The solution for axisymmetrical shells with abrupt curvature change (corrugated shells) by the finite element method

It has been noted in the present paper that the finite element method using linear elements for solving axisymmetrical shells cannot be applied to the analysis of axisymmetrical shells with abrupt curvature change, owing to the fact that the influence of the curvature upon the angular displacements has been neglected. The present paper provides a finite element method using linear elements in which the influence of curvature is considered and the angular displacements are treated as continuous parameters. This method has been applied to the calculation of corrugated shells of the type C, and compared with the experimental results obtained by Turner-Ford as well as with the analytical solution given by Prof. Chien Wei-zang. The comparisons have been proved that this theory is correct.     

A finite difference method for elastic wave scattering by a planar crack with contacting faces

This paper presents a finite difference time-domain technique for 2D problems of elastic wave scattering by cracks with interacting faces. The proposed technique introduces cracks into the finite difference model using a set of split computational nodes. The split-node pair is bound together when the crack is closed while the nodes move freely when open, thereby a unilateral contact condition is considered. The development of the open/close status is determined by solving the equation of motion so as to yield a non-negative crack opening displacement. To check validity of the proposed scheme, 1D and 2D scattering problems for which exact solutions are known are solved numerically. The 1D problem demonstrates accuracy and stability of the scheme in the presence of the crack-face interaction. The 2D problem, in which the crack-face interaction is not considered, shows that the proposed scheme can properly reproduce the stress singularity at the tip of the crack. Finally, scattered fields from cracks with interacting faces are investigated assuming a stick and a frictionless contact conditions. In particular, the directivity and higher-harmonics are investigated in conjunction with the pre-stress since those are the basic information required for a successful ultrasonic testing of closed cracks.     

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model

In this study, time-dependent fully discretized least-squares finite element model is developed for the transient response of Cosserat rod having inextensibility and unshearability constraints to simulate a surgical thread in space. Starting from the kinematics of the rod for large deformation, the linear and angular momentum equations along with constraint conditions for the sake of completeness are derived. Then, the α-family of time derivarive approximation is used to reduce the governing equations of motion to obtain a semi-discretized system of equations, which are then fully discretized using the least-squares approach to obtain the non-linear finite element equations. Newton׳s method is utilized to solve the non-linear finite element equations. Dynamic response due to impulse force and time-dependent follower force at the free end of the rod is presented as numerical examples.     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

Acoustic radiation from a submerged hollow FGM sphere

An exact study based on the linear theory of elasticity is presented for the steady-state sound radiation characteristics of an arbitrarily thick radially inhomogeneous elastic isotropic hollow sphere, immersed in and filled with ideal compressible fluids, and subjected to an arbitrary axisymmetric time-harmonic driving force at its internal surface. A modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their spherical harmonics by means of the laminated approximation approach. Taylor’s expansion theorem is subsequently employed to solve the modal state equation, ultimately calculating a global transfer matrix. Numerical results are presented for a water-submerged/air-filled steel/zirconia FGM hollow sphere under an axisymmetric distributed internal pressure force. The effects of shell wall thickness, the material compositional gradient, frequency, and subtended polar angle of the internal pressure force on the far-field radiated pressure directivity patterns as well as the total radiated power are examined. It is demonstrated that the material gradient can significantly change the acoustical characteristics of hollow inhomogeneous sphere, especially for thick shells at high excitation frequencies. Limiting cases are considered and good agreements with available results as well as with the computations made by using a finite element package are obtained.     

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper.A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given.Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass.The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.     

Shock-Wave–Vortex Interactions: Shock and Vortex Deformations, and Sound Production

In this paper we study the interaction of a shock wave with a cylindrical vortex. The objective of the study is to characterize the shock and vortex deformations and the mechanism of sound generation. The approach relies on the solution of the two-dimensional Euler equations by means of a high order finite volume weighted-ENO scheme. In order to provide some guidance into the analysis we have also developed an acoustic analogy of the problem by formulating a wave equation for the pressure disturbance that is solved analytically by means of Green's functions. A systematic study has been conducted by investigating the effects of vortex intensity and shock strength. Specifically, we have determined the dependence of shock distortion and vortex compression, and its subsequent nutation upon shock and vortex strengths. The acoustic field generated through shock--vortex interactions has been found to evolve in three stages and to exhibit a three sound quadrupolar directivity. In the early stages of the interaction the sound generated due to shock distortion shows a dipolar character, which then changes to a quadrupolar one due to a restoring mechanism that acts during the interaction of the shock with the rear part of the vortex. In the third stage secondary sounds are formed, which also show a quadrupolar directivity. The acoustic analogy applied to weak shock--vortex interactions also confirms the numerically predicted sound generation mechanism. Received 12 July 1999 and accepted 18 November 1999     

Dynamical equations of various types for finite deformable polar elastic media

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Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

Axisymmetric motion of a viscous fluid in the finite annular cavity of a rotating cylindrical vessel

The method of nets is used to investigate unsteady axisymmetric viscous flow in a cylindrical gap of finite height. This situation is characterized by vortex motion in a plane passing through the axis of the coaxial cylinders. These flows have previously been studied in relation to the case of stepwise variation of the angular velocity of the cylinders [1]. In the present case the angular velocity is varied linearly in the acceleration stage and the acceleration interval is a parameter of the problem. After acceleration the rotation rate is determined from the ordinary differential equation describing the process of deceleration of the system as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–42, March–April, 1988.The authors are grateful to E. M. Zhukhovitskii for his interest and valuable comments.     

复合材料尖劈和接头端部奇性场的反平面问题研究

提出了一个基于位移的分析尖劈端部奇性位移场和应力场反平面问题的非协调元特征法．该方法与过去原有求解裂纹尖端近似场的有限元特征法有几点不同：(1)导出虚功原理的出发点为二维扇区的散度原理;(2)有限元的单元形式为非协调元;(3)尖劈端部邻域内的位移场假定没有采用奇异变换技术．运用该方法给出了求解正交各向异性复合材料尖劈端部附近奇性应力指数、奇性位移和应力角分布函数的算例．计算结果表明,该方法较原来的有限元特征法所用的单元少而且精度高．     

一种新的计算各向异性材料裂纹尖端应力强度因子杂交元法

利用有限元特征分析法研究了平面各向异性材料裂纹端部的奇性应力指数以及应力场和位移场的角分布函数,以此构造了一个新的裂纹尖端单元。文中利用该单元建立了研究裂纹尖端奇性场的杂交应力模型,并结合Hellinger-Reissner变分原理导出应力杂交元方程,建立了求解平面各向异性材料裂纹尖端问题的杂交元计算模型。与四节点单元相结合,由此提出了一种新的求解应力强度因子的杂交元法。最后给出了在平面应力和平面应变下求解裂纹尖端奇性场的算例。算例表明,本文所述方法不仅精度高,而且适应性强。     

Nonlinear Dynamics of Shells: Theory,Finite Element Formulation,and Integration Schemes

The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.     

基于新型裂尖杂交元的压电材料断裂力学研究

提出了一种裂尖邻域杂交元模型，将其与标准杂交应力元结合来求解压电材料裂纹尖 端的奇性电弹场和断裂参数的数值解．裂纹尖端杂交元的建立步骤为：1) 利用高次内插有限元特征法求解特征问题，得到反映裂尖奇异性电弹场状况的特 征值和特征角分布函数；2) 利用广义Hellinger-Reissner变分泛函以及特征问题的解来建立裂尖邻域杂交元模型．该 方法求解电弹场时，摒弃了传统有限元方法中裂尖奇异性场需要借助解析解的做法，也避免 了单纯有限元方法中需要在裂尖端部进行高密度单元划分．采用PZT5板中心裂纹问题 作为考核例，数值结果显示了良好的精确性．作为进一步应用，求解了含中心界面裂纹 的PZT4-PZT5两相压电材料的应力强度因子和电位移强度因子．所有的算例都考虑 了3种裂纹面电边界条件．     

小口径轴对称收缩喷嘴射流冲击大平板噪声指向特性研究

实验研究了3毫米口径轴对称收缩喷嘴在各种压比下射流垂直冲击和倾斜冲击坚固大平板产生的噪声的指向特性。发现噪声在过平板法线和喷嘴轴线的平面内呈近似四瓣分布，当喷嘴与平板距离减小时，指向壁射流下游的瓣得到增强，反之，指向喷嘴上游的瓣得到增强。喷嘴压比增加时，指向壁射流下游的瓣得到增强，反之，指向喷嘴上游的瓣得到增强。根据自由射流噪声的基本指向特性、射流冲击噪声基本指向特性、声波在平板处发生镜面反射和声波能量叠加的设定，建立了一个冲击射流总体噪声指向特性的模型，成功解释了实验结果，并揭示了形成冲击射流总体噪声指向特性的内在机理。     

Wedge indentation into elastic-plastic single crystals. 2: Simulations for face-centered cubic crystals

The asymptotic stress and deformation fields associated with the contact point singularity of a nearly-flat wedge indenter impinging on a specially-oriented single face-centered cubic crystal are derived analytically in a companion paper. In order to investigate the extent of the asymptotic fields, the indentation process is simulated numerically using single crystal plasticity. The quasistatically translating asymptotic fields consist of four angular elastic sectors separated by plastically deforming sector boundaries, as predicted in the companion paper. The asymptotic stress distributions are in accord with the analytical predictions. In addition, simulations are performed for a wedge indenter with a 90° included angle in order to investigate the consequences of finite deformation and finite lattice rotation. Several salient features of the deformation field for the nearly-flat indenter persist in the deformation field for the 90° wedge indenter. The existence of the salient features is validated by comparison to experimental measurements of the lower bound on geometrically necessary dislocation (GND) densities.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

Unsteady flow of a second-order fluid between concentric cylinders

The unsteady motion of an incompressible second-order fluid contained between two finite coaxial cylinders is examined when the outer cylinder is held fixed while the inner cylinder is constrained to execute an arbitrary angular velocity. A solution is obtained in closed form with the aid of transforms and an expression is obtained for the couple experienced. The particular case of a periodic angular velocity is then examined and some numerical work done. There is a marked difference between the results obtained and their classical counterparts.     

Direct measurement of the vorticity field in digital particle images

A new method called direct measurement of vorticity (DMV) for digital particle images is described in this paper. Unlike previous methods for calculating the vorticity in particle image velocimetry (PIV), the vorticity is determined directly from the average angular displacement of rotation between two matched patterns. In order to improve the stability and precision of the angular displacement, polar coordinates are used instead of Cartesian coordinates to depict gray level patterns. The results of a Monte Carlo simulation of an Oseen-vortex flow indicate that the accuracy of the DMV method is independent of the spatial resolution of the velocity sampling, and the errors in the velocity field will not be propagated into the vorticity field as occurs with some finite difference methods. Therefore, the DMV method is a good method to be used to extract the vorticity field from velocity data that has higher levels of uncertainty. Received: 26 November 1999/Accepted: 30 October 2000