Mathematical Modelling of the Elliptical Vortex Ring in a Viscous Fluid with the Vortex Filament Method

The article deals with modelling an elliptical vortex ring in a viscous fluid using the Lagrangian vortex filament method. The novelty is that earlier only inviscid flows restricted vortex filament method application. The proposed viscosity model uses an analogue of the diffusion rate method, which is widely applied to simulate plane-parallel and axisymmetric flows of viscous fluid. A transfer of the formula of a diffusion rate from two-dimensional flows to the model of spatial vortex filament is due to assumption that swirling of vortex lines (helicity of vorticity) is unavailable. Despite the laxity of the diffusion rate model for general spatial flows, its application enables taking into account the effect of viscous diffusion of vorticity, which provides expansion of vortex tubes in space. The paper formulates the vortex filament method in which the filaments are broken into the vortex segments. Such discretization enables turning from the equation of vorticity evolution in partial derivatives to a system of ordinary differential equations with respect to the parameters of the segments. Formulas to calculate a filament system-induced flow rate as well as formulas to perform approximate calculation of an analogue of the diffusion rate are given.The objective is to propose the viscosity model as an application to the vortex filament method by the example of modelling the evolution of an elliptical vortex ring in viscous fluid. The calculation results obtained by the vortex method are compared with the existing experiment and with the calculation performed by the grid method in the OpenFOAM package. A feature of the problem is that there are zones of nonzero helicity of vorticity where the proposed model of viscosity, strictly speaking, is not correct. It is shown that the results of calculations are in good agreement with each other and are in complete agreement with experiment. This allows saying that the effects of swirling vortex lines do not significantly affect the results of modelling a specific example of the spatial flow of viscous fluid by the proposed modification of the vortex filament method.

Efficiency Analysis of OLAP-data Hypercube Decomposition for Exponential Computational Complexity Methods

The paper studies problems of reduction (decomposition) of OLAP-hypercube multidimensional data models. When decomposing large hyper-cubes of multidimensional data into sub-cube components the goal is to increase the computational performance of analytical OLAP systems, which is related to decreasing computational complexity of reduction methods for solving OLAP-data analysis problems with respect to the computational complexity of non-reduction methods, applied to data directly all over the hypercube. The paper formalizes the concepts of reduction and non-reduction methods and gives a definition of the upper bound for the change in the computational complexity of reduction methods in the decomposition of the problem of analyzing multidimensional OLAP-data in comparison with non-reduction methods in the class of exponential degree of computational complexity.The exact values of the upper bound for changing computational complexity are obtained for the hypercube decomposition into two sub-cubes on sets consisting of an even and an odd number of sub-cube structures, and its main properties are given, which are used to determine the decomposition efficiency. A formula for the efficiency of decomposition into two sub-cube structures for reduction of OLAP data analysis problems is obtained, and it is shown that with an increase in the dimension “n” of the lattice specifying the number of sub-cubes in the hypercube data structure, the efficiency of such a decomposition obeys an exponential law with an exponent “n/2”, regardless of the parity “n”. The examples show the possibility to use the values (found) of the upper bound for the change in computational complexity to establish the effectiveness criteria for reduction methods and the expediency of decomposition in specific cases.The paper results can be used in processing and analysis of information arrays of hypercube structures of analytical OLAP systems belonging to the Big-Data or super-large computer systems of multidimensional data.

Planar Five-link Biped Robot Movement over a Stepped Surface

Modeling the anthropomorphic robot movement is of great interest to researchers all over the world. At the same time, the movement control of a walking mechanism is always a high dimension challenge. The difficulty with the anthropomorphic robot control is also caused by the fact that such a mechanism has always a hybrid dynamics and represents a sequential change of two phases – the single support phase and the double support phase (phase of changing robot’s leg). At the single support phase and at another phase the behavior of the biped robot is described by a system of ordinary differential equations and by a system of linear algebraic equations, respectively.The task of biped robot movement control has been studied in detail for the case when the robot moves over the horizontal surface. Obstacles make the task significantly complicated. The paper considers the movement control of the biped robot over the surface that is a periodic alternation of horizontal sections and obstacles. The obstacles represent steps of the same height known. It is assumed that the lengths of horizontal sections and steps are known as well. The objective is to create a control that provides robot’s periodic movement over the specified surface according to inherent characteristics of a walking human.For the single support phase, the outputs are proposed, the equality of which to zero corresponds to the robot’s movement with a given set of characteristics. The paper presents the feedback controls that stabilize the proposed outputs for a finite amount of time. By choosing the feedback parameters, it is possible to adjust the stabilization time so that the outputs become equal to zero when reached the end of each step.It is shown that for the chosen control law, the problem of constructing the control of robot’s periodic movement is reduced to the solution of a nonlinear equation. In the paper, we discuss the approaches to solving this equation and present the results of numerical simulation.The results obtained can be used to solve the problem of providing control of the biped robot movement over the surfaces with obstacles of a more complicated shape.Modeling the anthropomorphic robot movement is of great interest to researchers all over the world. At the same time, the movement control of a walking mechanism is always a high dimension challenge. The difficulty with the anthropomorphic robot control is also caused by the fact that such a mechanism has always a hybrid dynamics and represents a sequential change of two phases – the single support phase and the double support phase (phase of changing robot’s leg). At the single support phase and at another phase the behavior of the biped robot is described by a system of ordinary differential equations and by a system of linear algebraic equations, respectively.The task of biped robot movement control has been studied in detail for the case when the robot moves over the horizontal surface. Obstacles make the task significantly complicated. The paper considers the movement control of the biped robot over the surface that is a periodic alternation of horizontal sections and obstacles. The obstacles represent steps of the same height known. It is assumed that the lengths of horizontal sections and steps are known as well. The objective is to create a control that provides robot’s periodic movement over the specified surface according to inherent characteristics of a walking human.For the single support phase, the outputs are proposed, the equality of which to zero corresponds to the robot’s movement with a given set of characteristics. The paper presents the feedback controls that stabilize the proposed outputs for a finite amount of time. By choosing the feedback parameters, it is possible to adjust the stabilization time so that the outputs become equal to zero when reached the end of each step.It is shown that for the chosen control law, the problem of constructing the control of robot’s periodic movement is reduced to the solution of a nonlinear equation. In the paper, we discuss the approaches to solving this equation and present the results of numerical simulation.The results obtained can be used to solve the problem of providing control of the biped robot movement over the surfaces with obstacles of a more complicated shape.

Multi-Output Regression for Analyzing Power System Random States in Reliability Assessment by the Monte Carlo Method

Increasing calculation speed of the electric power system (EPS) reliability of is one of the key issues in their operational management and long-term development planning. Analytical methods to assess the EPS reliability seem to be impossible due to large size of the problem and, as a consequence, essentially the only option for assessing is to use the Monte Carlo method. When it is used both the speed and the accuracy of calculation directly depend on the number of randomly generated system states and the complexity of their calculation in the model. Methods aimed at increasing computational efficiency can relate to two directions - reducing the states under consideration and simplifying the computational model for each state. Both options are performed provided that calculation accuracy is retained.The article presents research on using the machine learning methods and, in particular, the multi-output regression method to modernize the reliability assessment technique via the Monte Carlo method. Machine learning methods are used to determine the power deficit (realization of a random variable) for each random EPS state.The use of multi-output regression enables comprehensive determining of values of all the required variables. The experimental studies are based on the two test circuits of electric power systems: three-zone and IEEE RTS-96 with 24 zones of reliability.

Modal Analysis Problem Solution for a Mathematical Model Formed by the Extended Nodal Method

The article proposes an option for transforming a mathematical model of the object, formed by the extended nodal method in the time-domain solution for modal analysis. Since finding the eigenvalues ​​and eigenvectors for systems of ordinary equations given in the Cauchy normal form is possible, calculations are presented that allow us to obtain a system of equations in the Cauchy normal form from a mathematical model in a differential-algebraic form through linearization. The extended nodal method contains derivatives of state variables in the vector of unknown, and the Jacobi matrix obtained at each Newton iteration of each step of numerical integration can be used to obtain a linearized mathematical model, but the equilibrium equations, as a rule, contain several derivatives with respect to time. By introducing additional variables, it is possible to reduce the linearized mathematical model to the Cauchy normal form, while the Jacobi matrix structure remains essentially unchanged.The proposed solution is implemented in the mathematical core of the PRADIS Gen2 PA-8 software package, which made it possible to expand its functionality by an operator of modal analysis.The presented calculations of test schemes have shown the correctness of the method proposed.

Optimizing GOST R 34.12 "Magma" Algorithms for 8-Bit Microcontrollers

The paper concentrates on development of optimizing methods for the GOST R 34.12-2015 "Magma" cipher algorithm when it is implemented on 8-bit microcontrollers. There is a number of techniques in the paper, which being used, allow you to create the specialized implementations of the algorithm: 1) focused on the operation speed; 2) focused on reducing the memory used; 3) optimal which involves the best solutions based on two previous implementations. Each optimization method is represented by description and performance indicators of the results obtained in comparison with the direct implementation of the algorithm. So, in the case of optimal algorithm implementation the enciphering process is 11 times accelerated, and an amount of the occupied memory is 1/32 of the microcontroller's memory. The built-in compiler tools were used to optimize the software code. The techniques described are applicable to any 8-bit platform.

Computational Complexity Analysis of Decomposition Methods of OLAP Hyper-cubes of Multidimensional Data

The paper investigates the problems of reduction (decomposition) of multidimensional data models in the form of hypercube OLAP structures. OLAP data processing does not allow changes in the dimension of space. With the increase in data volumes, the productivity of computing cubic structures decreases. Methods for reducing large data cubes to sub-cubes with smaller volumes can solve the problem of reducing computing performance.The reduction problems are considered for cases when the cube lattice has already determined criteria aggregation, and the cube decomposition into smaller cubes is needed to reduce the computation time of the full lattice when dynamically changing data in the cube.The objective of the paper is to find conditions for reducing the computational complexity of solving data analysis problems by reduction methods, to obtain exact quantitative boundaries for reducing the complexity of decomposition methods from the class of polynomial degrees of complexity, to establish the nature of the dependence of computational performance on the structural properties of a hypercube, and to determine the quantitative boundaries of computational performance for solving decomposition problems of data aggregation .The study of the computational complexity of decomposition methods for the analysis of multidimensional hyper-cubes of polynomial-logarithmic and polynomial degrees of complexity is carried out. An exact upper limit is found for reducing the complexity of decomposition methods for analyzing the initial OLAP - data hypercube with respect to non-decomposition ones and based on them criteria are proved for the effective application of reduction methods for analyzing hypercube structures in comparison with traditional non-reduction methods.Examples of decomposition methods of cube structures are presented, both reducing and increasing computational complexity in comparison with calculations using the full model.The results obtained can be used in processing and analysis of information arrays of hypercube structures of analytical OLAP-systems belonging to the BigData class, or ultra-large computer multidimensional data systems.

The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

Algebraic Criteria of the Integral Separation for Solving Certain Classes of Ordinary Differential Equations Systems

One of the important directions of the qualitative theory of ordinary differential equations is to study the properties of linear systems that satisfy the condition of integral separation. Anyway, integral separation becomes apparent in all studies concerning the asymptotic behavior of the solutions for the linear systems under the action of small perturbations.The papers of V.M. Millionschikov, B.F. Bylov, N.A. Izobov, I.N. Sergeev et al. proved that the available integral separation is the main reason for the rough stability of the characteristic Lyapunov exponents, the rough stability of the highest Lyapunov exponent, and the rough diagonalizability of systems by Lyapunov transformations, and other fundamental properties of linear differential systems.The paper presents the basic properties of the set of linear systems with constant, periodic, reducible coefficients and proves the algebraic criteria for their property of integral separation of solutions to be available.The results can be used in modeling dynamic processes.