Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Analysis of Local Dynamics of Difference and Close to Them Differential-Difference Equations

We study the local dynamics of one class of nonlinear difference equations which is important for applications. Using perturbation theory methods, we construct sets of singularly perturbed differential-difference equations that are close (in a sense) to initial difference equations. For the problem on the stability of the zero equilibrium state and for certain infinite-dimensional critical cases, we propose a method that allows us to construct analogs of normal forms. We mean special nonlinear boundary value problems without small parameters, whose nonlocal dynamics describes the structure of solutions to initial equations in a small neighborhood of the equilibrium state. We show that dynamic properties of difference and close to them differential-difference equations considerably differ.     

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.     

Asymptotic analysis of three-dimensional dynamic equations of a thin plate : PMM vol. 38, n 6, 1974, pp. 1072–1078

Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.     

Synthesis of virtual measurements in nonlinear observation problem

The method of the observation problems reducing to the algebraic ones is considered for the systems, which are linear with respect to unknown components of the phase vector. The approach proposed is based on the methods of the controlled stabilization of nonlinear system with respect to the part of variables. The equations of the initial observable system are supplemented by the equations of its controlled prototype. Then control law synthesied in such way that any given manifold becomes an invariant for extended system. For ensuring of this manifold attracting property the partial differential equations are obtained. Finally, the chosen in such way algebraic relations are considered as additional virtual measurements of unknown state. As example the problem of the angular velocity determination of a rigid body is considered. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the application of genetic algorithms and gradient methods to the problem of the reconstruction of the initial stress field in a poroelastic inhomogeneous column

Initial stress fields play an important role in deformational processes in porous structures. Accurate modeling of dynamics behavior of the porous medium taking initial stresses into account requires an appropriate mathematical model. One of the common approaches for the construction of a mathematical model of the dynamic behavior of a medium taking the initial state into account is linearization. This paper presents equations of longitudinal vibrations of a poroelastic inhomogeneous body in the presence of the initial stress field. The influence of the initial stress rate and Biot’s modulus on the dynamic behavior of the environment is analyzed. The problem of reconstructing the initial stress field for the poroelastic inhomogeneous column based on the genetic algorithm and nonlinear optimization with the use of modern packages of numerical simulation is studied. A series of numerical experiments on the reconstruction of different initial stress field distribution laws using the described methods is carried out.     

An extension of the Hamilton-Jacobi method

A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.     

Real-time calculation of the current state estimates for a class of delay systems

The linear optimal observation problem is examined for one type of nonstationary delay system with an uncertainty in the initial state. A fast implementation of the dual method is proposed for calculating estimates of the initial state. This implementation is based on the quasi-reduction of the fundamental matrix of solutions to the mathematical model of delay systems. It is shown that an iteration step of the dual method only requires that auxiliary systems of ordinary differential equations be integrated on small time intervals. An algorithm is described for the real-time calculation of current state estimates. The results are illustrated by the optimal observation problem for a third-order stationary delay system.     

A study of the stable equilibrium of thin shells compliant to shear and compression

Using relations of the geometrically nonlinear theory of thin shells compliant to shear and compression (the six-modal variant), we have written the key equations for determining their initial post-critical state by the finite element method. A specific feature of this model lies in the semidiscretization of the vector of displacements of an elastic body with respect to its variable thickness, based on the Timoshenko–Mindlin kinematic hypotheses, with preservation of the total vector of rotations of a normal to the median surface. We have solved numerically the problem of the stability of a circular plate, clamped over its contour, under the action of radial compressive forces, distributed uniformly along the contour. We have also performed a comparative analysis of the numerical solutions obtained and data known from the literature.     

Point estimation and some applications to iterative methods

We consider one of the crucial problems in solving polynomial equations concerning the construction of such initial conditions which provide a safe convergence of simultaneous zero-finding methods. In the first part we deal with the localization of polynomial zeros using disks in the complex plane. These disks are used for the construction of initial inclusion disks which, under suitable conditions, provide the convergence of the Gargantini-Henrici interval method. They also play a key role in the convergence analysis of the fourth order Ehrlich-Aberth method with Newton's correction for the simultaneous approximation of all zeros of a polynomial. For this method we state the initial condition which enables the safe convergence. The initial condition is computationally verifiable since it depends only on initial approximations, which is of practical importance.     

The problem of recovering the temperature and moisture fields in a porous body from incomplete initial data

We propose a statement and computational scheme for the inverse problem of recovering the temperature field and the moisture distribution in a body with incompletely known initial conditions. We give additional relations on the integral values of the unknown functions and introduce a test for the choice of a unique solution of the problem from the set of admissible temperature and moisture functions. We state conditions for independence of the additional data and obtain systems of equations and conditions that close the initial indeterminate problem. We study in detail the example of heat-moisture conduction in a layer. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 66–73.     

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.     

On the convergence of Wang-Zheng's method

The choice of initial conditions ensuring safe convergence of the implemented iterative method is one of the most important problems in solving polynomial equations. These conditions should depend only on the coefficients of a given polynomial P and initial approximations to the zeros of P. In this paper we state initial conditions with the described properties for the Wang-Zheng method for the simultaneous approximation of all zeros of P. The safe convergence and the fourth-order convergence of this method are proved.     

Modeling via the internal energy balance and analysis of adhesive contact with friction in thermoviscoelasticity

In this paper we introduce and investigate a model for adhesive contact with friction between a thermoviscoelastic body and a rigid support.A PDE system, consisting of the evolution equations for the temperatures in the bulk domain and on the contact surface, of the momentum balance, and of the equation for the internal variable describing the state of the adhesion, is derived on the basis of a surface damage theory by M. Frémond.The existence of global-in-time solutions to the associated initial–boundary value problem is proved by passing to the limit in a carefully tailored time-discretization scheme.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.     

The problem of package guidance by a given time for a linear control system with delay

The problem of guaranteed closed-loop guidance by a given time under incomplete information on the initial state is studied for a dynamical control system with delay by means of the method of open-loop control packages. A solvability criterion is proved for this problem in the case of a finite set of admissible initial states. The proposed technique is illustrated by a specific linear control system of differential equations with delay.     

Regular and Irregular Solutions in the Problem of Dislocations in Solids

For an initial differential equation with deviations of the spatial variable, we consider asymptotic solutions with respect to the residual. All solutions are naturally divided into classes depending regularly and irregularly on the problem parameters. In different regions in a small neighborhood of the zero equilibrium state of the phase space, we construct special nonlinear distribution equations and systems of equations depending on continuous families of certain parameters. In particular, we show that solutions of the initial spatially one-dimensional equation can be described using solutions of special equations and systems of Schr¨odinger-type equations in a spatially two-dimensional argument range.     

On some properties of the Navier–Stokes system of equations

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.     

Interior local null controllability of one‐dimensional compressible flow near a constant steady state

We consider the one‐dimensional compressible Navier–Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd.     

非线性最优控制问题的保辛多层次求解方法

将非线性系统的最优控制问题导向Hamilton系统,提出了求解非线性最优控制问题的保辛多层次方法．首先,以时间区段两端状态为独立变量并在区段内采用Lagrange插值近似状态和协态变量,通过对偶变量变分原理将非线性最优控制问题转化为非线性方程组的求解．然后,在保辛算法的具体实施过程中提出了多层次求解思想,以2N类算法为基础由低层次到高层次加密离散时间区段,利用Lagrange插值得到网格加密后的初始状态与协态变量作为求解非线性方程组的初值,可提高计算效率．数值算例验证了算法在求解效率与求解精度上的有效性．