特征值摄动法在利用动响应结构小损伤检测中的应用

目前在使用遗传算法或神经网络方法进行结构动力学损伤检测,需要基于少量的在线测量损伤结构数据和大量的数值仿真数据来实现,其中通过有限元方法来获得仿真数据的巨大计算量是动力学结构损伤检测方法发展中所面临的一个重要问题。本文在建模方面应用近年来提出的调整单元刚度模拟损伤的先进方法,以保证在损伤前后结构自由度数目不变;在此基础上应用特征值摄动法来减少损伤检测中计算量,并通过对复合材料层合板响应信号的小波分析验证了使用一阶矩阵摄动在有效降低计算量的同时,可以获得对损伤检测而言足够准确的响应信号。     

Linear Potentials in Nonlinear Potential Theory

Pointwise gradient bounds via Riesz potentials, such as those available for the linear Poisson equation, actually hold for general quasilinear degenerate equations of p-Laplacean type. The regularity theory of such equations completely reduces to that of the classical Poisson equation up to the C ¹-level.     

A New Approach to Front Propagation Problems: Theory and Applications

In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii) stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics. (Accepted July 8, 1996)     

小波方法及其非线性力学问题应用分析

小波分析是近几十年来发展起来的重要数学分支,被誉为“数学显微镜”,其独具的多分辨分析和大量可供选择的,可兼具正交性、紧支性、对称性、低通滤波、线性相位及插值性等优良数学品质的小波基函数为强非线性微分方程的数值求解带来了新的契机。自上世纪90年代以来,诸如小波伽辽金法、小波配点法、小波有限单元法和小波边界单元法等数值方法被先后构建出来并成功应用于各类力学问题的定量研究之中。本文从小波提出的历史背景及作为其理论基础的多分辨分析出发,对现有基于小波理论的各类数值方法进行梳理,总结各自的优点、缺点和下一步可能的发展方向,为未来基于小波理论的定量分析方法的发展及其在复杂非线性力学问题中的应用研究提供参考。     

Dynamical-System Approach to Solving Linear Programming Problems and Applications in Economic Modeling

A class of dynamical systems on symplectic manifolds solving linear programming problems is described. The structure of an orbit space is analyzed within the framework of the Marsden–Weinstein reduction scheme. Some examples having applications in modern macroeconomic modeling are studied in detail.     

Nonlinear Spherical Caps and Associated Plate and Membrane Problems

This paper deals with the existence and multiplicity problem of the equilibrium solutions of an elastic spherical cap within nonlinear strain theory. We pose the problem in the form of a three parameter bifurcation problem, one parameter being related to the load, the others to the geometry. When the geometrical parameters are different from zero, the so-called generic case, we revisit the nonuniqueness results, and explore the solutions in the parameter space. Then we analyze the formal limits as the geometrical parameters tend to zero. When the curvature tends to zero, we obtain from the nonlinear shell a von Kármán plate, a new, although natural, result. When the thickness parameter tends to zero, we get a nonlinear membrane problem. A study of the latter gives infinitely many solutions, and we discuss the construction, shapes, and stability in detail. This revised version was published online in August 2006 with corrections to the Cover Date.     

Topical Problems and Applications of Creep Theory

A historical review of achievements in creep theory is given. Primary attention is focused on the phenomenological approach. Different constitutive equations are discussed for primary and secondary creep as well as for creep with damage. New creep problems are examined     

结构局部非线性修改灵敏度分析及其应用

以线性系统的非线性修改方法为基础，导出了局部非线性振动的灵敏度分析方法；提出了灵敏度为指导的优化设计思想，并将其应用于大型结构上的局部非线性因素的优化设计。文中介绍一些典型非线性特征的数学描述。最后，本文以火炮结构中的炮口振动为例，提出在炮口处增设炮口减振器的设计思想，并用本文方法对炮口减振器的参数进行优化设计，得到满意的结果。     

Global Continuation in Displacement Problems of Nonlinear Elastostatics via the Leray-Schauder Degree

We obtain global solution continua of forced displacement problems of nonlinear elastostatics via a Leray-Schauder scheme. We adopt strong ellipticity as a bsic constitutive hypothesis. The usual Leray-Schauder approach, based in part upon the reduction of the boundary value problem to an operator equation for the zeros of a compact vector field, apparently fails. On the one hand, strong ellipticity alone is not enough to insure “invertibility/rdquo; of the principal, quasilinear part of the differential operator. More importantly, the physical requirement of local injectivity of the deformation and the associated growth of the stored-energy function dictate that ellipticity is not uniform. We demonstrate how to overcome these difficulties. Finally, with additional, physically reasonable restrictions on the stored-energy function, we obtain unbounded branches of classical, globally injective solutions. Accepted: October 22, 1999     

Linear and Nonlinear Aerodynamic Theory of Interaction between Flexible Long Structure and Wind

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Development of an Internal Bone Remodelling Theory and Applications to Some Problems in Orthopaedic Biomechanics*

Bone tissue is a porous, heterogeneous and anisotropic material, which adapts its mechanical properties depending of the local stress level. This evolutive behaviour of the bone is normally known as bone remodelling. In this work, a bone-remodelling theory, based on the principles of continuum damage mechanics, is presented. The corresponding mathematical formulation has been implemented in a finite element code in order to predict the bone response after implantation of a prosthesis or fixation. Although the present model is not based on experimental verification, the model predicts important qualitative experimental results, being still necessary to test against experimental/clinical work. The main aim of this paper has been, therefore, the qualitative study of the long-term bone evolution, especially of the human femur when different types of implants are employed. A comparative analysis between two widely used hip prostheses (the Exeter and the SHP), has been performed. We have also studied the treatment of proximal femoral fractures by means of extramedullary and intramedullary implants.     

Applications of Nonlinear Dynamical Systems Theory in Developmental Psychology: Motor and Cognitive Development

Applications of nonlinear dynamical systems theory to psychology have led to recent advances in understanding neuromotor development and advances in theories of cognitive development. This article reviews published findings associated with a specific coherent and influential application from which a theory of adaptive, self-organized cognition has been derived and related to a theory of developmental dynamics of the neuromotor system. The review focuses on implications of the two theories for quantifying developmental phenomena, and suggests a method for quantifying the cognitive theory.     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

Some Problems of Linear Thermoelasticity in the Ginzburg–Landau Theory of Phase Transitions

The well-posedness of the problem of linear thermoelasticity in the Ginzburg–Landau theory of phase transitions is proved.     

Linear and Nonlinear Problems on the Elastic Deformation of Complex Shells and Methods of Their Numerical Solution

Some approaches to the solution of problems on the elastic deformation of thin-walled solids with a complex shape are analyzed on the basis of linear and geometrically nonlinear models. The general characteristic of the classical approaches to the solution of the problems is discussed. Approaches employing new classes of surfaces are considered. Solutions to some problems on the stress state of complex shell elements are presented     

A Sobolev-Hardy Inequality with¶Applications to a Nonlinear Elliptic Equation¶arising in Astrophysics

In this paper we analyze the existence and non-existence of cylindrical solutions for a nonlinear elliptic equation in ?³, which has been proposed as a model for the dynamics of galaxies. We prove a general integral inequality of Sobolev-Hardy type that allows us to use variational methods when the power p belongs to the interval [4, 6]. We find solutions in the range 4 < p≤ 6. The value p= 4 seems to have characteristics similar to those of the critical Sobolev exponent p= 6.     

Linear and Nonlinear Algorithms for Stress Separation in Photoelasticity

An experimental-numerical hybrid method for the stress separation in photoelasticity is proposed in this study. In the proposed method, boundary conditions for a local finite element model, that is, tractions along boundaries are inversely determined from photoelastic fringes. Two algorithms are proposed for determining the boundary condition. One is a linear algorithm in which the tractions are obtained by the method of linear least-squares from both principal stress difference and principal direction. Another is the nonlinear algorithm in which the tractions are determined only from the principal stress difference. After determining the boundary conditions for the local finite element model, the stresses can be obtained by finite element direct analysis. The effectiveness is demonstrated by applying the proposed method to a perforated plate under tension and contact problems. Results show that the boundary conditions of the local finite element model can be determined from the photoelastic fringes and then the individual stresses can be obtained by the proposed method. Furthermore, the stresses can be evaluated even if the boundary condition is complicated such as at the contact surface. It is expected that the proposed method can be powerful tool for stress analysis.     

New Transforms for the Resolving Equations in Elastic Theory and New Integral Transforms, with Applications to Boundary-Value Problems of Mechanics

This paper reviews results from an analysis of exact boundary-value solutions of static and dynamic elasticity obtained by the method of integral transforms. Consideration is given to the solutions of problems for a half-space and a blunted hollow cone, the heat conduction problem for a cone, and the diffraction problem for a cone with a hole along the generatrix. Solutions of mechanics problems are analyzed and the method of integral transforms is generalized     

C- METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

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