Solution Of Static Problems Research Articles

AN APPLICATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS FOR THE ELASTODYNAMIC PROBLEM

A method of the fundamental solutions (MFS) is employed for the initial boundary value problem for the elastodynamic equation in annular planar domains. Using the Laguerre transformation in time the non-stationary problem is reduced to a sequence of stationary Dirichlet problems with a known sequence of fundamental solutions. The solutions of stationary problems are approximated by a linear combination of the narrowing of the fundamental solutions. By the collocation method from the Dirichlet data, recursive linear systems are obtained for finding the unknown coefficients in the MFS approximations. Presented numerical results confirm the applicability of the proposed method.

НЕКОТОРЫЕ ОСОБЕННОСТИ РАСЧЕТА НЕСТАЦИОНАРНОГО ПОВЕДЕНИЯ РЕАКТОРА

To analyze the features of calculating the transient behavior of fast breeder reactors, both using the classical system of kinetic equations and using the system of partial neutron transfer equations, computational studies of the test model of the MET1000 reactor developed under the Generation-IV project were carried out. As transient processes, displacements (dumping) of control rods were simulated with the assessment of reactivity effects in the specified reactor through the solution of stationary problems, i. e. through the use of an asymptotic estimate of the reactivity obtained from solving stationary homogeneous neutron transfer equations and from processing by the Inverse Solution of the Kinetics Equation method of solving a transient problem. Test calculations were performed in the 26th and 28th group approximations using the ABBN-93 and ABBN-RF libraries and eight groups of delayed neutrons. It is shown that the features of the determination of both calculation and measurement of the effects of reactivity and efficiency of control rods in Fast Breeder Reactors are associated with the presence of both instantaneous and delayed neutrons in the reactor, with the working region of the neutron spectrum and with a change in the shape of the neutron flux in the process of introducing perturbation.

ИСПОЛЬЗОВАНИЕ ПАРЦИАЛЬНЫХ УРАВНЕНИЙ ДЛЯ АНАЛИЗА КИНЕТИКИ РЕАКТОРОВ НА БЫСТРЫХ НЕЙТРОНАХ

To analyze the possibilities of using the system of partial neutron transport equations in evaluating the kinetics of a fast breeder reactors compared to the classical system of equations, computational studies of test models of MET-1000 and MOX-1000 reactors developed in the framework of the Generation-IV. As transient processes, the displacements (dumping and lifting) of control rods were simulated, and the reactivity effects in the indicated reactors were evaluated through the solution of stationary problems, i.e. through the use of an asymptotic estimate of the reactivity obtained from the solution of the stationary eigenvalue problem and from the processing of the inverse solution kinetic equation method from solving the transient problem. Test calculations were carried out in 26 and 28 group approximations using the BNAB-93 and BNAB-RF libraries and eight groups of delayed neutrons. The article shows that the features of determining, both calculation and measurement, the effects of reactivity and efficiency of control rods in fast breeder reactors are related to the working region of the neutron spectrum.

Stability of corrugated expansion bellows: shell and rod models

The problem of stability of corrugated thin-wall expansion bellows under hydrostatic internal load is considered. The bellows is modeled by an elastic rod with equivalent tensile, bending, and shear stiffness. Equations for calculating the critical value of fluid pressure at which the bellows loses stability are analytically derived, using an expression for hydrostatic follower load. The equivalent stiffness of the bellows is determined further from the solution of static problems for the elastic corrugated shell. The numerical solutions of the boundary value problems and the critical values of the pressure are obtained by the finite difference method. Additionally, a computer model of the expansion bellows was developed by ANSYS software, and the bellows stability was analyzed using shell finite elements. Calculations confirm the necessity of accounting for the axial displacement of expansion joint support when determining the critical pressure.

Two-criteria technical and economic optimization of forest transportation problem

The systematic analysis of optimizing the producer-consumer structure transportation problem is carried out in paper. It is in demand for the timber industry complex. The well-known minimum transportation cost objective is solved as a static problem using the linear programming method. Due to complicated market development conditions, it is necessary to set up and solve the problem of minimizing transportation time (quick performance) as a dynamic problem. Solving it in general case, the transportation cost may be different from the minimum cost determined by the static problem solution. The dynamic problem solution is based on the carrier’s vehicle performance information used for determining the functional time spent per transported product unit. The rational decision making under market conditions is only possible by adopting a compromise obtained on the basis of the multi-criteria optimization theory. The task of determining the compromise area of two-criteria optimization (cost-time minimum ratio) is decided by experts. It is solved using the linear programming method for static and dynamic problems and their linear approximation. To develop a compromise transportation plan becomes the subject of solving the inverse transportation problem. The systemic “producer – transport – consumer” approach is necessary for sustainable timber industry complex development, when the producer-transport system makes up a single production process. Its parameters are determined on the basis of systems theory.

Scalarization of stationary semiclassical problems for systems of equations and its application in plasma physics

We propose a method for determining asymptotic solutions of stationary problems for pencils of differential (and pseudodifferential) operators whose symbol is a self-adjoint matrix. We show that in the case of constant multiplicity, the problem of constructing asymptotic solutions corresponding to a distinguished eigenvalue (called an effective Hamiltonian, term, or mode) reduces to studying objects related only to the determinant of the principal matrix symbol and the eigenvector corresponding to a given (numerical) value of this effective Hamiltonian. As an example, we show that stationary solutions can be effectively calculated in the problem of plasma motion in a tokamak.

Асимптотические решения несвязанных задач стационарной теплопроводности и термоупругости для двухслойных пластин с неклассическими граничными условиями.

Асимптотические решения несвязанных задач стационарной теплопроводности и термоупругости для двухслойных пластин с неклассическими граничными условиями.

Positive stationary solutions and threshold results for the non-homogeneous semilinear parabolic equation with Robin boundary conditions

Let Ω be a smooth bounded domain in Rn. Considering the following Robin problem for a semilinear parabolic equation (0.1){ut−Δu=up+f(x),(x,t)∈Ω×(0,T),∂u∂ν+βu=0,(x,t)∈∂Ω×[0,T),u(x,0)=u0(x),x∈Ω, we show that for any function f(x) satisfying (F) which will be given in the introduction, there exists a positive number βf⋆ such that problem (0.1) has no stationary solution if β∈(0,βf⋆), and has at least two stationary solutions when β>βf⋆. Moreover, among all stationary solutions of problem (0.1) there is a minimal one. We prove further that the minimal stationary solution of problem (0.1) is stable, whereas, any other stationary solutions of problem (0.1) are initial datum thresholds for the existence and non-existence of a global solution to problem (0.1).

On some stationary problems of magnetohydrodynamics in multi-connected domains

We prove maximal regularity estimates in the Sobolev spaces $$ W_{\text{p}}^2 $$ and in the Holder spaces C2+α for weak solutions of stationary problems of magnetohydrodynamics. We give precise formulations of these problems in multi-connected domains. Bibliography: 16 titles.

Parameter identification in quasistationary thermoelastic problems

Explicit expressions for gradients of residual functionals are obtained for the realization of gradient methods of parameter identification in thermoelastic problems. The gradients are expressed explicitly in terms of the solutions of state problems and the associated conjugate problems.

Solution of stationary heat-conduction problems for curvilinear regions using eigenfunction expansions (fundamental solutions)

Formulas for solution of stationary problems of heat conduction in bodies of a curvilinear shape have been obtained in explicit form using eigenfunction expansions; an analogous solution has been constructed for temperature fluctuations. An algorithm of computation of the boundary functions for classical regions has been proposed; these functions make it possible to reduce the boundary conditions of the problem to a homogeneous form. The exact fundamental solutions in the region of a rectangle with arbitrary smooth boundary conditions of the 1st kind have been constructed using them. These solutions are fundamental, since they can be used when boundary-value problems and inverse problems with unknown boundary conditions are considered for a wide range of curvilinear regions.

Axisymmetric Dynamic Stability of Rotating Sandwich Circular Plates

The axisymmetric dynamic stability of rotating sandwich circular plates with a constrained damping layer subjected to a periodic uniform radial loading along the outer edge of the host plate is studied in the present paper. The viscoelastic material in middle layer is assumed to be frequency dependent and incompressible, and complex representations of moduli are used. Equations of motion of the system are derived by the finite element method where the geometry stiffness matrices induced by rotation and external load are evaluated from solutions of static problems. Bolotin’s method is employed to determine the regions of dynamic instability while the eigenvalue problems with frequency dependent parameters are solved by the modified complex eigensolution method. Numerical results show that the effects of constrained damping layer tend to stabilize the circular plate system and the widths of unstable regions decrease with increasing of rotational speeds.

Axisymmetric vibration and damping analysis of rotating annular plates with constrained damping layer treatments

The axisymmetric vibration and damping of rotating annular plates with constrained damping layer treatments are analyzed in the present paper. The discrete layer annular finite element method is employed to derive the equations of motion for the rotating composite plate where the geometry stiffness matrix induced by rotation is evaluated from solutions of static problems. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved, and the frequencies and modal loss factors of the composite plate are extracted. The effects of stiffness and thickness of the viscoelastic and constraining layers on the natural frequencies and modal loss factors of rotating composite plates are presented. Also, the composite plates with various boundary conditions and partial treatments are discussed.

Minimax confidence intervals for the binomial parameter

The game-theoretic approach to setting and solution of statistical problems is considered by the example of confidence interval (CI) estimation problem for parameter θ of the binomial distribution B( n, θ). The confidence probability (CP) and the minimax decision function (MDF) are constructed for any set of CIs. This differs from the classical approach where CI is constructed for a given CP. CPs for fixed width CIs are given in tabulated form. One half of these CIs is shorter than classical CIs for the same CPs. The MDF under consideration enables observation number to be reduced by about 1.5 times from the classical approach for the same reliability and accuracy.

Stress analysis of laminated shells of revolution with an imperfect interlayer contact

An approach is proposed to solve problems on the contact interaction of layers in structurally inhomogeneous shells of revolution with allowance for both the normal and tangential stresses in the contact zones. A system of algebraic equations is constructed by using a method for solution of static problems for arbitrary shells of revolution. From this system, the contact zones and contact stresses are determined by the iteration method. The restricted behavior of the adhesion layer in cleavage and shear is taken into account. As an example, the stress state of a two-layer cylindrical shell under a concentrated load is determined

Numerical solution of static problems for multitiered structures

A linear stress—strain analysis is made of a structure consisting of two discretely reinforced cylindrical shells and an intermediate conical shell with large openings, which is simulated by a three-dimensional framework. This three-tier system is under longitudinal and local flexural loads. The calculations are made by the finite-difference method using modified equations of the mixed method for shells and by the deformation method for the framework. Two forms of the structure differing by specified parameters are studied

Solution of statistical problems for a class of exponential distributions of random variables

For a class of exponential-type distributions with the power index ranging from 0.25 to 8, a simple formula for an approximate evaluation of the distribution function is presented. The formula is then used for solving certain statistical problems.

On the solution of stationary problems for the thermal conductivity of heat-sensitive bodies in contact

A method is given for the solution of stationary problems for the heat conductivity of thermoelastic bodies in contact. The method is illustrated by an example of an n-layer conduit.

Quasistatic problem of a non-homogeneous elastic layer containing a crack

A nonhomogeneous elastic layer is weakened by an infinite, rectilinear crack separating two layers of different elastic materials. The boundary surfaces of the layer are rigidly clamped and the crack surfaces loaded by arbitrary forces satisfying the conditions of antiplane state of strain. Considered are two cases, the crack and its load propagating at a constant velocity along the horizontal axis, and the load being a harmonic function of time, respectively. In the both cases exact values of the stress intensity factor for arbitrary loading (arbitrary load amplitude) of the crack are given. In the limiting cases, solutions of static problems are obtained. The results are illustrated by particular solutions concerning the cases when the crack edge load (or its amplitude) is constant on its entire length.

Solution of statistical problems in elasticity theory in the singular approximation

A method of calculating elastic fields and effective moduli of microheterogeneous solids is developed in the random field theory. The solution is obtained in the form of an operator series, each term of which is constructed on the basis of the regular component of the second derivative tensor of the equilibrium Green function. The zeroth approximation of such a series consists of the local part of the interaction between inhomogeneity grains. The possibilities of the method are illustrated on the example of an isotropic mixture of two isotropic components.