Analysis and implementation of a new constant acceleration subcycling algorithm

An algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps. With integer time step ratios, the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle. The algorithm is shown to have good accuracy, and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms. The implementation of a generalised form of the algorithm with non-integer time step ratios is presented. © 1997 John Wiley & Sons, Ltd.     

On the accuracy of direct integration methods for structural dynamics

Abstract

This paper presents a method of eigen analysis to evaluate the accuracy of the time integration method in the structural dynamic analysis of both transient and steady state responses. Accuracy measure for transient response is evaluated by period elongation and amplitude decay after a complete cycle of response of an undamped system. The proposed method is a unique measure, independent of the initial conditions. Accuracy measure for steady state response is evaluated by the amplitude decay and phase angle of a frequency response. The accuracy measure proposed by this method can be applied to all ranges of ?t/T. The significance of the large range ?t/T is discussed. The proposed method is used to evaluate the accuracy of three commonly used time integration methods: the Wilson method, the Newmark method and the Houbolt method.     

精细直接积分法的积分方法选择

讨论了精细直接积分法中积分方法选择问题。通过理论推导和数值试验,指出为保持精细算法的高精度,应根据荷载的性质选择合适的积分方法,并得出激励为多项式形式时应选择代数精度高的积分方法的结论,指出科茨积分、高斯积分是保持精细算法高精度的较好积分方法。     

Joint statistics of combined first- and second-order random processes

Two-term Volterra series are often used to describe non-linear random processes in subject areas as diverse as communication theory and the dynamics of offshore structures. It is well known that the characteristic function, the moments and the cumulants of such processes can be calculated analytically and that the probability density function can be calculated accurately and efficiently. This paper considers the joint statistics between two such random processes by deriving: (i) an exact expression for the joint characteristic function; (ii) an efficient means for calculating the joint moments; (iii) an ‘exact’ numerical means for calculating the joint probability density function (jpdf). For the special case of a combined first- and second-order process and a pure first-order process it is shown that it is possible to derive analytical expressions for the characteristic function and to calculate the jpdf accurately and efficiently using saddle-point integration. In addition to the above, the maximum entropy principle is used to calculate the jpdf.     

Subcritical and supercritical bifurcations of the first- and second-order Benney equations

The problem of the stability threshold of thin-film dynamics as described by the Benney equation of both first and second orders is revisited. The main result is that the primary Hopf bifurcation of the Benney equation of first order is supercritical for smaller values of Reynolds number and subcritical for its larger values. This result is numerically validated and further investigated analytically to reveal coexisting stable and unstable traveling waves. However, the primary bifurcation of the second-order Benney equation is supercritical for any Reynolds numbers. Sideband instability of traveling-wave regimes whose amplitude and frequency arise from the corresponding complex Ginzburg-Landau equation (CGLE) is found for the Benney equation of both first and second orders.     

Inelastic post buckling behavior of very short column

Abstract

This paper presents an unconditionally stable implicit algorithm for the direct integration of a linear structural dynamic equation of motion. The algorithm is based on two simultaneous difference equations and a weighting factor G for solving displacement at the next time step. Unconditional stability is proven for the weighting factor G in the range from –0.466 to 0.140 in all undamped and damped cases. The unconditionally stable range of G can be extended, from –1.0 to 0.333, for certain types of structure. The amplitude decay and period variation are used as the basic parameters to compare the accuracy of the present algorithm with various other integration methods. A spring‐mass‐dashpot model is applied to illustrate the algorithm for transient and quasi‐static analyses.     

Discrete equivalent time integration methods for transient analysis

It was recently shown in a series of papers that the frequency response of temporally discretized finite element equations and, consequently, the achievable accuracy cannot be manipulated independently in different frequency ranges. In addition, there exist limitations on the achievable accuracy of a time integration method no matter what the order of accuracy of the method is. Motivated by this fact, a family of time integration methods is derived in the time domain based on the principle that the exact solution of the semidiscrete equation of a system and the solution of the time integration method match at discrete time steps. It is necessary to pursue an exact match at discrete time steps, i.e. discrete equivalence, since the solutions of semidiscrete equations are obtained only at the time steps. Two time integration methods, that are exact at the time steps, are obtained based on the impulse and ramp response invariance principles. Numerical examples are presented to show the advantage of the proposed methods and to compare the performance of them with the performance of some popular methods. Copyright © 2003 John Wiley & Sons, Ltd.     

车用制动空气压缩机结构辐射噪声特性研究

针对某公司生产的KYV480型号车用制动活塞式空气压缩机(简称:空压机)运行过程中,振动噪声较大,一定程度上影响了驾驶员和乘客的乘车舒适性和身心健康的情况,文章基于有限元和声学边界元联合求解的方法对空压机的结构表面辐射噪声特性进行研究.在计算得到空压机运行过程中壳体所受载荷的基础上,对空压机壳体表面的振动加速度响应进行...     

A theory of development and design of generalized integration operators for computational structural dynamics

A standardized formal theory of development/evolution, characterization and design of a wide variety of computational algorithms emanating from a generalized time weighted residual philosophy for dynamic analysis is first presented with subsequent emphasis on detailed formulations of a particular class relevant to the so‐called time integration approaches which belong to a much broader classification relevant to time discretized operators. Of fundamental importance in the present exposition is the evolution of the theoretical design and the subsequent characterization encompassing a wide variety of time discretized operators, and the proposed developments are new and significantly different from the way traditional modal type and a wide variety of step‐by‐step time integration approaches with which we are mostly familiar have been developed and described in the research literature and in standard text books over the years. The theoretical ideas and basis towards the evolution of a generalized methodology and formulations emanate under the umbrella and framework and are explained via a generalized time weighted philosophy encompassing single‐field and two‐field forms of representations of the semi‐discretized dynamic equations of motion. Therein, the developments first leading to integral operators in time, and the resulting consequences then systematically leading to and explaining a wide variety of generalized time integration operators of which the family of single‐step time integration operators and various widely recognized and new algorithms are subsets, the associated multi‐step time integration operators and a class of finite element in time integration operators, and their relationships are particularly addressed. The generalized formulations not only encompass and explain a wide variety of time discretized operators and the recovery of various original methods of algorithmic development, but furthermore, naturally inherit features for providing new avenues which have not been explored an/or exploited to‐date and permit time discretized operators to be uniquely characterized by algorithmic markers. The resulting and so‐called discrete numerically assigned [DNA] markers not only serve as a prelude towards providing a standardized formal theory of development of time discretized operators and forum for selecting and identifying time discretized operators, but also permit lucid communication when referring to various time discretized operators. That which constitutes characterization of time discretized operators are the so‐called DNA algorithmic markers which essentially comprise of both (i) the weighted time fields introduced for enacting the time discretization process, and (ii) the corresponding conditions these weighted time fields impose (dictate) upon the approximations (if any) for the dependent field variables in the theoretical development of time integrators and the associated updates of the time discretized operators. Furthermore, a single analysis code which permits a variety of choices to the analyst is now feasible for performing structural dynamics computations on modern computing platforms. Copyright © 2001 John Wiley & Sons, Ltd.     

A methodology for the formulation of error estimators for time integration in linear solid and structural dynamics

In this article, we present a novel methodology for the formulation of a posteriori error estimators applicable to time‐stepping algorithms of the type commonly employed in solid and structural mechanics. The estimators constructed with the presented methodology are accurate and can be implemented very efficiently. More importantly, they provide reliable error estimations even in non‐smooth problems where many standard estimators fail to capture the order of magnitude of the error. The proposed methodology is applied, as an illustrative example, to construct an error estimator for the Newmark method. Numerical examples of its performance and comparison with existing error estimators are presented. These examples verify the good accuracy and robustness predicted by the analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

An effective data parallel self-starting explicit methodology for computational structural dynamics on the connection machine CM-5

This paper discusses the implementation aspects and our experiences towards a data parallel explicit self-starting finite element transient methodology with emphasis on the Connection Machine (CM-5) for linear and non-linear computational structural dynamic applications involving structured and unstructured grids. The parallel implementation criteria that influence the efficiency of an algorithm include the amount of communication, communication routing, and load balancing. To provide simplicity, high level of accuracy, and to retain the generality of the finite element implementation for both linear and non-linear transient explicit problems on a data parallel computer which permit optimum amount of communications, we implemented the present self-starting dynamic formulations (in comparison to the traditional approaches) based on nodal displacements, nodal velocities, and elemental stresses on the CM-5. Data parallel language CMFortran is employed with virtual processor constructs and with:SERIAL and:PARALLEL layout directives for arrays. The communications via the present approach involve only one gather operation (extraction of element nodal displacements or velocities from global displacement vector) and one scatter operation (dispersion of element forces onto global force vector) for each time step. These gather and scatter operations are implemented using the Connection Machine Scientific Software Library communication primitives for both structured and unstructured finite element meshes. The implementation aspects of the present self-starting formulations for linear and elastoplastic applications on serial and data parallel machines are discussed. Numerical test models for linear and non-linear one-dimensional applications and a two-dimensional unstructured finite element mesh are then illustrated and their performance studies are discussed.     

Properties of time integration with first order filter damping

Introduction of algorithmic damping by increasing the parameter values in the Newmark algorithm leads to undesirable low‐frequency damping and reduced order of accuracy. It is demonstrated, how these effects can be removed by introducing an extra damping term in the form of a first order linear filter. When the linear filter is discretized in time, the state variable associated with the filter can be eliminated, leading to a weighted average of the equations of motion at two consecutive times. The filter procedure contains the known versions of alpha weighted Newmark methods as special cases, but gives a different and improved weighting of the excitation terms. A complete analysis of the properties of the algorithm when used on linear systems is given, including the frequency response properties. It is demonstrated that the effect of ‘overshoot’ is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system, and analytic results are presented for the magnitude of the effect. Copyright © 2005 John Wiley & Sons, Ltd.     

A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact,joints, and friction

We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second‐order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial‐grade virtual prototyping software. Copyright © 2005 John Wiley & Sons, Ltd.     

样条配点法计算动响应

本文以3次B样条函数作为时域的试函数,用双配点法建立起一个分析结构动响应的单步计算格式.通过对算法精度和稳定性的详尽分析,找到了两个值得推荐的方案:条件稳定的四阶算法(C1)及无条件稳定的二阶算法(C2).这两个方案的算法特性与一些熟知算法作了比较.     

Ti/Cu扩散溶解层的组织结构和形成规律

采用镶嵌式扩散偶,在不同退火处理条件下,对Ti/Cu扩散溶解层进行了研究.利用扫描电子显微镜和电子探针显微分析仪观察和分析了扩散溶解层的组织结构和形成规律.结果表明,随着加热温度的升高和保温时间的延长,在Ti/Cu界面处会形成相界面依附于扩散偶组元Cu丝、形态各不相同、层数以及总层与单层厚度逐渐增加的"环状"扩散溶解层;当进行700℃、100小时真空退火热处理时,扩散溶解层厚度为93μm;其中一层呈"锯齿状"朝向Cu,分别有两层处于同一个层区域内,并以"竹笋状"方式互相交错重叠;结构为Cu/Cu4Ti/Cu2Ti/Cu3Ti2/Cu4Ti3/CuTi/CuTi2/Ti,而且其结构与Cu-Ti相图上各个相的左右排列顺序一致.不同的扩散温度和时间,Ti/Cu相界面处将几乎同时结晶出不同层数、厚度和结构的扩散溶解层.     

计算动响应的若干无条件稳定的二阶算法的比较

本文分析比较若干二阶的直接积分算法,主要是Zienkiewicz,Wood等的SS32算法(1984)和邵慧萍,蔡承文的三参数算法(1987)。文中指出,当参数符合一定对应关系时两族算法具有几乎相同的精度。但在参数空间中的无条件稳定区,后者的大于前者,而且三参数法避免了SS32所固有的严重超越现象。     

State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

A fourth‐order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase‐space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth‐order. An optional high‐frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth‐order in the frequency. Thus, the low‐frequency response is virtually undamped, and the algorithm remains third‐order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front. Copyright © 2005 John Wiley & Sons, Ltd.     

Effects of stochasticity on the length and behaviour of ecological transients

There is a growing recognition that ecological systems can spend extended periods of time far away from an asymptotic state, and that ecological understanding will therefore require a deeper appreciation for how long ecological transients arise. Recent work has defined classes of deterministic mechanisms that can lead to long transients. Given the ubiquity of stochasticity in ecological systems, a similar systematic treatment of transients that includes the influence of stochasticity is important. Stochasticity can of course promote the appearance of transient dynamics by preventing systems from settling permanently near their asymptotic state, but stochasticity also interacts with deterministic features to create qualitatively new dynamics. As such, stochasticity may shorten, extend or fundamentally change a system’s transient dynamics. Here, we describe a general framework that is developing for understanding the range of possible outcomes when random processes impact the dynamics of ecological systems over realistic time scales. We emphasize that we can understand the ways in which stochasticity can either extend or reduce the lifetime of transients by studying the interactions between the stochastic and deterministic processes present, and we summarize both the current state of knowledge and avenues for future advances.