Original System Of Equations Research Articles

An improved implicit direct-forcing immersed boundary method (DF-IBM) around arbitrarily moving rigid structures

An improved implicit direct-forcing immersed boundary method (DF-IBM) is presented for simulating incompressible flows around complex rigid structures undergoing arbitrary motion. The current approach harnesses the pressure implicit with splitting of operators algorithm to handle the fluid–solid system's dual constraints in a segregated manner. As a result, the divergence-free condition is preserved throughout the Eulerian domain, and the no-slip velocity boundary condition is exactly enforced on the immersed boundary. A new pressure Poisson equation (PPE) is derived, incorporating the boundary force where the no-slip condition is already met, enabling the use of fast iterative PPE solvers without modifications. The improvement involves integrating Lagrangian weight methods having better reciprocity over the IBM-related linear operators with the implicit formulation. An additional force initialization scheme is introduced to further boost the algorithm's performance. The method's accuracy, efficiency, and capability are verified through various stationary and moving immersed boundary cases. The results are validated against experimental and numerical data from the literature. The proposed improvements seamlessly integrate into existing incompressible fluid solvers with minimal adjustments to the original system equations, highlighting their ease of implementation.

Dynamical analysis of a new 3D chaotic system: non-equilibrium point, conservative, rotationally hidden attractor

A novel three-dimensional chaotic system is proposed in this paper. The nonlinear features are enhanced by adding e-exponent function to make the system behavior more complex and diverse. Through calculation and experimental verification, it is found that the system is a conservative system without equilibrium point, and this kind of system may produce hidden dynamic behavior. Therefore, four different dynamic analysis tools are chosen to simulate the possible hidden dynamics. The rotation factor is used to create some new models on the original system equation. With constantly adjustments of the rotation angle, the different resulting rotational hidden attractors are used to further prove that such chaos exists in the new proposed system. In the end, the realization of DSP shows the feasibility of this system in practical applications.

Areas for improving regional policy tools in the service sector based on cross-impact analysis

Subject. This article discusses the issues of strategic planning and development of the service sector in the regions of Russia. Objectives. The article aims to analyze the factors that determine the specifics of the regional policy for the development of the service sector. Methods. For the study, we used economic and mathematical research methods. Results. The article finds that the strategies of socio-economic development of some regions of Russia are characterized by an incomplete representation of the interrelations between priorities, goals and objectives, as well as the lack of quantitative assessments of threats to various sectors of the economy. The article proposes an original system of equations that help taking into account the influence of various factors on the development of the service sector. Relevance. The results of the study can be taken into account when forming and adjusting a regional socio-economic policy.

Разрешимость одномерной задачи движения жидкости в пороупругой среде с проницаемыми границами

The initial-boundary value problem of a one-dimensional viscous fluid flow in a deformable viscous porous medium with permeable boundaries is considered. The governing equations are the equations of mass conservation for each phase, the equation of momentum conservation for a liquid phase in terms of Darcy’s law, the equation of momentum conservation for the whole system, and the rheological equation for porosity. The original system of equations in the Lagrange variables is reduced to a third-order equation for the porosity function. The first part of this paper presents the formulation of the problem, the definition of the classical solution to the considered problem, and the existence and uniqueness theorem for the problem of Hölder classes. In the second part of this paper, the local theorem of existence and uniqueness for the problem of Hölder classes is proved for an incompressible fluid using the Tikhonov-Schauder fixed-point theorem. The physical principle of the maximum porosity function is determined.

Hydrodynamic Substitution and Dynamics of Non–Hamiltonian Systems

The possibility of transforming the system of ordinary differential equations to the system of hydrodynamic Euler-type equations with application of hydrodynamic substitution for the Liouville transport equation for phase density equivalent to the original system of equations. Conditions for integrating of Euler-type equations are obtained by modifying HamiltonJacobi method for non-Hamiltonian dynamical systems.

Parametric excitation of plane waves and traveling nonlinear surface charge structures by an oscillating electric field at the plasma boundary

With the help of a single equation, to which it was possible to reduce the original system of equations, it turned out to be possible to describe the excitation of parametric instability on plane surface waves in an alternating electric field, its saturation due to the nonlinearity of the surface charge and the generation of nonlinear structures of this charge propagating along the boundary and depending on the parameters of external influence. The amplitude of saturation of the increasing plane waves was found as a result of stabilization of parametric instability by the nonlinear effect of the surface charge. An example of a nonlinear wave structure propagating along the surface of the boundary as a result of the interaction of an external electric field with a nonlinear surface charge is presented.

A family of inverse characteristics methods

Introduction. The main idea of the grid-characteristic method is to take into account the characteristic properties, systems of hyperbolic equations, and the finite velocity of propagation of perturbations in the simulated media.Materials and methods. The simplest hyperbolic equation is a one-dimensional linear transfer equation. To increase the order of approximation of the grid-characteristic scheme to the second, you can use the Bim-Warming scheme. If we use a four-point pattern, we get a central Lax-Vendroff scheme. Difference schemes for the linear transfer equation can be obtained using the method of indefinite coefficients.Results. The grid-characteristic scheme admits a conservative variant, which is relevant if there are discontinuities (shock waves, shock waves) inside the integration domain, while the original system of equations for a matrix with constant coefficients, in partial derivatives, should be presented in a divergent form.Discussion and conclusions. When numerically solving a three-dimensional problem, the construction is performed similarly, in the case of upper and lower bounds, after scalar multiplication of the scheme by eigenvectors, relations approximating the compatibility conditions with the first order of accuracy are obtained.

Hyperbolic equivalent k-ϵ and k-ω turbulence models for moment-closures

The ten-moment equations are considered as a first-order alternative of Navier-Stokes equations when the effect of heat transfer is negligible. This model takes the form of first-order hyperbolic conservation laws, which carry many numerical advantages. However, the applicability of this model is still limited due to the lack of appropriate turbulence models. Applying the Reynolds-averaging concept to the ten-moment model, a set of governing equations for turbulent flow can be obtained, which is referred to as the Reynolds-averaged ten-moment equations. The traditional turbulence models designed for the Reynolds-averaged Navier-Stokes (RANS) equations are not ideal for the Reynolds-averaged ten-moment equations, as the extra partial differential equations (PDEs) introduce second-order derivatives. These terms destroy the pure hyperbolic nature of the original system of equations, which consequently removes all numerical advantages of first-order systems. To maintain the first-order hyperbolic form, the desired turbulence model should remain in the same form. Recently, a hyperbolic-relaxation turbulence model has been proposed by the authors, which is developed by hyperbolizing Prandtl's one-equation model using a relaxation method known as the Chen-Levermore-Liu p-system. Unfortunately, developing a hyperbolic version of two-equation models using the same method is very difficult. This is because the diffusion coefficients of the two-equation models are more complicated than in the one-equation model. In this paper, another relaxation method, the Cattaneo-Vernotte approach, is used to develop the hyperbolic-relaxation form of classical two-equation models. The solution of the resulting equations exhibits dispersive wave behaviour. To study this feature, a dispersion analysis of the Reynolds-averaged ten-moment equations with the new turbulence models is presented. Several numerical experiments are studied to investigate the effect of the relaxation parameters. The derived turbulence models are then coupled to the Reynolds-averaged ten-moment equation and further validated by solving a canonical two-dimensional turbulent plane mixing-layer problem, planar free-jet problem, and circular free-jet problem.

A scaled boundary finite element partitioning based reduced order algorithm for the elastic analysis of cyclically symmetric structures

AbstractAn efficient reduced order algorithm is proposed for the elastic SBFE analysis for 2‐D cyclic symmetric structures with or without a common node. The general stiffness matrices of scaled boundary finite element (SBFE) is proved to be block‐circulant, and can be constructed via the basic region, instead of the whole computing domain. Thus, the expense on eigenvalue analysis required in generating stiffness matrix can be significantly reduced, and the solution scale can be reduced by partitioning the system equation into a series of small independent subproblems. Furthermore, the presented algorithm is combined with the Woodbury formula to reduce the computational cost on the analysis of incomplete cyclically symmetric structures, the original system equation is transformed into the equations with block‐circulant coefficient matrices, which is efficiently solved by partitioning algorithm. Four numerical examples are provided to demonstrate the effectiveness and advantages of the proposed approach.

Numerical methods to evaluate Koopman matrix from system equations* *Contribution to the special collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.

Local Solvability of the Flow Problem for the Equations of Motion of Two Interpene Crating Fluids

The paper is dedicated to the one-dimensional problem of the nonisothermal flow of a two-phase mixture of viscous incompressible fluids with inhomogeneous boundary conditions. The mathematical model describing the two viscous fluids mixture flow is based on the equations of mass conservation, momentum conservation for each phase, and on the energy conservation equation, in the large. Local in time solvability of the initial boundary value problem in S.L.Sobolev and Helder spaces is proved. Section 1 sets the problem set up and provides the short literature review on the topic close papers and the main result formulation. Section 2 explains the transformation of the original system of equations. Sections 3, 4 prove the existence of the strong and classic solutions on a small time interval with constant true density using the Bubnov-Galerkin method. Notionally, the proof of the theorem is based on the similar result proof for viscous heatconducting gas (Antonsev S.N., Kazhihov A.V., Monahov V.N. Boundary value problems of heterogeneous fluid mechanic). The particularity of the considered problem is the presence of inhomogeneous boundary conditions.

A mathematical model of an epidemic with an arbitrary law of recovery

The paper proposes a mathematical model of the epidemic process, taking into account the dependence of the rates of cure and loss of immunity on time. Today, mathematical models of epidemics based on the basic Kermack–McKendrick model have become widespread. The most famous models are Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR). These models are based on dividing the population into separate groups that are in different epidemic conditions. The description of the models is based on differential equations similar to the equations of birth and death in the process of radioactive transformations of elements in a radioactive chain. However, this approach does not take into account the dependence of the probabilities of the transition of the population from group to group on time spent in the treatment process and in the process of loss of acquired immunity. The known models do analyze the nature of the course of the epidemic for long periods of time, when the process can enter a stationary state. The paper proposes a mathematical model based on dividing the population into separate groups. The first group consists of healthy people susceptible to infection due to contact with members of the second group, which includes the infected population. Members of the third group are being treated, the fourth group includes members of the society who have recovered with antibodies and are vaccinated. The fifth group consists of deceased members of society. In contrast to the SIR and SEIR models, the proposed approach takes into account that immunity is lost over time, and people who survived again move to the group susceptible to infection. The dependences of the probabilities of transition from group to group on time spent both during the treatment process and in the loss of acquired immunity have been taken into account. Thus, the proposed mathematical model is based on five integro-differential equations, two of which are partial differential equations. A new mathematical model has been formulated that makes it possible to take into account the dependence of the cure rate and the probability of transition from the vaccinated state to the initial state on time spent in the corresponding state. It is shown that the proposed model is autocatalytic. With increasing time, a state of bistability is observed, when, under the same boundary conditions, two stable states coexist. Switching between states is determined by the epidemic spread control parameter found in the work. One of the stable states is stationary and leads to the end of the epidemic, the second one leads to the population’s extinction. It has been shown that, for the stationary regime, the form of the distribution function in terms of treatment time and exposure time in the vaccinated state does not affect the final result in any way. The conditions for suppressing the epidemic for managing the process of its development are formulated. Numerical experiments were carried out to simulate the epidemic spreading process, taking into account the constancy of all transition probabilities. Integration of the original system of equations was carried out using the Radaus algorithm for stiff differential equations. The results of numerical simulations have confirmed that the experimental data agree with the theory of the control parameter. The results of the work can be used to organize the management of the epidemic spreading process in order to suppress it as soon as possible by changing the value of the control parameter.

Calculation of steam drop mixture flows with boiling explosion mechanism using a multidimensional nodal method of characteristics

A hyperbolic model of a vapor-drop mixture is presented which takes into account the evaporation of drops by highlighting an explosive mechanism. The model is developed on the basis of the generalized-equilibrium model of a mixture previously proposed by the author. In this early model, the equations were hyperbolized by adding interfractional-interaction forces, and the liquid fraction was assumed to be incompressible. In the model proposed here, we assume that the phase transition at intense droplet evaporation occurs in an overheated state, when the fluid temperature exceeds the saturation temperature. A characteristic analysis of the model equations is carried out and their hyperbolicity is shown. An analytical formula is obtained for calculating the speed of sound in a vapor-drop mixture. It is noted that the speed of sound in the mixture in the presence of phase transformations turns out to be somewhat lower than that given by wood's formula. The multidimensional nodal method of characteristics for integrating hyperbolic systems, which is based on splitting the original system of equations into a number of one-dimensional subsystems, is described. Differential relations that hold true along the characteristic directions for each of the subsystems are derived. When calculating one-dimensional problems, an iterative algorithm of the inverse method of characteristics is applied. The computational method has been tested on a number of problems with self-similar solutions. Using the described method of integrating a multidimensional system of equations, the flow of a vapor-droplet flow near the disk was studied. It is shown that, in a number of cases, the explosive boiling of liquid droplets should be taken into account because it can significantly change the pattern of the flow around a disk.

Parameterized Periodic Steady-State Analysis via Reduced-Order Modeling

This article presents a new approach to construct reduced-order models for large nonlinear circuits excited by periodical or quasi-periodical sources. The key feature in the constructed reduced model is that it enables, through solving a much smaller system, tracing the variations in the periodical steady-state response of the original circuit with regards to variations in selected design parameters. The proposed approach is based on the notion of using discrete empirical interpolation to project the nonlinear operator of the circuit equation onto a smaller subspace. In addition, the original system equations are projected onto the subspace spanned by the first few derivatives of the solution with respect to the selected design parameters. Computational savings are made possible through solving a reduced system of equations instead of the full system. Numerical examples are presented to validate the accuracy and efficiency of the proposed approach.

A block forward substitution method for solving the hypercomplex finite element system of equations

The hypercomplex finite element method, ZFEM, allows the analyst to compute highly-accurate arbitrary-order shape, material property, and loading derivatives by augmenting the traditional finite element method with multiple imaginary degrees of freedom. In ZFEM, the real variables are converted to hypercomplex variables such as multicomplex, multidual, or quaternions. By uplifting the real variables to hypercomplex, derivatives are computed in an automated fashion using a standard finite element formulation. The use of multicomplex or multidual numbers provides higher-order derivatives. The drawback of ZFEM is that it increases the number of degrees of freedom of the real variable system by a factor 2n, where n is the order of the required derivative. In consequence, ZFEM increases the memory consumption and the solution time of the system of equations compared to the real variable system. The block forward substitution method (BFS), proposed in this work, addresses the memory and runtime issues. This new method solves the original real-valued FEM system once. Then, the derivatives are computed using pseudo-loads with the original system of equations. In contrast with the conventional solution method of ZFEM, BFS computes the hypercomplex contributions to the stiffness matrix element-wise, and it never assembles nor solves the full hypercomplex system of equations. In effect, the BFS method generalizes the first-order semi-analytical complex variable method to any order derivative. The BFS method provides the capability to allow a combination of real-variable and hypercomplex-variable elements within the same model. The numerical results indicate that a first-order derivative can be obtained with 1% to 8% additional computational time of the real-variable analysis. This allows the computation of multiple first-order derivatives by post-processing of a single FEM analysis. Additionally, it was shown that fourth-order shape sensitivities can be computed in less than 5% additional runtime of the real-variable FEM analysis.

Inverse Problems of the Holling-Tanner Model Identification with Incomplete Information

In this paper we present a novel method for numerical parameter identification of the Holling-Tanner model with incomplete information. It means that information about predator or prey is unavailable, or only particular information about these species is available. The proposed method is based on elimination of variable characterizing unknown population from the original system of equations and obtaining a new nonlinear ordinary differential equation. In this equation, the dependent variable characterizes dynamics of the known population and new set of parameters functionally depends on the original unknown parameters. In the case of the Holling-Tanner model, the number of new parameters is higher than the number of original unknown parameters. Hence, there exist several constraints between new unknown parameters, which must be taken into consideration in the process of the parameter identification. The conventional method, based on the Lagrange constraint minimization of a goal function gives a nonlinear system of equations where the number of equations is equal to the sum of new unknown parameters and constraints. In this novel method, proposed in this paper, the number of equation is exactly equal to the number of constraints which substantially simplifies solution of the problem.

Cosmological models based on a statistical system of scalar charged degenerate fermions and an asymmetric Higgs scalar doublet

On the basis of the general relativistic statistical and kinetic theory, a consistent closed cosmological model is formulated. It is based on a statistical system of scalar charged fermions interacting by means of classical and phantom scalar fields. Based on the study of the microscopic dynamics of scalar charged particles, within the framework of the Lagrangian as well as Hamiltonian formalism, a function of the dynamic mass of scalar charged particles was constructed and it was shown that for the consistency of the theory, it is necessary to remove the nonnegativity condition for this function. On the basis of the Lagrangian formalism, equations of gravitational and scalar fields with singular sources are formulated and microscopic conservation laws are obtained. Within the framework of the general relativistic kinetic theory, macroscopic equations of gravitational and scalar fields are formulated and macroscopic conservation laws are obtained. These equations' full correspondence to microscopic equations with singular sources is shown. Further, on the basis of the obtained equations, a cosmological model for a degenerate system of scalarly charged fermions is formulated. An exact solution of the constitutive equations for a degenerate scalar-charged plasma in the cosmological model is obtained, which made it possible to significantly simplify the original system of equations. On the basis of the obtained solution of the constitutive equations, two fundamentally different cosmological models are formulated, one of which has two types of singly scalarly charged fermions, while the second has one kind of fermions charged with two charges of various nature. A qualitative analysis of the obtained 6-dimensional dynamic system for a two-component model is carried out.

Time-delayed positive position feedback control of nonlinear vibrations of continuous rotating shafts

The objective of the present study is to suppress the large amplitudes of a continuous spinning shaft under an active vibration control. For this purpose, the Positive Position Feedback (PPF) control with time-delays are applied to the system via Macro-Fiber-Composite (MFC) sensors and actuators. The nonlinear dynamics of the system is explored analytically utilizing the perturbation method in the primary and one-to-one internal resonances. The steady state solutions of the system and controller in the Cartesian and polar forms are obtained. The effects of the controller gains and time delays on the stability and loci of bifurcations of the system are investigated. The controller shows its feasibility in controlling the system vibrations. Also, it is demonstrated that by increasing the controller gain coefficients the minimum amplitude regions of the system are increased. Furthermore, the safe boundary of the loop delays is studied. The results of the analytical approach are in good agreement with the numerical results of the original system of equations.

The Existence of Pullback Attractors for Non-autonomous Stochastic Plate Equations with Linear Memory

Consider the existence of a non-autonomous two-dimensional stochastic plate equation with linear memory term pullback the attractor on . Apply the Ornstein-Uhlenbeck process to deal with the random term, transform the original equation into a deterministic equation containing random variables, and then estimate its consistency by replacing the system solution with variables, and prove that the random dynamic system corresponding to the original system equation pullback the absorption set Existence, and finally proves the system's pullback asymptotically compaction , which leads to the existence of the pullback attractor of the original system.

Solution of the problems of quasi-statics for an elastic body with double porosity

The construction of solutions in explicit form is especially important from the point of view of its application, since it makes it possible to effectively carry out a quantitative analysis of the problem under study. This paper investigates the processes of deformation of solids in the quasi-static case. Two-dimensional boundary value problems of Dirichlet and Neumann for an elastic body with double porosity are considered. In Using the Laplace transform, these problems are reduced to auxiliary boundary value problems. Special representations of solutions to auxiliary boundary value problems are constructed using elementary functions that allow reducing the original system of equations to equations of a simple structure and facilitate the solution of the original problems. Auxiliary boundary value problems are solved for a specific elastic body - a porous disk. Solutions to these problems are obtained in the form of series. Conditions are provided that ensure the absolute and uniform convergence of these series and the use of the inverse Laplace theorem. It is proved that the inverse transforms provide a solution to the initial problems.