数值摄动算法及其CFD格式

作者提出的数值摄动算法把流体动力学效应耦合进NS方程组和对流扩散（CD）方程离散的数学基本格式（MBS）,特别是耦合进最简单的MBS即一阶迎风和二阶中心格式之中,由此构建成一系列新格式,称呼方便和强调耦合流体动力学起见,称它们为流体力学基本格式（FMBS）。构建FMBS的主要步骤是把MBS中的通量摄动重构为步长的幂级数,利用空间分裂和导出的高阶流体动力学关系式,把结点变量展开成Taylor级数,通过消除重构格式修正微分方程的截断误差诸项求出幂级数的待定系数,由此获得非线性FMBS。FMBS的公式是MBS与 （及 ）之简单多项式的乘积, 和 分别是网格Reynolds数和网格CFL数。FMBS和MBS使用相同结点,简单性彼此相当,但FMBS精度高稳定范围大,例如FMBS包含了许多绝对稳定和绝对正型、高阶迎风和中心有限差分（FD）格式和有限体积（FV）格式,这些格式对网格Reynolds数的任意值均为不振荡格式。可见对不振荡CFD格式的构建,数值摄动算法提供了不同于调节数值耗散等常见的人为构建方法,而利用流体力学自身关系以及把迎风机制通过上、下游摄动重构引入中心MBS的解析构建方法,FMBS除了直接应用于流体计算外;对于通过调节数值耗散、色散和数值群速度特性重构高分辨率格式的研究,最简单FMBS提供了比最简单MBS更精确、但同样简单的基础和起步格式。FMBS用于计算不可压缩流,可压缩流,液滴萃取传质,微通道两相流等,均获得良好数值结果或与已有Benchmark解一致的数值结果。已有文献称数值摄动算法为新型高精度格式和高的算法和高的格式;本文FMBS比数值摄动格式的称呼可更好反映FMBS的物理内容。文中也讨论了值得进一步研究的一些课题,该法亦可用于其它一些数学物理方程（例如,简化Boltzmann方程、磁流体方程、KdV-Burgers方程等）MBS耦合物理动力学效应的重构。      

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

The finite element absolute nodal coordinate formulation (ANCF) is often used in modeling very flexible bodies in multibody system (MBS) applications. This formulation leads to a constant mass matrix, allowing for an efficient sparse matrix implementation. Nonetheless, the use of the ANCF finite elements to model stiff structures can lead to high frequencies associated with ANCF coupled deformation modes, as discussed in the literature. Implicit numerical integration methods can be effectively used to develop efficient procedures for the solution of MBS differential/algebraic equations. Most existing implicit integration algorithms, however, require numerical differentiation of the equations of motion, and some of these integration methods do not ensure that the kinematic algebraic constraint equations are satisfied at all levels (position, velocity, and acceleration). Because of these limitations, existing implicit integration methods can be less accurate and less efficient when used to solve large scale MBS applications. In order to circumvent this problem, the two-loop implicit sparse matrix numerical integration (TLISMNI) method was proposed for the solution of MBS differential/algebraic equations. The TLISMNI method does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation. This paper discusses TLISMNI implementation issues including the step size selection, the error control, and the effect of the numerical damping. The relation between the step size selection and the structure stiffness is also discussed. The use of the computer implementation described in this paper is demonstrated by solving very stiff structure problems using the Hilber?CHughes?CTaylor (HHT) method, which includes numerical damping. An eigenvalue analysis and Fast Fourier Transform (FFT) are performed in order to identify the fundamental modes of deformation and demonstrate that the contributions of these fundamental modes can be erroneously damped out when some other implicit integration methods are used. The TLISMNI method, on the other hand, captures the contributions of these fundamental modes. The results, obtained using the TLISMNI method, are compared with the results obtained using other methods including the implicit HHT-I3 and the explicit Adams integration methods. The results obtained show that the TLISMNI method can be five times faster than the other two methods when no numerical damping is considered.     

Nonstructural geometric discontinuities in finite element/multibody system analysis

Existing multibody system (MBS) algorithms treat articulated system components that are not rigidly connected as separate bodies connected by joints that are governed by nonlinear algebraic equations. As a consequence, these MBS algorithms lead to a highly nonlinear system of coupled differential and algebraic equations. Existing finite element (FE) algorithms, on the other hand, do not lead to a constant mesh inertia matrix in the case of arbitrarily large relative rigid body rotations. In this paper, new FE/MBS meshes that employ linear connectivity conditions and allow for arbitrarily large rigid body displacements between the finite elements are introduced. The large displacement FE absolute nodal coordinate formulation (ANCF) is used to obtain linear element connectivity conditions in the case of large relative rotations between the finite elements of a mesh. It is shown in this paper that a linear formulation of pin (revolute) joints that allow for finite relative rotations between two elements connected by the joint can be systematically obtained using ANCF finite elements. The algebraic joint constraint equations, which can be introduced at a preprocessing stage to efficiently eliminate redundant position coordinates, allow for deformation modes at the pin joint definition point, and therefore, this new joint formulation can be considered as a generalization of the pin joint formulation used in rigid MBS analysis. The new pin joint deformation modes that are the result of C ⁰ continuity conditions, allow for the calculations of the pin joint strains which can be discontinuous as the result of the finite relative rotation between the elements. This type of discontinuity is referred to in this paper as nonstructural discontinuity in order to distinguish it from the case of structural discontinuity in which the elements are rigidly connected. Because ANCF finite elements lead to a constant mass matrix, an identity generalized mass matrix can be obtained for the FE mesh despite the fact that the finite elements of the mesh are not rigidly connected. The relationship between the nonrational ANCF finite elements and the B-spline representation is used to shed light on the potential of using ANCF as the basis for the integration of computer aided design and analysis (I-CAD-A). When cubic interpolation is used in the FE/ANCF representation, C ⁰ continuity is equivalent to a knot multiplicity of three when computational geometry methods such as B-splines are used. C ² ANCF models which ensure the continuity of the curvature and correspond to B-spline knot multiplicity of one can also be obtained. Nonetheless, B-spline and NURBS representations cannot be used to effectively model T-junctions that can be systematically modeled using ANCF finite elements which employ gradient coordinates that can be conveniently used to define element orientations in the reference configuration. Numerical results are presented in order to demonstrate the use of the new formulation in developing new chain models.     

Partition method and experimental validation for impact dynamics of flexible multibody system

The impact problem of a flexible multibody system is a non-smooth, high-transient, and strong-nonlinear dynamic process with variable boundary. How to model the contact/impact process accurately and efficiently is one of the main difficulties in many engineering applications. The numerical approaches being used widely in impact analysis are mainly from two fields: multibody system dynamics (MBS) and computational solid mechanics (CSM). Approaches based on MBS provide a more efficient yet less accurate analysis of the contact/impact problems, while approaches based on CSM are well suited for particularly high accuracy needs, yet require very high computational effort. To bridge the gap between accuracy and efficiency in the dynamic simulation of a flexible multibody system with contacts/impacts, a partition method is presented considering that the contact body is divided into two parts, an impact region and a non-impact region. The impact region is modeled using the finite element method to guarantee the local accuracy, while the non-impact region is modeled using the modal reduction approach to raise the global efficiency. A three-dimensional rod-plate impact experiment is designed and performed to validate the numerical results. The principle for how to partition the contact bodies is proposed: the maximum radius of the impact region can be estimated by an analytical method, and the modal truncation orders of the non-impact region can be estimated by the highest frequency of the signal measured. The simulation results using the presented method are in good agreement with the experimental results. It shows that this method is an effective formulation considering both accuracy and efficiency. Moreover, a more complicated multibody impact problem of a crank slider mechanism is investigated to strengthen this conclusion.     

Reference-free mixed FE-MBS approach for beam structures with constraints

In this paper, an efficient way to simulate elastic beam structures with large translations and deformations is derived. The formulation is close to classical multi-body system (MBS) formulations and makes use of two coordinate sets: One providing equations of motion in a compact form and one for the coupling of single elements to one finite-element structure. Therefore, a coordinate transformation is needed. Constraints within the structure are considered following the common formulations of MBSs. Both uni- and bilateral constraints can be modelled. The system is extended to impacts on the structure. The model is validated by a comparison to a classical FEM model. A beam under different loads is given as an example especially for the treatment of constraints and impacts.     

An approach for the coupled simulation of machining processes using multibody system and smoothed particle hydrodynamics algorithms

The prediction accuracy of a simulation method is limited by its theoretical background. This fact can lead to disadvantages regarding the simulation quality when investigating systems of high complexity, e.g. containing components showing a fairly different behavior. To overcome this limitation, co-simulation approaches are used more and more, combining the advantages of different simulation disciplines. That is why we propose a new strategy for the dynamic simulation of cutting processes. The method couples Lagrangian particle methods, such as the smoothed particle hydrodynamics (SPH) method, and multibody system (MBS) tools using co-simulations. We demonstrate the capability of the new approach by providing simulation results of an orthogonal cutting process and comparing them with experimental data.     

Comparison of Results in Flexible Multibody Dynamics Using Various Approaches

Throughout the past decades a considerable amount of work has been dedicated to the development and application of formalisms to simulate flexible bodies in a multibody system (MBS). The two most common approaches to describe the deformation of a flexible body with respect to their floating frame of reference are to apply the linear finite-element formulation and the linearmodal approach using global mode shapes.On the field of flexible body modelling, this paper discusses two topics: (a) the consideration of quadratic terms in the equations of deformation and (b) the presentation of methods for global mode preparation to reduce the computer time for dynamic simulations without loss of accuracy regarding deformations.The proposed methods are applied in two examples, for the deformation of a stabilisation linkage of a car's front suspension and for the simulation of the deployment of a flexible solar array.     

A locomotive’s dynamic response to in-service parameter variations of its hydraulic yaw damper

An improved hydraulic yaw damper model with series in-service clearance and comprehensive stiffness was proposed by Wang et al. (Nonlinear Dyn 65(1–2):13–34 2011). In order to study how in-service parameter variations to the hydraulic yaw damper affect the dynamics of a Chinese \(\hbox {SS}_{9}\) locomotive, this study continued that research by establishing a multibody system (MBS) model of the \(\hbox {SS}_{9}\) locomotive–rail coupling system, and then validating the MBS model using field test data from the \(\hbox {SS}_{9}\) . Extensive simulations were performed, and the results demonstrated that both the effective stiffness and the small clearance accumulated between two ends of the damper due to wear and lack of maintenance had remarkable impacts on the locomotive’s critical speed and on its normal operation. The results also influenced the locomotive’s ride comfort, but the effect of the small clearance was more remarkable than that of the effective stiffness in this regard, and these parameters had little to no influence on the locomotive’s curve-negotiation performance. The small clearance and effective stiffness are usually omitted or simplified in engineering, and so it was important to apply the proposed in-service nonlinear damper model with series clearance and stiffness to a vehicle dynamics study and improve the accuracy of vehicle design. The study was also useful for setting pertinent vehicle maintenance standards in engineering to control the influence of such in-service parameter variations.     

A comparative study of joint formulations: application to multibody system tracked vehicles

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.