On the numerical solution of the turbulence energy equations for wave and tidal flows

This paper deals with the numerical solution, using finite difference methods, of the hydrodynamic and turbulence energy equations which describe wind wave and tidally induced flow. Calculations are performed using staggered and non-staggered finite difference grids in the vertical, with various time discretizations of the production and dissipation terms in the turbulence energy equations. It is shown that the time discretization of these terms can significantly influence the stability of the solution. The effect of time filtering on the numerical stability of the solution is also considered. The form of the mixing length is shown to significantly influence the bed stress in wind wave problems. A no-slip condition is applied at the sea bed, and the associated high-shear bottom boundary layer is resolved by transforming the equations onto a logarithmic or log-linear co-ordinate system before applying the finite difference scheme. A computationally economic method is developed which remains stable even when a very fine vertical grid (over 200 points) is used with a time step of up to 30 min.     

Bending problems of rectangular Reissner plate with free edges laid on tensionless Winkler foundations

In this paper,an analytical method for solving the bending problems of rectangularReissner plate with free edges under arbitrary loads laid on tensionless Winkler foundationsis proposed.By assuming proper form of Fourier series with supplementary terms,whichmeet derivable conditions,for deflection and shear force functions,the basic differentialequations with given boundary conditions can be transformed into a set of simple infinitealgebraic equations.For common Winkler foundations,this set of equations can be solveddirectly and for tensionless Winkler foundations,it is a set of weak nonlinear algebraicequations,the solution of which can be obtained easily by using iterative procedures.     

Meshfree weak–strong (MWS) form method and its application to incompressible flow problems

A meshfree weak–strong (MWS) form method has been proposed by the authors' group for linear solid mechanics problems based on a combined weak and strong form of governing equations. This paper formulates the MWS method for the incompressible Navier–Stokes equations that is non‐linear in nature. In this method, the meshfree collocation method based on strong form equations is applied to the interior nodes and the nodes on the essential boundaries; the local Petrov–Galerkin weak form is applied only to the nodes on the natural boundaries of the problem domain. The MWS method is then applied to simulate the steady problem of natural convection in an enclosed domain and the unsteady problem of viscous flow around a circular cylinder using both regular and irregular nodal distributions. The simulation results are validated by comparing with those of other numerical methods as well as experimental data. It is demonstrated that the MWS method has very good efficiency and accuracy for fluid flow problems. It works perfectly well for irregular nodes using only local quadrature cells for nodes on the natural boundary, which can be generated without any difficulty. Copyright © 2004 John Wiley & Sons, Ltd.     

Delay equations with fluctuating delay related to the regenerative chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.     

On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in Axisymmetric Cold Plasmas

For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.     

固体介质中SPH方法的拉伸不稳定性问题研究进展

光滑粒子流体动力学法(smoothed particle hydrodynamics, SPH)是一种基于核估计的无网格Lagrange数值方法.它用粒子方程离散流体动力学的连续方程, 既可以处理有限元难于处理的大变形和严重扭曲问题, 又可以处理有限差分法不易处理的自由边界和材料界面的问题, 在固体力学中的冲击、爆炸和裂纹模拟中具有广阔的发展前景.但是, 该算法的拉伸不稳定性(tensile instability)问题是它在固体力学领域中应用的最大障碍.对SPH稳定性分析表明, 算法不稳定性的条件仅与应力状态和核函数的2阶导数有关.目前, 应力点法(stress points)、Lagrange核函数法、人工应力法(artificialstress)、修正光滑粒子法(corrective smoothed particle method, CSPM)和守恒光滑法(conservativesmoothing)以及其他一些方法成功地改善了SPH的拉伸不稳定性, 但是每一种方法都不能彻底解决SPH的拉伸不稳定性问题.本文介绍了SPH法的方程和Von Neumann稳定性分析的思想, 以及国内外在这几个方面的研究成果及其最新进展, 同时指出目前研究中存在的问题和研究的方向.      

贮腔类三维自由液面动力学问题数值研究

讨论了贮腔类三维自由液面动力学问题的数值研究,将任意的拉格朗日-欧拉运动学描述关系引入到系统的控制方程中,采用任意的拉格朗日-欧拉描述跟踪自由液面,推导了自由面上结点的法向矢量计算公式。采用Galerkin余量法推导了Navier-Stokes方程的空间离散有限元方程,采用三维自由液面上微分几何理论推导了表面张力计算公式。数值研究中考虑了接触角效应,最后进行了三维数值算例分析。     

两个新型的八结点块体单元—三维离散算子差分法

给出了弹性力学三维问题的离散算子差分法 ,讨论离散算子差分法在三维问题中的特点 ,意在为该方法的进一步发展提供依据 ,为应用弱形式进行数值求解的研究提供参考。本文从弹性力学平衡方程更为一般的弱形式出发 ,给出了含边界参数的弱形式方程。由该方程不仅可以得到有限元法 ,还可得到离散算子差分法。给出了两个八结点块体单元 ,虽然单元中位移函数是非协调的 ,不需特殊处理便可保证离散格式收敛 ,并对单元位移有十分好的反映能力。     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

Application of higher-order flux-limited methods in compositional simulation

A higher-order flux-limited finite-difference scheme has been implemented into a compositional simulator to discretize the convection terms of the component conservation equations and the relative permeability terms of the phase fluxes. Harten's total variation diminishing criteria are imposed directly to the finite-difference equations and the bounds of the flux limiters which are suitable for larger Courant numbers and nonuniform grid systems are obtained. A time-correction technique is applied to increase the time accuracy and improve the stability condition.The scheme has been tested for miscible and immiscible flow problems in one and two dimensions, and the results were compared with those using a third-order method without flux limiting and some available analytical solutions.     

Hydroelastic stability of a rectangular plate interacting with a layer of ideal flowing fluid

The three-dimensional formulation of the problem on the natural vibrations and stability of an elastic plate which interacts with a quiescent or flowing fluid is represented and a finite element algorithm of its numerical implementation is proposed. The governing equations, which describe vortex-free ideal fluid dynamics in the case of small perturbations, are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The plate strains are determined on the basis of the Timoshenko theory. The variational principle of virtual displacements which takes into account the work done by inertial forces and the hydrodynamic pressure is used for the mathematical formulation of the dynamic problem of elastic structure. The solution of the problem is reduced to calculations and an analysis of complex eigenvalues of a coupled system of two equations. The effect of the fluid layer height on the eigenfrequencies and the critical velocities of the loss of stability is estimated numerically. It is shown that there exist different types of instability determined by combinations of the kinematic boundary conditions prescribed at the plate edges.     

Propagation of discontinuities against a static background

The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.     

Review: Formulating efficient finite-element codes for flows in regular domains

Many fluid flow problems of current interest occur in domains that are mappable to a rectangle or a box; conformal mappings are particularly useful in this regard. We are concerned here with the efficient solution of such problems using finite elements. The central issue is the element choice, and this issue is addressed in terms of operation counts, computer memory and I/0 requirements, and the extent to which code vectorization is possible. It is concluded that rectangular (box) elements generally lead to more efficient algorithms that triangular (tetrahedral) elements. A synthesis of algorithms, based on bilinear (trilinear) elements, is presented. The algorithms have the attributes of simplicity, accuracy, stability and straightforward incorporation of boundary conditions. For bilinear and trilinear elements, it is found that product and first-derivative terms are well-handled by the Galerkin FE method, but that it is advantageous to go outside of the Galerkin framework when treating second-derivative terms. It is particularly important to consider the form of the governing equations, vis-à-vis the choice of staggered, non-staggered and/or mixed-order elements, and to choose an appropriate time scheme. The described techniques have been successfully applied to a variety of problems in regular domains, including the solution of the three-dimensional time-dependent hydrostatic primitive equations; these are stiff and include first and second derivative terms, non-linearities and variable coefficients due to a conformal mapping.     

A quasi-velocity-based nonlinear controller for rigid manipulators

The problem of position control in the operational space of a robot manipulator is addressed in the paper. The proposed controller is based on equations of motion expressed in terms of normalized generalized velocity components (NGVC) which result from decomposition of the manipulator inertia matrix. The sufficient conditions for global exponential stability of the system under the controller are given. It is shown that using the controller an further insight into the system dynamics is possible. The proposed control algorithm is tested via simulation on a spatial manipulator with three degrees of freedom.     

Theoretical treatment of fluid flow for accelerating bodies

Most computational fluid dynamics simulations are, at present, performed in a body-fixed frame, for aeronautical purposes. With the advent of sharp manoeuvre, which may lead to transient effects originating in the acceleration of the centre of mass, there is a need to have a consistent formulation of the Navier–Stokes equations in an arbitrarily moving frame. These expressions should be in a form that allows terms to be transformed between non-inertial and inertial frames and includes gravity, viscous terms, and linear and angular acceleration. Since no effects of body acceleration appear in the inertial frame Navier–Stokes equations themselves, but only in their boundary conditions, it is useful to investigate acceleration source terms in the non-inertial frame. In this paper, a derivation of the energy equation is provided in addition to the continuity and momentum equations previously published. Relevant dimensionless constants are derived which can be used to obtain an indication of the relative significance of acceleration effects. The necessity for using computational fluid dynamics to capture nonlinear effects remains, and various implementation schemes for accelerating bodies are discussed. This theoretical treatment is intended to provide a foundation for interpretation of aerodynamic effects observed in manoeuvre, particularly for accelerating missiles.     

Dynamics of a spacecraft with large flexible appendage constrained by multi-strut passive damper

This paper is concerned with the dynamics of a spacecraft with multi-strut passive damper for large flexible appendage.The damper platform is connected to the spacecraft by a spheric hinge,multiple damping struts and a rigid strut.The damping struts provide damping forces while the rigid strut produces a motion constraint of the multibody system.The exact nonlinear dynamical equations in reducedorder form are firstly derived by using Kane’s equation in matrix form.Based on the assumptions of small velocity and small displacement,the nonlinear equations are reduced to a set of linear second-order differential equations in terms of independent generalized displacements with constant stiffness matrix and damping matrix related to the damping strut parameters.Numerical simulation results demonstrate the damping effectiveness of the damper for both the motion of the spacecraft and the vibration of the flexible appendage,and verify the accuracy of the linear equations against the exact nonlinear ones.     

Verification of deforming polarized structure computation by using a closed-form solution

Various engineering systems exploit the conversion between electromagnetic and mechanical work. It is important to compute this coupling accurately, and we present a method for solving the governing equations simultaneously (at once) without a staggering scheme. We briefly present the theory for coupling the elecgoverning equations as well as the variational formulation that leads to the weak form. This weak form is nonlinear and couples various fields. In order to solve the weak form, we use the finite element method in space and the finite difference method in time for the discretization of the computational domain. Numerical problems are circumvented by selecting the field equations carefully, and the weak form is assembled using standard shape functions. In order to examine the accuracy of the method, for the case of a linear elastic material under small deformations, we present and use an analytic solution. Comparison of the computation to the closed-form solution shows that the computational approach is reliable and models the jump of the electromagnetic fields across the interface between two different materials.     

Stability case study of the ACROBOTER underactuated service robot

The dynamics of classical robotic systems are usually described by ordinary differential equations via selecting a minimum set of independent generalized coordinates. However, different parameterizations and the use of a nonminimum set of (dependent) generalized coordinates can be advantageous in such cases when the modeled device contains closed kinematic loops and/or it has a complex structure. On one hand, the use of dependent coordinates, like natural coordinates, leads to a different mathematical representation where the equations of motion are given in the form of differential algebraic equations. On the other hand, the control design of underactuated robots usually relies on partial feedback linearization based techniques which are exclusively developed for systems modeled by independent coordinates. In this paper, we propose a different control algorithm formulated by using dependent coordinates. The applied computed torque controller is realized via introducing actuator constraints that complement the kinematic constraints which are used to describe the dynamics of the investigated service robotic system in relatively simple and compact form. The proposed controller is applied to the computed torque control of the planar model of the ACROBOTER service robot. The stability analysis of the digitally controlled underactuated service robot is provided as a real parameter case study for selecting the optimal control gains.     

Non-linear dynamics of curved beams. Part 2, numerical analysis and experiments

The aim of this work is to formulate a model for the study of the dynamics of curved beams undergoing large oscillations. In Part 1, the interest was oriented to the formulation of a consistent analytical model and to obtain the equations of motion in weak form. In Part 2, a case-study is considered and the response for various initial curved configurations, obtained by varying the initial curvature, is analyzed. Both the free and the forced problems are considered: the linear free dynamics are studied to detect how the initial configuration affects the modal properties and to enlighten the typical phenomena of frequency coalescence and avoidance; the forced dynamics are then studied for different internal resonance conditions to enlighten the phenomenon of the dynamic instability under a shear periodic tip follower force and to describe the various classes of post-critical motion. The results of experimental tests conducted on a slightly imperfect straight beam prototype are eventually discussed.