. The second method involves mathematical and computer modeling. A numerical model implemented in the ANSYS finite element modeling program was used. A mathematical model of radiation based on the Discrete Ordinates method was used to display thermal processes. When comparing the results of the calculation of heat emission by two methods, it was revealed that the relative error of the diagram calculation method in two-dimensional space compared to modeling averaged 34.9 %, in three-dimensional — 22.5 %. This is due to the fact that when calculating by the diagram method, radiation sources and irradiation objects (in a two-dimensional formulation), as well as the radiating surface of sources (in a three-dimensional formulation) are considered not as three-dimensional objects (as in modeling), but as point objects, which effects the accuracy of the method.

FEATURES OF THE INITIAL DEFORMED STATE OF PYROPEZOELECTRICS OF TRIGONAL SYSTEM

A model of a prestressed thermoelectroelastic medium under conditions of initial mechanical stress, electrostatic field and temperature is proposed. Within the framework of the coupled theory of thermoelectroelasticity, linearized constitutive relations and equations of motion of a prestressed thermoelectroelastic medium are constructed. It is assumed that the initial deformed state induced in the material is homogeneous, the initial temperature does not exceed the temperature of phase transitions, and the initial electrostatic field is specified by the electric field strength vector. Within the framework of the assumptions made, a three-dimensional formulation of dynamic problems for prestressed semi-bounded media made of the materials under consideration is given. Using operational calculus methods, the problem is reduced to a system of ordinary differential equations, the solution of which is presented in matrix form and allows one to analyze the influence of external factors of various natures on the dynamics of a prestressed thermoelectroelastic medium. Particular attention is paid to a detailed study of the initial deformed state of the material under conditions of separate and combined exposure to mechanical stress, electrostatic field and temperature. The presence of piezo- and pyro-effects, characteristic of the class of pyroelectrics under consideration, is shown: the appearance of electrical induction under the action of mechanical stresses along the axes, and the appearance of electrical induction under thermal influence in the absence of mechanical stresses. The types of mechanical influences leading to maximum values of electrical induction have been established. It is shown that under combined temperature and mechanical influences, mechanical stresses play a decisive role in the nature of the induced deformations. The presence of an initial temperature can either weaken or strengthen the influence of the electrostatic field. With a combined mechanical and electrostatic effect, the presence of an initial electrostatic field of high intensity, depending on its direction, leads to qualitative changes in the nature of the stress-strain state. The research results are presented in dimensionless parameters, presented in the form of graphs and may be of particular interest in the development, design and optimization of pyropiezoelectric materials used in the creation of new micro- and nano-sized devices and devices for general purposes.

Outflow of a supersonic overexpanded air jet into a water

Numerical simulation of the underwater outflow of a supersonic overexpanded air jet is considered. The calculations are carried out in an unsteady three-dimensional formulation with a hybrid approach using Reynolds-averaged Navier–Stokes equations for nozzle flow and large eddy-simulation for jet flow. The calculation of the interaction of a gas with a liquid is performed with a volume of fluid, which takes into account the surface tension and compressibility of water. The structural features of turbulence when a supersonic air jet exhausted into water are analyzed. An unsteady formation of a supersonic jet and a gas cavity in a water is obtained, the shock-wave structure of the flow is visualized and the level of pressure fluctuations in water is reported. A comparison of the flow pattern when an air jet exhausted into air and water is made on a qualitative level. The results of numerical simulation of the jet velocity field are used to analyze the Kelvin–Helmholtz and Rayleigh–Taylor instability mechanisms and to reveal the dominant instability mechanism in different jet propagation regions. The results obtained are compared with experimental high-speed photographic recording data and numerical calculations At a qualitative level in literature.

Application of the Double Approximation Method for Constructing Stiffness Matrices of Volumetric Finite Elements

Introduction. When numerically solving problems of elasticity theory in a three-dimensional formulation by the finite element method, finite elements (FE) in the form of parallelepipeds, prisms and tetrahedra are used. Regularly, the construction of stiffness matrices of volumetric FE is based on the principle of isoparametricity, which involves the Lagrange polynomials to approximate the geometry and displacements. In computational practice, the most widespread FE are the so-called multilinear isoparametric FE with a linear law of approximation of displacements. The main disadvantage of these elements lies in the “locking” effect when modulating bending deformations. Moreover, the error of the numerical solution increases drastically in the case when the structure, in comparison to conventional deformations, undergoes significant displacements as a rigid whole. Long-term experience in solving problems of deformable solid mechanics by the finite element method has shown that existing volumetric FE have slow convergence, specifically, when modeling bending deformations of plates and shells. This study aims at constructing stiffness matrices of multilinear volumetric FE of increased accuracy allowing for rigid displacements based on the double approximation method.Materials and Methods. The mathematical apparatus of the double approximation method based on the principle of a separate representation of the distribution functions of displacements and deformations inside the element, was used to construct the stiffness matrices of volumetric FE. The storage and processing of the resulting system of equations was implemented in algorithmic terms of sparse matrices. Software development and computational experiments were carried out using the Microsoft Visual Studio 2013 64-bit computing platform and the Intel ® Parallel Studio XE 2019 compiler with the integrated Intel ® Visual Fortran Composer XE 2019 text editor. Visualization of the calculation results was performed using the descriptor graphics of the MATLAB computer mathematics package. A large eight-node SOLID185 CE of the ANSYS Mechanical software complex was used as a test sample.Results. Mathematical tool and software were developed to study the stress-strain state of massive structures under various types of external actions. The authorized application software package was verified on test examples with known analytical solutions. It has been shown that the constructed FE accurately satisfy the basic requirements for finite element modeling of spatial problems of elasticity theory.Discussion and Conclusion. The performed testing of the developed mathematical and program toolkit has shown that the finite elements constructed on the basis of the double approximation method can successfully compete with similar SOLID185 volumetric elements of the ANSYS Mechanical software complex. The proposed elements can be integrated into domestic import-substituting software systems that implement the finite element method in the form of the displacement method.

Refined three-dimensional pile element formulation for second-order analysis of pile-supported structures accounting for complex loading conditions

The second-order analysis method is a simulation-based design method but less utilized for pile-supported structures due to the current inaccurate and unstable analysis methods for piles, especially for those under complex loading conditions in pile-supported structures. A refined Three-dimensional Pile Element (3DPE) formulation is proposed to promote the application of the second-order analysis method in pile-supported structures. The proposed refined 3DPE formulation can efficiently capture the nonlinear Soil-Structure Interaction (SSI) that is over-simplified in the conventional analysis method. Compared with the existing Pile Element formulation, the proposed refined 3DPE formulation can more accurately capture the structural behaviors of piles under complex loading conditions, including the shear effect and axial-torsional coupled response. Furthermore, three soil stiffness matrices are firstly proposed to improve the numerical stability of the spatial analysis using the existing element formulation. Extensive examples are provided to validate the effectiveness of the refined 3DPE formulation, indicating its applicability in extending the second-order analysis to pile-supported structures.

PROGRAMMABLE BEHAVIOR OF THE METAMATERIAL BY KINDS OF UNIT CELLS CONNECTION

The paper is devoted to the numerical analysis of uniaxial loading of a sample of the mechanical metamaterial. The structure of the mechanical metamaterial is constructed of a ring and four ligaments, which make up a tetrachiral structure. The peculiarity of such a structure consists in torsion of the sample under the force load. Two methods of connecting cells in the metamaterial are considered: "adjoining" and "overlapping". The "adjoining" method of joining cells increases the thickness of the internal structure of the sample, which allows us to consider it as a topological defect of the metamaterial. The "overlapping" method saves the base material from which the tetrachiral structure is constructed. Differences in the structure when constructing a three-dimensional sample result in a significant change in the characteristics of the sample. Numerical solution of the problem is performed in a three-dimensional formulation using the finite element method. The constitutive relation describing the behavior of the model corresponds to Hooke's law. Numerical simulation of uniaxial loading made it possible to obtain the results of mechanical response and to analyze the effective properties of metamaterials. A topological defect in the form of a thickening of the internal structure elements led to a difference in the linear dimensions of the two samples. The increased thickness of the unit cell connection elements resulted in a decrease in the effective density of the metamaterial sample. The same sample showed a triple increase in the value of the elastic modulus. The greater ability to resist deformation resulted in a reduced twist effect compared to the sample whose cells were joined by the "overlapping" method. The results obtained will make it possible to program the mechanical behavior and properties of the metamaterial sample.