Guaranteed and computable bounds of the limit load for variational problems with linear growth energy functionals

The paper is concerned with guaranteed and computable bounds of the limit (or safety) load, which is one of the most important quantitative characteristics of mathematical models associated with linear growth functionals. We suggest a new method for getting such bounds and illustrate its performance. First, the main ideas are demonstrated with the paradigm of a simple variational problem with a linear growth functional defined on a set of scalar valued functions. Then, the method is extended to classical plasticity models governed by vonMises and Drucker-Prager yield laws. The efficiency of the proposed approach is confirmed by several numerical experiments.     

Statistical Studies of the Electric Breakdown in Nitrogen in the Duration Range of 3 ms–60 min

The statistical characteristics of an electric breakdown in the nitrogen in the spike (cathode)–plane gap in the duration range of (3 × 10^–3)–3600 s at voltages close to a static breakdown have been studied. It has been found that a probability of a gap breakdown is nonmonotonously distributed over time. The presence of maxima in the probability distribution confirms a contribution of some processes that both stimulate and suppress a breakdown. The typical times of the processes are ~30 ms, ~10^–1 s, and ~300 s.     

A simple approach to T2 imaging in MRI

A simple method for obtaining images whose contrast depends only on T₂ is described and tested both on phantoms and in vivo. The method works reliably and effectively under clinically realistic operating conditions using standard imaging protocols. It can result in a substantial reduction in imaging times for T₂ weighted images.     

A priori error estimates of variational-difference methods for Hencky plasticity problems

In this article, convergence of equilibrium finite-element approximations for variational problems of the Hencky plasticity is analyzed. To obtain a priori error estimates, two regularized problems are considered and additional differentiability properties of their solutions are investigated. This allows us to prove that there is a relation between the parameters of regularization and sampling such that equilibrium approximations of the regularized problems produce a sequence of tensor-functions converging to the solution of the perfectly elasto-plastic problem. Convergence estimates are established. Bibliography:11 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 226–234. Translated by S. I. Repin.     

A posteriori error estimation for nonlinear variational problems by duality theory

In this paper, we use the duality theory of the calculus of variations to derive a posteriori error estimates. We obtain a general form of this (duality) error estimate and show that the known classes of a posteriori error estimates are its particular cases. Bibliography: 21 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 201–214. Translated by S. I. Repin.     

Estimates of deviations from exact solutions of variational inequalities based upon Payne–Weinberger inequality

A new method for obtaining computable estimates for the difference between exact solutions of elliptic variational inequalities and arbitrary functions in the respective energy space is suggested. The estimates are obtained by transforming the corresponding variational inequality without the use of variational duality arguments. These estimates are valid for any function in the energy class and contain no constants depending on the mesh used to find an approximate solution. This method for linear elliptic and parabolic problems was earlier suggested by the author. The guaranteed error bounds we derive can be of two types. Estimates of the first type contain only one global constant, which is a constant in the Friedrichs type inequality. Estimates of the second type are based on the decomposition of Ω into convex subdomains and the Payne–Weinberger inequalities for these subdomains. Bibliography: 20 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 81–90.     

Chemical modification of polyolefin blends

The effect of crosslinking on the properties of low density polyethylene / polypropylene blends was investigated. It was found that crosslinking results in much higher deformability of the blends compared to uncrosslinked materials. A decrease in crystalline portion determined by DSC leads to lower modulus in crosslinked samples. High drawability of crosslinked samples is mainly attributed to peculiar behaviour of PE part of the crosslinked blend where thicker lamellae are formed during drawing.