Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations

This investigation is concerned with the use of an implicit integration method with adjustable numerical damping properties in the simulation of flexible multibody systems. The flexible bodies in the system are modeled using the finite element absolute nodal coordinate formulation (ANCF), which can be used in the simulation of large deformations and rotations of flexible bodies. This formulation, when used with the general continuum mechanics theory, leads to displacement modes, such as Poisson modes, that couple the cross section deformations, and bending and extension of structural elements such as beams. While these modes can be significant in the case of large deformations, and they have no significant effect on the CPU time for very flexible bodies; in the case of thin and stiff structures, the ANCF coupled deformation modes can be associated with very high frequencies that can be a source of numerical problems when explicit integration methods are used. The implicit integration method used in this investigation is the Hilber–Hughes–Taylor method applied in the context of Index 3 differential-algebraic equations (HHT-I3). The results obtained using this integration method are compared with the results obtained using an explicit Adams-predictor-corrector method, which has no adjustable numerical damping. Numerical examples that include bodies with different degrees of flexibility are solved in order to examine the performance of the HHT-I3 implicit integration method when the finite element absolute nodal coordinate formulation is used. The results obtained in this study show that for very flexible structures there is no significant difference in accuracy and CPU time between the solutions obtained using the implicit and explicit integrators. As the stiffness increases, the effect of some ANCF coupled deformation modes becomes more significant, leading to a stiff system of equations. The resulting high frequencies are filtered out when the HHT-I3 integrator is used due to its numerical damping properties. The results of this study also show that the CPU time associated with the HHT-I3 integrator does not change significantly when the stiffness of the bodies increases, while in the case of the explicit Adams method the CPU time increases exponentially. The fundamental differences between the solution procedures used with the implicit and explicit integrations are also discussed in this paper.     

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.     

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.     

Integration of B-spline geometry and ANCF finite element analysis

The goal of this investigation is to introduce a new computer procedure for the integration of B-spline geometry and the absolute nodal coordinate formulation (ANCF) finite element analysis. The procedure is based on developing a linear transformation that can be used to transform systematically the B-spline representation to an ANCF finite element mesh preserving the same geometry and the same degree of continuity. Such a linear transformation that relates the B-spline control points and the finite element position and gradient coordinates will facilitate the integration of computer aided design and analysis (ICADA). While ANCF finite elements automatically ensure the continuity of the position and gradient vectors at the nodal points, the B-spline representation allows for imposing a higher degree of continuity by decreasing the knot multiplicity. As shown in this investigation, a higher degree of continuity can be systematically achieved using ANCF finite elements by imposing linear algebraic constraint equations that can be used to eliminate nodal variables. The analysis presented in this study shows that continuity of the curvature vector and its derivative which corresponds in the cubic B-spline representation to zero knot multiplicity can be systematically achieved using ANCF finite elements. In this special case, as the knot multiplicity reduces to zero, the recurrence B-spline formula causes two segments to automatically blend together forming one cubic segment defined on a larger domain. Similarly in this special case, the algebraic constraint equations required for the C ³ continuity convert two ANCF cubic finite elements to one finite element, demonstrating the strong relationship between the B-spline representation and the ANCF finite element representation. For the same order of interpolation, higher degree of continuity at the finite element interface can lead to a coarser mesh and to a lower dimensional model. Using the B-spline/ANCF finite element transformation developed in this paper, the equations of motion of a finite element mesh that represents exactly the B-spline geometry can be developed. Because of the linearity of the transformation developed in this investigation, all the ANCF finite element desirable features are preserved; including the constant mass matrix that can be used to develop an optimum sparse matrix structure of the nonlinear multibody system dynamic equations.     

Lie Algebraic Control for the Stabilization of Nonlinear Multibody System Dynamics Simulation

It is well known that when equations of motion are formulated using Lagrange multipliers for multibody dynamic systems, one obtains a redundant set of differential algebraic equations. Numerical integration of these equations can lead to numerical difficulties associated with constraint violation drift. One approach that has been explored to alleviate this difficulty has been contraint stabilization methods. In this paper, a family of stabilization methods are considered as partial feedback linearizing controllers. Several stabilization methods including the range space method, null space method, Baumgarte's method, and the damping and stiffness penalty methods are examined. Each can be construed as a particular partial feedback linearizing controller. The paper closes by comparing several of these constraint stabilization methods to another method suggested by construction: the variable structure sliding (VSS) control. The VSS method is found to be the most efficient, stable, and robust in the presence of singularities.     

带约束非线性多体系统动力学方程数值分析方法

Lagrange方法是建立带约束多体系统动力学方程的普遍方法之一 ,其方程的形式为微分 代数方程组 ,数值计算与数值分析是研究多体系统动力学特性的重要方法。本文利用缩并法给出了带约束多体系统动力学方程的隐式数值计算方法和Lyapunov指数的计算方法。将数值仿真、Lya punov指数计算和Poincare映射有机结合 ,分析非线性多体系统动力学行为。通过一个算例 ,说明该方法的有效性     

精细时程积分法及其数值衍生格式应用评估

旋翼气动弹性耦合动力学方程本质上是一组刚性比较大的非线性偏微分方程。在有限元结构离散后,可改写为非齐次微分方程组,其中非齐次项是桨叶运动量（位移与速度）和气动载荷的函数。针对这类方程,本文尝试引入精细积分法及其衍生格式,借助数值方法计算Duhamel积分项。从积分精度与数值稳定性方面比较研究具有代表性的精细库塔法和高精度直接积分法。结合隐式积分算法,评估精细积分法应用于旋翼动力学方程的可行性。算例表明,精细积分法对矩形直桨叶动力学方程具有足够的求解精度。     

非线性阻尼动力方程的复合积分法

非线性动力方程直接积分法的基础是构造$t$时刻与t+\Delta t时刻状态量间的关系, 由此形成基本量的非线性方程组, 再在每个时间步内采用 Newton-Raphson或BFGS等迭代方法求解. 该文基于Bathe复合积分法(composite implicit time integration), 提出了非线性阻尼系统基于速度变量的复合时间 积分迭代格式. 以非线性黏滞阻尼Sdof系统为例, 按上述方法以及基于BFGS迭代 的Newmark-\beta法编制Fortran程序, 结果与Adina软件对比, 验证了该文方法的有效性.     

病态代数方程的精细积分解法

基于精细积分思想,提出了一种有效的病态代数方程组求解方法。类似于稳态热传导方程可视为瞬态热传导方程的极限形式,将具有正定对称实系数矩阵的病态代数方程组归结为一个常微分方程组初值问题的极限形式,并在此基础上建立了病态代数方程组的精细积分解法。该方法不仅精度高,而且能以指数速度收敛,具有较高的效率。本文还讨论了病态代数方程...     

DYNAMIC ANALYSIS OF SPINNING DEPLOYMENT OF A SOLAR SAIL COMPOSED OF VISCOELASTIC MEMBRANES

近年来, 可用于航天器推进的太阳帆自旋展开技术引起人们广泛关注. 这类太阳帆可视为由中心旋转毂轮、若干柔性绳索、太阳帆薄膜和集中质量等组成的刚柔耦合多体系统.为了对系统中的太阳帆薄膜进行建模, 提出了基于绝对节点坐标方法描述的黏弹性薄板单元, 并对其有效性进行了验证.针对简化的"IKAROS"自旋展开太阳帆系统, 采用结合自然坐标方法与绝对节点坐标方法的绝对坐标方法对其进行建模, 采用广义-α方法对大规模系统动力学方程进行求解.研究了黏弹性太阳帆薄膜自旋展开过程的动力学特性, 讨论了薄膜的黏弹性阻尼对自旋展开过程的影响规律.     

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

α-GMRES: A new parallelizable iterative solver for large sparse non-symmetric linear systems arising from CFD

Linearization of the non-linear systems arising from fully implicit schemes in computational fluid dynamics often result in a large sparse non-symmetric linear system. Practical experience shows that these linear systems are ill-conditioned if a higher than first-order spatial discretization scheme is used. To solve these linear systems, an efficient multilevel iterative method, the α-GMRES method, is proposed which incorporates a diagonal preconditioning with a damping factor α so that a balanced fast convergence of the inner GMRES iteration and the outer damping loop can be achieved. With this simple and efficient preconditioning and damping of the matrix, the resulting method can be effectively parallelized. The parallelization maintains the effectiveness of the original scheme due to the algorithm equivalence of the sequential and the parallel versions.     

结构动力方程的增维精细积分法

对线性定常结构动力系统提出的精细积分方法,能够得到在数值上逼近于精确解的结果,但对于非齐次动力方程涉及到矩阵求逆的困难。提出采用增维的办法,将非齐次动力方程转化为齐次动力方程,在实施精细积分过程中不必进行矩阵求逆,这种方法对于程序实现和提高数值稳定性十分有利,而且在大型问题中计算效率较高,从而改进了精细积分方法的应用,数值例题显示了本文方法的有效性。     

On the Use of Implicit Integration Methods and the Absolute Nodal Coordinate Formulation in the Analysis of Elasto-Plastic Deformation Problems

In the general theory of continuum mechanics, the state of rotation and deformation of material points can be uniquely defined from the displacement field by using the nine independent components of the displacement gradients. For this reason, the use of the absolute rotation parameters as nodal coordinates, without relating them to the displacement gradients, leads to coordinate redundancy that leads to numerical and fundamental problems in many existing large rotation finite element formulations. Because of this fundamental problem, special measures that require modifications of the numerical integration methods were proposed in the literature in order to satisfy the principle of work and energy. As demonstrated in this paper, no such measures need to be taken when the finite element absolute nodal coordinate formulation is used since the principle of work and energy are automatically satisfied. This formulation does not suffer from the problem of coordinate redundancy and ensures the continuity of stresses and strains at the nodal points. In this study, the use of the implicit integration methods with the consistent Lagrangian elasto-plastic tangent moduli is examined when the absolute nodal coordinate formulation is used. The performance of different numerical integration methods in the dynamic analysis of large elasto-plastic deformation problems is investigated. It is shown that all these methods, in the case of convergence, yield a solution that satisfies the principle of work and energy without the need of taking any special measures. Semi-implicit integration methods, however, can lead to numerical difficulties in the case of very stiff problems due to the linearization made in these methods in order to avoid the iterative Newton--Raphson procedure. It is also demonstrated that the use of the consistent Lagrangian-plastic tangent moduli derived in this investigation using the absolute nodal coordinate formulation leads to better convergence of the iterative Newton--Raphson procedure used in the implicit integration methods.     

多柔体系统数值分析的模型降噪方法

多柔体系统的动力学方程通常是一组刚性微分方程, 目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量, 其求解效率难以令人满意. 为了降低多柔体系统动力学方程的刚性, 从而可采用ODE45等常规微分方程求解器进行求解, 研究了在建模过程中滤除高频振荡分量的方法. 在以当前时刻为起点的短时间内对柔性体的应力进行均匀化, 用均匀化后的应力计算柔性体的变形虚功率, 由此得到的系统动力学方程的解中不含过高频率的弹性振动, 并且可以通过调节均匀化时间区间的长度参数控制滤波的范围. 数值算例表明: 这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器, 并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率. 本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.      

基于边界值方法的微分动力系统数值计算方法

对高维非线性初值问题,微分求积法在每一步的积分过程中需要求解一个更高维的非线性方程组,因而计算量巨大。基于微分求积法与边界值方法两者之间的关系,可以将广义向后差分方法和扩展的隐式梯形积分方法看作是经典微分求积法的稀疏表达形式。将广义向后差分方法以及扩展的隐式梯形积分方法这两类边界值方法应用于微分动力系统的数值计算,提出了一类新的数值计算方法。理论分析及算例结果表明,对高维非线性微分初值问题的数值计算,本文方法相对于经典的微分求积法具有更高的计算效率。     

High-order time integration applied to metal powder plasticity

The present article treats two objectives. In the first investigation attention is focused on the application of time-adaptive finite elements formulated on the basis of a high-order time integration procedure on a constitutive model for compressible finite strain viscoplasticity for metal powder. In this connection, it has to be emphasized that the integration procedure is not only applied to the evolution equations on Gauss-point level but on the total system of differential–algebraic equations resulting from the application of the vertical line method on the quasi-static finite element equations. The specific application emerges from the field of metal powder compaction. Particular studies are carried out using stiffly accurate, diagonally implicit Runge–Kutta methods in combination with the Multilevel-Newton algorithm for solving the DAE-system. In this respect, the effort vs. accuracy behavior is investigated which is also related to order reduction known in elastoplasticity. The second topic treats the local stress algorithm for taking into account the yield function based finite strain viscoplasticity model, where the classical Newton–Raphson method fails. This is the reason why most constitutive models of powder materials are implemented into explicit finite element codes. Thus, the proposed investigations compare different methods in view of a stable and efficient integration process in implicit finite element formulations.     

求解非比例阻尼体系复模态的实模态摄动法

根据工程结构的实际情况,建立了非比例阻尼结构体系复模态特性的近似求解方 法------实模态摄动法. 这一方法以复Ritz向量展开原理为基础, 把非比例阻尼结构体系复模态特性的分析过程分解为两个基本步骤,首先以结构体系的实模 态向量构建复Ritz向量的求解子空间,然后通过非线性复代数方程组的求解代替扩阶后的复 特征值方程的求解,从而简化了计算过程.通过两个算例表明: 这一方法不仅计算简便，而且具有较高的计算精度和执行效率, 对于复杂的非比例阻尼系统是很适用的，具有一定的工程应用价值.     

带约束多体系统动力学方程的隐式算法

研究了带约束多体系统隐式算法,用子矩阵的形式推导出了多体系统正则方程的Ｊａｃｏｂｉ矩阵,它适用于多种隐式算法并给出了隐式Ｒｕｎｇｅ－Ｋｕｔｔａ算法,最后用一算例表明了隐式算法的计算效率和精度明显优于算法。