Numerical Solution Of Problem Research Articles

Modelling and numerical solution of problems of structural mechanics with unilateral constraints and friction

The article deals with the problems of calculation of elastically deformable systems with unilateral constraints and Coulomb friction. For the numerical solution of the problem, the finite element method is used, the contact interaction is modeled by means of the contact finite elements (CFE) of the frame-rod type. The basic relations and statements of static and dynamic contact problems for the interacting elastic bodies are given. To comply with the limitations under the conditions of ultimate friction-sliding, the method of compensating loads is applied. On the basis of the proposed discrete contact model and step-by-step loading schemes, the numerical algorithms are developed, which allows calculating structures with unilateral constraints and Coulomb friction, both under static and dynamic loads, as well as taking into account the different contact conditions that approximate the calculation scheme to the real operating conditions of a structure or construction. With the help of the proposed approach, numerical solutions of the problem of contact of the structure with the base under dynamic load are obtained and analyzed.

Construction of the equations of state for polycrystalline solids for the purpose of the numerical solution of problems of continuous medium mechanics

The equations of state for polycrystalline solids which are in region of compression are developed on experimentally defined shock Hugoniot adiabat and dependence of specific heat on density and temperature. These equations are convenient for the numerical solution of problems of continuous medium mechanics. The numerical method for definition of specific cold energy and the Gruneizen function at T = 0 K depending on compression ratio is proposed. Results of comparison for temperatures behind shock wave in sapphire by means of the developed equations of state and other methods are given.

A new boundary element strategy for modeling and simulation of three-temperature nonlinear generalized micropolar-magneto-thermoelastic wave propagation problems in FGA structures

In this paper, we introduce a new theory called nonlinear generalized micropolar-magneto-thermoelasticity involving three temperatures. Because of strong nonlinearity of wave propagation of functionally graded anisotropic (FGA) structures problems associated with this theory, the analytical or numerical solutions for problems associated with this theory are always difficult and require the development of new numerical techniques. So, we propose a new boundary element technique for the solution of such problems. The numerical results are depicted graphically to show the effect of magnetic field and confirm the validity and accuracy of our proposed technique.

MATHEMATICAL AND COMPUTER SIMULATION OF THE PROCESSES OF TWO-PHASE JOINT GAS FILTRATION AND WATER IN A POROUS ENVIRONMENT

A mathematical model, methods and algorithms for the numerical solution of problems of joint gas-water filtration in porous media are considered. The mathematical model of the process of non-stationary joint gas-water filtration in a porous medium is described by a system of nonlinear differential equations of parabolic type. In the numerical solution of the boundary value problem of gas displacement by water in a porous medium, the differential sweeping method is used for systems of differential-difference equations. The system of differential-difference equations with respect to the gas pressure function is nonlinear, therefore, an iterative method is used for it, based on the method of quasilinearization of nonlinear terms. Computational algorithms and software have been developed for conducting computational experiments to study unsteady processes of gas filtration in porous media. The results of the developed software, as well as the results of computational experiments in a graphical form in visual form are given.

NON-CONSERVATIVENESS OF FORMULAS AS AN OBSTACLE TO THE MOTION SIMULATION

The paper considers the specifics of connection setting between deformations and stresses as applied to the problem of large non-linear oscillations, when deformations are considered arbitrarily large. It is noted that the introduction of some arbitrary physical relations in the computational model can lead to the establishment of a non­conservative system. It is shown, for example, that distribution of the ordinary Hooke’s law formulas to the area of large deformations leads to the establishment of material with the non-conservative properties. The examples are given for the numerical solution of problems with the nonlinear oscillations, where an increase in the oscillation am­plitudes or occurrence of unauthorized internal friction is shown. The simplest version of the material properties free from the indicated deficiencies is given. One of the paper conclusions is that when specifying the elastic properties of the material, it is necessary to ensure that the resulting system is conservative.

Лингвостатистические и ассоциативно-вербальные параметры в исследовании идиолекта американского ученого-математика (на материале кинодиалога фильма “A Beautiful Mind”)

The study is focused on an American mathematician’s idiolect by analysis of the protagonist’s direct speech in the film A Beautiful Mind. The ideal scientist’s generalized image followed by the stylization of his idiolect contains the research capacity: this modelled language personality refers not only and not so much to its prototype, but to the stereotype of a mathematician’s idiolect circulated to American English speakers. The study was done by the methodology based on counting the word frequency by using software TextSTAT and comparing the results with the word frequency in the Corpus of Contemporary American English to assess the degree of deviation of the mathematician’s stylized lexicon from an average American English native speaker’s lexicon objectively. The associative semantic analysis of the studied personality’s main metonymic characteristic expressed by the phrase “A Beautiful Mind” located the associative semantic fields of the concept MIND in the native English speakers’ mass consciousness and set the main directions to research the associative semantic fields of the American mathematician’s general lexicon as follows: “Cognition”, “Mental condition” and “Eccentricity”. The further research is focused on the scientist’s vocabulary which is directly related to his professional activity. The structuring the field “Professional communication” (terms, professional slang, jargonisms, nomenclature), studying the quantitative ratios of these vocabulary groups and their contextual links revealed the specific scientific interests, isolation in a team and deviant language behavior. The analysis of the field “Cognition” indicated several nouns and verbs (work, time, number, to work, to win, to lose and others) as its semantic center. The research of their contextual links showed that the language personality’s main value is a dedicated cognitive process consisting in numerical solutions of problems to determine the causal relationships of reality. The solutions are not only thought, but imagined; feelings are also involved in the cognitive process. Multiplicity of solution makes choice important. The associative semantic fields structuring the mathematician’s lexicon are interrelated. There are lexical units (love, bicycle) that carry information in the intersections of the fields implicitly. Such lexical units form symbols which connect the series of heterogeneous concepts in the studied language personality’s mind.

Efficiency of Two Approaches to Computing the Flow around an Airfoil with Flaps in the Case of Flow Separation

Model equations describing convective transport are used to analyze the approximation errors of an explicit numerical scheme and various implicit schemes with the same approximation of spatial derivatives. It is shown that, under time step constraints determined by the Courant–Friedrichs–Lewy condition, the implicit scheme is inferior in accuracy to the explicit one and, with a further increase in the time step, the accuracy of simulated convective processes degrades substantially. Two methods for implementing the marching procedure in time are considered, namely, a fractional step in the case of an explicit scheme and a dual step in the case of an implicit scheme. It is shown that the fractional step method is efficient only on grids with a scatter of cell sizes of 100–1000. For the numerical solution of problems with no-slip conditions on solid walls (scatter of cell sizes of 104–105), two approaches are proposed: an implicit scheme with a dual step in all cells and an zonal approach, in which a dual step is used in a thin near-wall domain (about 3% of the thickness of a developed turbulent boundary layer), while a fractional step is applied in the rest of the domain. These two approaches are used to compute the flow over an airfoil with flaps. Numerical and experimental data are compared. The accuracy of the numerical results is estimated. The causes of error formation are examined. The domain of efficient application is determined for each of the indicated approaches.

Analytical-Numerical Approach to Describing Time-Periodic Motion of Fronts in Singularly Perturbed Reaction–Advection–Diffusion Models

The paper presents an analytical-numerical approach to the study of moving fronts in singularly perturbed reaction–diffusion–advection models. A method for generating a dynamically adapted grid for the efficient numerical solution of problems of this class is proposed. The method is based on a priori information about the motion and properties of the front, obtained by rigorous asymptotic analysis of a singularly perturbed parabolic problem. In particular, the essential parameters taken into account when constructing the grid are estimates of the position of the transition layer, as well as its width and structure. The proposed analytical-numerical approach can significantly save computer resources, reduce the computation time, and increase the stability of the computational process in comparison with the classical approaches. An example demonstrating the main ideas and methods of application of the proposed approach is considered.

Simulating Flows of Viscous Incompressible Fluid on Graphics Processors Using the Splitting Scheme and Multigrid Method

The use of general purpose graphics processors for the numerical solution of problems in dynamics of viscous incompressible fluid is discussed. Specific features of the parallel implementation of the splitting scheme (projection method) are considered. The system of difference equations produced by the discretization of Poisson’s equations for pressure is solved using the multigrid method. A number of benchmark problems are solved on graphics processors, and approaches to the optimization of program code by using different memory types are discussed. The speedup of computations on graphics processors is compared with the computations on the central processor using grids with different resolution and different decompositions of the initial data into blocks.

Time-based information sharing approach for employed foragers of artificial bee colony algorithm

Collective foraging and information sharing behaviors of honey bees have lead to emerge different swarm intelligence-based optimization techniques. Within these swarm intelligence-based optimization techniques, Artificial Bee Colony (ABC) algorithm has a special position due to its less control parameters, robust, phase-divided and easily implementable structures. Although standard workflow of ABC algorithm is capable of producing optimal or near optimal solutions for numerous problems, there are still some intelligent operations that are not directly modeled for the ABC algorithm in order to maintain the reduced complexity of the implementation and small number of control parameters. In this study, ABC algorithm is tried to be powered with a more realistic dancing approach called time-based information sharing, for short tb, model. The proposed model is integrated into the workflow of the standard ABC algorithm and its well-known variants. Experimental studies carried out on both classical and bound constrained single-objective CEC2015 benchmark functions showed that the proposed model in which the dancing durations of the employed bees are determined by the fitness values of the memorized sources significantly improved the performance of the standard and other variants of the ABC algorithm.

Modelling of Contact Interaction with Allowance for Nonlinear Compliance in Unilateral Constraints

The finite element model and numerical solution of the contact problem in the presence of nonlinear deformable layer between interacting bodies are considered. To discretize the contact layer, frame-rod contact finite elements are used, which also model different contact states (separation, clutch, friction-slip, etc.). The use of frame-rod contact elements allows the physical nonlinearity of the contact layer to be reduced to the internal nonlinearity of the system of contact elements, while the nonlinear properties of the seam will be set through the nonlinear characteristics of the individual rods of the contact elements. The numerical algorithm for solving the contact problem is based on the method of step-by-step analysis with iterative refinement of the solution for the current load level. With its help, the numerical solutions of the problem of contact of the structure with the base under different conditions of the contact seam have been obtained and analyzed.

English

The paper deals with formulation and numerical solution of problems of identification of material parameters for continuum mechanics problems in domains with heterogeneous microstructure. Due to a restricted number of measurements of quantities related to physical processes, we assume additional information about the microstructure geometry provided by CT scan or similar analysis. The inverse problems use output least squares cost functionals with values obtained from averages of state problem quantities over parts of the boundary and Tikhonov regularization. To include uncertainties in observed values, Bayesian inversion is also considered in order to obtain a statistical description of unknown material parameters from sampling provided by the Metropolis-Hastings algorithm accelerated by using the stochastic Galerkin method. The connection between Bayesian inversion and Tikhonov regularization and advantages of each approach are also discussed.

The generalized finite difference method for in-plane crack problems

This paper documents the first attempt to apply the generalized finite difference method (GFDM), a recently-developed meshless method, for the numerical solution of problems with cracks in general anisotropic materials. To solve the resulting second-order elliptic partial differential equations with mixed boundary conditions, the explicit formulae for the partial derivatives of unknown functions in the equations are derived by using the Taylor series expansions combining with the moving-least squares approximation in this meshless GFD method. To deal with the strong discontinuous crack-faces, some special treatments are applied to modify this GFDM. The node distributions are locally refined in the vicinity of the crack-tips. By dividing the crack domain into two parts, the sub-domain method is also used for comparing with the single-domain method. The direct displacement extrapolation method, the path-independent J-integral and the interaction integral methods are, respectively, used to compute the stress intensity factors and compared. Finally, some classical crack examples are presented to show the effectiveness and accuracy of the proposed meshless method for crack problems.

Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains

Mathematical analysis and numerical solutions of problems with unknown shapes or geometrical domains is a challenging and rich research field in the modern theory of the calculus of variations, partial differential equations, differential geometry as well as in numerical analysis. In this series of three review papers, we describe some aspects of numerical solution for problems with unknown shapes, which use tools of asymptotic analysis with respect to small defects or imperfections to obtain sensitivity of shape functionals. In classical numerical shape optimization, the boundary variation technique is used with a view to applying the gradient or Newton-type algorithms. Shape sensitivity analysis is performed by using the velocity method. In general, the continuous shape gradient and the symmetric part of the shape Hessian are discretized. Such an approach leads to local solutions, which satisfy the necessary optimality conditions in a class of domains defined in fact by the initial guess. A more general framework of shape sensitivity analysis is required when solving topology optimization problems. A possible approach is asymptotic analysis in singularly perturbed geometrical domains. In such a framework, approximations of solutions to boundary value problems (BVPs) in domains with small defects or imperfections are constructed, for instance by the method of matched asymptotic expansions. The approximate solutions are employed to evaluate shape functionals, and as a result topological derivatives of functionals are obtained. In particular, the topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, defects, source terms and cracks. This new concept of variation has applications in many related fields, such as shape and topology optimization, inverse problems, image processing, multiscale material design and mechanical modeling involving damage and fracture evolution phenomena. In the first part of this review, the topological derivative concept is presented in detail within the framework of the domain decomposition technique. Such an approach is constructive, for example, for coupled models in multiphysics as well as for contact problems in elasticity. In the second and third parts, we describe the first- and second-order numerical methods of shape and topology optimization for elliptic BVPs, together with a portfolio of applications and numerical examples in all the above-mentioned areas.

Shifted equivalent sources and FFT acceleration for periodic scattering problems, including Wood anomalies

This paper introduces a fast algorithm, applicable throughout the electromagnetic spectrum, for the numerical solution of problems of scattering by periodic surfaces in two-dimensional space. The proposed algorithm remains highly accurate and efficient for challenging configurations including randomly rough surfaces, deep corrugations, large periods, near grazing incidences, and, importantly, Wood-anomaly resonant frequencies. The proposed approach is based on use of certain “shifted equivalent sources” which enable FFT acceleration of a Wood-anomaly-capable quasi-periodic Green function introduced recently (Bruno and Delourme (2014) [4]). The Green-function strategy additionally incorporates an exponentially convergent shifted version of the classical spectral series for the Green function. While the computing-cost asymptotics depend on the asymptotic configuration assumed, the computing costs rise at most linearly with the size of the problem for a number of important rough-surface cases we consider. In practice, single-core runs in computing times ranging from a fraction of a second to a few seconds suffice for the proposed algorithm to produce highly-accurate solutions in some of the most challenging contexts arising in applications.

Numerical models in applied plasmadynamic and plasmastatic problems

We present a review of mathematical models and computations in the two plasma physic projects: creation of powerful plasma accelerators and researches of equilibrium plasma configurations in magnetic traps. Steady-state transonic plasma flows in accelerators are investigated in the plasmadynamic models. The recent time works deal with studies of the longitudinal magnetic field influence on the plasma acceleration process in the channels of various geometry. Equilibrium magnetoplasma configurations in the traps are researched in plasmastatic models based on the numerical solution of problems with the Grad-Shafranov equation. Configurations are investigated in the galateya-traps with current carrying conductors immersed into the plasma. In the recent time works, configurations in toroidal magnetic traps are studied in comparison with their analogues straightened into the cylinder.

ОЦЕНКА ПЕРСПЕКТИВ РАЗВИТИЯ ТУРИСТСКОГО КЛАСТЕРА АСТРАХАНСКОЙ ОБЛАСТИ

Tourist and recreational clusters have become an effective tool for regional development, as they allow using collective opportunities to create tourism products, as well as contribute to the development of related industries. In the Astrakhan region, the tourism cluster was created with the participation of government agencies and is currently in an active stage of its development. The Astrakhan regional tourist cluster comprises interacting economic agents of the tourism sector and related activities, administrative bodies and coordinations, public organizations and research institutes. Despite the existing measures of state support, further effective development of the tourism cluster is associated with the solution of numerous problems. To study the mechanisms of interaction among the participants of the tourism cluster of the Astrakhan region, a survey was conducted, the results of which confirm the presence of a number of problems, including those in the field of cluster management. There have been shown unregulated issues of interdepartmental cooperation on execution of responsibilities for implementation of state policy in the field of tourism, as well as the low level of formalized interaction of cluster members, problems with stuffing, partial use of tourism and recreational potential of the region and increase of ‘shadow’ market participants. Nowadays, in the Astrakhan region there has been adopted the Strategy of tourism cluster development for the period up to 2020. Systematic realization of the Strategy will create conditions for the formation of a competitive and actively developing tourism cluster.

Boundary Element Modeling of Dynamics of a Bubble in Contact with a Solid Surface at Low Reynolds Numbers

In the present study, the dynamics of a bubble attached to the surface and driven by the acoustic field at low Reynolds numbers are considered. The approach is based on the boundary element method (BEM) for Stokes flows, which is especially effective for the numerical solution of problems in the three-dimensional case. However, the dynamics of computing compressible bubbles are difficult to formulate due to the degeneration of the conventional BEM for Stokes equations. In the present approach, an additional relation based on the Lorenz reciprocity principle is used to resolve the problem. To describe the contact line dynamics a semiempirical law of motion is used. Such an approach allows us to bypass the known issue of nonintegrability stresses in the moving triple point. The behavior of a bubble attached to the surface in the cases of a pinned or moving contact line is studied. The developed method can be used for the detailed study of bubble dynamics in contact with a solid wall in order to determine the optimal conditions and parameters of surface cleaning processes.

A NUMERICAL TECHNIQUE FOR THE SOLUTION OF GENERAL EIGHTH ORDER BOUNDARY VALUE PROBLEMS: A FINITE DIFFERENCE METHOD

In this article, we present a novel finite difference method for the numerical solution of the eighth order boundary value problems in ordinary differential equations. We have discretized the problem by using the boundary conditions in a natural way to obtain a system of equations. Then we have solved system of equations to obtain a numerical solution of the problem. Also we obtained numerical values of derivatives of solution as a byproduct of the method. The numerical experiments show that proposed method is efficient and fourth order accurate.

A numerical approach to joint continuous and impulsive control of Markov chains

Abstract This work considers the continuous and impulse control of the finite state Markov chains in continuous time. In general continuous control governs the transition rates between the states of Markov chain (MC), so the instants and directions of the state changes are random. Meanwhile sometimes there is an urgent necessity to realise the transition which leads to immediate change of the state. Since such transitions need different efforts and produce different effects on the function of the MC itself, one can consider this situation as an impulse control, and if at the same time there is a possibility of graduate control we are coming to the problem of joint continuous and impulse control. In the article we develop the martingale representation of the MC governed by joint continuous and impulse control and give an approach to the solution of the optimal control problem on the basis of the dynamic programming equation. This equation has a form of quasivariational inequality which in the case of finite state MC can be reduced to the solution of the system of ordinary differential equation with one switching line. We give the proof of the verification theorem and find the numerical solution for problems with deterministic and stochastic impulse controls.