Stable numerical integration of dynamical systems subject to equality state-space constraints

Numerical simulation of constrained dynamical systems is known to exhibit stability problems even when the unconstrained system can be simulated in a stable manner. We show that not the constraints themselves, but the transformation of the continuous set of equations to a discrete set of equations is the true source of the stability problem. A new theory is presented that allows for stable numerical integration of constrained dynamical systems. The derived numerical methods are robust with respect to errors in the initial conditions and stable with respect to errors made during the integration process. As a consequence, perturbations in the initial conditions are allowed. The new theory is extended to the case of constrained mechanical systems. Some numerical results obtained when implementing the numerical method here developed are shown.     

A recursive method for the dynamic analysis of mechanical systems in spatial motion

Summary. In the present study, a recursive method for generating the equations of motion of mechanical systems that undergo spatial motion is presented. The method uses the force and moment equations to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotation matrices. For the open loop case, the equations of motion are generated recursively along the serial chains. Closed loop systems are transformed to open loop systems by cutting suitable kinematic joints and introducing cut-joint constraints. The method is simple and suitable for computer implementation. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

Sensitivity analysis for dynamic mechanical systems with finite rotations

This paper presents a sensitivity analysis for dynamic systems with large rotations using a semi‐analytical direct differentiation technique. The choice of a suitable time integration strategy for the rotation group appears to be critical for the development of an efficient sensitivity analysis. Three versions of the generalized‐α time integration scheme are considered: a residual form, a linear form, and a geometric form. Their consistency is discussed, and we show that the residual form, which is the most direct extension of the generalized‐α algorithm defined in structural dynamics, should not be used for problems with large rotations. The sensitivity analysis is performed and close connections are highlighted between the structure of the sensitivity equations and of the linearized dynamic equations. Hence, algorithms developed for the original problem can be directly reused for the sensitivities. Finally, a numerical example is analysed in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

A form of the translational dynamical equations for relative motion in systems of many non-rigid bodies

Summary The translational dynamical equations for an arbitrary system of interconnected non-rigid bodies is written in terms of the relative translations between bodies, the barycentric vectors describing attachment points between bodies, and the deformational deflections of those points. The results are derived in terms of the characteristics of the system graph, in the spirit of the Roberson-Wittenburg formulation of the rotational dynamical equations.

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Eine Darstellung der die translatorische Relativbewegung in Systemen vieler verformbarer Körper charakterisierenden Differentialgleichungen

Zusammenfassung Die Differentialgleichungen der translatorischen Relativbewegung eines beliebigen Systems von miteinander verbundenen verformbaren Körpern werden in den Relativverschiebungen, in den die Anordnung des Systems beschreibenden baryzentrischen Vektoren und in den durch Deformation verursachten Verschiebungen der Befestigungspunkte der Teilkörper angegeben. Die Ergebnisse werden in den Charakteristiken des Systemgraphen hergeleitet.

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Dedicated to Professor Kurt Magnus in honor of his sixtieth birthday.     

A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non‐smooth boundaries

The aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and efficient collocation method based on Chebyshev polynomials of the first kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the efficiency of the numerical approach we propose. Copyright © 2003 John Wiley & Sons, Ltd.     

含界面相三维全五向编织复合材料拉伸行为模拟及失效机制研究

基于三维全五向（Q5D）编织复合材料的细观结构模型,通过引入界面相单元,建立了含界面相Q5D编织复合材料单轴拉伸损伤失效分析模型。应用Python语言实现对ABAQUS的二次开发,将Linde等提出的失效准则和Von-Mises应力准则分别用于纱线和基体的渐进损伤判断,并确定材料的整体失效模式;对于界面相,采用Quads准则进行损伤判断。利用周期性位移边界条件,对含界面相Q5D编织复合材料的纵向拉伸应力-应变行为进行了渐进损伤数值模拟,详细讨论了在纵向拉伸载荷作用下材料的细观损伤起始、扩展和最终失效的演化过程,分析了材料的细观损伤失效机制,预测了材料的极限破坏强度,并研究了界面相性能对材料整体力学行为的影响规律。研究结果表明,数值模拟结果与实验值吻合较好,验证了渐进损伤模型的有效性,为该类材料的力学分析和优化设计奠定了基础。     

Determination of the linear equations of position-sensing detectors for small motion measurement systems

Small motion measurement systems are widely used in industry measurement fields to measure small positional/angular motions. These systems usually consist of two parts: a measuring assembly and a reference assembly. The position-sensing detectors (PSDs) are embedded in either measuring assembly or reference assembly to sense the variations of laser light incidence points when there are any small positional/angular motions. To use these systems, it is necessary to determine the linear equations of PSD readings, which relate the six-degrees-of-freedom small positional/angular motions and PSD readings. The purpose of this paper is to derive these equations based on the paraxial raytracing method. Two measurement systems are used as illustrative examples to validate the proposed methodology. The methodology of this study will be useful for system design of PSD-based measurement systems and their applications.     

Comparison of several numerical methods for mechanical systems with impacts

The modellization of mechanical systems with non‐linearities of impact type leads to discontinuities in the velocity. In this paper, we study some numerical methods adapted to the occurrence of such discontinuities from a single‐degree‐of‐freedom vibro‐impact system. Theoretical results of consistency are given for numerical methods valid for smooth ordinary differential equations, with several kinds of procedures for approximating impact times. The algorithms are applied to classical Newmark and Runge–Kutta schemes, and the numerical behaviour is investigated for two categories of periodic response, either with finite or infinite number of impacts per cycle. These numerical methods are compared to a scheme without explicit computation of impact times. We show that high orders of convergence can be obtained with appropriate schemes, and we discuss the time of computation needed in each case. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparison of direct and preconditioned iterative techniques for sparse,unsymmetric systems of linear equations

In this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric, sparse systems of linear algebraic equations. These problems occur in finite difference and finite element simulations of semiconductor devices, and fluid flow problems. We consider five iterative methods that appear to be the most promising for this class of problems: the biconjugate gradient method, the conjugate gradient squared method, the generalized minimal residual method, the generalized conjugate residual method and the method of orthogonal minimization. Each of these methods was tested using similar preconditioning (incomplete LU factorization) on a set of large, sparse matrices arising from finite element simulation of semiconductor devices. Results are shown where we compare the computation time and memory requirements for each of these methods against one another, as well as against a direct method that uses LU factorization to solve these problems. The results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large, sparse systems of equations, and can offer significant savings in storage and CPU time.     

Microscopic derivation of the hydrodynamic equations for the superfluid Fermi system: Application to the two-fluid hydrodynamic equations in superfluid 3He-B

A microscopic theory for deriving hydrodynamic equations for Fermi superfluids is developed and applied to the two-fluid hydrodynamic equations in superfluid ³He-B near the transition temperature. It is shown how the two-fluid hydrodynamic equations as well as the Boltzmann-type transport equations for the Bogoliubov-Valatin quasiparticles are derived from the matrix kinetic equation. Special attention is paid to the derivation of the particle number conservation law and also to the definition of the chemical potential when the system deviates from local equilibrium.     

基于工作流的医学图像三维可视化平台

针对当今医学影像领域的软件工具包普遍缺乏有效的算法集成机制和友好的用户图形界面的情况,采用Java2平台企业版（J2EE）的体系结构开发了一种基于工作流的医学图像三维可视化平台——POMI。在该平台上实现了一个切片数据重建及可视化模块,该模块以工作流机制集成了一系列算法和可视化工具。利用该模块完成了对虚拟中国人的部分切片数据分析、三维重建以及结果的可视化。基于POMI平台这种易于集成数据源和算法工具的高可扩展性结构,可以为医学影像技术领域的研究人员提供一个理想的集成分析平台。     

Squeezing of the mechanical motion and beating 3 dB limit using dispersive optomechanical interactions

We study an optomechanical system consisting of an optical cavity and movable mirror coupled through dispersive linear optomechanical coupling (LOC) and quadratic optomechanical coupling (QOC). We work in the resolved side band limit with a high quality factor mechanical oscillator in a strong coupling regime. We show that the presence of QOC in the conventional optomechanical system (with LOC alone) modifies the mechanical oscillator’s frequency and reduces the back-action effects on mechanical oscillator. As a result of this the fluctuations in mechanical oscillator can be suppressed below standard quantum limit thereby squeeze the mechanical motion of resonator. We also show that either of the quadratures can be squeezed depending on the sign of the QOC. With detailed numerical calculations and analytical approximation we show that in such systems, the 3 dB limit can be beaten.     

A method for the random analysis of linear systems subjected to non-stationary multi-correlated loads

The paper shows a procedure for evaluating the correlation matrix and the evolutionary power spectral density matrix of the response of linear structural systems subjected to random non-stationary multi-correlated vector processes. The approach reduces this problem to the solution of some corresponding stationary problems. It is shown that the assumption of a modulating matrix function, whose elements are the sum of exponential functions, allows to transform the initial non-stationary problem into a stationary one. This stationary problem can be solved by well-known unconditionally stable step-by-step numerical procedures in the time domain, and, in closed form, in the frequency domain. The comparison of the results obtained using the proposed approach with those obtained by Monte Carlo Simulations (MCS) has revealed a very good level of accuracy.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.