工程力学中的互补问题:模型

对互补问题在非线性力学中的一些应用情况给出了较概括的介绍，重点放在了静态接触问题、弹塑性问题以及结构优化问题的互补模型上。将力学问题写成互补模型的好处主要有两点：一是使力学问题有更自然、更精确的数学描述；二是可以考虑用互补问题的有效和强健的数值算法来求解。     

Stefan-like problems

This paper presents an overview of the free boundary problems connected to the problem of Stefan type or, more generally, to problems describing a change of phase. After a sketch of the historical development of the research in the area, some specific questions are addressed, such as classical solvability of the problem in several space dimensions, regularization of supercooling, dynamical contact angle.

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Sommario Il lavoro presenta una visione generale dei problemi a frontiera libera del tipo di Stefan o ad esso collegati, e più in generale dei problemi che descrivono cambiamenti di fase. Dopo aver riassunto lo sviluppo storico delle ricerche in questo campo, viene dedicata maggiore attenzione ad alcuni problemi specifici come la risoluzione classica di problemi multidimensionali, la regolarizzazione dei problemi con sovraraffreddamento, il problema dell'angolo di contatto.

     

On the determination of missing boundary data for solids with nonlinear material behaviors,using displacement fields measured on a part of their boundaries

The paper is devoted to the derivation of a numerical method for expanding available mechanical fields (stress vector and displacements) on a part of the boundary of a solid into its interior and up to unreachable parts of its boundary (with possibly internal surfaces). This expansion enables various identification or inverse problems to be solved in mechanics. The method is based on the solution of a nonlinear elliptic Cauchy problem because the mechanical behavior of the solid is considered as nonlinear (hyperelastic or elastoplastic medium). Advantage is taken of the assumption of convexity of the potentials used for modeling the constitutive equation, encompassing previous work by the authors for linear elastic solids, in order to derive an appropriate error functional. Two illustrations are given in order to evaluate the overall efficiency of the proposed method within the framework of small strains and isothermal transformation.     

黄河下游拟建堤防截渗墙的工程地质问题分析

根据黄河下游堤防区的工程地质勘察资料 ,分析和阐述了堤防区拟建截渗墙段的工程地质条件 ,针对堤基土渗透变形破坏、堤基土液化、不均匀沉陷、大堤隐患等主要工程地质问题进行了专门研究 ,并提出了有意义的处理措施 ,对黄河下游堤防工程的实施具有重要的实际意义。     

Three-dimensional solutions for general anisotropy

The Stroh formalism is extended to provide a new class of three-dimensional solutions for the generally anisotropic elastic material that have polynomial dependence on x₃, but which have quite general form in x₁,x₂. The solutions are obtained by a sequence of partial integrations with respect to x₃, starting from Stroh's two-dimensional solution. At each stage, certain special functions have to be introduced in order to satisfy the equilibrium equation. The method provides a general analytical technique for the solution of the problem of the prismatic bar with tractions or displacements prescribed on its lateral surfaces. It also provides a particularly efficient solution for three-dimensional boundary-value problems for the half-space. The method is illustrated by the example of a half-space loaded by a linearly varying line force.     

A higher-order boundary element method for three-dimensional potential problems

A method of eliminating the singularities involved in boundary element methods for three-dimensional potential problems is presented and the non-singular expressions of integrals on an element on which the singular point is situated are given for linear and quadratic interpolation functions. Numerical examples are compared with analytical solutions to show that the higher-order interpolations have better precision.     

Control Problems for the Steady-State Equations of Magnetohydrodynamics of a Viscous Incompressible Fluid

Control problems for a steady-state model of the magnetohydrodynamics of a viscous incompressible fluid in a bounded domain with an impermeable, perfectly conducting boundary are formulated. The resolvability of the problems is studied, the use of the Lagrange principle is justified, and optimality systems are analyzed.     

Exponential numerical methods for one-dimensional one-phase Stefan problems

Summary Three exponential iterative methods for one-dimensional one-phase Stefan problems based on the transformation of the moving boundary problem into a mixed one, the discretization of the time variable, and the piecewise linearization of the resulting two-point boundary-value problem at each time step are proposed. Two of the methods are based on the strong conservation-law form of the governing equation and analytically solve a piecewise advection–diffusion equation, whereas the third exponential technique accounts for transient, advective, and diffusive effects when determining the solution. These exponential methods provide piecewise-analytical (exponential) solutions, which, by imposing continuity conditions, are globally continuous throughout the domain, and one of them provides globally smooth solutions. The methods have been applied to the classical one-phase Stefan problem and solutions in excellent agreement with the exact ones have been obtained for several Stefan numbers. In addition, it is shown that the method that accounts for transient, advective, and diffusive effects preserves the similarity of the analytical solution to Stefan problems, yields a tridiagonal matrix, and exhibits a spatial accuracy of, at least, fourth order. Application of this method to a forced one-phase Stefan problem indicates that it provides solutions in excellent agreement with those obtained by means of explicit finite difference and nodal integral techniques, and that the melting-front location exhibits some oscillations in the initial stages whose amplitude decreases as the Stefan number is decreased and as time increases, but which increases as the amplitude of the forcing temperature is increased. It is also shown that the temperature profiles in the liquid are affected by the amplitude and frequency of the forcing and the Stefan number.This research was partially financed by Project BFM2001–1902 from the Ministerio de Educación y Ciencia of Spain and Fondos FEDER.The author gratefully acknowledges the comments made by the referees,which have resulted in a clearer presentation.     

A Note on the Removal of Further Breakdowns in Classical Solutions of Laplace"s Equation on Sectorial Regions

This note considers solutions to Laplace"s equation on sectors with varying vertex angles. Under Neumann conditions on the radial boundaries, there are two critical vertex angles for which classical separable solutions break down. These breakdowns have been noted in the literature and resolved. Here Dirichlet and mixed conditions are also treated. For all three boundary conditions in combination, nine further critical angles are identified and valid corresponding solutions found. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical issues related to the modelling of electrochemical impedance data by non-linear least-squares

Non-linear least-squares (NLS) fitting is the typical approach to the modelling of electrochemical impedance spectroscopy (EIS) data. In general the application of NLS to EIS models can give rise to ill-posed problems. On the one side, with ill-posed problems it is not possible to prove a priori that a unique solution exists. On the other side, the relevant numerical approximations cannot ensure that a unique solution exists even a posteriori. It is therefore basically pointless to endeavour to achieve one absolute minimum of any objective function for an EIS model in an NLS problem. A lack of awareness of the above-mentioned factors might render numerical approaches tending to locate the absolute minimum questionable.     

Large-scale statistical inverse computation of inelastic accretion in transient granular flows

In this work, the accretion of fine-grained particulate matter into larger objects in high-speed granular flows, due to sudden disturbances in their mean velocity field, is investigated. A multibody collision model is developed whereby the coefficients of restitution and friction, as well as quantities such as the contact area and collision duration time, are implicit functions of the relative collision velocities and surfacial thermochemical reactions during impact. A recursive fixed-point multilayered staggering scheme is developed to simulate the resulting coupled non-linear system. Inverse problems are then constructed whereby transient flow conditions, reaction rates, particulate volume fractions, hardnesses, etc., are sought which deliver prespecified aggregate growth from a base starting particulate size. Classical gradient-based methods perform poorly, to this class of problems due to the fact that the associated objective functions depend in a non-convex and non-differentiable manner on the mentioned starting-state parameters. Furthermore, the results are very sensitive to the size of the control volumes selected. Therefore, due to the lack of robustness of classical gradient-based minimization schemes, a statistical genetic algorithm is developed whereby (I) the starting state-variables are represented by a “genetic string”, and concepts of evolutionary behavior, such as selective reproduction, are applied to a population of such strings in order to determine an optimal set of starting state-parameters and (II) sequences of control volumes, each containing a finite number of particles, are adaptively computed until the sequential change in the ensemble average of a population of control volumes all fall below a given tolerance. Three-dimensional numerical examples are given to illustrate the behavior of the model and the overall solution process.     

一种新型的求解扭转问题的边界元方法

证明了柱体自由扭转的边界积分方程被积函数的散度等于零,将翘曲函数表示为翘曲势函数在边界点的数值计算,避免求解奇异的数值积分。实例计算表明,该表度较高。     

Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids

Methods are presented for obtaining exact analytical representations of supercoiled equilibrium configurations of impenetrable elastic rods of circular cross-section that have been pretwisted and closed to form rings, and a discussion is given of applications in the theory of the elastic rod model for DNA. When, as here, self-contact is taken into account, and the rod is assumed to be inextensible, intrinsically straight, transversely isotropic, and homogeneous, the important parameters in the theory are the excess link Δℒ (a measure of the amount the rod was twisted before its ends were joined), the ratio ω of the coefficients of torsional and flexural rigidity, and the ratio d of cross-sectional diameter to the length of the axial curve C. Solutions of the equations of equilibrium are given for cases in which self-contact occurs at isolated points and along intervals. Bifurcation diagrams are presented as graphs of Δℒ versus the writhe of C and are employed for analysis of the stability of equilibrium configurations. It is shown that, in addition to primary, secondary, and tertiary branches that arise by successive bifurcations from the trivial branch made up of configurations for which the axial curve is a circle, there are families of equilibrium configurations that are isolas in the sense that they are not connected to bifurcation branches by paths of equilibrium configurations compatible with the assumed impenetrability of the rod. Each of the isolas found to date is connected to a bifurcation branch by a path which, although made up of solutions of the governing equations, contains regions on which the condition of impenetrability does not hold. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some Stratigraphic Control Problems

In this paper, we are interested in lithology diffusion models applied in the field of stratigraphic basin simulations for large-scale depositional transport processes of sediments. Such models describe erosion-sedimentation processes and take into account limited weathering via nonstandard unilateral problems. Various theoretical results, illustrations, and numerical solutions are presented for the monolithologic column case. A new conservation law involved in modeling is formulated, and mathematical tools for solving the problem are described.     

动态有限元程序评介

Ⅰ.引言二体和多体碰撞问题越来越受到广泛重视。这不仅因为军事应用的需要,还由于如下几个方面的兴趣:交通安全;车辆碰撞;核反应装置在冲击载荷下的安全问题;装甲系统的设计;由液体和固体微粒碰撞引起的材料的腐蚀和断裂;空间飞行器对流星体碰撞的防护;爆炸成型和焊接,等等。      

Dynamic deformation of a thermoviscoelastic rod of triangular cross section in a coupled formulation

For the coupled model of a thermoviscoelastic rod of equilateral triangular cross section, two exact solutions are obtained for the cases where a normal displacement and a shear stress or a tangential displacement and a normal stress are specified on the lateral surface of the rod. A dimensionless parameter R₀ is introduced to judge the appropriateness of taking into account the coupling in the formulation of the problem. Formulas are given for the velocities and lengths of the temperature, shear, and longitudinal waves, which can be used in experiments to determine the physical properties of thermoviscoelastic materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 128–143, July–August, 2007.     

On some aspects of Stefan problems

A short discussion of three new unsolved Stefan-like problems is presented.

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Sommario Viene presentata una breve discussione di tre nuovi problemi del tipo di Stefan non risolti.

     

符拉索夫板弯曲问题的通解

 针对符拉索夫板弯曲问题的基本方程, 给出了两个形式简洁 的通解定理.     

域外奇点法在弹性问题及其物性值反问题中的应用

本文采用域外奇点法中的位移法和应力法求解弹性问题，并对弹性物性值进行反分析，由有限个观测点的位移值。同时反算出材料的纵向弹性模量Ｅ和泊桑比Ｖ。本文方法属于边界型解法，可有效地避免解的奇异性，不需要数值积分，具有输入数据少，精度高，计算时间短和程序编制容易等优点。