Complete systems of doubly- and triply-periodic particular solutions of the equations of elasticity theory in spherical coordinates

Institute of Superhard Materials, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 8, pp. 29–34, August, 1995.     

平面问题极坐标下几何方程的一种解析法

经典弹性力学教材中都是通过单元的几何变形来推导平面问题极坐标下的几何方程. 这里, 用解析法从直角坐标系下的几何方程推出极坐标下的几何方程.     

Construction of n-sided polygonal spline element using area coordinates and B-net method

In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.     

极坐标系下的Legendre谱元方法求解Poisson-型方程

r=0处的坐标奇异性是求解极坐标下Poisson-型方程的关键。本文提出一种极坐标系下基于Galerkin变分的Legendre谱元方法用于求解圆形区域内的Poisson-型方程,物理区域的径向和周向划分若干单元,计算单元均采用Legendre多项式展开;圆心所在单元的径向使用LGR(Legendre Gauss Radau)积分点,其他单元径向使用LGL(Legendre Gauss Lobatto)积分点,从而避免了极点处1/r坐标奇异性,周向单元均采用LGL积分点。利用区域分解技术,可以避免节点在极点附近聚集;最后求解了多个Dirichlet或Neumann边界条件下的Poisson-型方程算例。数值结果表明,谱元方法具有很高的精度。     

随机振动问题的广义坐标合成法

工程中的随机振动分析多采用完全二次型组合法(CQC)及其改进算法,如虚拟激励法(PEM)和谐波激励法(HEM)。广义坐标合成法(GCS)提出了一种计算随机响应的新思路,其基本原理是将物理空间的计算转移到自由度较小的振型空间进行,从而缩减了计算量和计算规模。对于较大规模结构体系的随机振动问题,GCS方法计算响应协方差矩阵的计算量只相当于PEM的2r/sT,其中s、r和T分别为激励功率谱矩阵的秩、振型总数和离散频率点数。此外,对于给定激励时程的问题,由于GCS方法直接求解广义坐标运动方程,因而可以方便得出响应时程。通过对几种方法的详细对比,说明对于大多数只需要求解响应方差的随机振动问题,GCS是最优的计算方法。PEM只有在s相似文献   

基于线性多点逼近方法的结构局部非线性损伤识别

结构损伤通常伴随一定的非线性响应特征,当非线性特征较为明显时,单纯的有限元更新方法无法克服本身所固有的非线性特征的限制,在动力特性测试中也存在许多困难.本文研究时域内损伤识别方法,将局部非线性结构损伤等效为一个的附加子结构,基于线性多点逼近方法,将一个非线性损伤识别问题转化为作用于线性模型的载荷识别问题.通过结构非线性响应识别出等效附加力的时间历程,利用附加子结构的输入输出特性识别实际结构的单元损伤特性.在所提出的识别方法理论算法基础上,对单元刚度线性损伤和非线性损伤形式进行了数值模拟,算例显示这种方法的有效性.     

一种曲折裂纹尖端单元位移场的构造方法

在扩展有限元的框架内,本文发展了一种构造裂尖单元位移场的方法。整个裂纹沿程两侧的非连续位移场只通过富集变换的阶梯函数表征,在裂尖单元,通过调整形函数使得非连续性严格地消失于裂纹尖端。在避免混合区单元引入不满足单位分解的附加位移项的同时,实现了裂纹尖端单元位移场部分非连续特性的表达。还对裂尖单元的形函数调整原则进行了分析,以平面四节点单元为例提出了两种调整方式。文中裂尖单元中含有曲折裂纹的算例说明了本文方法的有效性和适用性。     

Residual strain measurement in composites using the cure-referencing method

This paper describes the details of a novel procedure called the cure-referencing method (CRM) to measure the strains associated with residual stresses on the surface of composite panels. The CRM involves the replication of diffraction gratings onto the surface of composite specimens during the autoclave during cycle. Residual strains associated with the curing process are measured using moiré interferometry at room temperature after the specimens have been taken out of the autoclave. The procedures for both the grating replication and the moiré interferometry experiment are described in detail. A method of high-temperature moiré interferometry was developed to resolve the residual strains due to thermal expansion from those due to chemical matrix shrinkage and stress relaxation. These procedures are demonstrated on unidirectional and multidirectional laminates and on woven textile composites.     

Solving material nonlinear structural problems by a finite analytic method in local coordinates

The purpose of this paper is to develop a finite analytic (FA) numerical solution for the elasto-plastic problem of the total theory. Schemes for the FA method in local coordinates for solving non-linear governing equations in the form of Navier equations are derived, which can be utilized to solve the problem in a domain of arbitrary geometry. Numerical illustration shows that the schemes are effective and practical.     

Vibration characteristics of a spherical–cylindrical–spherical shell by a domain decomposition method

A domain decomposition method is used to analyze the free and forced vibration characteristics of a spherical–cylindrical–spherical shell, based on Reissner–Naghdi's thin shell theory. The joined shell is divided into some cylindrical and spherical shell segments along the meridional (longitudinal) direction. Double mixed series, i.e., Fourier series and Chebyshev polynomials, are employed as the admissible displacement functions to obtain the discretized equation of motion for the joined shell. Numerical comparisons with the results derived by FEM and those available in the previous literature are made to validate the present method. Moreover, the effects of length-to-radius and radius-to-thickness ratios on the natural frequencies are also investigated.     

Gasdynamic equations of higher approximations of the chapman-enskog method. Review

Some features of the derivation of the gasdynamic equations of higher (Burnett and super-Burnett) approximations of the Chapman-Enskog method are discussed for non-dense monatomic gases in the absence of external forces. The equations are given, and their properties and the results of applying them to various problems of gas dynamics are described. The problems of deriving, justifying, and testing macroscopic models are among the fundamental problems of not only continuum mechanics but physical kinetics in general. In the latter case it is a question of reducing more general molecular-kinetic models to simpler macroscopic ones. Historically, the classical Chapman-Enskog method of solution (as the Knudsen number Kn → 0) of the kinetic Boltzmann-Maxwell equation is of primary importance. Its application yields a chain of Euler and Navier-Stokes equations and equations of higher (Burnett, super-Burnett, etc.) approximations of the method. The publication of this review is a result of the increasing interest in the equations of higher approximations of the Chapman-Enskog method, in particular, in the Burnett equations. Primarily, this is a consequence of advances in the application of the latter equations to the problem of shock wave structure and the possible (by virtue thereof) expansion of the region of applicability of the macroscopic rarefied gas flow models. In addition, using the Burnett approximation, a series of new interesting effects has been established. In particular, it has been shown that in the classical problem of heat transfer between bodies heated to different temperatures it is necessary to take into account not only the temperature creep but also the Burnett temperature stresses. At the same time, a number of fundamental problems concerning the status of these equations has not yet been solved. An extensive literature is devoted to this problem. In the present paper, we shall mainly consider studies carried out since the end of the sixties and concerned with the best developed case of neutral monatomic (structureless) non-dense gas flow in the absence of external forces. Purely mathematical problems are not considered. Preference is given to final or review studies appearing in the most accessible publications. In Section 1 we consider the problem of taking out-of-order terms in the Chapman-Enskog method into account and make a brief comparison with other perturbation methods. Section 2 is devoted to the expressions for the transport properties in the Burnett and super-Burnett approximations. The relations between the Burnett approximation and axiomatic continuum mechanics theory, the thermodynamics of irreversible processes, and nonequilibrium statistical mechanics are discussed in Section 3. New effects due to specific terms of the formulas for the Burnett transport properties as Kn → 0 are considered in Section 4. The basic features of formulations of the boundary value problems for the systems of gasdynamic equations considered are outlined in Section 5. Finally, we describe the results of applying these equations to particular problems: some exact results of the kinetic theory (Section 6), the propagation of sound waves and cylindrical Couette flow (Section 7), shock wave structure and hypersonic hydrodynamics (Section 8). Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–28, July–August, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-01244).     

Residual stress by blind-hole method with off-center hole

Application of the blind-hole drilling technique to measurement of residual stresses in areas too restricted for drilling guides is now possible by using a set of equations derived to account for off-center drilled holes. Prior technique required the hole to be drilled concentric with the gage center within ±0.001 in (0.0254 mm), thus requiring the use of a precision locating and drilling guide. The equations apply to commercially available 0-, 90- and 225-deg strain-gage rosettes, but can be easily modified or scaled for different arrays. Details of a hand-drilling procedure are presented, along with examples. Potential errors and data interpretation are discussed.     

A numerical comparison of two different approximations of the error in a meshless method

Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. In the Element Free Galerkin (EFG) method, it is obviously important that the error of approximation should be estimated, as it is in the Finite Element Method (FEM).In this paper we compare two different procedures to approximate the a posteriori error for the EFG method, both procedures are recovery based errors. The performance of the two different approximations of the error is illustrated by analysing different examples for 2-D potential and elasticity problems with known analytical solutions, using regular and irregular clouds of points. For irregular clouds of points, it is recommended to use smooth transition of nodes, thus creating areas of decreasing nodal densities.     

The normal flux method at the boundary for multidimensional finite volume approximations in CFD

This paper presents a general method for imposing boundary conditions in the context of hyperbolic systems of conservation laws. This method is particularly well suited for approximations in the framework of Finite Volume Methods in the sense that it computes directly the normal flux at the boundary. We generalize our approach to nonconservative hyperbolic systems and discuss both the characteristic and the noncharacteristic cases. We present several applications to models occurring in Computational Fluid Mechanics like the Euler equations for compressible inviscid fluids with real equation of state, shallow water equations, magnetohydrodynamics equations and two fluid models.     

Domain decomposition method with hybrid approximations applied to solve problems of elasticity

To solve two-dimensional boundary-value problems of elasticity, two iteration algorithms of the domain decomposition method are proposed: parallel Neumann–Neumann and sequential Dirichlet–Neumann. They are based on the hybrid boundary–finite-element approximations. The algorithms are proved to converge. The optimal parameters are selected using the minimum-residual and steepest-descent methods. Some plane problems of elasticity are solved as examples, and stationary and nonstationary iteration algorithms in these examples are analyzed for efficiency Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 18–29, November 2008.