Asymptotic analysis of three-dimensional dynamic equations for thin two-layer elastic plates

In accordance with the method described in /1–3/, a derivation of two-dimensional equations of motion is given for a thin two-layer (non-symmetric) elastic plate. The mean values of the bending stiffness, the density, and Poisson's ratio are found, and the position of the middle plane is determined. In the coordinate system attached to this plane, the system of equations is separated into quasistatic equations for the longitudinal motion and a dynamic equation (of the ordinary kind) for the transverse component of the displacement. Unlike /1–3/, only one characteristic dimension in the longitudinal direction is introduced, which turns out to be sufficient and simplifies the analysis. Formulae of the complete field of stresses are provided. Stresses, which are of secondary importance for homogeneous plates, may be essential when the strength of the joint of the layers is considered.     

A problem of the bending of a plate for a doubly-connected domain bounded by polygons

The problem of the bending of an isotropic elastic plate, bounded by two convex polygons is considered. It is assumed that the internal boundary of the plate is simply supported and normal bending moments act on each section of the external contour in such a way that the angle of rotation of the middle surface of the plate is a piecewise-constant function. With respect to the complex potentials, which express the bendings of the middle surface (Goursat's formula), the problem is reduced to a Riemann-Hilbert boundary-value problem for a circular ring, the solution of which is constructed in analytic form. Estimates are given of the behaviour of these potentials in the neighbourhood of the corner points.     

解任意四边形板弯曲问题的样条有限元法

关于用样条函数解板的弯曲问题,[1]在1979年讨论了矩形板和菱形板的弯曲;[2]在1981年对简支边界条件的矩形板,用振动梁函数和B样条函数组合作为插值函数,得到了效率更高的算式;[3]在1984年对[2]作了补充,采用拉格朗日乘子法,得到了在各种边界条件下平板弯曲的近似解,但所讨论的仍然是矩形板.     

Geometric quadrisection in linear time, with application to VLSI placement

We consider the following problem: given a set of points in the plane, each with a weight, and capacities of the four quadrants, assign each point to one of the quadrants such that the total weight of points assigned to a quadrant does not exceed its capacity, and the total distance is minimized.

This problem is most important in placement of VLSI circuits and is likely to have other applications. It is NP-hard, but the fractional relaxation always has an optimal solution which is “almost” integral. Hence for large instances, it suffices to solve the fractional relaxation. The main result of this paper is a linear-time algorithm for this relaxation. It is based on a structure theorem describing optimal solutions by so-called “American maps” and makes sophisticated use of binary search techniques and weighted median computations.

This algorithm is a main subroutine of a VLSI placement tool that is used for the design of many of the most complex chips.     

板弯曲问题的具两组高阶基本解序列的MRM方法

讨论了双参数地基上薄板弯曲问题.利用两组高阶基本解序列,即调和及重调和基本解序列,采用多重替换方法(MRM方法),得到了板弯曲问题的MRM边界积分方程.证明了该方程与边值问题的常规边界积分方程是一致的.因此由常规边界积分方程的误差估计即可得到板弯曲问题MRM方法的收敛性分析.此外该方法还可推广到具多组高阶基本解序列的情形.     

A generalized Cauchy problem for the linear differential equations of coupled physical - mechanical fields

A generalized Cauchy problem for a partial differential equation with constant coefficients, which is encountered in the study of physical processes in continuous media with widened physical - mathematical fields (see /1/) (generalized coupled thermoeleasticity /2/, coupled thermoeleasticity, porous media saturated with a viscous fluid /5/, mass and heat transfer /6/, linearized magnetoelasticity /7/, etc.) is considered. The characteristic properties of the solution of the problem, under certain constraints imposed on an equation by the stability condition, are studied. The presence of waves of higher and lower order is characteristic for the solution; in the course of time the lower-order waves are maintained and take a characteristic form. In the general case, the solution is represented in the form of integrals over the segments which link the singular points of Fourier - Laplace transforms with respect to time of the solution under consideration. The methods proposed enable an exact investigation to be made of the processes described by the equation for any time constants, and they also enable one to isolate the singularities at the fronts of propagating perturbations. As an application, the dynamic processes taking place in a thermoelastic subsapce (2) as a result of applying a mechanical and a thermal input at the boundary is studied. It is shown that in the case of unit perturbation of the boundary, the stress and temperature waves in the course of time assume a bell-shaped form and propagate with adiabatic velocity. A numerical analysis of the process which occurs due to sudden application of the force and of the thermal shock at the boundary is given.     

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

The method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.     

On generalized formulation of the equilibrium problem of an elastic strip

A “nonenergetic” formulation of the boundary value problems of statics of an elastic strip based on the principle of admissible displacements, is studied. The formulation makes possible, in particular, the study of problems concerning the strips of infinite energy, while retaining the external form of the “energetic” formulation /1–3/, and produces unique solvability of the problem under weaker restrictions imposed on the external loads. Such a formulation is also possible for other problems of the theory of elasticity.     

On synthesis in a differential game

The control problem is considered with minimization of the guaranteed result for a system described by an ordinary differential equation in the presence of uncontrolled noise. The concepts and formulation of the problem in /1/ are used. It is shown that, when forming the optimal control by the method of programmed stochastic synthesis /1–3/, the extremal shift at the accompanying point /1, 4/ can be reduced to extremal shift agianst the gradient of the appropriate function. This explains the connection between the programmed stochastic synthesis and the generalized Hamilton-Jacobi equation /5, 6/ in the theory of differential games.     

Deformation of a composite elastic plane weakened by a periodic system of the arbitrarily loaded slits

The paper deals with the problems of periodic system of cuts distributed along the boundary of a bond connecting two elastic half-planes and acted upon by nonperiodic loads. In one problem it is assumed that the cuts are open, with normal and tangential stresses applied to their edges, while in another problem the edges touch each other and are loaded by tangential stresses. The method of solution is based on the simultaneous use of the discrete Fourier transformation and the theory of boundary value problems for automorphous analytic functions. The solutions are otained in quadratures. Other classes of problems to which the proposed methods can be applied, are described.

Generally speaking, in the case of irregular loads, the solution is usually based on the theory of representation of the symmetry groups /1,2/, and in the case of certain types of symmetry, particularly the translational, on the discrete Fourier transforms /3– 6/. However the objects of transformation may be different in one and the same problem, and their choice affects significantly the solvability of the boundary value problem for the transformed quantities in the cell of periods. Below two problems of the theory of cracks are solved in quadratures to illustrate the effective simultaneous use of the discrete Fourier transformation and the Muskhelishvili method.     

Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ?₊, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.     

On bending an elastic plate with a delaminated thin rigid inclusion

Under study is the problem of bending an elastic plate with a thin rigid inclusion which may delaminate and form a crack. We find a system of boundary conditions valid on the faces of the crack and prove the existence of a solution. The problem of bending a plate with a volume rigid inclusion is also considered. We establish the convergence of solutions of this problem to a solution to the original problem as the size of the volume rigid inclusion tends to zero.     

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.     

Study of the strength of plastic sandwich construction

Results are given for edge-supported sandwich panels with a honeycomb core tested in bending in two directions. The experimental data are compared with theoretical results obtained from equations proposed by the authors. It is established that in designing honeycomb panels for bending in two directions it is possible to treat the panel material as virtually isotropic in the plane of the laminations and to use the authors' equations for determining the deflections and the normal and shear stresses.Mokhanika Polimerov, Vol. 1, No. 5, pp. 45–50, 1965     

Monotone-iterative method for mixed boundary value problems for generalized difference equations with “maxima”

In this paper the authors investigate special type of difference equations which involve both delays and the maximum value of the unknown function over a past time interval. This type of equations is used to model a real process which present state depends significantly on its maximal value over a past time interval. An appropriate mixed boundary value problem for the given nonlinear difference equation is set up. An algorithm, namely, the monotone iterative technique is suggested to solve this problem approximately. An important feature of our algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima”, and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. Several numerical examples are considered to illustrate the practical application of the suggested algorithm.     

On the integrable case of a Riemann-Hilbert boundary value problem for two functions and the solutions of certain mixed problems for a composite elastic plane

A method for solving the Riemann-Hilbert boundary value problem with piecewise-constant coefficients is generalized /1/. It is shown that the following static problems of a composite elastic plane with three kinds of connection conditions allow of exact solutions: 1) the splicing line is weakened by a system of loaded slots and a transverse shear crack or the edges of one of the slots are partially contacting, or one of the slots is cleaved by a rigid insert; 2) the splicing line is reinforced by a system of thin rigid inclusions and there is one arbitrarily located delamination zone; 3) the elastic half-planes are contacting (with slip) on a certain section of their boundaries, and mixed boundary conditions in the displacements and stresses are given on the rest of the boundaries.

In the general case the Riemann-Hilbert boundary value problem for many functions reduces to the problem of a linear conjugation, and then to Fredholm integral Eqs./2/. Closed solutions are obtained in certain special cases /3–5/. For applications we mention the papers /6, 7/, where problems are considered concerning slits at the interface of two elastic media with two kinds of physical boundary conditions taken into account simultaneously.     

From the classical to the generalized von Kármán and Marguerre–von Kármán equations

In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre–von Kármán shallow shells.

First, we briefly review the “classical” von Kármán and Marguerre–von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations.

In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data.

Following recent joint works, we then reduce these more general equations to a single “cubic” operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation.     

Superfast algorithms for Cauchy-like matrix computations and extensions

An effective algorithm of [M. Morf, Ph.D. Thesis, Department of Electrical Engineering, Stanford University, Stanford, CA, 1974; in: Proceedings of the IEEE International Conference on ASSP, IEEE Computer Society Press, Silver Spring, MD, 1980, pp. 954–959; R.R. Bitmead and B.D.O. Anderson, Linear Algebra Appl. 34 (1980) 103–116] computes the solution to a strongly nonsingular Toeplitz or Toeplitz-like linear system , a short displacement generator for the inverse T⁻¹ of T, and det T. We extend this algorithm to the similar computations with n×n Cauchy and Cauchy-like matrices. Recursive triangular factorization of such a matrix can be computed by our algorithm at the cost of executing O(nr²log³ n) arithmetic operations, where r is the scaling rank of the input Cauchy-like matrix C (r=1 if C is a Cauchy matrix). Consequently, the same cost bound applies to the computation of the determinant of C, a short scaling generator of C⁻¹, and the solution to a nonsingular linear system of n equations with such a matrix C. (Our algorithm does not use the reduction to Toeplitz-like computations.) We also relax the assumptions of strong nonsingularity and even nonsingularity of the input not only for the computations in the field of complex or real numbers, but even, where the algorithm runs in an arbitrary field. We achieve this by using randomization, and we also show a certain improvement of the respective algorithm by Kaltofen for Toeplitz-like computations in an arbitrary field. Our subject has close correlation to rational tangential (matrix) interpolation under passivity condition (e.g., to Nevanlinna–Pick tangential interpolation problems) and has further impact on the decoding of algebraic codes.     

Vibrations of an annular plate under the influence of a force rotating at a constant velocity

An exact solution is obtained for a dynamic problem concerning the bending of circular and annular plates under the influence of a force rotating at a constant angular velocity. It is shown that this problem is related to the problem of the bending of a plate under the influence of a harmonic load. Velocities leading to resonance are found and their first values are calculated for circular hinged and rigidly fastened plates.Translated from Dinamicheskie Sistemy, No. 4, pp. 62–65, 1985.     

The uniform heating and bending of a two-layer plate

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic layer considered (a soft plate) is in ideal contact with a rigid isotropic thin elastically deformed layer. The ends of the plate are load-free. A boundary layer of the soft plate (a thin contact layer) is introduced, which enables the boundary conditions on the ends of the plate to be formulated in such a way that the problem has a bounded smooth solution [1]. The two-layer plate, generally speaking, is bounded along the axis perpendicular to the axes directed along the length and thickness of the plate. The resultant force and the resultant moment, applied to the end transverse sections, are equal to zero. The exact solution of the temperature problem is sought using the equations of the theory of elasticity. The plane problem of the bending of a two-layer plate acted upon by a uniformly distributed pressure applied to the side surface of an anisotropic layer is solved by a similar method. The ends of the rigid isotropic layer are clamped.