A mesh stability study of FEM explicit algorithms for structural large-scale deformation impact responses

To FEM explicit algorithms for structural large-scale deformation impact responses, algorithm stability is discussed in the present paper. Algorithm stability is thought to include two aspects: One is called difference pattern stability and the other is called mesh stability. A self-adaptive adjusting method is proposed to ensure mesh stability with little amount of computation increased.     

Unified classification of stability of pin-jointed bar assemblies

Based on the energy criterion and geometrical nonlinearity theory, this paper broadens conventional concepts of structural stability to explain some non-generic stability phenomena of pin-jointed bar assemblies in a unified and coherent way. A novel classification for stability conditions of such kind of structures is put forward, using analysis of the constitution of the tangential stiffness matrix. Some classical issues, including geometrical stability and stability of mechanisms, are re-investigated under this new concept as part of the formal theoretical development. Effects of bars stiffness are introduced into the necessary and sufficient conditions of intrinsic stability (stability of structure devoid of internal forces). The stability conditions for mechanisms, whether they acquire stiffness from self-stressing or external loading, are also probed. The stability of infinitesimal mechanism is expounded through consideration of high-order variations of the potential energy. Some discussions are provided at the end to build up an integrated understanding of stability of pin-jointed bar assemblies.     

Effect of the steam compressibility on the stability of a phase interface in geothermal systems with constant temperature

The stability of vertical steady-state flows in geothermal water-above-steam systems is considered. The steam compressibility and the capillary forces on a phase interface are taken into account. A domain of the physical parameters of the geothermal system, where vertical steady-state flows exist, is found. The linear stability of these flows is analyzed, namely, the dispersion relation is obtained, the stability diagram is constructed, and the possible types of loss of stability are found using asymptotic and numerical methods. The effect of the steam compressibility and other physical parameters on vertical steady-state flows and their stability is analyzed.     

Mean stability of stochastic difference systems

In this paper the mean stability of linear and non-linear stochastic difference systems is considered. For linear systems the relationship between mean stability and other stability definitions is explored. For the non-linear system explicit criteria for mean stability are derived when the non-linear term satisfies a certain realistic condition.     

Robust stability for a type of uncertain time-delay systems

ＮｏｔａｔｉｏｎＲ　ｒｅａｌ＿ｎｕｍｂｅｒｆｉｅｌｄｘ　ｖｅｃｔｏｒ;ｘ=(ｘ1 ,ｘ2 ,… ,ｘｎ) Ｔ,ｘｉ∈ＲｘＴ　ｔｒａｎｓｐｏｓｅｏｆｖｅｃｔｏｒｘＡＴ　ｔｒａｎｓｐｏｓｅｏｆｍａｔｒｉｘＡλｍａｘ(Ａ)　ｍａｘｉｍｕｍｅｉｇｅｎｖａｌｕｅｏｆＡλｍｉｎ(Ａ)　ｍｉｎｉｍｕｍｅｉｇｅｎｖａｌｕｅｏｆＡ‖ｘ‖　Ｅｕｃｌｉｄｅａｎｎｏｒｍｏｆｖｅｃｔｏｒｘ;　　　‖ｘ‖ =(ｘＴｘ) 1 / 2‖Ａ‖　ｍａｔｒｉｘｎｏｒｍｏｆＡ ;　　　　　　‖Ａ‖ =λ1 / 2ｍａｘ(ＡＴＡ)μ(Ａ)　ｍａｔｒｉｘｍｅａｓｕｒｅ;　　　 μ(…     

A stability analysis of an oscillating body located in fluid flow using automatic differentiation

This paper presents a stability analysis of an oscillating body subjected to fluid forces located in a transient incompressible viscous flow. If the body is supported by elastic springs, oscillation will begin. If the characteristic period of the body and the excited oscillating period due to fluid forces match each other, resonance can occur. Stability analysis is therefore needed to determine the nonlinear behavior of the body. This paper presents an analysis of the changing stability of bodies by the numerical computation. To implement the computation, the motion of fluid around a body is expressed by the Navier–Stokes equation described in the arbitrary Lagrangian–Eulerian form. The fluid influence on the body is discretized by the finite element method based on a mixed interpolation by the bubble function in space. The motion of the body is assumed to be expressed by the equations of motion. To evaluate stability, stability function is defined by the total energy of the oscillating body. The stability is judged according to a stability index, obtained by the use of the automatic differentiation (AD) of the stability function. AD is a derivative computation method that gives high accuracy. By the use of AD, the second‐order derivative matrix, which is needed to compute the stability index, can be obtained exactly. For the numerical studies, analyses of one degree of freedom and two degrees of freedom (2DOF) for a circular cylinder and 2DOF for a rectangular cylinder are carried out. A combination of a cylinder and supporting elastic spring can produce stable, neutral and unstable states. It is shown that the stability of the cylinder can be determined by the stability index. This paper shows new possibilities for stability analysis of bodies located in a fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Hydrodynamic stability of boundary layers with surface mass transfer

The linear stability of the flat plate boundary layer with surface blowing and suction is investigated by the application of numerical techniques. Complete neutral stability curves, critical Reynolds numbers and wave numbers, and other stability characteristics are determined for a wide range of surface mass transfer intensities. The critical Reynolds number, based on the displacement thickness, is found to vary from 59 to 32500 between the extreme limits of blowing and suction that are investigated. Comparisons are made between the present results and available linear stability information for boundary layers with surface mass transfer and with free-stream pressure gradients. The universal stability bound of Joseph is evaluated and compared with the corresponding numerically exact neutral stability curve.     

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples. The project supported by the National Science Foundations of Russia and China (10072012)     

On the local stability analysis of the approximate harmonic balance solutions

Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.     

黏塑性本构计算的稳定性分析

提出本构方程计算方法的稳定性问题,针对黏塑性本构计算的显式精确算法的稳定性进行分析,发现该算法并非无条件稳定,使用小扰动方法给出了其计算稳定的必要条件,稳定性条件对数值计算中的时间步长提出限制要求。通过有限元算例验证了分析的正确性,计算结果也表明理论推导得到的稳定性公式能够准确预测满足计算稳定性条件要求的最大时间步长与各参数之间关系。     

Stability of fast parallel MHD shock waves in polytropic gas

The stability of plane shock waves in Magnetohydrodynamics for an ideal medium is studied. Stability results are obtained for the special case of fast parallel shock waves in a polytropic gas. Linear stability is proved for a polytropic gas with arbitrary γ. The domain of structural (nonlinear) stability, where the uniform Lopatinsky condition is fulfilled for the stability problem, is found.     

The influence of geostrophic force on the stability of an heterogeneous conducting fluid with a radial gravitional force

The linear and nonlinear stability of a heterogeneous incompressible inviscid perfectly conducting fluid between two cylinders is investigated in the presence of a radial gravitational force and geostrophic force. The stability for linear disturbances is investigated using the normal mode method, while the nonlinear stability is investigated by applying the energy method. In the case of linear theory, it is found that a necessary condition for in stability is that the algebraic sum of hydrodynamic, hydromagnetic and rotation Richardson number is less than one quarter somewhere in the fluid. A semi-circle theorem similar to that of Howard is also obtained. In the case of nonlinear disturbances a universal stability estimate namely a stability limit for motions subject to arbitrary nonlinear disturbances is obtained in the form $$E \leqslant E_0 \exp ( - 2M\tau ).$$ The motion is asymptotically stable if $$\delta \leqslant 1 + J_m + J_H $$ somewhere in the fluid. This asymptotic stability limit is improved using the calculus of variation technique. We also find that whenδ=1/4, andJ _R=1, both the linear and nonlinear stability criteria coincide and in that particular case, we have a necessary and sufficient condition for stability.     

Temporal Stability of Boundary-Free Shear Flows: The Effects of Diffusion

The stability of boundary-free shear flow is studied for the case of variable viscosity due to binary diffusion across the shear layer. This leads to the main difficulty of this investigation, the direct coupling of the momentum and species equations in both the base state calculations as well as the stability analysis. Linear stability analysis is used to examine the effect of a nonuniform concentration profile on the stability of the flow. It is found that for the flow to be stable for all disturbance wave numbers the Reynolds number has to be zero. This is in agreement with constant viscosity free shear flow stability theory. Increasing the magnitude of concentration gradient (increasing the Schmidt number) destabilizes the flow.     

Delay-dependent stability of linear multistep methods for differential systems with distributed delays

This paper deals with the stability of linear multistep methods for multi-dimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.     

Dynamic stability analysis of finite element modeling of piezoelectric composite plates

The dynamic stability of negative-velocity feedback control of piezoelectric composite plates using a finite element model is investigated. Lyapunov’s energy functional based on the derived general governing equations of motion with active damping is used to carry out the stability analysis, where it is shown that the active damping matrix must be positive semi-definite to guarantee the dynamic stability. Through this formulation, it is found that imperfect collocation of piezoelectric sensor/actuator pairs is not sufficient for dynamic stability in general and that ignoring the in-plane displacements of the midplane of the composite plate with imperfectly collocated piezoelectric sensor/actuator pairs may cause significant numerical errors, leading to incorrect stability conclusions. This can be further confirmed by examining the complex eigenvalues of the transformed linear first-order state space equations of motion. To overcome the drawback of finding all the complex eigenvalues for large systems, a stable state feedback law that satisfies the second Lyapunov’s stability criteria strictly is proposed. Numerical results based on a cantilevered piezoelectric composite plate show that the feedback control system with an imperfectly collocated PZT sensor/actuator pair is unstable, but asymptotic stability can be achieved by either bonding the PZT sensor/actuator pair together or changing the ply stacking sequence of the composite substrate to be symmetric. The performance of the proposed stable controller is also demonstrated. The presented stability analysis is of practical importance for effective design of asymptotically stable control systems as well as for choosing an appropriate finite element model to accurately predict the dynamic response of smart piezoelectric composite plates.     

The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability

A brief historical overview is given which discusses the development of classical stability concepts, starting in the seventeenth century and finally leading to the concept of Lyapunov stability at the beginning of the twentieth century. The aim of the paper is to find out how various scientists thought about stability and to which extent their work is related to the stability concepts bearing their names, i.e. Lagrange, Poisson and Lyapunov stability. To this end, excerpts of original texts are discussed in detail. Furthermore, the relationship between the various works is addressed.     

Stability analysis of abnormal multiplication of plankton using parameter identification technique

This paper presents the mathematical approach for the abnormal multiplication of plankton. An abnormal multiplication can be expressed as an unstable problem and the stability of the system is investigated by introducing eigenvalues of a mathematical equation. The stability of the system can be judged by an eigenvalue based on the Lyapunov's stability theory. In this paper, the Arnoldi‐QR method is used to obtain eigenvalues and eigenvectors of the system. The mode superposition method is employed to create spatial distribution needed to analyse the stability. To obtain the objective eigenvalue, the parameter identification technique is employed. The finite element method is used for the discretization in space. Lake Kasumigaura, which is located in Ibaraki Prefecture in Japan, is selected and actual data in 1975, 1976, 1991 and 2000 are used in order to investigate the stability of the specified lake in Japan. Copyright © 2004 John Wiley & Sons, Ltd.     

Thermoelastic stability of first strain gradient solids

The problem of thermoelastic stability of a first strain gradient solid under noncons rvative loadings is studied. The Liapunov-Movchan method of elastic stability analysis is reviewed. An energy Liapunov functional is constructed and the sufficiency criteria for the stability of a loaded equilibrium configuration are derived. The effects of motion dependent surface tractions and couples are discussed. The special case of isothermal elastic stability of solids with couple stress is also considered.     

Magnetohydrodynamic stability of heterogeneous incompressible nondissipative conducting liquids

Summary This discussion which is restricted to the flow of heterogeneous, incompressible, inviscid, perfect conducting liquids between two rotating or nonrotating coaxial cylinders is divided into three parts. In the first part the stability of the liquid in question between two coaxial nonrotating cylinders with an applied magnetic field perpendicular to the flow is investigated. A sufficient condition for stability is found and if the motion is unstable an upper bound for the amplification factor is given. As a particular case the stability of the liquid with uniform steady velocity, and density varying as a function of a distance from the axis of the cylinders is discussed and it is found that the effect of the magnetic field makes the flow more stable.In part two the stability of the liquid in question between two rotating coaxial cylinders with an applied magnetic field in the tangential direction is discussed. A necessary and sufficient condition for stability is derived.In part three the stability of the same liquid between two rotating coaxial with an applied magnetic field in the axial direction is treated. A sufficient condition for stability is derived. In general, we infer that in the case of parallel flow the magnetic field plays the same role in the liquid as gravitational field in Synge's) discussion.     

First integral and stability of motion in the critical cases

In this paper, we indicate that after the Liapunov function by using linear combination of mechanical first integral was suggested by Chetayev in 1946. He and his students solved stability of conservative system by means of this method. But he had trouble to solve the problems by means of cut and try. Moreover, the condition of stability is imperfect. Solution by this method is limited for problems of purely imaginary roots. The cases of zero roots have not been considered. Condition of stability secured is more strict.This paper suggests that the differential equation can be transformed into standard form by method of cancellation of cyclic coordinates (method of lowering degree of order). and condition of stability can be determined by energy integral. By this method not only the computation is clear and concise. But also zero roots can be considered. Therefore the problems of two cyclic coordinates can be transformed into second-order system, and we get new conclusion of the condition of stability simply. As for problems of single cyclic coordinate, infact, Chetayev and his students did not solve the stability of the gyroscope of auter-gimbal with horizontal axis or arbitrary angle. In this paper, it shows that the method suggested here is useful for stability of these problems. The condition of conditional stability and instability were derived.