一类非耗散的显式时间积分方法

针对大型复杂结构的动力学问题，为了得到高效的计算方法，以泰勒公式为基础，通过调整参数改善算法的稳定性，得到了一类具有非耗散特性且以加速度为基本变量的显式数值积分方法。对提出的方法进行了稳定性和精度分析，并通过多个算例对提出的方法、蛙跳式中心差分法、Newmark-β法和精确解进行比较。结果表明，无阻尼情况下，所提出的方法的稳定性条件与中心差分法相同；提出的方法的振幅衰减率为0，具有非耗散特性，其周期误差约为隐式Newmark-β法的一半；所提出的方法给出了蛙跳式中心差分法和无耗散特性的翟方法的统一格式，并可以衍生出更多的显式时间积分方法。     

结构动力方程求解中隐式格式向显式格式的转换

摘 要：以广义加速度隐式算法为例,指出常用隐式算法可转换为显式算法。转换过程主要包括三个步骤：首先是对隐式求解的耦联线性方程组的系数矩阵取逆,然后将此逆矩阵在单位矩阵附近进行级数展开,最后进行级数截断。以常加速度方法的显式化过程为例,详细阐述隐式算法转换为显式算法的过程,着重介绍级数展开的项数对计算精度的影响,以及转换过程中时间步长必须同时满足算法的稳定性和级数收敛条件。揭示隐式算法与显式算法之联系,使可按高精度隐式算法得到与其相伴随的高精度显式算法。     

一族无条件稳定的显式结构动力学算法

利用离散控制理论分析HHT-α算法,提出了一族具有可控数值阻尼的无条件稳定显式结构动力学算法—显式HHT-α法,用于线性和非线性结构动力学分析。新算法采用显式的位移、速度递推式。研究了所提算法的精度,稳定性,数值色散和能量耗散特性。研究表明该算法对于线弹型和刚度软化型非线性系统是无条件稳定的,算法数值阻尼由单个参数控制,对于特定的参数值,所提算法不会产生数值能量耗散。此外所提出的显式算法的数值色散和能量耗散特性与隐式HHT-α算法相同。数值算例验证了理论分析的正确性。     

基于显式有限元的薄壁结构吸能特性预测

为了探讨轴向载荷冲击作用下试验参数对薄壁结构吸能特性的影响规律，预测和分析某一类型的薄壁结构吸能特性，基于显式有限元技术建立了这一类型薄壁结构的BP人工神经网络吸能特性预测模型。以方形薄壁结构为例，通过改变结构质量、冲击速度、横截面尺寸、壁厚等影响结构吸能特性的因素对这一类型的薄壁结构进行了大量的数值试验，获得不同试验条件下的结构吸能特性参数，然后建立了这一类型薄壁结构的吸能特性预测模型，并进行自主训练学习, 当训练步数为2035时，网络模型达到误差要求。结果表明：该BP神经网络模型的输出样本与目标值十分接近, 比吸能误差值为-2.53％，有效撞击力误差值为4.67％，有效撞击行程误差值为-3.90％，说明该模型具有较好的精度。     

基于速度的显式等效力控制方法的研究

该文采用等效力控制(EFC)来求解在实时子结构试验中的速度差分方程,采用反馈控制取代数学迭代来求解非线性动力方程。谱半径分析表明,由于准确地模拟了速度响应,结合显式Newmark-β算法的等效力控制方法的稳定界限与系统的阻尼系数无关,在不同阻尼比下稳定界限保持为Ω=2,具有良好的数值特性。而直接预测速度的中心差分法和采用线性插值模拟速度的平均加速度等效力控制方法,其稳定界限随着系统阻尼比的增大而降低,对于过阻尼结构(阻尼比大于1),原本无条件稳定的平均加速度等效力控制方法变为条件稳定:对于阻尼比为1.05的动力系统,其稳定界限为Ω=1.45。最后,采用该文方法对安装磁流变(MR)阻尼器的单自由度结构的地震响应进行数值分析,结果表明该文方法能正确跟踪结构速度和位移命令,因而对于速度相关型结构体系,具有良好的适用性和精确性。     

对流扩散方程的一种新的显式方法

基于作者建立的一类新的指数型非对称半显式差分方法,提出了一种数值求解对流扩散方程的指数型交替组显方法。该方法具有固有并行性,而且无条件稳定,数值结果表明,该方法精度优于Evans和Abdullah在1985年提出的交替组显格式。     

基于加速度泰勒展开的动力学方程显式积分方法

该文旨在提出兼顾适用性、可靠性与高效性的结构振动时域积分算法。基于加速度的泰勒展开式,引入截断系数考虑高阶项的影响,提出了具有4阶精度的加速度公式;通过积分并考虑典型时间步初始时刻系统动力平衡条件,建立了位移和速度的单步递推公式,运用终止时刻系统运动方程修正加速度。与多步积分法相比,单步积分法无需记录当前时间步以外时刻响应。稳定性分析表明,临界步长相比中心差分法增加40%。通过线性系统振动响应计算发现,当步长-系统固有周期（荷载周期）比达到0.2时,该文方法的振幅衰减率和周期延长率均小于5%;对于非线性系统,为降低算法阻尼和周期误差的影响,需控制步长周期比小于0.1。     

基于动力特征分区的显式-精细混合积分法

大型动力系统中常因局部的高频振动及非线性等特性限制了系统的积分步长而导致整体计算量激增，针对此问题提出一种分区域异步长显式-精细混合积分方法。在特性复杂的局部区域采取显式积分法，根据精度和稳定性要求取较小的时间步长求解；在其余常规区域则应用精细积分方法，采取可以跨越显式积分区周期的大积分步长求解。对于精细积分区域边界荷载，提出一种基于离散FFT变换的线性项与主频谐波项的组合表示方法，并给出了此种荷载形式下的精细积分计算格式。数值算例结果表明该法能够明显提高计算效率，在显式积分区域和精细积分区域都有很高的精度。     

基于显式动力学的焊点冲击失效分析

组件级高速剪切测试是用来研究芯片封装中Sn-Ag-Cu焊点冲击可靠性问题的一个重要手段。实验研究表明：随着冲击速度的增加,焊点封装结构的失效会由焊锡母材的韧性破坏向界面金属间化合物(IMC)的脆性断裂过渡;同时,其荷载-位移响应曲线形态也会发生显著的改变。为了能够更详细地了解封装结构的冲击失效行为,并进一步改进其结构设计,该文提出结合焊锡材料应变率相关的动态硬化特性,利用渐进损伤模型来模拟其动态损伤过程;同时,引进一种能够有效表征复合型裂纹扩展的内聚力模型来模拟IMC的脆性动态断裂。与实验结果的对比表明：该文提出的方法能够较为有效地表征焊点封装结构在不同冲击速度下的失效行为。     

阻尼弹性结构动力计算的显式差分法

本文在时域有限元离散的基础上,导出了集中质量阻尼弹性结构动力模型的显式差分法,数值稳定条件同中心差分法。文中还分析了用中心差分法结合单边差方法的不足之处。     

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

A multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain-to-subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one-dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one-dimensional heterogeneous bar and in heterogeneous plain strain problems.     

An explicit time integration technique for dynamic analyses

A simple explicit solution technique for problems in structural dynamics, based on a Modified Trapezoidal rule Method (MTM) approximation of the governing ordinary differential equations, is developed. The resulting conditionally stable explicit method (MTM) can be easily implemented and is extremely simple to use. Particular attention is focused herein on the concept of numerical stability of the proposed method for a free-vibrational response of a linear undamped Single-Degree-Of-Freedom system (SDOF). To examine the effectiveness, strengths, and limitations of MTM, error analyses for the natural period, the displacement, the velocity and the associated phase angle for a free undamped simple mass–spring system are derived and compared with Modified Euler Method (MEM) and the well-known Newmark Beta Method (NBM). Numerical examples for a SDOF system and a Multi-Degree-Of-Freedom (MDOF) system are presented to illustrate the strengths and the limitations of the proposed method.     

Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies

We reformulate the traditional velocity based vector‐space Newmark algorithm for the rotational dynamics of rigid bodies, that is for the setting of the SO(3) Lie group. We show that the most naive re‐write of the vector space algorithm possesses the properties of symplecticity and (almost) momentum conservation. Thus, we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms, but which curiously does not seem to have been considered in the open literature so far. Copyright © 2005 John Wiley & Sons, Ltd.     

A simple explicit single step time integration algorithm for structural dynamics

In this article, a new single-step explicit time integration method is developed based on the Newmark approximations for the analysis of various dynamic problems. The newly proposed method is second-order accurate and able to control numerical dissipation through the parameters of the Newmark approximations. Explicitness and order of accuracy of the proposed method are not affected in velocity-dependent problems. Illustrative linear and nonlinear examples are used to verify performances of the proposed method.     

Explicit momentum‐conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

We reformulate the midpoint Lie algorithm, which is implicit in the torque calculation, to achieve explicitness in the torque evaluation. This is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Thus, we obtain two related variants, both of which are explicit in the torque calculation, but only first order in the time step. Both variants are momentum conserving and both are symplectic. Consequently, drawing on the properties of the composition of maps, we introduce another algorithm that combines the two variants in a single time step. The resulting algorithm is explicit, momentum conserving, symplectic, and second order. Its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators. Copyright © 2005 John Wiley & Sons, Ltd.     

Large-step explicit time integration via mass matrix tailoring

A method for tailoring mass matrices that allows large time-step explicit transient analysis is presented. It is shown that the accuracy of the present tailored mass matrix preserves the low-frequency contents while effectively replacing the unwanted higher mesh frequencies by a user-desired cutoff frequency. The proposed mass tailoring methods are applicable to elemental, substructural as well as global systems, requiring no modifications of finite element generation routines. It becomes most computationally attractive when used in conjunction with partitioned formulation as the number of higher (or lower) modes to be filtered out (or retained) are significantly reduced. Numerical experiments with the proposed method demonstrate that they are effective in filtering out higher modes in bars, beams, plain stress, and plate bending problems while preserving the dominant low-frequency contents.     

An explicit family of time marching procedures with adaptive dissipation control

In this work, an explicit family of time marching procedures with adaptive dissipation control is introduced. The proposed technique is conditionally stable, second‐order accurate, and has controllable algorithm dissipation, which adapts according to the properties of the governing system of equations. Thus, spurious modes can be more effectively dissipated and accuracy is improved. Because this is an explicit time integration technique, the new family is quite efficient, requiring no system of equations to be dealt with at each time step. Moreover, the technique is simple and very easy to implement. Numerical results are presented along the paper, illustrating the good performance of the proposed method, as well as its potentialities. Copyright © 2014 John Wiley & Sons, Ltd.     

A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics

A comprehensive study of A-stable linear two-step time integration methods for structural dynamics analysis is presented in this paper. An optimal A-stable linear two-step (OALTS) time integration method is revealed with desirable performance on low-frequency accuracy and high-frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second-order equations of structural dynamics; is implicit, A-stable, and second-order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.     

Multi‐model Arlequin method for transient structural dynamics with explicit time integration

This paper addresses the explicit time integration for solving multi‐model structural dynamics by the Arlequin method. Our study focuses on the stability of the central difference scheme in the Arlequin framework. Although the Arlequin coupling matrices can introduce a weak instability, the time integrator remains stable as long as the initial kinematic conditions of both models agree on the coupling zone. After showing that the Arlequin weights have an adverse impact on the critical time step, we present two approaches to circumvent this issue. Computational tests confirm that the two approaches effectively preserve a feasible critical time step and show the efficiency of the Arlequin method for structural explicit dynamic simulations. Copyright © 2017 John Wiley & Sons, Ltd.