TWO SIMPLE FAST INTEGRATION METHODS FOR LARGE-SCALE DYNAMIC PROBLEMS IN ENGINEERING

A new simple explicit two-step method and a new family of predictor–corrector integration algorithms are developed for use in the solution of numerical responses of dynamic problems. The proposed integration methods avoid solving simultaneous linear algebraic equations in each time step, which is valid for arbitrary damping matrix and diagonal mass matrix frequently encountered in practical engineering dynamic systems. Accordingly, computational speeds of the new methods applied to large system analysis can be far higher than those of other popular methods. Accuracy, stability and numerical dissipation are investigated. Linear and nonlinear examples for verification and applications of the new methods to large-scale dynamic problems in railway engineering are given. The proposed methods can be used as fast and economical calculation tools for solving large-scale nonlinear dynamic problems in engineering.     

Modeling railroad track structures using the finite segment method

In the finite segment method, the dynamics of a deformable body is described using a set of rigid bodies that are connected by elastic force elements. This approach can be used, as demonstrated in this investigation, in the simulation of some rail movements. In order to ensure that the rail geometry is not distorted as the result of the finite segment displacements, a new track model that consistently integrates the absolute nodal coordinate formulation (ANCF) geometry and the finite segment method is developed. ANCF finite elements define the track geometry in the reference configuration as well as the change in the geometry due to the movement of the finite segments of the track. Using ANCF geometry and the finite segment kinematics, the location of the wheel/rail contact point is predicted online and used to update the creepage expressions due to the finite segment displacements and rotations. The location of the wheel/rail contact point and the updated creepage expressions are used to evaluate the creep forces. A three-dimensional elastic contact formulation (ECF-A) which allows for wheel/rail separation is used in this investigation. The rail displacement due to the applied loads is modeled by a set of rigid finite segments that are connected by a set of spring-damper elements. Each rail finite segment is assumed to have six rigid body degrees of freedom. The equations of motion of the finite segments are integrated with the railroad vehicle system equations of motion in a sparse matrix formulation. The resulting dynamic equations are solved using a predictor–corrector numerical integration method that has a variable order and variable step size. The finite segments may be used to model specific phenomena that occur in railroad vehicle applications, including rail rotations and gauge widening. The procedure used in this investigation to implement the finite segment method in a general purpose multibody system (MBS) computer program is described. Two simple models are presented in order to demonstrate the implementation of the finite segment method in MBS algorithms. The limitations of using the finite segments approach for modeling the track structure and rail flexibility are also discussed.     

Improved numerical solutions of nonlinear oscillatory systems

We consider numerical solutions of nonlinear oscillatory systems where closed-form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stability, and speed of the generalized Newmark scheme of Zienkiewicz and Taylor using one-step multiple-value algorithms. Previously, a Laplace transform-based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higher order Newton-Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's, and Mathieu equations. Comparisons are carried out regarding speed, accuracy, and stability, of results from the Runge-Kutta and one-step multiple-value methods with remainder (truncation error) terms, previously not considered. Results compare favorably. A system of two-degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple-degree of freedom systems.     

Process of the Deposition of Charged Polydisperse Gas Suspension on the Plate Surface in an Electrical Field

The process of powder sputtering onto a flat surface in an electric field is described on the basis of the numerical solution of a system of equations of the dynamics of a polydisperse gas suspension. The model includes equations for the motion of the carrier and disperse phases under the action of the aerodynamic friction force and the Coulomb force, taking into account the interphase exchange of momentum and energy. The system is solved by the explicit predictor–corrector method with splitting over the spatial directions and the nonlinear correction scheme. The numerical model is used to obtain the velocity and density fields of the gas suspension in the interelectrode space and on the surface of the target electrode.     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

考虑桥面等级退化影响的风-车流-桥梁耦合振动分析

综合考虑了车流随机性和桥面等级退化等因素,提出了一种新的分析模型,该模型能分析在役桥梁不同服役期间在交通荷载及风耦合作用下的桥梁动态响应。基于已有车辆模型,建立了一个可考虑车辆纵向振动的18自由度空间车辆模型,考虑邻近车辆影响基础之上的改进CA(Cellular Automation-元胞自动机)模型和桥面退化模型,并引入等效车轮荷载的方法,通过风、桥梁和车辆相互作用关系,建立了风-车流-桥梁系统的耦合振动分析模型。数值计算表明:该文所提出的方法能够合理的模拟风-车流-桥梁系统的耦合振动。     

Assessment of explicit and semi‐explicit classes of model‐based algorithms for direct integration in structural dynamics

The ‘model‐based’ algorithms available in the literature are primarily developed for the direct integration of the equations of motion for hybrid simulation in earthquake engineering, an experimental method where the system response is simulated by dividing it into a physical and an analytical domain. The term ‘model‐based’ indicates that the algorithmic parameters are functions of the complete model of the system to enable unconditional stability to be achieved within the framework of an explicit formulation. These two features make the model‐based algorithms also potential candidates for computations in structural dynamics. Based on the algorithmic difference equations, these algorithms can be classified as either explicit or semi‐explicit, where the former refers to the algorithms with explicit difference equations for both displacement and velocity, while the latter for displacement only. The algorithms pertaining to each class are reviewed, and a new family of second‐order unconditionally stable parametrically dissipative semi‐explicit algorithms is presented. Numerical characteristics of these two classes of algorithms are assessed under linear and nonlinear structural behavior. Representative numerical examples are presented to complement the analytical findings. The analysis and numerical examples demonstrate the advantages and limitations of these two classes of model‐based algorithms for applications in structural dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Matrix differential equation and higher-order numerical methods for problems of non-linear creep,viscoelasticity and elasto-plasticity

The constitutive equation is assumed in a very general form which includes as special cases non-linear creep, incremental elasto-plasticity as well as viscoelasticity represented by a chain of n standard solid models. Subdividing the structure into N finite elements, the problem of structural analysis is formulated with a system of 6N(n + 1) ordinary non-linear first-order differential equations in terms of the components of stresses and strains in the elements. This formulation enables one to apply Runge–Kutta methods or the predictor–corrector methods.     

Numerical integration of non-linear elastic multi-body systems

This paper is concerned with the modelling of nonlinear elastic multi-body systems discretized using the finite element method. The formulation uses Cartesian co-ordinates to represent the position of each elastic body with respect to a single inertial frame. The kinematic constraints among the various bodies of the system are enforced via the Lagrange multiplier technique. The resulting equations of motion are stiff, non-linear, differential-algebraic equations. The integration of these equations presents a real challenge as most available techniques are either numerically unstable, or present undesirable high frequency oscillations of a purely numerical origin. An approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system, whereas the forces of constraint are discretized so that the work they perform vanishes exactly. The combination of these two features of the discretization guarantees the stability of the numerical integration process for non-linear elastic multi-body systems. Examples of the procedure are presented.     

Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

列车通过小半径反向曲线桥梁的动力相互作用分析

以新建的某铁路货运专线上的反向曲线上的小半径曲线桥梁为例,考虑列车曲线通过的特点及轮轨非线性相互作用,建立空间的列车和曲线桥梁的振动方程,运用模态叠加法求解振动方程。通过车桥耦合振动专用分析程序VBC2.0的数值分析,得出列车通过该反向曲线上小半径曲线梁桥的一些振动规律,并给出我国货运列车通过该桥梁的合理行车速度及其他建议。     

High‐order predictor–corrector algorithms

New predictor–corrector algorithms are presented for the computation of solution paths of non‐linear partial differential equations. The predictors and the correctors are based on perturbation techniques and Padé approximants. This extends the Asymptotic Numerical Method (ANM), which is an efficient high‐order continuation technique without corrector. The efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier–Stokes equations. Many variants have been tested to establish an optimal algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

An improved predictor/multi-corrector algorithm for a time-discontinuous Galerkin finite element method in structural dynamics

This work presents an improved predictor/multi-corrector algorithm for linear structural dynamics problems, based on the time-discontinuous Galerkin finite element method. The improved algorithm employs the Gauss–Seidel method to calculate iteratively the solutions that exist in the phase of the predictor/multi-corrector of the numerical implementation. Stability analyses of iterative algorithms reveal that such an improved scheme retains the unconditionally stable behavior with greater efficiency than another iterative algorithm. Also, numerical examples are presented, demonstrating that the proposed method is more stable and accurate than several commonly used algorithms in structural dynamic applications. Received 18 June 1999     

Discontinuous variational time integrators for complex multibody collisions

The objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time t_k, representing predictor and corrector configurations of the system. The resulting sequence of configuration pairs then represents a discontinuous—or non‐classical—trajectory. Continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times. In particular, in systems subject to one‐sided contact constraints simulated via discontinuous variational time integrators, the predictor configuration is not required to satisfy the one‐sided constraints, whereas the corrector configuration is obtained by a closest‐point projection (CPP) onto the admissible set. The resulting trajectories are generally discontinuous, or non‐classical, but are expected to converge to classical or continuous solutions for decreasing time steps. We account for dissipation, including friction, by means of a discrete Lagrange–d'Alembert principle, and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems, and the right rate of energy decay in dissipative systems. The structure, range and scope of the discontinuous variational time integrators, and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics. Copyright © 2014 John Wiley & Sons, Ltd.     

A GENERALIZED VARIABLE FORMULATION FOR GRADIENT DEPENDENT SOFTENING PLASTICITY

A mesh-independent finite element method for elastoplastic problems with softening is proposed. The regularization of the boundary value problem is achieved introducing in the yield function the second order gradient of the plastic multiplier. The backward-difference integrated finite-step problem enriched with the gradient term is given a variational formulation where the consitutive equations are treated in weak form as well as the other field equations. A predictor–corrector scheme is proposed for the solution of the non-linear algebraic problem resulting from the finite element discretization of the functional. The expression of the consistent tangent matrix is provided and the corrector phase is formulated as a Linear Complementarity Problem. The effectiveness of the proposed methodology is verified by one- and two-dimensional tests.     

A computational approach for predicting the hydroelasticity of flexible structures based on the pressure Poisson equation

This work is concerned with the modeling of the interaction of fluid flow with flexible solid structures. The flow under consideration is governed by the Navier–Stokes equations for incompressible viscous fluids and modeled with low‐order velocity–pressure finite elements. The motion of the fluid domain is accounted for by the arbitrary Lagrangian–Eulerian formulation. The structure is represented by means of an appropriate standard finite element formulation. The spring smooth analogy is used to mesh control. The time integrating algorithm is based on the predictor–multi‐corrector algorithm. An important aspect of the present work is the introduction of a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve the hydroelasticity problem of an incompressible viscous fluid with an elastic body that is vibrating due to flow excitation. The PPE is derived to be consistent with the coupled system equation for the fluid–structure interaction (FSI). Based on this approach, an efficient monolithic method is adopted to simulate hydroelasticity between the flexible structure and the flow. The fluid pressure is implicitly derived to satisfy the incompressibility constraint, and the other unknown variables are explicitly derived. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite. To demonstrate the performance of the proposed approach, two working examples, a beam immersed in incompressible fluid and a guide vane of a Francis turbine passage, were used. The results show the validity of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Continuous casting of steels in Japan

The effective pair potentials of molten AgBr, CuBr, CuI and RbBr have been obtained from measured structural information by solving the modified hypernetted-chain (MHNC) equation coupled with a predictor–corrector method, recently proposed for a binary system. The usefulness of the pair potentials obtained in this work was suggested by reproducing the particular pre-peak in the partial structure factor of Cu–Cu pair in molten CuBr and CuI, because such pre-peak couldnot be given while using the model potentials previously proposed for these two molten salts. Some characteristic features in these four molten salts were also detected in the bridge function, which was introduced to the MHNC equation as a many-body interaction contribution.     

An optimized predictor–corrector scheme for fast 3D crack growth simulations

An optimized predictor–corrector scheme for the accelerated simulation of 3D fatigue crack growth is presented. Based on experimental evidence, it is assumed that the crack front shape ensures a constant energy release rate. Starting from a crack front satisfying this requirement a predictor step is performed. Usually, the new crack front does not fulfill the requirement of a constant energy release rate. Therefore, several corrector steps are needed. Within the new predictor–corrector scheme the history of crack growth is taken into account to reduce the number of corrector steps. The efficiency of the new scheme is shown on two numerical examples providing a speed up of a factor above three.     

基于面接触的车桥耦合振动研究

现有分析车-桥耦合振动的研究中都假设移动车辆与路面的接触关系为点接触。事实上,轮胎与路面是通过面接触的。通过建立新的三维车轮模型,分析了面接触对车-桥耦合振动的影响,将车轮与路面的接触面模拟为长方形,通过接触面间的位移协调条件和力相互作用建立车-桥耦合振动方程。对车速、车轮刚度与阻尼、接触面尺寸大小、车辆数目等进行参数分析,研究了接触面对车-桥耦合振动的影响。数值计算与试验结果验证表明:所提出的模型能更准确合理的研究车桥耦合振动。     

Decomposition contact response (DCR) for explicit finite element dynamics

We propose a new explicit contact algorithm for finite element discretized solids and shells with smooth and non‐smooth geometries. The equations of motion are integrated in time with a predictor‐corrector‐type algorithm. After each predictor step, the impenetrability constraints and the exchange of momenta between the impacting bodies are considered and enforced independently. The geometrically inadmissible penetrations are removed using closest point projections or similar updates. Penetration is measured using the signed volume of intersection described by the contacting surface elements, which is well‐defined for both smooth and non‐smooth geometries. For computing the instantaneous velocity changes that occur during the impact event, we introduce the decomposition contact response method. This enables the closed‐form solution of the jump equations at impact, and applies to non‐frictional as well as frictional contact, as exemplified by the Coulomb frictional model. The overall algorithm has excellent momentum and energy conservation characteristics, as several numerical examples demonstrate. Copyright © 2005 John Wiley & Sons, Ltd.