Numerical-perturbation analysis of edge effect in bending laminated plate

In this paper, we consider a bending laminated plate. At first, the dimensionless variables are used to transform the equilibrium equations of any layer to perturbation differential equations. Secondly, the composite expansion is used and the solution domain is divided into interior and boundary layer regions and the mathematical models for the outer solution and the inner solution are yielded respectively. Then, the inner solution is expressed with the boundary intergral equation.Project Supported by the National Science Foundation of China.     

On one model of a transversally isotropic elastic foundation

The method of expansion of the stress and displacement components into series in terms of functions of the transverse coordinate and the method of variation with respect to the state being determined are used to construct an iterative model of a transversally isotropic elastic foundation. This model allows for all the SSS components. The order of the equations does not depend on the number of terms of the expansions, which approximate the displacements and stresses. As an example, the problem on a concentrated force applied to an elastic foundation is considered. Pridneprovskaya State Academy of Building and Architecture, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 130–134, June, 2000.     

中厚板统计分析的随机边界元法

本文建立了分析含随机材料参数并具厚度不均匀性的中厚板问题的随机边界元法,基于Taylor级数展开技术,分析和到广义位移的均值和一阶偏差的积分方程,其中将材料参数的随机性和厚度的不均匀性作为等效荷载处理,从而得到广义边界位移或面力的均值和协方差,并进一步求出部点广义位移和内力的均值和协方差,最后用本文方法计算了两个数例,并对所得结果进行了分析,探讨。     

3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loadings

Summary  The problem of a piezoceramic hollow sphere is investigated analytically based on the 3D equations of piezoelasticity. The functionally graded property of the material along the radial direction can be taken arbitrarily in the paper. Displacement and stress functions are introduced, and two independent state equations with variable coefficients are derived. By employing the laminate model, the two state equations are transformed into ones with constant variables from which the state variable solution is easily obtained. Two linear relationships between the state variables at the inner and outer spherical surfaces are established. Numerical calculations are performed for different boundary conditions imposed on the spherical surfaces. Received 28 February 2001; accepted for publication 26 June 2001     

Thermally induced dynamic instability of laminated composite conical shells

Thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method. The laminated composite conical shells are subjected to static and periodic thermal loads. The linear instability approach is adopted in the present study. A set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs. The formulation begins with three-dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible. Using the method of differential quadrature and Bolotin's method, and imposing the orthonormality and solvability conditions on the present asymptotic formulation, we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner. The principal instability regions of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.     

Solution of Dynamic Problems for Thin-Walled Structural Elements with Nonuniform Dynamic Boundary Conditions

The characteristics of the stress–strain state of thin-walled structural elements are determined in the case where dynamic boundary loads or displacements described by pulse functions are specified. A general scheme for realization of the method of natural-mode expansion is stated as applied to differential equations with unknown functions of one spatial coordinate and time. Theoretical relations for rods, plates, and shells are given. The potential of the approach developed is illustrated by solving specific problems     

薄板统计分析的随机边界元法

本文建立了分析含随机材料参数的薄板弯曲问题的随机边界元法。基于Ｔａｙｌｏｒ级数展开技术，分别得到了位移的均值和一阶偏差的边界积分方程，发现材料参数的随机性可作为一个等效的随机荷载处理，从而得到边界位移或边界力的均值和协方差，并进一步求出内点位移和力矩的均值和协方差，最后用本文方法计算了两个算例，并对结果进行了必要的分析。     

Stressed state of a composite cylindrical shell with an elliptical hole

The results of investigations of the stress-strained state (SSS) of isotropic cylindrical shells with an elliptical hole are represented in monograph [4]. The modified method of expansion in terms of minor parameter [3] is suggested for calculation of orthotropic shells. The method does not consider, however, lateral shear strains introducing a significant contribution in SSS of composite shells. The procedure for solving problems of SSS calculation near curvilinear holes in shells of arbitrary shape with variable geometrical and physical characteristics is suggested in [1] on the basis of variational-difference method (VDM). Here the relations of the linear theory of anisotropic inhomogeneous shells and the hypothesis of a straight line are taken as the initial ones for all the packet of laminated composite shell as a whole. In the present work we present the numerical results obtained according to the procedure given in [1] for an orthotropic cylindrical shell with an elliptical hole loaded by the axial force and complaint for the lateral shear.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 11, pp. 57–62, November, 1993.     

A boundary element formulation using the simple method

A new boundary element method is described for calculation of the steady incompressible laminar flows. The method is based on the well-known SIMPLE algorithm. The new boundary element method allows one to find the fields of the pressure and velocity corrections without inner iterations, thus reducing the computational time drastically. This makes it different from the method developed by Patankar and Spalding.³² However, the new method demands a much larger computer strorage. The boundary integral equations are discretized with the help of constant boundary elements and constant cells. The values of the integrals along the boundary elements and the cells for the two-dimensional domain are found analytically. To preserve the stability in the iteration process, under-relaxation for the convection terms is used. This paper gives the results of calculations of the flows between two plane parallel plates at Re = 20 and Re = 200, the flows in a square cavity with a moving upper lid at Re = 1 and Re = 100 and the flow in a plane channel with sudden symmetric expansion at Re =46·6.     

Stochastic boundary element method in elasticity

The stochastic boundary element method is developed to analyze elasticity problems with random material and/or geometrical parameters and randomly perturbed boundaries. Based on the first-order Taylor series expansion, the boundary integration equations concerning the mean and deviation of the displacements are derived, respectively. It is found that the randomness of material parameters is equivalent to a random body force, so the mean and covariance matrices of unknown boundary displacements and tractions can be obtained. Furthermore, the mean and covariance of displacements and stresses at inner points can also be obtained. Numerical examples show that the proposed stochastic boundary element method gives satisfactory solutions, as compared with those obtained by theoretical analysis or other numerical methods. The project supported by the National Natural Science Foundation of China and the State Education Commission Foundation of China     

High-order theory for arched structures and its application for the study of the electrostatically actuated MEMS devices

A high-order theory for arched rods and beams based on expansion of the two-dimensional (2D) equations of elasticity into Legendre’s polynomials series has been developed. The 2D equations of elasticity have been expanded into Legendre’s polynomials series in terms of a thickness coordinate. Thereby, all equations of elasticity including Hooke’s law have been transformed to corresponding equations for coefficients of Legendre’s polynomials expansion. Then system of differential equations in term of displacements and boundary conditions for the coefficients of Legendre’s polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in details. For obtained boundary-value problems, a finite element method has been used and numerical calculations have been done with COMSOL Multiphysics and MATLAB. Developed theory has been applied for study pull-in instability and stress–strain state of the electrostatically actuated micro-electro-mechanical Systems.     

Dynamic system for reference signal transmission and reconstruction in distributed MIMO radars

The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton–Maclaurin expansion. Several basic theorems on the renormalization method are proven. Some interesting applications are given, including asymptotic solutions of quantum anharmonic oscillator and discrete boundary layer, the reductions and invariant manifolds of some discrete dynamics systems. Furthermore, the homotopy renormalization method based on the Newton–Maclaurin expansion is proposed and applied to those difference equations including no a small parameter. In addition, some subtle problems on the renormalization method are discussed.     

Conjugate free convection over a vertical slender hollow cylinder embedded in a porous medium

A numerical study of the steady conjugate free convection over a vertical slender, hollow circular cylinder with the inner surface at a constant temperature and embedded in a porous medium is reported. The governing boundary layer equations for the fluid-saturated porous medium over the cylinder along with the one-dimensional heat conduction equation for the cylinder are cast into dimensionless form, by using a non-similarity transformation. The resulting non-similarity equations with their corresponding boundary conditions are solved by using the Keller box method. Emphasis is placed on the effects caused by the wall conduction parameter, p, and calculations have covered a wide range of this parameter. Heat transfer results including the temperature profiles, the interface temperature profiles and the local Nusselt number are presented. Received on 17 November 1997     

On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported

The symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. The basic equations for the plates are first transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained. Since only the basic elasticity equations of the plates are used, this method eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

港口非线性波浪耦合计算模型研究

建立了外域用差分法求解高阶Boussinesq方程、内域用边界元法求解Laplace方程的二维船 非线性波浪力时域计算的耦合模型. 研究了该类耦合模型的匹配条件、耦合求解过程和内域、 外域公共区域长度的确定. 该耦合模型计算结果与只用边界元求解Laplace方程模型的计算 结果和实验结果对比表明，该耦合模型不仅计算精度高，而且计算效率快，适用于研究较大 区域内波浪对物体的非线性作用.     

A combined perturbation/finite-difference procedure applied to temperature effects and stability in a laminar boundary layer

Summary By means of a combined method it is demonstrated for regular perturbation problems how the higher order terms of an asymptotic expansion may be determined from numerical solutions of the non-expanded basic equations.The method is applied to heat transfer effects in a laminar boundary layer and to the analysis of its stability. All first- and second-order coefficients of the problem are determined from numerical solutions of the basic set of equations.     

A class of two-dimensional dual integral equations and its application

Because exact analytic solution is not available,we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics.The integral equations by this procedure are reduced to infinite algebraic equations.The accuracy of the solution lies in the boundary collocation technique.The application of which for some complicated initial- boundary value problems in solid mechanics indicates the method is powerful.     

Development of circular function‐based gas‐kinetic scheme (CGKS) on moving grids for unsteady flows through oscillating cascades

In this paper, the circular function‐based gas‐kinetic scheme (CGKS), which was originally developed for simulation of flows on stationary grids, is extended to solve moving boundary problems on moving grids. Particularly, the unsteady flows through oscillating cascades are our major interests. The main idea of the CGKS is to discretize the macroscopic equations by the finite volume method while the fluxes at the cell interface are evaluated by locally reconstructing the solution of the continuous Boltzmann Bhatnagar–Gross–Krook equation. The present solver is based on the fact that the modified Boltzmann equation, which is expressed in a moving frame of reference, can recover the corresponding macroscopic equations with Chapman–Enskog expansion analysis. Different from the original Maxwellian function‐based gas‐kinetic scheme, in improving the computational efficiency, a simple circular function is used to describe the equilibrium state of distribution function. Considering that the concerned cascade oscillating problems belong to cases that the motion of surface boundary is known a priori, the dynamic mesh method is suitable and is adopted in the present work. In achieving the mesh deformation with high quality and efficiency, a hybrid dynamic mesh method named radial basic functions‐transfinite interpolation is presented and applied for cascade geometries. For validation, several numerical test cases involving a wide range are investigated. Numerical results show that the developed CGKS on moving grids is well applied for cascade oscillating flows. And for some cases where nonlinear effects are strong, the solution accuracy could be effectively improved by using the present method. Copyright © 2017 John Wiley & Sons, Ltd.     

On the energy transfer equation for weakly interacting waves

The derivation of the transfer equation based on analysis of the equations for spectral semi-invariant and not invoking equations for realization of the random wave field is presented. Uniformly valid asymptotic expansions for the third and the fourth spectral semi-invariant are constructed using the multiple scale method and the matched asymptotic expansion method. This approach makes it possible to investigate the boundary layer in a neighbourhood of the resonant surface where intensive growth in time of the third spectral semi-invariant occurs. This boundary layer defines the form of the transfer equations. An analogous boundary layer for the fourth spectral semiinvariant and its influence on the second and the third spectral semi-invariants are also investigated.