Exact solution of the problem for dynamic field of displacements in rods of variable length

Initial boundary value problems are considered for rods that change length over time. These problems are transformed to problems about rods with mobile ends. A special method which allows one to obtain exact solutions of such problems is developed. This method is a generalization of a method of reflections for rods and strings of constant length. As an application of this class of problems, the hoisting of ropes in mining lifts, including their problems, are considered. Exact expressions for displacements in rods of variable length are obtained. The same results can be applied to cross oscillations of strings of variable length.     

Determining the natural frequencies of highly inhomogeneous shells of revolution with transverse strain

A method of studying the natural vibrations of highly inhomogeneous shells of revolution is developed. The method is based on a nonclassical theory of shells that allows for transverse shear and reduction. By separating variables, the two-dimensional problem is reduced to a sequence of one-dimensional eigenvalue problems. The inverse iteration method is used to reduce these problems to a sequence of inhomogeneous boundary-value problems solved by the orthogonal sweep method. The capabilities of the method are illustrated by solving certain representative problems and comparing their solutions with those obtained using the three-dimensional theory of elasticity, the classical theory of shells, and the refined Timoshenko model __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 38–47, September 2007.     

Mathematical Modeling of Inverse Problems of Forming Taking into Account the Incomplete Reversibility of Creep Strain

Functionals of direct and inverse problems of forming structural components are constructed taking into account the theory of incomplete reversibility of deformations. Formulations of these problems are given, and the uniqueness of their solutions is proved. An iterative method for solving inverse problems of forming structural components is proposed. Numerical solutions of these problems are obtained using a finite-element method.     

Homotopy solution of the inverse generalized eigenvalue problems in structural dynamics

The structural dynamics problems, such as structural design, parameter identification and model correction, are considered as a kind of the inverse generalized eigenvalue problems mathematically. The inverse eigenvalue problems are nonlinear. In general, they could be transformed into nonlinear equations to solve. The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations. This method had no requirements for initial value essentially because of the homotopy path to solution. Numerical examples were presented to illustrate the homotopy method.     

Study of half-space elastic contact problems by caustics

The optical method of caustics has been successfully applied to several two dimensional problems of elasticity. Up to now, no complicated three dimensional problems of elasticity have ever been treated by this method. In this paper, the experimental technique of caustics is developed, the caustics are obtained by annealing the stress-frozen epoxy slices. In applying this technique to Boussinesq's problem of a normal force and Cerruti's problem of a tangential force on the plane surface of a half-space, the experimentally obtained caustics for these problems are seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of the rather complicated three dimensional elasticity problems, including crack problems, by the author's method is also possible.     

The solution of elasticity problems for the half-space by the method of Green and Collins

The paper reviews the method of complex potential functions developed by Green and Collins as applied to axisymmetric mixed boundary value problems in elasticity for the half-space. It is shown how the method can be applied to problems in several coupled potential functions such as adhesive and frictional contact problems, to problems involving annular regions and to problems in thermoelasticity. Attention is given to the question of choosing a formulation which leads to a well-behaved numerical solution. Tables are given of the most commonly needed inversion formulae and of expressions for total load and stress intensity factor.     

弱不连续问题高阶有限元离散系统的GAMG法

弱不连续问题(如含夹杂问题)是固体力学计算中的一类重要问题。高阶有限元方法由于其具有更好的逼近效果,是确保数值解在界面保持较高精度的计算方法之一。但与线性元相比,高阶单元需要更多的计算机存储单元,具有更高的计算复杂性。本文利用两水平算法的思想,将高阶有限元离散系统化归于线性元离散系统的求解,为弱不连续问题高阶有限元离散系统设计了一种新的基于几何与分析信息的代数多重网格(GAMG)法,并应用于圆形求解域含单夹杂问题的高阶有限元离散系统的求解。数值试验结果表明,相比于常用GAMG法,新方法的迭代次数基本不依赖于问题规模、单元阶次以及杨氏模量的间断性,CPU计算时间得到明显改善,具有更好的计算效率和鲁棒性,可大大提高弱不连续问题有限元分析的整体效率。     

A new approach to inverse problems of wave equations

We introduce a multi-cost-functional method for solving inverse problems of waveequations.This method has its simplicity,efficiency and good physical interpretation.It hasthe advantage of being programmed for two-or three-(space)dimensional problems as wellas for one-dimensional problems.     

无网格方法的研究进展与展望

目前正在发展的无网格方法采用基于点的近似,可以彻底或部分地消除网格,因此在处理不连续和大变形问题时可以完全抛开网格重构。无网格方法是目前科学和工程计算方法研究的热点,也是科学和工程计算发展的趋势。本文首先简单地阐述了无网格方法,然后详细叙述了目前提出的各种无网格方法的研究进展,最后对目前无网格方法存在的问题进行了探讨,提出了今后的研究方向。     

A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes

Recently, a discontinuous Galerkin method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution of the Helmholtz equation in the mid-frequency regime. This method was fully developed however only for regular meshes, and demonstrated only for interior Helmholtz problems. In this paper, we extend it to irregular meshes and exterior Helmholtz problems in order to expand its scope to practical acoustic scattering problems. We report preliminary results for two-dimensional short wave problems that highlight the superior performance of this discontinuous Galerkin method over the standard finite element method.     

Asymptotic analysis of perforated plates and membranes. Part 1: Static problems for small holes

Static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi-scale homogenization method. Using the asymptotic homogenization method the original boundary-value problem is reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. The combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems. Taking into the account small size of holes the method of perturbation of the shape of the boundary and the Schwarz alternating method are used. The problems of torsion of a rod with perforated cross-section; deflection of the perforated membrane loaded by a normal load; and bending of perforated plates with circular and square holes are considered consecutively. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated.     

On the application of pseudospectral FFT techniques to non-periodic problems

The reduction-to-periodicity method using the pseudospectral fast Fourier transform (FFT) technique is applied to the solution of non-periodic problems, including the two-dimensional incompressible Navier–Stokes equations. The accuracy of the method is explored by calculating the derivatives of given functions, one- and two-dimensional convective-diffusive problems, and by comparing the relative errors due to the FFT method with a second-order finite difference (FD) method. Finally, the two-dimensional Navier–Stokes equations are solved by a fractional step procedure using both the FFT and the FD methods for the driven cavity flow and the backward-facing step problems. Comparisons of these solutions provide a realistic assessment of the FFT method.     

Block element method for solving integrated equations of contact problems in wedge-shaped domains

This paper describes the block element method for spatial integral equations with a difference kernel in the boundary-value problems of continuum mechanics and mathematical physics. The basis of the proposed method is the Wiener — Hopf method, whose generalization for a spatial case is called integral factorization method. The block element method is applied to solve problems in domains with piecewise smooth boundaries containing corner points. The developed method was used to solve the contact problem for a wedge-shaped stamp occupying the first quadrant. This paper describes in detail the methods of obtaining various characteristics of the solution constructed by reversing the system of one-dimensional linear integral equations typical for dynamics and static contact problems for stamps in the form of a strip.     

Boundary element calculation of shallow water waves

A boundary element method is proposed for studying periodic shallow water problems. The numerical model is based on the shallow water equation. The key feature of this method is that the boundary integral equations are derived using the weighted residual method and the fundamental solutions for shallow water wave problems are obtained by solving the simultaneous singular equations. The accuracy of this method is studied for the wave reflection problem in a rectangular tank. As a result of this test, it has been shown that the number of element divisions and the distribution of nodes are significant to the accuracy. For numerical examples of external problems, the wave diffraction problems due to single cylindrical, double cylindrical and plate obstructions are analysed and compared with the exact and other numerical solutions. Relatively accurate solutions are obtained.     

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

A collocated discrete least squares meshless method for the solution of the transient and steady‐state hyperbolic problems is presented in this paper. The method is based on minimizing the sum of the squared residuals of the governing differential equation at some points chosen in the problem domain as collocation points. The collocation points are generally different from nodal points, which are used to discretize the problem domain. A moving least squares method is employed to construct the shape functions at nodal points. The coefficient matrix is symmetric and positive definite even for non‐symmetric hyperbolic differential equations and can be solved efficiently with iterative methods. The proposed method is a truly meshless method and does not require numerical integration. Advantages of the collocation points are shown to be threefold: First, the collocation points are shown to be responsible for stabilizing the method in particular when problems with shocked solution are attempted. Second, the collocation points are also shown to improve the accuracy of the solution even for problems with smooth solutions. Third, the collocation points are shown to contribute to the efficiency of the method when solving steady‐state problems via faster convergence of the resulting algorithm. The ability of the method and in particular the effect of collocation points are tested against a series of one‐dimensional transient and steady‐state benchmark examples from the literature and the results are presented. A sensitivity analysis is also carried out to investigate the effect of the base polynomials on the accuracy and convergence characteristics of the method in solving steady‐state problems. The results show the ability of the proposed method to accurately solve difficult hyperbolic problems considered. The method is also shown to be particularly stable for problems with shocked solution due to the inherent stabilizing mechanism of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic analysis of perforated plates and membranes. Part 2: Static and dynamic problems for large holes

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained.     

带源参数的二维热传导反问题的无网格方法

利用无网格有限点法求解带源参数的二维热传导反问题，推导了相应的离散方程. 与 其它基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不 需要划分网格，用配点法离散控制方程，可以直接施加边界条件，不需要在区域内部求积分. 用有限点法求解二维热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点. 最后通过算例验证了该方法的有效性.     

Construction of Basis Functions and Their Application to Boundary-Value Problems of Mechanics of Continuous Media

A unified method for constructing basis (eigen) functions is proposed to solve problems of mechanics of continuous media, problems of cubature and quadrature, and problems of approximation of hypersurfaces. Numerical-analytical methods are described, which allow obtaining approximate solutions of internal and external boundary-value problems of mechanics of continuous media of a certain class (both linear and nonlinear). The method is based on decomposition of the sought solutions of the considered partial differential equations into series in basis functions. An algorithm is presented for linearization of partial differential equations and reduction of nonlinear boundary-value problems, which are reduced to systems of linear algebraic equations with respect to unknown coefficients without using traditional methods of linearization.     

土性参数各向异性随机场的表征与建模方法

土性参数在空间上的相关性具有各向异性,因此对各向异性随机场表征与建模方法的研究具有重要的意义.本文首先通过对相关函数的分析,将各向异性相关的问题归结到相关距离函数的探讨上,给出了一种描述参数各向异性相关的方法;其次分析了 目前常用的两种随机场反演方法在处理各向异性问题所面临的问题,(1)局部平均划分法只适合横观各向同性随机场的生成,难以处理任意的各向异性问题,(2)矩阵分解法反演随机场时,随机场网格数会受到计算机计算能力的限制,难以处理大型的随机场反演问题.针对以上两点问题,基于相关函数的快速衰减特性改进了矩阵分解法,并将改进的矩阵分解法与局部平均划分法进行了反演精度对比,同时用改进的矩阵分解法反演了旋转横观各向同性随机场,结果表明用改进的矩阵分解法处理各向异性问题是适用的.     

边界元特征值分析及Burton-Miller法探究BEM eigenvalue analysis and the investigation of the Burton-Miller method

运用围道积分方法将边界元非线性特征值问题转化为规模很小的广义特征值问题,从而构造出一种边界元特征值分析方法。数值算例验证了该方法的求解精度。针对外声场问题,通过对常规、法向导数和Burton‐Miller边界积分方程的虚假特征频率的计算和比较,揭示了Burton‐Miller法规避虚假特征频率的本质,并对其中的叠加常数的最优取值给出了一种新的解释。