R-function Method Research Articles

Free Vibration Analysis of Porous Functionally Graded Material Plates with Variable Thickness on an Elastic Foundation Using the R-Functions Method

Free vibrations of porous functionally graded material (FGM) plates with complex shapes are analyzed by using the R-functions method. The thickness of the plate is variable in the direction of one of the axes. Two types of porosity distributions through the thickness are considered: uniform (even) and non-uniform (uneven). The elastic foundation is defined by two parameters (Winkler and Pasternak). To obtain the mathematical model of the problem, the first-order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction are modeled by means of a power law. Variational Ritz’s method joined with the R-functions theory is used for obtaining a semi-analytical solution of the problem. The approach is applied to a number of case studies and validated by means of comparative analyses carried out on rectangular plates with a traditional finite element approach. The proof of the efficiency of the approach and its capability to handle actual engineering problems is fulfilled for FGM plates having complex shapes and various boundary conditions. The effect of different parameters, such as porosity distribution, volume fraction index, elastic foundation, FGM types, and boundary conditions, on the vibrations is studied.

Analysis of Stability and Oscillations of Porous Power and Sigmoid Functionally Graded Sandwich Plates by the R-Function Method

In this paper, the R-function method is used for the first time to study the stability and oscillations of porous functionally graded (FG) sandwich plates with a complex geometric shape. It is assumed that the outer layers of the plate are made of functionally graded materials (FGM), and the filler is isotropic, namely ceramic. The differential equations of motion were obtained using the usual first-order shear deformation theory with a given shear coefficient (FSDT). Two models of porosity distribution according to the power (P-law) and sigmoid (S-law) laws were studied. Analytical expressions for calculating the effective mechanical characteristics of FG materials with even and uneven porosity distribution were obtained. The proposed approach takes into account the fact that the subcritical state of the plate can be heterogeneous, and therefore, first of all, the stresses in the middle plane of the plate are determined, and then the eigenvalue problem is solved in order to find the critical load. To determine the critical load and plate frequencies, the Ritz method was used along with the R-function method. The developed algorithms and software are tested on test examples and compared with known results obtained using other methods. A number of problems of stability and oscillations of the FG of porous sandwich plates with a complex geometric shape for various layer stacking schemes, various boundary conditions and laws of porosity distribution have been solved.

Bending of Plates with Complex Shape Made from Materials that Differently Resist to Tension and Compression

A new numerical-analytical method for solving physically nonlinear bending problems of thin plates with complex shape made from materials that differently resist to tension and compression is developed. The uninterrupted parameter continuation method is used to formulate and linearize the problem of physically nonlinear bending. For the linearized problem, a functional in the Lagrange form, given on the kinematically possible displacement rates, is constructed. The main unknown problems (displacements, strains, stresses) were found from the solution of the initial problem, which was solved by the Runge-Kutta-Merson method with automatic step selection, by the parameter related to the load. The initial conditions are found from the solution of the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method in combination with the R-function method, which allows to submit an approximate solution in the form of a formula – a solution structure that exactly satisfies the boundary conditions and is invariant with respect to the shape of the domain where the approximate solution is sought. The test problem for the nonlinear elastic bending of a square hinged plate is solved. Satisfactory agreement with the three-dimensional solution is obtained. The bending problem of the plate of complex shape with combined fixation conditions is solved. The influence of the geometric shape and fixation conditions on the stress-strain state is studied. It is shown that failure to take into account the different behavior of the material under tensile and compression can lead to significant errors in the calculations of the stress-strain state parameters.

PLACEMENT AND CONSTRUCTION OF 3D OBJECTS IN THE SYSTEM OF THE NATIONAL VIRTUAL UNIVERSITY BASED ON 3D TECHNOLOGIES

The article discusses the fields of application and content of virtual reality systems, the results of a comparative analysis of virtual educational environment platforms, and their role in presenting educational content in a virtual electronic format. The subject of research is geometric modeling, visualization methods and software tools of 3D models of objects in virtual environments. The purpose of the research is to develop algorithms for the representation and visualization of 3D models of exterior and interior objects in the organization of a virtual university, as well as a software tool for creating 3D models in the organization of a virtual university. Scientific novelty of the research, we offer new methods for geometric modeling of 3D objects in visualization, i.e. constructive logic-algebraic R-function method (RFM). The practical results of the research are aimed at increasing the activity of learners in modern virtual 3D environments in educational institutions based on the software tool, and thereby supporting the teaching process in the virtual three-dimensional environment designed on the basis of modern national landscape design in the Uzbek language “vAcademia.uz” software was developed.

Ritz R-Function Method for the Analysis of Variable-Stiffness Plates

This paper presents a novel approach for analyzing the free vibration response of variable-stiffness panels. A method is proposed herein, which relies upon a combination of the R-functions and the Ritz method. A powerful and peculiar aspect of this approach consists in the possibility of handling any complex geometry with reduced modeling effort and few degrees of freedom to be handled. The so-obtained formulation is of particular interest for the early design studies of plate- and shell-like structures, especially when topological features are of concern. Specifically, the possibility of studying configurations of arbitrary shape with reduced effort is of crucial importance for exploiting the potential offered by composites with nonstraight fibers. Exemplary results are presented for structures with different geometries and shapes. The comparison against results from the literature and finite element calculations reveals the potential of the approach as a valuable mean for assisting the design of innovative variable-stiffness configurations.

About the application of the R-function for geometric modeling of 3D objects of complex shapes in a virtual educational environment

This article is devoted to the creation of a national virtual university platform, geometric modeling of the design of exteriors, interiors and characters in the field of information technology based on 3D technologies. We know that visualization uses geometric splines and polygonal mesh construction methods. In virtual reality systems, each object is represented by a three-dimensional model. The real challenge now is to create custom models that control them. A three-dimensional model of a character is represented by a depth map, dots, a polygonal model, a parametric model describing anthropometric, ansaphic and profile features of a human face. The research process includes the study of the rules of visualization of virtual three-dimensional objects through internal and external models, the theory of geometric modeling, algorithms, methods and algorithms of computer modeling, the use of virtual reality algorithms in education. Today, when engineering and technology are rapidly developing in our country, traveling to the virtual world is of great interest to many. As a result, three-dimensional content and landscape design expand the human imagination and serve to capture our knowledge about the subject and object in our memory. It would be more effective to transfer practical classes in computer halls of educational institutions to the virtual world and organize them in this virtual environment using virtual objects. In addition, all aspects of the subject can be explained and taught, and students will be able to use it virtually. Creating a toolbar that includes all virtual objects becomes a priority. Object-oriented programming technologies and testing methods were used. This article proposes new methods of geometric modeling of three-dimensional objects mentioned above, that is, the constructive logical-algebraic method of R-functions (RFM). This method allows you to depict 3D objects of high complexity.

Free vibration analysis of FGM plates and shallow shells by the R-functions method

Research of free vibration of porous functionally graded material (FGM) plates with complex shape is carried out. It is supposed that thickness of the plate changes in direction of one of the axes. Two types of porosity distributions through the thickness even and uneven are considered. Elastic foundation is two parameters (Winkler and Pasternak). To obtain the mathematical model of the given problem the first order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction is modelled by the power law. Variational Ritz’s method joined with the R-functions theory is used for solution of the formulated problem. The developed approach is tested on a lot of problems. Comparative analysis of the obtained results for rectangular plates confirms their verification. Demonstration of the efficiency of the proposed approach is fulfilled for FGM plate with complex shape and various boundary conditions. Effect of the different parameters on vibration characteristics such as porosity, volume exponent, elastic foundation, types of FGM, boundary conditions is studied.

Nonlinear bending of polygonal functionally graded plates resting on elastic foundation

To investigate nonlinear bending of the functionally graded (FGM) plates with complex shape and resting on elastic foundation a variational-structural method (RFM) is proposed. Mathematical statement of nonlinear boundary value problems of plate bending is carried out in the framework of the classical geometrically nonlinear plate theory. To solve a sequence of linear boundary value problems obtained as a result of the linearization of the original nonlinear system of equations by the method of successive loadings and the Newton method, the method of R-functions was applied. Testing, as well as comparison of the obtained results with the results of other authors, made it possible to establish the reliability and effectiveness of the developed approach and apply it to study the stress-strain state (SSS) of thin plates with complex shape. A computational experiment was carried out for thin hexagonal plates with mixed boundary conditions for various types of external load and elastic foundation characteristics. The relationship between the greatest deflection of the plate and the load is obtained. The results are presented in the form of graphs.

Method of Solving Geometrically Nonlinear Bending Problems of Thin Shallow Shells of Complex Shape

A new numerical analytical method for solving geometrically nonlinear bending problems of thin shallow shells and plates of complex shape is given in the paper. The problem statement is performed within the framework of the classic geometrically nonlinear formulation. The parameter continuation method was used to linearize the nonlinear bending problem of shallow shells and plates. An increasing parameter t related to the external load, which characterizes the shell loading process, is introduced. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible movement speeds, is constructed. To find the main unknowns of the problem of nonlinear bending of the shell (displacement, deformation, stress), the Cauchy problem was formulated by the parameter t for the system of ordinary differential equations, which was solved by the fourth order Runge-Kutta-Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometric linear deformation. The right-hand sides of the differential equations at fixed values of the parameter t, corresponding to the Runge-Kutta-Merson scheme, were found from the solution of the variational problem for the functional in the Lagrange form. Variational problems were solved using the Ritz method combined with the R-function method, which allows to accurately take into account the geometric information about the boundary value problem and provide an approximate solution in the form of a formula - a solution structure that exactly satisfies all (general structure) or part (partial structure) of boundary conditions. The test problem for the nonlinear bending of a square clamped plate under the action of a uniformly distributed load of different intensity is solved. The results for deflections and stresses obtained using the developed method are compared with the analytical solution and the solution obtained by the finite element method. The problem of bending of a clamped plate of complex shape is solved. The effect of the geometric shape on the stress-strain state is studied

MATHEMATICAL MODELING OF THE QUASI-STATIONARY PROCESSES OF VISCOUS MIXTURE MIXING IN A RECTANGULAR AREA BY THE R-FUNCTIONS METHOD

Mixing processes are found in the chemical, pharmaceutical and food industries. Fluid mixing is one of the fundamental scientific problems associated with modern concepts of regular and chaotic dynamics. The paper considers the problem of mathematical modeling of the quasi-stationary process of mixing a viscous mixture. This problem consists of two sub-problems: determination of the velocity field in the flow region (Eulerian formalism) and investigation of the trajectories of individual fluid particles (Lagrange formalism). To solve the first subproblem, it is proposed to jointly use the principle of superposition, the structural method (method of R-functions) and the Ritz variational method. The methods of nonlinear dynamics and qualitative theory of differential equations are used to solve the second subproblem. A plane quasi-steady flow is considered in a rectangular region and it is assumed that the side walls are at rest, and the upper and lower walls move alternately according to the given laws. According to the method of R-functions, the structures of the solutions were built and the use of the Ritz variational method for the approximation of the uncertain components of the structures was justified. The operation of the proposed method is illustrated by the results of a computational experiment, which was conducted for different modes of wall motion. The practical interest of the considered regimes is due to the fact that they lead to the emergence of chaotic behavior when mixing occurs most efficiently. Using the methods of nonlinear dynamics, the location of periodic (hyperbolic and elliptical) points was investigated and the Poincaré section was constructed. Further research with the help of the method proposed in the work can be related to the consideration of flows in more geometrically complex regions and more complex mixing regimes, as well as in the application to the calculation of industrial problems.

Mathematical and Computer Modeling of Convective Heat Transfer in Fuel Cartridges of Fuel Elements with Different Shapes and Packing of Rods

The paper consists of three sections and is of an informational and generalizing nature, indicating promising areas for further research. The first section "R-functions method in mathematical modeling of convective heat transfer in fuel cartridges with fuel elements" is devoted to the use of new constructive tools of the R-functions method for mathematical and computer modeling of fuel elements packings with different types of symmetry, as well as the study of convective heat transfer in fuel elements grids and the effect of the type of packing on the distribution of velocity and temperature. An octahedral cartridge with 37 fuel elements packed according to three patterns (cyclic, checkerboard and in-line) is considered. It is noted that when constructing the equations of a cartridge with bundles of fuel elements using the new method, the number of R-operations and, accordingly, the calculation time are significantly reduced. An analysis of the obtained results allows to conclude that the maximum temperature is obtained with cyclic packing. The scheme of the reactor, the cartridges of which are hexagonal casings, with 91 fuel elements placed in each of them both with checkerboard and cyclic packing, is also considered. In the second section "Thermal-hydraulic calculation of fuel elements cartridges in case of violation of the rods packing symmetry", a hexagonal fuel cartridge with 169 fuel elements and checkerboard packing is considered. An increase in temperature is analyzed in case of violation of the packing symmetry while maintaining the parallelism of the rods, as well as in case of a curvature of one of them. The third section "R-functions, fuel element with polyzonal finning of the shell and heat transfer during fluid motion" is focused on the construction of equations for various finning surfaces of fuel elements and the study of hydrodynamic and temperature fields in case of polyzonal finning of the shell. At the same time, using the apparatus of tensor analysis, a transition to a curvilinear non-orthogonal (helical) coordinate system was made. It is noted that mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult to conduct for one reason or another. In addition, working with mathematical model of the process and the computational experiment make it possible to investigate the properties and behavior of the process in various situations relatively quickly and without significant expenses. The reliability of the methods, results and conclusions is confirmed by comparison with the information given in the references, the results of the analysis of the numerical convergence of solutions and the calculation of the residual.

Solution of the Problems of Stability of Laminated Plates with Holes by the Method of R-Functions

We propose a method for the evaluation of the critical loads for laminated plates with holes. Plates and holes made in these plates may have different geometric shapes and conditions of their fastening. It is assumed that the plate is compressed by static forces in its middle plane. The mathematical statement of the problem is formulated within the framework of the refined first-order theory based on the hypothesis of straight line (Timoshenko-type theory). To solve the problem, we use the Ritz method and the R-functions theory. The solution of the problem of elasticity theory takes into account the inhomogeneous subcritical state of the laminated plate. The reliability of the proposed approach is confirmed by good agreement of the obtained numerical results with the available data. Its efficiency is illustrated by an example of investigation of the problem of stability of laminated plates with one and two quadrangular holes. We analyze the influence of the arrangement of holes, their sizes, and the method of fastening on the critical load and natural frequencies.

Mathematical modeling of fluid flows through the piecewise homogeneous porous medium by R-function method

The stationary fluid flow through a piecewise homogeneous porous medium is considered under the assumption that Darcy's law holds. The mathematical model of this problem is defined as an elliptic equation for the stream function, supplemented by the second-type boundary conditions at the water boundaries and the first-type boundary conditions at the impervious to liquid boundaries. The problem statement also includes the conditions of conjugation at the separation line between two soils and the unknown value of fluid discharge, which can be established from the additional integral ratio. It is proposed to use the structure-variational method of R-functions in order to numerically analyze and solve the current problem. The complete solution structure for the boundary value problem of stream function regarding the R-functions method is established, moreover, the application of the Ritz method for approximating an unspecified structural formula component is substantiated. Then, the approximate value of the fluid discharge and the approximate solution of the original problem are found from the additional integral ratio. The computational experiment was carried out with different coefficients of permeability within the area, which has the shape of the lower half ring. It is established that as the number of coordinate functions increases, the value of fluid discharge becomes constant, indicating the convergence of the proposed method.

Application of V.L. Rvachev R-function method to solving mathematical modeling problems

Application of V.L. Rvachev R-function method to solving mathematical modeling problems

Vibrations of Plates with Complex Shape: Experimental Modal Analysis, Finite Element Method, and R-Functions Method

In this paper, the dynamic behavior of 3D-printed plates with different shapes and boundary conditions is investigated. The natural frequencies and mode shapes were determined using three different methods: the experimental analysis, the finite element method, using Nastran, and the R-functions method. The experimental and theoretical results are compared. The specimens tested included four cases. The test procedure is deeply described, and the material properties of the plates are given. The fixed-fixed configuration shows a better agreement both in the rectangular plate and in the plate with rectangular cuts, and the R-functions method gives better convergence with respect to the experimental and finite element analysis. The simply supported arrangement indicates some uncertainty in the boundary realization of the specimen.

Thermal Accelerometer Simulation by the R‑Functions Method

As well as many modern devices, thermal accelerometers (TAs) need a sophisticated mathematical simulation to find the ways for their performance optimization. In the paper, a novel approach for solving computational fluid dynamics (CFD) problems in the TA’s cavity is proposed (MQ-RFM), which is based on the combined use of Rvachev’s R-functions method (RFM) and the Galerkin technique with multiquadric (MQ) radial basis functions (RBFs). The semi-analytical RFM takes an intermediate position between traditional analytical approaches and numerical methods, such as the finite-element method (FEM), belonging to the family of the so-called meshless techniques which became popular in the last decades in solving various CFD problems in complex-shaped cavities. Mathematical simulation of TA by using the MQ-RFM was carried out with the purpose to simulate the temperature response of the device and to study and improve its performance. The results of numerical experiments were compared with well-known analytical and numerical benchmark solutions for the circular annulus geometry and it demonstrated the effectiveness of the MQ-RFM for solving the convective heat-transfer problem in the TA’s cavity. The use of solution structures allows one to take a relatively small number of expansion terms to achieve an appropriate accuracy of the approximate solution satisfying at the same time the given boundary conditions exactly. The application of the MQ-RFM gives the possibility to obtain semi-analytical solutions to the diffusion-convection problems and to identify the main thermal characteristics of the TA, that allows one to improve the device performance.

ANALYTICAL MODELLING OF PERFORATED GEOMETRICAL DOMAINS BY THE R-FUNCTIONS

This paper deals with the construction of boundary equations for geometric domains with perforation. Different types of perforated geometric domains are considered. The R-functions method for analytical modelling of perforated geometrical domains is used. For all constructed equations, function plots are obtained.

Mathematical model and computational experiments for the calculation of three-layer plates of complex configuration

Mathematical model and computational experiments to solve the problems of bending of three-layer plates of complex configuration are considered in the paper. Mathematical model and computational algorithm given in the paper are developed on the basis of the Hamilton-Ostrogradky’s variation principle, the Bubnov-Galerkin methods and the Rvachev R-functions. Numerical results of the problem are presented. Results are given in tabular and graphical form. Numerical studies of the results obtained are analyzed using the R-function method in combination with the Bubnov-Galerkin variational method to solve the problem of calculating the vibration of three-layer plates of complex configuration.

Features of Constructing an Orthonormal Basis Based on Atomic Functions as Applied to Solving Boundary Value Electrodynamics Problems for Regions of Complex Shape

A method for orthonormalizing atomic functions is proposed and substantiated. The efficiency of using a basic system to solve a number of boundary value electrodynamics problems is shown. Numerical experiment and analysis of the physical results showed that the R-functions method combined with basis functions is universal for waveguides and resonators of arbitrary shape.

APPLICATION OF R-FUNCTIONS METHOD AND SMOOTHED PARTICLE HYDRODYNAMICS FOR FLUID SIMULATION

Context. Existing fluid simulation methods have several disadvantages and can be improved with the help of new approaches to the solution of problems of computational fluid dynamics, which confirms the relevance of the work. Objective. The goal of the work is to improve existing methods of mathematical modeling of fluid based on smoothed particle hydrodynamics and R-functions method. Method. A new approach of joint use of smoothed particle hydrodynamics, marching cubes and R-functions method is proposed. Smoothed particle hydrodynamics helps to simulate fluid movement in real time. The method considers fluid as a discrete number of sample points (particles), which have mass, velocity, position and physical field quantities (pressure, temperature, mass-density, etc.). The R-functions method allows to solve the inverse problem of analytic geometry: finding an analytical equation of a 2D (3D) object based on its geometrical representation. Using the obtained equation, one can simply detect a particle collision with the object boundary and plot the object surface with the help of marching cubes algorithm. The suggested method allows to achieve good simulation quality and to perform all needed calculations and rendering in real time. Results. Computational experiments for the problem of fluid simulation were carried out. Various numbers of particles were used. Different kinds of objects were put into the considered region in order to investigate the fluid behavior. Conclusions. The results of visual simulations allow us to say that the obtained approach works as expected. Therefore, this method can be applied to several problems of fluid simulation where the collision detection with arbitrary objects is considered. Further research may be devoted to the optimization of neighbor-search algorithm, to performing all calculations in graphics processing unit or to taking into account other physical quantities.