Projection methods in flexible multibody dynamics. Part II: Dynamics and recursive projection methods

In Part I of this paper the kinematic relationships between the absolute, elastic and joint accelerations are developed. In this paper, these kinematic equations are used with the generalized Newton-Euler equations and the relationship between the actual and generalized reaction forces to develop a recursive projection algorithm for the dynamic analysis of open-loop mechanical systems consisting of a set of interconnected rigid and deformable bodies. Optimal matrix permutation, partitioning and projection methods are used to eliminate the elastic accelerations while maintaining the inertia coupling between the rigid body motion and the elastic deformation. Recursive projection methods are then applied in order to project the inertia of the leaf bodies onto their parent bodies. This leads to an optimal symbolic factorization which recursively yields the absolute and joint accelerations, and the joint reaction forces. The method presented in this paper avoids the use of Newton-Raphson algorithms in the numerical solution of the constrained dynamic equations of open-loop kinematic chains since the joint accelerations are readily available from the solution of the resulting reduced system of equations. Furthermore, the method requires only the inversion or decomposition of relatively small matrices and the numerical integration of a minimum number of co-ordinates. Open-loop multibody robotic manipulator systems are used to compare the results and efficiency of the recursive methods with that of the augmented formulations that employ Newton-Raphson algorithms.     

A bimodal plasticity theory of fibrous composite materials

Summary It is shown that elastic-plastic response of metal matrix composites reinforced by aligned continuous fibers can be described in terms of two distinct modes. In the matrix-dominated mode, the composite deforms primarily by plastic slip in the matrix, on planes which are parallel to the fiber axis. In the fiber-dominated mode, both phases deform together in the elastic and plastic range. Constitutive equations are derived for the matrix-dominated mode of deformation in composites with elastic-perfectly plastic matrices. Response in the fiber-dominated mode is approximated by the self-consistent and Voigt models. The two deformation modes give different branches of the overall yield surface which identify the state of stress that activates a particular mode, and indicate the conditions for mode transition in a given composite system. The matrix-dominated mode is found to exist in systems reinforced by fibers of large longitudinal shear stiffness, such as boron or silicon carbide. Systems reinforced by more compliant fibers, such as graphite, appear to deform exclusively in the fiber-dominated mode. The results show good agreement with experimental data, and with predictions obtained from a more accurate material model. They also help to reconcile several different plasticity theories of fibrous composites, and suggest limits of their validity.With 9 FiguresPrepared for the Symposium on Plasticity: Foundations and Future Directions. In Memory of Aris Phillips. January 28–30, 1987. University of Florida, Gainesville, U.S.A.     

基于图非负矩阵分解的图像配准

本文提出一种新的利用图的谱对应绝对值特征向量的非负矩阵分解图像配准方法.首先利用图像特征构造了无向权图的非负权矩阵,通过非负矩阵分解得到了包含原始图像全部特征的特征基图像;然后将非负权矩阵谱对应绝对值特征向量作为非负矩阵分解的初始值进行迭代,既能反映图的结构特征信息,又能提高图像的匹配率;最后在特征基向量空间找到了两图...     

Field‐consistent higher‐order free‐interface component mode synthesis

A new free‐interface component mode synthesizing technique based on the notion of higher‐order field consistency is proposed. The present technique employs higher‐order residual attachment modes in addition to kept normal modes while consistency in matching field variables at the substructure interface is maintained. The present field‐consistency approach does not increase the size of the synthesized system even if higher‐order residual attachment modes are included. Higher‐order analyses greatly improves solution accuracy while the final system size always remains the same as the size of the kept normal modes regardless of the order of residual attachment modes. A general procedure of deriving higher‐order residual attachment modes is first described and then a new field‐consistent higher‐order synthesis technique is presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Decoupling joint and elastic accelerations in deformable mechanical systems using non‐linear recursive formulations

The aim of this paper is to develop non‐linear recursive formulations for decoupling joint and elastic accelerations, while maintaining the non‐linear inertia coupling between rigid body motion and elastic deformation in deformable mechanical systems. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large. Consequently, the use of these recursive formulations for solving the joint and elastic accelerations becomes less efficient. In this paper, the non‐linear recursive formulations have been used to decouple the elastic and joint accelerations in deformable mechanical systems. The relationships between the absolute, elastic and joint variables and generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. By using the inertia matrix structure of deformable mechanical systems and the fact that joint reaction forces associated with elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The use of the approaches developed in this investigation is illustrated using deformable open‐loop serial robot and closed‐loop four‐bar mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of Component Mode Synthesis to Protein Structure for Dynamic Analysis

This paper concerns the application of component mode synthesis for biomolecule modeling to understand protein dynamics. As for protein dynamics, eigenvalue problem should be formulated to obtain eigenvalue, eigenvector and thermal fluctuation. To describe the thermal fluctuation of protein, normal mode analysis is introduced and normal modes identify the dynamic behavior of protein very well. Component mode synthesis considers the given complex structure as an assembly of smaller components. The selection of a component may be arbitrary. When the component mode synthesis is applied to formulate the eigenvalue problem of protein structure, we selected a protein which may be composed of two and/or four domains. The domain of protein can be considered as a component. In this sense the number of component is increased as necessary, and the size of each component is decreased for fast calculation. The component mode synthesis widely used in engineering was well applied to understand protein dynamics in present study.     

Optimal component mode synthesis for medium frequency problem

The component mode synthesis (CMS) with fixed interface (denoted Craig–Bampton) method uses a combination of static and dynamic modes. The usual definition of this CMS leads to a coupling between static and dynamic modes which are not orthogonal with respect to the stiffness matrix. This type of basis is not well suited for dynamic explicit computations, because the resulting mass matrix is not diagonal. If one keeps the same basis mode set but uses an orthogonalization process with respect to the mass matrix, the quality of the reduced Craig–Bampton system is kept but the basis vectors are combined differently. The aim of this paper is to propose a new way to control the accuracy of the reduced dynamic system for a specific frequency domain. Thus a new CMS is defined in order to be accurate in the medium frequency range. Copyright © 2010 John Wiley & Sons, Ltd.     

复振型导数计算的可变移频值方法

实际结构系统由于存在多种不同性质的阻尼其动态特性很复杂,振型导数的计算也比较困难.采用模态加速和移频的思想发展了一种基于模态叠加的复振型导数计算方法.首先对控制方程进行移频处理,利用广义幂级数展开式获得模态迭代公式,并利用迭代结果与各阶振型表示复振型导数;然后把系统的广义动柔度矩阵表示为已知的低阶模态与截断的高阶模态之和,高阶模态部分采用多个矩阵多项式与一个广义幂级数的乘积表示,并利用系统的低阶模态和系统矩阵进行计算;各阶移频值表示为相应的移频系数与复特征值的乘积,它们仅与最低阶模态移频值的模和本阶模态的单位复特征值有关,而最低阶模态的移频系数通过精度分析获得.给出了合适的模态加速迭代次数.该方法仅需进行一次系统矩阵的分解就可获得高精度的多个复振型导数.算例表明方法正确、高效.     

具有弹性耦合结构振动系统的自由界面模态综合法

本文给出了以弹性连结件为耦合件的振动系统自由界面模态综合技术,把弹性连结件定义为“软子结构”单独处理,其余部件自然划分为若干子结构。利用有限单元法获取各子结构自由界面模态信息,用子结构的截断主模态及其高阶剩余柔度修正项作为结构 RayLeigh-Ritz 分析的假设模态,从而大幅度压缩了结构分析自由度。实例计算表明,该方法具有良好的综合精度,是分析具有弹性耦合结构振动问题的有效方法。     

Simulating deformation of objects with multi‐materials using boundary element method

In this paper, an algorithm for simulating the elastostatic deformation of multiple objects with different material properties using boundary element method is introduced. By tessellating the surface of a geometric model into elements and classifying all the element nodes into different groups with different attributes, and partitioning the stiffness matrix into several sub‐matrices according to these attributes, a compact expression for the unknown variables is obtained. Comparing with the direct matrix inversion method, the dimension of the system matrix in the expression has been effectively reduced. Besides, this expression shows that the deformation of a multi‐component object can be simulated in a way similar to that of a single‐component object. The capacitance method is adopted for evaluating the deformation of the object. Experimental results illustrate that the proposed method is practical and efficient. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved series expansion method of frequency response function under medium and high frequency excitations

In view of the computational divergence in the series expansion method of frequency response function under medium and high frequency excitations, a new improved algorithm for dynamics system response is proposed. Structural modes are divided into available low order modes and truncated high order modes. The frequency response function of truncated high order modes is expanded by the application of Taylor series on the basis of power series expansion and modal superposition method. According to the coupling characteristics between low order and high order modes to mass and stiffness matrices, the contribution of truncated high order modes to the frequency response function is expressed as the low order mode matrix and system matrix. The present method considers the relation between structure frequency and excitation frequency. The results show that the improved algorithm expands the series expansion method to the range of medium and high frequency excitations, and the calculation accuracy of the frequency response function is improved under the incomplete modal conditions. Numerical results validate that this method is feasible and has good convergence.     

A fast,memory efficient and robust sparse preconditioner based on a multifrontal approach with applications to finite‐element matrices

In this article, we introduce a fast, memory efficient and robust sparse preconditioner that is based on a direct factorization scheme for sparse matrices arising from the finite‐element discretization of elliptic partial differential equations. We use a fast (but approximate) multifrontal approach as a preconditioner and use an iterative scheme to achieve a desired accuracy. This approach combines the advantages of direct and iterative schemes to arrive at a fast, robust, and accurate preconditioner. We will show that this approach is faster (～2×) and more memory efficient (～2–3×) than a conventional direct multifrontal approach. Furthermore, we will demonstrate that this preconditioner is both faster and more effective than other preconditioners such as the incomplete LU preconditioner. Specific speedups depend on the matrix size and improve as the size of the matrix increases. The preconditioner can be applied to both structured and unstructured meshes in a similar manner. We build on our previous work and utilize the fact that dense frontal and update matrices, in the multifrontal algorithm, can be represented as hierarchically off‐diagonal low‐rank matrices. Using this idea, we replace all large dense matrix operations in the multifrontal elimination process with O(N) hierarchically off‐diagonal low‐rank operations to arrive at a faster and more memory efficient factorization scheme. We then use this direct factorization method at low accuracies as a preconditioner and apply it to various real‐life engineering test cases. Copyright © 2016 John Wiley & Sons, Ltd.     

A ‘nodeless’ dual superelement formulation for structural and multibody dynamics application to reduction of contact problems

A ‘nodeless’ superelement formulation based on dual‐component mode synthesis is proposed, in which the superelement dynamic behavior is described in terms of modal intensities playing the role of intrinsic variables. A computational scheme is proposed to build an orthogonal set of static modes so that the system matrices can have a diagonal or nearly diagonal form, providing thus high computational efficiency for application in the context of structural dynamics as well as flexible multibody dynamics. Connection to adjacent components is expressed through kinematic relationships between intrinsic variables and local displacements. The efficiency of the method is demonstrated on a simple example involving multiple unilateral contact. Copyright © 2015 John Wiley & Sons, Ltd.     

一种利用子结构综合技术的模型修正方法

提出了基于改进自由界面子结构模态综合法的结构模型修正方法。首先给出了一种改进的自由界面子结构模态综合法,该方法通过构造一组与系统低阶模态加权正交的向量集,有效地解决了含有刚体模态时系统剩余柔度矩阵的求解问题。然后利用摄动法对每个子结构进行摄动,并计算灵敏度,通过子结构综合技术得到由摄动量表示的综合系统方程和灵敏度方程,给出了基于灵敏度分析的修正计算方法。该方法仅需对综合方程进行修正,极大的减缩了待修正问题的计算规模,提高了修正效率。最后,以典型数值算例验证了该方法的有效性。     

Transient non-linear simulation with component mode synthesis

A method is presented to incorporate non-linear material behavior in a transient simulation with component mode synthesis reduction. This method is used in conjunction with rigid body solution techniques to expand the useful range of rigid body design tools for the development of complex mechanisms. A fixed interface component mode synthesis technique is enhanced with an algorithm to approximate the effect of plastic deformation during a dynamic simulation. The plastic strain is determined from the elastic modal response using classical plasticity theory and applied to the modal solution by projecting an effective nodal force vector on the modal coordinates to induce the necessary plastic deformation. This method can be used during the design process to approximate the non-linear dynamic response of complex mechanisms and offer significant computational savings over a full fidelity, non-linear dynamic solution.     

Projection methods in flexible multibody dynamics. Part I: Kinematics

In this paper a recursive projection method for the dynamic analysis of open-loop mechanical systems that consist of a set of interconnected deformable bodies is presented. The configuration of each body in the system is identified using a coupled set of reference and elastic co-ordinates. The absolute velocities and accelerations of leaf or child bodies in the open-loop system are expressed in terms of the absolute velocities and accelerations of the parent bodies and the time derivatives of the relative co-ordinates of the joints between the bodies. The dynamic differential equations of motion are developed for each link using the generalized Newton-Euler equations. The relationship between the actual joint reactions and the generalized forces combined with the kinematic relationships and the generalized Newton-Euler equations are used to develop a system of loosely coupled equations which has a sparse matrix structure. Using matrix partitioning and recursive projection techniques based on optimal block factorization an efficient solution for the system accelerations and joint reaction forces is obtained. This solution technique yields a much smaller operations count and can more effectively exploit vectorization and parallel processing. It also allows a systematic procedure for decoupling the joint and elastic accelerations.     

一种面向平面多刚柔系统的冲击响应建模和计算方法

结合传递矩阵方法建模灵活和计算效率高的优点,提出了一种基于“传递矩阵”概念的多体系统冲击响应建模和计算方法。以受基础冲击的平面多刚柔系统为研究对象,采用Newmark-β法对元件的方程高阶项进行线性化,用模态方法处理柔体的变形,建立了一般刚体和典型刚体(刚性均质矩形板、带弹性支撑的刚性均质矩形薄板)、一般柔体和典型柔体(Euler-Bernouni梁)的冲击扩展传递矩阵,冲击激励包含平动和转动两种成分,给出了基于Newmark-β法的系统响应数值迭代求解算法程序。用一个工程实例,通过与有限元方法的对比,验证了方法的准确性,得出了转动冲击激励成分对总体响应的贡献不能忽略的结论。方法的研究对象虽然只是平面多刚柔系统,但很容易推广到三维的情况。     

非对称粘滞阻尼系统的约束部件模态综合

本文对具非对称粘滞阻尼的振动系统提出一种约束部件模态综合法。首先消除状态空间部件运动方程内不独立的界面位移与界面速度，然后以截断的约束部件复模态、位移约束模态及速度约束模态作为里兹基，减缩总体方程。文末以算例说明了方法的有效性。     

A cellular Potts model for the MMP-dependent and -independent cancer cell migration in matrix microtracks of different dimensions

Cell migration is fundamental in a wide variety of physiological and pathological phenomena, among other in cancer invasion and development. In particular, the migratory/invasive capability of single metastatic cells is fundamental in determining the malignancy of a solid tumor. Specific cell migration phenotypes result for instance from the reciprocal interplay between the biophysical and biochemical properties of both the malignant cells themselves and of the surrounding environment. In particular, the extracellular matrices (ECMs) forming connective tissues can provide both loosely organized zones and densely packed barriers, which may impact cell invasion mode and efficiency. The critical processes involved in cell movement within confined spaces are (i) the proteolytic activity of matrix metalloproteinases (MMPs) and (ii) the deformation of the entire cell body, and in particular of the nucleus. We here present an extended cellular Potts model (CPM) to simulate a bio-engineered matrix system, which tests the active motile behavior of a single cancer cell into narrow channels of different widths. As distinct features of our approach, the cell is modeled as a compartmentalized discrete element, differentiated in the nucleus and in the cytosolic region, while a directional shape-dependent movement is explicitly driven by the evolution of its polarity vector. As outcomes, we find that, in a large track, the tumor cell is not able to maintain a directional movement. On the contrary, a structure of subcellular width behaves as a contact guidance sustaining cell persistent locomotion. In particular, a MMP-deprived cell is able to repolarize and follow the micropattern geometry, while a full MMP activity leads to a secondary track expansion by degrading the matrix structure. Finally, we confirm that cell movement within a subnuclear structure can be achieved either by pericellular proteolysis or by a significant deformation of cell nucleus.