The fundamental equations of dynamics using representation of quasi-coordinates in the space of non-linear non-holonomic constraints

ThesystemofdynamicsconsistsofNparticles,thevectorrlofithparticlemaybeexpressedbythefunctionofthegeneralizedcoordinatesq,andtimeI;itsconfigurationisdeterminedbygeneralizedcoordinatesq,;itisacteduponbygnon-linearnon-hoIonomicconstraints'inoneorder,i.e.j'(q.…     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

An automatic constraint violation stabilization method for differential/ algebraic equations of motion in multibody system dynamics

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A method for finding the invariants and exact solutions of coupled non-linear differential equations with applications to dynamical systems

The polynomial invariants of (a set) non-linear differential equations are found by using a direct approach. The integrability of these invariants deserves the integrability of the given set of coupled differential equations. As applications, the Lorenz and Rikitake sets, among others, are studied. New invariants are obtained.     

A comparison between a collocation and weak implementation of the rigid‐body motion constraint on a particle surface

In this work, we implemented and compared two different methods to impose the rigid‐body motion constraint on a solid particle moving inside a fluid. We consider a fictitious domain method to easily manage the particle motion. As the solid as well as the fluid inertia are neglected, the particle can be discretized through its boundary only. The rigid‐body motion is imposed via Lagrange multipliers on the boundary. In the first method, such constraints are imposed in discrete points on the boundary (collocation), whereas in the second the constraint is imposed in a weak way on elements dividing the particle surface. Two test problems, that is, a spherical and an ellipsoidal particle in a sheared Newtonian fluid, are chosen to compare the methods. In both cases, the analysis is carried out in 2D as well as in 3D. The results show that for the collocation method an optimal number of collocation points exist leading to the smallest error. However, small variations in the optimal value can generate large deviations. In the weak implementation, the error is only mildly affected by the number of elements used to discretize the particle boundary and by the Lagrange multiplier's interpolation space. A further analysis is carried out to study the effect of an approximated integration of weak constraints. A comparison between the two methods showed that the same accuracy can be achieved by using less constraints if the weak discretization is used. Finally, the rigid‐body motion imposed via weak constraints leads to better conditioned linear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

考虑局部切向约束接触问题的直接刚度法

针对工程中大量存在的切向滑移受到约束的接触问题,提出了基于Lagrange乘子的点-面及点-点接触直接刚度法,该方法生成的接触协调条件可以直接组装到结构刚度阵中,从而可直接用于考虑存在结构几何非线性及材料非线性的接触问题分析中。采用这一算法进行了考虑拱坝横缝张合效应的地震响应分析,结果表明考虑切向滑移约束对接触中的张开度有明显影响。     

多体系统动力学方程的无违约数值计算方法

多体系统动力学方程为3阶微分代数方程,已有的约束违约稳定法存在位移违约问题,数值仿真准确性和稳定性不足。本文将求解高阶微分代数方程的降阶理论、ε嵌入处理方式与隐式龙格库塔法相结合,提出了直接满足位移约束条件的多体系统动力学方程的无违约算法,避免了约束违约问题。该方法先将多体动力学方程转化为2阶微分代数方程,并与位移约束方程联立;再应用ε嵌入隐式龙格库塔法进行数值求解。应用两种方法分别对单摆机构进行数值仿真,结果表明本文的方法不仅能适应较大步长,且准确性和稳定性均优于约束违约稳定法。     

THE LAGRANGE DYNAMIC EQUATIONS OF MULTI－RIGIDBODY SYSTEMS WITH EXTERNAL SHOCKS

ＴＨＥＬＡＧＲＡＮＧＥＤＹＮＡＭＩＣＥＱＵＡＴＩＯＮＳＯＦＭＵＬＴＩ－ＲＩＧＩＤＢＯＤＹＳＹＳＴＥＭＳＷＩＴＨＥＸＴＥＲＮＡＬＳＨＯＣＫＳ￥ＺｈａｎｇＤｉｎｇｇｕｏ（章定国）（ＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏ...     

用拟压缩性方法求解不可压Euler方程

用拟压缩性方法和Ｊａｍｅｓｏｎ的有限体积算法求解了二维和三维定常可可压Ｅｕｌｅｒ方程。分别采用显、隐式时间离散推进求解;分析了人工粘性的阶数对定常解收敛性的影响,应用该方法计算了单个翼型和翼身组合体的低速绕流,结果与实验吻合较好。     

Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements

The theory of Doi and Edwards for entangled polymers has been recently modified for the case of fast flows to account for convective contributions to molecular dynamics. The flow-induced relative motion between neighboring chains removes constraints and speeds up relaxation. Convective constraint release (CCR) may thus explain why the shear stress is seen to approach a plateau at high shear rates instead of decreasing as predicted by the basic theory. In slow flows, as well as in step strain, another discrepancy between theory and observations can be found in the normal stress ratio in shear Ψ=−N₂/N₁. The theoretical value for Ψ at low deformations is 1/7 whereas measured values for well-entangled systems are systematically larger. We have recently considered the possibility that this discrepancy arises because force balance requirements at the entanglement nodes are ignored in the classical theory. Accordingly, we have proposed a change in the orientational tensor Q. Here, we sum up on these recent findings by proposing single-relaxation-time constitutive equations of the integral or rate type incorporating those concepts in a simple way. Such equations should be suitable for numerical simulation of complex flows. Received: 1 January 2000 Accepted: 8 August 2000     

Meshless Galerkin analysis of Stokes slip flow with boundary integral equations

This paper presents a novel meshless Galerkin scheme for modeling incompressible slip Stokes flows in 2D. The boundary value problem is reformulated as boundary integral equations of the first kind which is then converted into an equivalent variational problem with constraint. We introduce a Lagrangian multiplier to incorporate the constraint and apply the moving least‐squares approximations to generate trial and test functions. In this boundary‐type meshless method, boundary conditions can be implemented exactly and system matrices are symmetric. Unlike the domain‐type method, this Galerkin scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns. The convergence and abstract error estimates of this new approach are given. Numerical examples are also presented to show the efficiency of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solutions of the shallow water equations with discontinuous bed topography

A simple scheme is developed for treatment of vertical bed topography in shallow water flows. The effect of the vertical step on flows is modelled with the shallow water equations including local energy loss terms. The bed elevation is denoted with z_b^‐ for the left and z_b⁺ for the right values at each grid point, hence exactly representing a discontinuity in the bed topography. The surface gradient method (SGM) is generalized to reconstruct water depths at cell interfaces involving a vertical step so that the fluxes at the cell interfaces can accurately be calculated with a Riemann solver. The scheme is verified by predicting a surge crossing a step, a tidal flow over a step and dam‐break flows on wet/dry beds. The results have shown good agreements compared with analytical solutions and available experimental data. The scheme is efficient, robust, and may be used for practical flow calculations. Copyright © 2002 John Wiley & Sons, Ltd.     

Stokes equations with penalised slip boundary conditions

We consider the finite-element approximation of Stokes equations with slip boundary conditions imposed with the penalty method. In the case of a smooth curved boundary, our numerical results suggest that curved finite elements, regularised normal vectors or reduced integration techniques can be used to avoid a Babuska’s-type paradox and ensure the convergence of finite-element approximations to the exact solution. Convergence orders with these remedies are also compared.     

Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

小内力平板弹性波散射问题的摄动方法

基于平板小挠度弯曲波动方程,采用摄动方法具有纵向内力作用下的平板开孔弹性波的散射问题进行了研究,得么了传播稳态波时此种平板弯曲波动问题的分析解,分析了均匀纵向内力对弹性波散射结果的影响,作为算例,本文给出了平板圆形开孔的动应力集中系数的数值结果,并对计算结果进行了讨论。     

Explicit solutions to the coupled KdV equations with variable coefficients

1　IntroductionandtheProblemPresentedSeekingtheexplicitsolutionofthenonlinearpartialdifferentialequation (NPDEs)isanimportantsubjectinsolitontheoryanditsapplication .Formanyyears,themainattentionwaspaidtotheconstantcoefficientNPDEs[1～ 7],manypowerfulmethodshavebeenproposedanddeveloped.Inrecentyears,moreandmoreattentionshavebeenpaidtovariablecoefficientNPDEs[8～ 13].Manymethodssuchassimilarityreductionmethod ,truncatedexpansionmethodandhomogeneousbalancemethodhavebeenextendedtosolvevaria…     

A lattice Boltzmann model for the compressible Euler equations with second‐order accuracy

In this paper, we propose a new lattice Boltzmann model for the compressible Euler equations. The model is based on a three‐energy‐level and three‐speed lattice Boltzmann equation by using a method of higher moments of the equilibrium distribution functions. In order to obtain second‐order accuracy, we employ the ghost field distribution functions to remove the non‐physical viscous parts. We also use the conditions of the higher moment of the ghost field equilibrium distribution functions to obtain the equilibrium distribution functions. In the numerical examples, we compare the numerical results of this scheme with those obtained by other lattice Boltzmann models for the compressible Euler equations. Copyright © 2008 John Wiley & Sons, Ltd.