一种基于Hamel形式的无条件稳定动力学积分算法

时间积分算法是求解动力学系统的一个核心问题.动力学方程的时间积分经常会出现数值不稳定现象,有限元空间离散也通常会造成伪高频振荡,因而,发展解决上述问题的数值积分算法具有重要的理论价值.本文基于Hamel场变分积分子,通过新的数值积分算法的构造方法,提出了一种无条件稳定的Hamel广义α方法,具体内容包括:构造特殊的变分形式,利用变分积分子等工具,建立无条件稳定的数值积分算法;在相同框架下,提出更高精度的数值格式;结合活动标架法的特性,将算法的一般形式推广到李群空间,得到Hamel广义α文所提出的Hamel广义α方法是无条件稳定的,具有二阶精度并且能够快速过滤掉虚假的高频振荡.数值算例的结果显示,本文所提方法具备了传统方法的精度、耗散和稳定性优势,既适合一般的线性空间,也适用于李群空间,同时还可以发展高阶精度算法.本文发展了构造变分积分子的新模式.     

移动简谐荷载作用下桥梁响应的高效计算

在计算移动荷载过桥问题中广泛使用的Newmark方法必须在每一时间步内限制荷载的大小和作用位置都不能改变。精细积分法虽然允许荷载的大小在每一时间步长内发生变化,但是仍假定其作用位置是不变的,未能采取措施以描述荷载沿着桥面的连续移动性。本文提出三种精细积分格式,在每一时间步内不但允许移动荷载的大小按简谐规律连续变化,而且模拟了简谐荷载在空间域的连续移动。通过与Newmark方法和简单问题的解析解进行数值比较,表明用本文提出的方法可以用较粗的结构单元和较大的时间步长而获得很高的计算精度。在精度相同的前提下,计算效率比Newmark方法可提高1~2个数量级。     

列车荷载作用下黏弹性半空间体的随机动力响应Dynamic response of viscoelastic half-space subjected to train loads

针对列车荷载作用下黏弹性半空间体响应的问题,利用虚拟激励法将系统的随机分析转化为确定性分析。根据列车荷载构造了相应的虚拟激励形式,通过傅里叶积分变换法把半空间体控制方程转入波数‐频率域,并推导获得了系统虚拟响应的积分形式解。当相速度接近或大于瑞利波速时,积分形式解中被积函数往往具有奇异性和高振荡性,使得数值计算相当困难。对此,将被积函数图形化以确定函数的积分限,并通过自适应数值积分算法解决被积函数的振荡性。数值算例中,进行了随机列车荷载作用下半空间体的响应分析,讨论了荷载移动速度及频率等参数变化对响应的影响,给出了响应的时间和空间分布规律。本文方法可进一步推广至移动矩形荷载等载荷模型,对移动荷载作用下环境振动行为预测具有很好的借鉴意义。     

非局部模型与变形局部化数值模拟

基于非线性身体场论,建立了材料非局部连续模型、变分方程及相应的实时拖带系大变形有限元数值模型,设计了这一模型的数值卷积算法,由于广义函数弱收敛定理和卷积理论,证明所提出的非局部连续模型具备收敛性和稳定性。并阐明了材料特征尺度数学物理意义,统计加权函数的选择原则,数值结果表明非局部模型描述变形局部化问题是适当的。     

Acoustics in anisotropic poroelasticity: space–time variational formulation

A variational space–time formulation to treat acoustics problems in porous medium is developed. At first, no separation of the space and time has been applied in the approximation of the field variables. This produces a matrix variational formulation for more general numerical treatments. Finally, separation of variables has been applied to produce a simpler space–time formulation, in which the numerical treatment of the space domain and the time integration algorithms can be identified. The numerical treatment has been oriented to find symmetry in the space and time domains. Numerical simulation in the frequency domain and in the time domain have been performed to validate the variational formulation with separation of variables.     

考虑结构自重的基于NURBS插值的3D拓扑描述函数法

在许多如大坝、桥梁等大型土木工程结构中,结构的自重是初始设计阶段必须考虑的重要载荷之一,因此研究自重载荷作用下的结构拓扑优化设计问题具有十分重要的意义.针对考虑自重载荷作用的拓扑优化问题所面临的主要困难,总结了现有处理考虑自重载荷的拓扑优化问题的三类主要方法;提出一种基于非均匀有理B样条(non-uniform rational B-splines,NURBS)基函数插值的拓扑描述函数方法,基于此方法研究了考虑设计依赖自重载荷作用的2D/3D结构优化设计问题.在列式下,高阶NURBS基函数被同时用于三维NURBS实体片中的几何场、位移场及设计变量场插值,实现了几何模型、分析模型和优化模型的有效统一,确保了位移场及设计变量场的高阶连续性;详细推导了基于NURBS基函数插值的考虑自重载荷作用的三维结构拓扑优化模型及其灵敏度列式,并采用移动渐进线方法(method of moving asymptotes,MMA)进行了优化求解;多个算例验证了方法的有效性和稳定性,结果表明,优化迭代过程稳健,收敛快,能够有效地克服自重载荷作用下连续体结构拓扑优化中经常遇到的低密度区域材料的寄生效应及目标函数的非单调性等问题.     

Effects of the dynamic wheel–rail interaction on the ground vibration generated by a moving train

Based on Biot’s fully dynamic poroelastic theory, the dynamic responses of the poroelastic half-space soil medium due to quasi-static and dynamic loads from a moving train are investigated semi-analytically. The dynamic loads are assumed to be generated from the rail surface irregularities. The vehicle dynamics model is used to simulate the axle loads (quasi-static loads) and the dynamic loads from a moving train. The compatibility of the displacements at wheel–rail contact points couple the vehicle and the track–ground subsystem, and yield equations for the dynamic wheel–rail loads. A linearized Hertzian contact spring between the wheel and rail is introduced to calculate the dynamic loads. Using the Fourier transform, the governing equations for the poroelastic half-space are then solved in the frequency–wavenumber domain. The time domain responses are evaluated by the fast inverse Fourier transform. Numerical results show that the dynamic loads can make important contribution to dynamic response of the poroelastic half-space for different train speed, and the dynamically induced responses lie in a higher frequency range. The ground vibrations caused by the moving train can be intensified as the primary suspension stiffness of the vehicle increases.     

A fast algorithm for soil dynamics calculations by wavelet decomposition

Summary The article presents a fast numerical algorithm for calculating the response of a halfspace under any surface loads. Under certain conditions there exists an analytical solution to the problem in the Fourier domain. To get the desired response, a numerical inverse Fourier transform of this analytic solution has to be made. By using a wavelet decomposition, the proposed algorithm can reduce the calculation time significantly, thus allowing the computation of complex problems. As an example, the response of the beam-halfspace coupled system under moving load is presented. Received 6 March 1997; accepted for publication 20 May 1997     

Dynamic response of plates on elastic foundations due to the moving loads

This paper discusses the dynamic response of thin plates on the elastic foundations due to the moving loads by means of the variational calculus. In the text we take the mass of moving loads into account, treat a series of questions such as the forced oscillations, the influence surfaces of the flexions and the influence surfaces of the inner forces, resonance conditions and critical speed and so forth.     

AN IMPLICIT SURFACE TENSION ALGORITHM FOR PICARD SOLVERS OF SURFACE–TENSION–DOMINATED FREE AND MOVING BOUNDARY PROBLEMS

One of the methods for solving a free or moving boundary problem is the use of Picard solvers which solve the geometry and the velocity field successively. When, however, the kinematic condition is used for updating the geometry in this technique, numerical stability problems occur for surface-tension-dominated flow. These problems are shown here to originate from the unstable integration of the local smoothing of the surface by surface tension. By an extension of the surface tension contribution to the flow field an implicit treatment of surface tension is obtained which overcomes these stability problems. The algorithm is applicable to both free and moving boundary problems, as will be shown by examples in this paper.     

高压储气罐泄漏喷射时冲击载荷数值计算

采用作者研发的用于含运动可变形物体流场计算的自适应非结构网格生成技术、动边界条件及数值处理方法,对地下高压储气罐发生泄漏喷射时罐盖（或破片）运动过程以及周围环境目标点上的冲击载荷进行了数值计算,讨论了在不同的罐盖质量、喷射孔径条件下目标点上冲击载荷的变化情况。研究结果表明,罐盖（或破片）的运动对爆炸场冲击载荷的影响是显著的。     

Experimental Testing of a Moving Force Identification Bridge Weigh-in-Motion Algorithm

Bridge weigh-in-motion systems are based on the measurement of strain on a bridge and the use of the measurements to estimate the static weights of passing traffic loads. Traditionally, commercial systems employ a static algorithm and use the bridge influence line to infer static axle weights. This paper describes the experimental testing of an algorithm based on moving force identification theory. In this approach the bridge is dynamically modeled using the finite element method and an eigenvalue reduction technique is employed to reduce the dimension of the system. The inverse problem of finding the applied forces from measured responses is then formulated as a least squares problem with Tikhonov regularization. The optimal regularization parameter is solved using the L-curve method. Finally, the static axle loads, impact factors and truck frequencies are obtained from a complete time history of the identified moving forces.     

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.     

An enriched finite element algorithm for the implicit simulation of the Stefan problem

An implicit enriched finite element algorithm is proposed to simulate heat transfer involving isothermal phase changes. This technique is based on a mixed variational formulation discretized by means of an enriched finite element approximation of the enthalpy in space. The interface is implicitly described without coupling with an interface-capturing technique. The time integration is carried out with an implicit (backward) Euler algorithm in time. Two examples in 1D and 2D clearly evidence the efficiency of the method developed.     

非线性Schrdinger方程的保辛数值求解

首先将非线性Schrdinger方程化为Hamilton正则方程形式,而后建立Hamilton体系下的变分原理。再用有限元法离散空间坐标,同时对时间坐标进行精细积分,最后运用混合能变分原理,提出非线性Schrdinger方程保辛数值解法。这种解法在保辛的同时,可以让能量和质量在积分格点上亦全部达到守恒。数值算例验证了该方法的有效性。     

On the incremental singular value decomposition method to support unsteady adjoint-based optimization

This paper proposes and evaluates an approximation model based on an incremental Singular Value Decomposition (iSVD) algorithm, for unsteady flow field reconstructions, needed for integrating the unsteady adjoint equations backward in time, within a gradient-based optimization loop. Due to the iSVD algorithm, the computational cost of solving the unsteady adjoint equations is reduced considerably, without practically affecting the accuracy of the computed gradient. Approximations to the unsteady flow fields are constructed while solving the time-varying flow equations (moving forward in time) and used to reconstruct these fields during the backward-in-time integration of the continuous adjoint equations. Optimization results obtained using the proposed method are compared to those computed using the binomial checkpointing technique, which acts as the reference method. Test cases for both flow control and shape optimization problems are presented.     

基于变分方法的工字型悬臂梁尺寸优化

以变分方法为工具,研究了分布载荷下工字型悬臂梁的尺寸优化问题,给出了在强度和位移约束下梁缘条沿展向的宽度函数,并简要介绍了优化准则法和多岛遗传算法. 以平直机翼为工程实例,将之简化为一个工字型悬臂梁,分别采用该3 种方法进行优化设计. 结果表明变分法不仅效率高,而且给出了尺寸分布的函数,这对飞行器概念设计阶段的机翼结构设计具有较大的指导意义.     

An Alternate Method to the Alternating Time-Frequency Method

A variational approach is developed which permits the calculation of thesteady-state time domain response of nonlinear ordinary differentialequations (ODE). Unlike numerical integration, transient calculationsare avoided and unlike harmonic balance, all calculations are performedin a single domain, namely the time domain. The relationships areestablished between the developed method and existing techniques such asfinite element in time (FET), standard finite differences (FD) and theharmonic balance method (HBM). The proposed technique also includes anarclength continuation algorithm allowing efficient parametric studiesto be performed. Local stability of the solutions is also assessed.     

Hamel框架下几何精确梁的离散动量守恒律

Hamel场变分积分子是一种研究场论的数值方法, 可以通过使用活动标架规避几何非线性带来的计算复杂度, 同时数值上具有良好的长时间数值表现和保能动量性质. 本文在一维场论框架下, 以几何精确梁为例, 从理论上探究Hamel场变分积分子的保动量性质. 具体内容包括: 利用活动标架法对几何精确梁建立动力学模型, 通过变分原理得到其动力学方程, 利用其动力学方程及Noether定理得到系统动量守恒律; 将几何精确梁模型离散化, 通过变分原理得到其Hamel场变分积分子, 利用Hamel场变分积分子和离散Noether定理得到离散动量守恒律, 并给出离散动量的一阶近似表达式; Hamel场变分积分子可在计算中利用系统对称性消除系统运动带来的非线性问题, 但此框架中离散对流速度、离散对流 应变及位形均不共点, 而这种错位导致离散动量中出现级数项, 本文对几何精确梁的离散动量与连续形式的关系及其应 用进行了讨论, 并通过算例验证了结论. 上述证明方法也同样适用一般经典场论场景下的Hamel场变分积分子. Hamel场变分积分子的动量守恒为进一步研究其保结构性质提供了参考依据.      

Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws

A physically sound three-dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented. It is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of Mittag–Leffler function type. A time integration scheme, based on the Generalized Midpoint rule together with the Grünwald algorithm for numerical fractional differentiation, for integration of the constitutive response is developed. The predictive capability of the viscoelastic model for describing creep, relaxation and damped dynamic responses is investigated both analytically and numerically. The algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code Abaqus. The algorithm is then used together with an explicit difference scheme for integration of structural responses. In numerical examples, the quasi-static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated.