Extension of the variational formulation of the problem for a rigid-plastic medium to velocity fields with slip-type discontinuities

Sets of velocity fields containing slip-type discontinuities at the boundary of the rigid-plastic medium, as well as within it, and the functionals defined on these sets, are described. It is shown that the exact lower bounds of the variational problems for these functionals are equal to the coefficient of the critical load. The minimax problem with saddle point constructed here is regarded as an extension of the classical minimax problem of the theory of critical loads.     

On Plane Dilatational Plasticity

The plane plastic deformation of a generally anisotropic rigid-plastic material which possesses a yield condition dependent upon mean triaxial stress and which, through the classical associated flow rule, exhibits plastic dilatation, is considered. This model is used to represent the behavior of micro-porous ductile metals in which the micro-cavities may be strongly aligned due to large prior plastic strains, as for example the material surrounding the tip of an extending notch in a ductile metal. It is shown that the stress and velocity fields are hyperbolic where a line of vanishing extension rate may be found in the plane of deformation, and that the characteristics of both the stress and velocity fields coincide with the lines of vanishing extension rate. Coincidence of the characteristics of stress and velocity fields in general anisotropic plastic bodies seems not to have been expected in earlier writings, but is a natural consequence of the associated flow rule. Simple means of determining whether a given stress state at yield lies in a hyperbolic or elliptic field are discussed. The role of characteristics in providing ductile fracture nuclei is discussed.     

On the approximation of the unsteady Navier–Stokes equations by finite element projection methods

This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of finite element approximations. The analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm. Error estimates in finite time are given. An extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method. Received October 2, 1995 / Revised version received July 9, 1997     

一类广义非线性强阻尼扰动发展方程的行波解

研究了一类非线性强阻尼广义扰动发展方程问题.它们在数学、力学、物理学等领域中广泛出现.首先,引入一个行波变换,把相应的偏微分方程问题转化为行波方程问题并求出原典型问题的精确解.再用小参数方法和引入伸长变量构造了问题的渐近解.最后, 用泛函分析的不动点理论证明了原非线性强阻尼广义扰动发展方程初值问题渐近行波解的存在性,并证明渐近解具有较高的精度和一致有效性.该文求得的渐近解是一个解析展开式, 所以它还可继续进行解析运算, 而单纯用数值模拟的方法是不行的.     

The Newton–Wigner Problem in the Relativistic Quantum Mechanics of Free Particles

We discuss the old Newton–Wigner problem, which is understood as the problem of a correct coordinate interpretation of the relativistic quantum mechanics of free particles. This problem is still relevant for quantum field theory because the S-matrix approach assumes that asymptotic fields describe relativistic free quantum-mechanical particles. From the modern standpoint, the original solution of this problem by Newton and Wigner already cannot be considered sufficient because it admits the smearing of wave packets with a superlight velocity. We discuss a possibility of overcoming this difficulty. This possibility is connected with relativistic deformations of the standard Heisenberg algebra. We describe situations in which a sort of desingularization of the effective free Hamiltonian occurs for some special deformations, which possibly allows preserving sublight velocity in the theory.     

Solvability of a quasi-steady rolling problem for porous materials

A quasi-steady rolling problem with nonlocal friction, for porous rigid-plastic, strain-rate-sensitive and strain hardening materials, is considered. A variational formulation is derived, consisting of a variational inequality and two evolution equations, coupling the velocity, strain hardening and relative density variables. The convergence of a variable stiffness parameters method is proved, and existence and uniqueness results are obtained. An algorithm, combining this method with the finite element method, is proposed and used for solving an illustrative rolling problem.     

极限分析的广义界限定理

本文研究了非连续流动场中,刚塑性介质极限分析完全解的界限问题.提出了一个包括界面条件及间断面条件在内的混合边值问题的广义变分原理,建立了极限载荷乘子的变分解析公式.并证明了一个新的界限定理,其中的场变量将不再受到屈服条件、不可压缩条件等约束的限制.此定理的推论给出了变分解与完全解之间的关系.初步应用表明,对于简单选取的场变量,由本文公式可以得到准确解的较佳界限值,结果具有较好的稳定性.     

Limit analysis decomposition and finite element mixed method

This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver, using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776-value to be compared to the best published lower bound 3.7752-by succeeding in solving a nonlinear optimization problem with millions of variables and constraints.     

Minimal p-extensions and the embedding problem

We investigate the ultrasolvability problem for minimal p-group extensions of odd order: for the factorgroup of such extension, there exists a Galois extension of number fields such as corresponding embedding problem is ultrasolvable (i.e. this embedding problem is solvable and all its solutions are fields).     

Extrusion of a plastic material from a circular sector with a small apex angle and a sink at the vertex

The plane inertialess extrusion of a perfect rigid-plastic material from a circular sector, the angle of which serves as a small parameter, is investigated using the asymptotic integration. The flow was initiated by the approach of the two sides of the sector and the existence in it of a sink of specified power. The principal velocity and stress approximations are obtained and the domains of applicability of the asymptotic expansions and the inertialess (quasistatic) solutions are found. Analogies are drawn with the solutions of the classical Prandtl problem and several of its extensions.     

Rank-One Approximation of Positive Matrices Based on Methods of Tropical Mathematics

Low-rank matrix approximation finds wide application in the analysis of big data, in recommendation systems on the Internet, for the approximate solution of some equations of mechanics, and in other fields. In this paper, a method for approximating positive matrices by rank-one matrices on the basis of minimization of log-Chebyshev distance is proposed. The problem of approximation reduces to an optimization problem having a compact representation in terms of an idempotent semifield in which the operation of taking the maximum plays the role of addition and which is often referred to as max-algebra. The necessary definitions and preliminary results of tropical mathematics are given, on the basis of which the solution of the original problem is constructed. Using the methods and results of tropical optimization, all positive matrices at which the minimum of approximation error is reached are found in explicit form. A numerical example illustrating the application of the rank-one approximation is considered.     

On the direct and inverse problems in fluid-structural interaction

This paper is concerned with the direct and inverse scattering problems in fluid-structure interaction. The scattering problem in the fluid-structure interaction can be simply described as follows: an acoustic wave propagates in the fluid domain of infinite extent where a bounded elastic body is immersed. The direct problem is to determine the scattered pressure and velocity fields in the fluid domain as well as the displacement fields in the elastic body, while the inverse problem is to reconstruct the shape of the elastic scatterer from a knowledge of the far field pattern of the fluid pressure or from the measured scattered fluid pressure field. As is well known, the inverse problems are generally nonlinear and highly ill-posed. For treating inverse problem of this kind, we reformulate the problem as a nonlinear optimization problem including special regularization terms. The precise formulation of the nonlinear objective functional will depend on the approaches of the direct problem. In this paper, the direct problem is reformulated by introducing an artificial boundary and the corresponding inverse problem will be analyzed. Some of the basic results are summarized without proofs. The latter are available in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Selection of optimal parameters for multipurpose controllable systems: General formulation of the problem and solution algorithms

This paper describes an approach to the selection of optimal parameters for multipurpose controllable systems intended to perform a series of maneuvers. The paper is an extension of the writer's previous papers on the subject (Refs. 1–6). It states the problem in general terms and discusses the selection of a functional. Depending on the amount of available data on the parameters of maneuver, the problem reduces to either minimization of a function of many variables or a game problem. Algorithms for the solution of the problems formulated are developed. A model problem of space flight mechanics with low-thrust engines is used for illustration purposes. Several applications of the approach developed are contained in Refs. 7–9. In addition, Refs. 8–9 suggest one more method for solving the problem of Section 3, the so-called method of optimal coverings. Another direction of research is described in Ref. 10.     

An efficiency estimate for a hierarchical reconstruction method in the problem of reconstructing the locations of discrete scattering centers from a set of projections

Under consideration is the problem of determining a maximal set for a family of points from a restricted collection of their two-dimensional projections. This problem arises naturally in the applications of physical hydroaerodynamics to optical diagnostics of real liquid and gas flows by measuring the instantaneous velocity fields in a flow volume. We propose some methods for reconstructing the original set and determining the sufficiency of measurement for solving uniquely the inverse problem for parallel and perspective projections. We statistically evaluate the efficiency of the reconstruction method.     

Two-parameter extremum problems of boundary control for stationary thermal convection equations

Two-parameter extremum problems of boundary control are formulated for the stationary thermal convection equations with Dirichlet boundary conditions for velocity and with mixed boundary conditions for temperature. The cost functional is defined as the root mean square integral deviation of the desired velocity (vorticity, or pressure) field from one given in some part of the flow region. Controls are the boundary functions involved in the Dirichlet condition for velocity on the boundary of the flow region and in the Neumann condition for temperature on part of the boundary. The uniqueness of the extremum problems is analyzed, and the stability of solutions with respect to certain perturbations in the cost functional and one of the functional parameters of the original model is estimated. Numerical results for a control problem associated with the minimization of the vorticity norm aimed at drag reduction are discussed.     

On stability of solutions of the coefficient inverse extremal problems for the stationary convection-diffusion equation

The coefficient inverse extremal problems are studied for the stationary convectiondiffusion equation in a bounded domain under mixed boundary conditions on the boundary of the domain. The role of control is played by the velocity vector of a medium and the functions that are involved in the boundary conditions for temperature. The solvability of the extremal problems is proven both for an arbitrary weakly lower semicontinuous quality functional and for the particular quality functionals. On the basis of analysis of the optimality system some sufficient conditions are established on the initial data providing the uniqueness and stability of optimal solutions under sufficiently small perturbations of both the quality functional and one of the functions involved in the original boundary value problem.     

LOCAL PRESSURE CORRECTION FOR THE STOKES SYSTEM

Pressure correction methods constitute the most widely used solvers for the timedependent Navier-Stokes equations.There are several different pressure correction methods,where each time step usually consists in a predictor step for a non-divergence-free velocity,followed by a Poisson problem for the pressure(or pressure update),and a final velocity correction to obtain a divergence-free vector field.In some situations,the equations for the velocities are solved explicitly,so that the numerical most expensive step is the elliptic pressure problem.We here propose to solve this Poisson problem by a domain decomposition method which does not need any communication between the sub-regions.Hence,this system is perfectly adapted for parallel computation.We show under certain assumptions that this new scheme has the same order of convergence as the original pressure correction scheme(with global projection).Numerical examples for the Stokes system show the effectivity of this new pressure correction method.The convergence order O(k^2)for resulting velocity fields can be observed in the norm l^2(0,T;L^2(Ω)).