The Timoshenko beam model of the differential quadrature element method

A new numerical approach for solving Timoshenko beam problems is proposed. The approach uses the differential quadrature method (DQM) to discretize the Timoshenko beam equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of Timoshenko beam structures. The resulting overall discrete equation can be solved by using a solver of the linear algebra. Numerical results of the DQEM Timoshenko beam model are presented. They demonstrate the DQEM numerical method.     

Differential quadrature element method for static analysis of shear deformable cross‐ply laminates

In this paper, the two‐dimensional differential quadrature element method (DQEM) is developed for the static analysis of symmetric cross‐ply laminates using the first‐order shear deformation plate theory. In this study, the laminated plate, which may contain different discontinuities in loading, geometry, material, and boundary conditions, is first divided into several simple plate elements and then the differential quadrature method (DQM) is applied to each simple element. Compatibility conditions are derived to connect the plate elements so that the overall matrix equation system for the whole plate is obtained and solved. The reliability of the DQEM for solving the titled problems is examined carefully through convergence and accuracy studies and finally some numerical test examples are given to demonstrate the applicability and flexibility of this method for practical use. The methodology presented here has overcome some critical drawbacks of the global DQM but is different from the Quadrature Element Method (QEM) since only one grid point is employed to represent the interface point. Copyright © 1999 John Wiley & Sons, Ltd.     

DQEM for free vibration analysis of Timoshenko beams on elastic foundations

 A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions. Received: 11 October 2002/Accepted: 26 November 2002     

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

Free and forced vibrations of plates by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media

Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.     

Dynamics of charged and conducting drops via the hybrid finite-boundary element method

A numerical study on the dynamic behaviour of a charged and conducting drop, with net electrical charge , is presented here, that is valid for arbitrary initial disturbances. It employs the integral form of Laplace's equation for the calculation of the velocity and electrostatic potentials, which only requires discretization and solution on the surface of the drop. Thus a hybrid method results with the integral equations solved via the boundary element technique, while the Galerkin finite element formulation is used for the kinematic and dynamic condition at the interface as well as for the net charge conservation equation. Recently, the authors followed this approach in their study on the free nonlinear oscillations of inviscid drops, and they were able to optimize time and space discretization as well as the treatment of the integral equation with excellent results.     

求解圆柱壳稳态谐响应的微分求积单元法

本文以Donnell经典壳体振动微分方程为基础,研究微分求积单元法(DQEM)在圆柱壳稳态谐响应计算中的应用。研究结果表明：微分求积单元法可较为方便的处理多种边界条件;与有限元法相比,微分求积单元法直接面向问题的微分方程,可用较少的节点获得较高的计算精度,计算效率较高。本文结果可为微分求积单元法在结构动力响应问题求解中的应用提供参考。     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Three-dimensional vibration analysis of multilayered piezoelectric composite plates

We investigate the free vibration of multilayered piezoelectric composite plates. A brief review of the theoretical developments in piezoelasticity that are relevant to smart composite structures is presented. An analysis is performed using the differential quadrature (DQ) technique to solve three-dimensional equations of piezoelasticity, and technical issues are thoroughly discussed. Solutions for piezoelastic laminated plates are made possible with the development and implementation of the DQ layerwise modeling technique, which allows different boundary conditions to be imposed at the edges of the plates. The numerical results of different plate problems are presented, and the effects of the piezoelasticity of these problems are investigated. The DQ model predictions are validated against existing results, with good agreement found.     

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.     

Boundary integral equation supported differential quadrature method to solve problems over general irregular geometries

Based on the interpolation technique with the aid of boundary integral equations, a new differential quadrature method has been developed (boundary integral equation supported differential quadrature method, BIE-DQM) to solve boundary value problems over generally irregular geometries. The quadrature rule of the BIE-DQM is that the first and the second derivatives of a function with respect to independent variables are approximated by a weighted linear combination of the function values at all discrete nodal points and the corresponding normal derivatives at all boundary points. Several numerical examples are considered to verify the feasibility and effectiveness of the proposed algorithm.     

DRBEM solution of exterior nonlinear wave problem using FDM and LSM time integrations

The nonlinear wave equation is solved numerically in an exterior region. For the discretization of the space derivatives dual reciprocity boundary element method (DRBEM) is applied using the fundamental solution of Laplace equation. The time derivative and the nonlinearity are treated as the nonhomogenity. The boundary integrals coming from the far boundary are eliminated using rational and exponential interpolation functions which have decay properties far away from the region of interest. The resulting system of ordinary differential equations in time are solved using finite difference method (FDM) with a relaxation parameter and least squares method (LSM). The proposed methods are examined with numerical test problems in which the behaviours of solutions are known. Although it gives almost the same accuracy with the DRBEM+FDM procedure, DRBEM+LSM solution procedure is preferred, since it is a direct method without the need of a parameter.     

3D multidomain BEM for a Poisson equation

This paper deals with the efficient 3D multidomain boundary element method (BEM) for solving a Poisson equation. The integral boundary equation is discretized using linear mixed boundary elements. Sparse system matrices similar to the finite element method are obtained, using a multidomain approach, also known as the ‘subdomain technique’. Interface boundary conditions between subdomains lead to an overdetermined system matrix, which is solved using a fast iterative linear least square solver. The accuracy, efficiency and robustness of the developed numerical algorithm are presented using cube and sphere geometry, where the comparison with the competitive BEM is performed. The efficiency is demonstrated using a mesh with over 200,000 hexahedral volume elements on a personal computer with 1 GB memory.     

模拟复杂流动的一种隐式直接力浸入边界方法

为避免复杂贴体网格的生成,该文采用一种隐式直接力浸入边界法模拟复杂边界流动问题。借助求解不可压缩N-S方程组的分步投影方法的思想,来求解基于浸入边界法的耦合系统方程。其中固体边界离散点的作用力密度通过强制满足固体边界的无滑移条件导出,进而通过δ光滑函数对固体壁面附近速度场进行二次修正。在空间离散上,对流项采用QUICK迎风格式,扩散项采用中心差分格式,采用二阶显式Adams-Bashforth法离散时间项。以雷诺数为25、40和300的圆柱绕流为基准数值算例,通过与实验结果和其他文献数值结果的对比,验证数值计算方法的可靠性。     

Some improvements of accuracy and efficiency in three dimensional direct boundary element method

Numerical analysis with the Boundary Element Method (BEM) has been used more and more in various engineering fields in recent years. In numerical techniques, however, there are some problems which have not been fully solved even now. The most essential one is the drop in the accuracy of results for internal points near the boundary of the structure, where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method. For the boundary integral equation of stress, this problem became more serious, and the accuracy can be improved only partly, even though very refined boundary elements are used. In this paper, the boundary integral equation is newly formulated using a relative quantity of displacement. In this way, the singularity of boundary integrals is reduced by the order of 1/r, and the accuracy of solution is improved significantly. Furthermore, in order to integrate it more accurately, two kinds of numerical integral methods are newly developed. By using these methods, both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty, although the discretization may be comparatively coarse. The generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems.     

地震作用下水-轴对称柱体相互作用的子结构分析方法

针对水-轴对称柱体动力相互作用问题,提出了一种地震作用下水-结构相互作用的时域子结构分析方法.基于三维不可压缩水体的波动方程和边界条件,利用分离变量法将其转换为环向解析、竖向和径向数值的二维模型;基于比例边界有限元推导了截断边界处无限域水体的动力刚度方程,并将水体内域有限元方程和人工边界处的动水压力进行耦合,从而得到结...