Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: The n-dimensional scalar case

In this work we are going to prove the functional J defined by

is weakly lower semicontinuous in W^1,p(Ω) if and only if W is separately convex. We assume that Ω is an open set in and W is a real-valued continuous function fulfilling standard growth and coerciveness conditions. The key to state this equivalence is a variational result established in terms of Young measures.     

Numerical solution of some unstable problem

While modeling nonlinear wakefield excited by a short laser pulse in a plasma channel, the pulse group speed is normally less than the speed of light, whereas they are usually considered to be equal. This paper shows that consideration of the speed difference leads to an exponential instability of the model. It is also proved that this unstable problem is equivalent to a specific stable problem. This feature allows us to develop stable numerical schemes for their solution.     

Numerical computation of stress induced microstructure

The mesh transformation method is applied on a two-dimensional elastic crystal model to study the formation of laminated microstructure in austenite-martensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched needle-like laminates can also be obtained by certain loads.     

A control problem governed by a second-order differential inclusion

This article studies some Bolza-type problems governed by second-order differential inclusions with two boundary conditions, where the controls are Young measures.     

Numerical solution procedures for the morning commute problem

This paper discusses solution techniques for the morning commute problem that is formulated as a discrete variational inequality (VI). Various heuristics have been proposed to solve this problem, mostly because the analytical properties of the path travel time function have not yet been well understood. Two groups of “non-heuristic” algorithms for general VIs, namely projection-type algorithms and ascent direction algorithms, were examined. In particular, a new ascent direction method is introduced and implemented with a heuristic line search procedure. The performance of these algorithms are compared on simple instances of the morning commute problem. The implications of numerical results are discussed.     

A class of stable energy minimisers for the rotating drop problem

We implement variational techniques and an implicit function theorem to derive constraints on angular velocity under which we may verify the existence, boundary regularity, and stability of an energy-minimising family of rotating liquid drops in a neighbourhood of the closed unit ball in Rn+1.     

晶体微观结构的数值计算

晶体微观结构是晶体材料在特定物理条件下其多个能量极小平衔态在空间形成的某种微尺度的规则分布．几何非线性的连续介质力学理论可以用能量极小化原理来解释晶体微观结构的形成，并用Young测度来刻画平衡态各变体在空间的概率分布．定性的理解与定量地分析和计算晶体材料的微观结构对于发展和改进高级晶体功能材料，如形状记忆合金、铁电体、磁至伸缩材料等，有重要的意义．本文回顾了近年来晶体微观结构数值计算方面的最新进展．介绍了计算晶体微观结构的几种数值方法及有关的数值分析结果。     

Numerical solution of a quasilinear parabolic problem

A combined approach of linearisation techniques and finite difference method is presented for obtaining the numerical solution of a quasilinear parabolic problem. The given problem is reduced to a sequence of linear problems by using the Picard or Newton methods. Each problem of this sequence is approximated by Crank-Nicolson difference scheme. The solutions of the resulting system of algebraic equations are obtained by using Block-Gaussian elimination method. Two numerical examples are solved by using both linearisation procedures to illustrate the method. For these examples, the Newton method is found to be more effective, especially when the given nonlinear problem has steep gradients.     

Numerical solution of a linear bilevel problem

The linear bilevel programming problem in the optimistic formulation is studied. It is reduced to an optimization problem with a nonconvex constraint in the form of a d.c. function (that is, the difference of two convex functions). For this problem, local and global search methods are developed. Numerical experiments performed for numerous specially generated problems, including large-scale ones, demonstrate the efficiency of the proposed approach.     

Numerical solution of a Rayleigh-Taylor instability problem

The numerical solution of a Rayleigh-Taylor instability problem where an inviscid liquid of finite depth is accelerated into a gas of semi-infinite extent is obtained by transforming the irregular flow domain into a rectangular domain by a coordinate transformation. The free surface equation is solved by a Crank-Nicolson procedure. The boundary condition at the free surface for the velocity potential ø which contains the time derivative of ø is also treated by an implicit scheme. The numerical results agree well with those obtained by higher order perturbation analysis.     

An elementary method of calculating an explicit form of Young measures in some special cases

We present an elementary method of explicit calculation of Young measures for a certain class of functions. This class contains, in particular, functions of a highly oscillatory nature which appear in optimization problems and homogenization theory. In engineering such situation occurs, for instance, in nonlinear elasticity (solid–solid phase transition in certain elastic crystals). Young measures associated with oscillating minimizing sequences gather information about their oscillatory nature and therefore about underlying microstructure. The method presented in this article makes no use of functional analytic tools. There is no need to use a generalized version of the Riemann–Lebesgue lemma and to calculate weak* limits of functions. The main tool is the change of variable theorem. The method applies both to sequences of periodic and nonperiodic functions.     

Numerical solution of the inverse problem for the wave equation

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 6–9, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Numerical solution to a stokes interface problem

A numerical algorithm is described for solving the generalized eigenvalue problem arising in the study of the spectrum of a preconditioned operator in the pressure equation derived from a Stokes interface problem. The algorithm is implemented for two finite element schemes. It is tested for a problem with an analytical solution and is applied to spectrum computations in the case of a piecewise constant viscosity. A large number of numerical experiments are analyzed, and recommendations are given for solving the Stokes interface problem in practice.     

Numerical solution of a problem in the theory of epidemics

We propose a numerical algorithm for a model of the spread of infection in the theory of epidemics. This model involves a threshold for becoming infective and leads to a system of delay-differential equations with the delay function which depends on the values of the unknowns over an interval. The numerical algorithm is based on diagonally implicit multistage integration formulas with stage order equal to the order of the method. This makes possible the efficient evaluation of past values to the accuracy compatible with the requested error tolerance. This algorithm is appicable to the general model without any simplifying assumptions on the parameters of the delay-differential system. Numerical results illustrate the effect of varying various parameters of the model on the spread of infection in a constant population.     

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.