A SYMBOLIC COMPUTATION METHOD TO DECIDE THE COMPLETENESS OF THE SOLUTIONS TO THE SYSTEM OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

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Symbolic computations in flexible multibody systems

The need for computer aided engineering in the analysis of machines and mechanisms led to a wide variety of general purpose programs for the dynamical analysis of multibody systems. The use of more lightweight structures and an increasing demand of high-precision mechanisms, such as robots, led to the incorporation of flexible bodies in this methodology. This paper presents a formalism for flexible multibody systems based on a minimum set of generalized coordinates and symbolic computation. A standardized object oriented data model is used for the time-invariant system matrices describing the elastodynamic behaviour of the flexible bodies. Consequently, the equations of motion are derived in a form independent of the chosen modelling technique for the elastic bodies. They are generated in a symbolic form using the symbolic formalismNEWEUL and the computer algebra systemMAPLE. Two examples, a rotating beam and a flexible robot, are presented in this paper in order to demonstrate the formalism.     

A stable high-order method for the heated cavity problem

A fourth-order method, without using extrapolation, is developed for the steady-state solution of a non-linear system of three simultaneous partial differential equations for the flow of a fluid in a heated closed cavity. The method is a finite difference method which has converged for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. The results are presented and compared with some of the accurate results available in de Vahl Davis and Jones, Shay and Schultz, and Dennis and Hudson. The method used to develop the fourth-order method presented in this paper can be used to develop high-order methods for other partial differential equations. The method was developed to be stable without using the upwinding technique.     

Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.     

不可压气流中二元机翼的分叉分析

本文分析了不可压气流中带有非线性俯仰刚度二元机翼的分叉问题.分析采用了工程实用的等效线化法和作为比较标准的数值积分法,并借助计算机代数系统按Hassard渐近展开算法及平均化算法求得解析解进行比较.从而论证等效线化法的可用性.     

Computer algebra-perturbation solution to a nonlinear wave equation

In this paper, the higher-order asymptotic solution to the Cauchy problem of a nonlinear wave equation is found by using a computer algebra-perturbation method. The secular terms in the solution from straightforward expansions are eliminated with the straining of characteristic, coordinates and the use of the renormalization technique, and the four-term uniformly valid solution is obtained with the symbolic computation by using a computer algebra system. The comparison of the derived asymptotic solution and the numerical solution shows that they coincide with each other for smaller ε and agree quite well for larger ε (e. g., ε=0.25) Project supported by the National Natural Science Foundation of China and Shanghai Municiple Natural Science Foundation     

Order reduction of retarded nonlinear systems – the method of multiple scales versus center-manifold reduction

We compare two approaches for determining the normal forms of Hopf bifurcations in retarded nonlinear dynamical systems; namely, the method of multiple scales and a combination of the method of normal forms and the center-manifold theorem. To describe and compare the methods without getting involved in the algebra, we consider three examples: a scalar equation, a single-degree-of-freedom system, and a three-neuron model. The method of multiple scales is directly applied to the retarded differential equations. In contrast, in the second approach, one needs to represent the retarded equations as operator differential equations, decompose the solution space of their linearized form into stable and center subspaces, determine the adjoint of the operator equations, calculate the center manifold, carry out details of the projection using the adjoint of the center subspace, and finally calculate the normal form on the center manifold. We refer to the second approach as center-manifold reduction. Finally, we consider a problem in which the retarded term appears as an acceleration and treat it using the method of multiple scales only. Communicated by G. Rega     

改进的移动最小二乘法

近年来发展的无网格方法大多采用移动员小二乘法来构造试函数，而应用移动最小二乘法形成的方程组有时会是病态的甚至奇异的，从而限制了它的发展和应用。本文采用带权正交函数作为基函数对移动最小二乘法做了改进，避免出现病态方程组，且在计算过程中不需要进行短阵求逆运算，提高了计算速度。之后，借鉴牛顿法、平衡法和摄动法对由移动最小二乘法得到的非线性代数方程组提出了新的求解方法。     

Dual Mesh Method for Upscaling in Waterflood Simulation

Detailed geological models typically contain many more cells than can be accommodated by reservoir simulation due to computer time and memory constraints. However, recovery predictions performed on a coarser upscaled mesh are inevitably less accurate than those performed on the initial fine mesh. Recent studies have shown how to use both coarse and fine mesh information during waterflooding simulations. In this paper, we present an extension of the dual mesh method (Verdière and Guérillot, 1996) which simulates water flooding injection using both the coarse and the original fine mesh information. The pressure field is first calculated on the coarse mesh. This information is used to estimate the pressure field within each coarse cell and then phase saturations are updated on the fine mesh. This method avoids the most time consuming step of reservoir simulation, namely solving for the pressure field on the fine grid. A conventional finite difference IMPES scheme is used considering a two phase fluid with gravity and vertical wells. Two upscaling methodologies are used and compared for averaging the coarse grid properties: geometric average and the pressure solve method. A series of test cases show that the method provides predictions similar to those of full fine grid simulations but using less computer time.     

Coupling of the methods of successive approximations and undetermined coefficients for the prediction of the thermal behaviour of uniform circumferential fins

This study addresses a distinct, unsophisticated computational procedure for solving approximately, but analytically, the one-dimensional heat equation for circumferential fins of uniform thickness with constant properties. This differential equation with variable coefficients, called the modified Bessel equation of zero order, is subject to a prescribed temperature at the base and zero heat rejection at the tip. Approximate temperature distributions and companion heat transfer rates of excellent quality have been obtained by adequately blending a polynomial curve fit, the method of successive approximations and the method of undetermined coefficients. Detailed error distributions are also presented for real uniform circumferential fins using the exact solution by modified Bessel functions as the baseline case. The calculations of analytic character were carried out with a symbolic algebra software, Maple V, on a personal computer. Received on 23 October 1997     

Computer‐algebra multiple‐timescale method for spatially periodic thin‐film viscous‐flow problems

A flexible, fully automated, computer‐algebra algorithm is developed for solving a class of non‐linear partial‐differential evolution equations arising frequently in the modeling of two‐dimensional transient free‐surface viscous thin‐film flows. The method, which is formulated for solving spatially periodic problems, is based upon an explicit multiple‐timescale asymptotic approximation of the thin‐film thickness. It admits the resolution of diverse physical phenomena by employing a finite geometric progression of increasingly slow timescales. The method is implemented on a challenging test problem comprising the evolution of an annular film of viscous liquid, with a free surface, adhering to the exterior of a horizontal rotating circular cylinder; as a model for numerous industrially motivated coating flows, this benchmark problem has been analyzed in diverse numerical and theoretical studies, against whose results those of the present method are compared. The explicit algebraic form of the solution admits a study of large‐time evolutionary dynamics that lies beyond the reach of considerably more expensive conventional numerical solvers, thereby shedding new light on the hitherto‐undiscovered explicit dependence of large‐time evolutionary fluid dynamics in terms of independent parameters describing gravitational and capillary effects. The results obtained from the new computer‐algebra procedure are demonstrated to be in good agreement with those obtained from a bespoke efficient numerical integration method that is spectrally accurate in space and 8th‐order (Runge–Kutta) in time. Newly discovered mechanisms describing the decay of free‐surface wave modes, from arbitrary initial conditions to the steady state, are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Free vibrations of a thin elastica by normal modes

In this work we investigate the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system. The system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes. Using a variety of perturbation techniques in conjunction with the computer algebra system MACSYMA, we obtain approximate expressions for a diversity of periodic motions, including nonlinear normal modes, elliptic orbits and non-local modes. The latter motions, which involve both bending and torsional motions in a 2:1 ratio, correspond to behavior previously observed in experiments by Cusumano.     

Kinematic analysis of Bricard’s mechanism

We show how the tools of computational algebra can be used to analyze the configuration space of multibody systems. One advantage of this approach is that the mobility can be computed without using the Jacobian of the system. As an example, we treat thoroughly the well-known Bricard’s mechanism, but the same methods can be applied to a wide class of rigid multibody systems. It turns out that the configuration space of Bricard’s system is a smooth closed curve, which can be explicitly parametrized. Our computations also yield a new formulation of constraints which is better than the original one from the point of view of numerical simulations.     

1:2 Internal Resonance of Coupled Dynamic System with Quadratic and Cubic Nonlinearities

The1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in1:2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the4-dimension bifurcation equations were reduced to3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on4-dimension center manifolds. Paper from Chen Yu-shu, Member of Editorial Commuttee, AMM Foundation item: the National Natural Science Foundation of China (1990510); the National Key Basic Research Special Fund (G1998020316); the Doctoral Point Fund of Education Committee of China (D09901) Biography: Chen Yu-shu (1931-)     

Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems

In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.     

具有初始构型的MEMS驱动器的跳跃和吸合现象研究

在曲梁变形后以弧长为参数的自然坐标系中,利用曲梁大变形分析理论,建立了具有任意初始构型的微电驱动器大变形电动力学分析的数学模型,并采用微分求积法(DQM)进行空间离散,得到了一组具有强非线性的微分-代数系统方程,运用Petzold-Gear BDF方法进行时间域内的求解。研究了MEMS驱动器在电场力作用下的瞬态动力学特性,包括跳跃(snap-through)和吸合(pull-in)现象,并与已有实验结果进行了比较。     

Symbolic generation of the kinematics of multibody systems in EasyDyn: From MuPAD to Xcas/Giac

In the EasyDyn multibody open source project, computer algebra has been used from the beginning to generate the expressions of velocities and accelerations of the bodies, by symbolic differentiation of their position. Originally, the MuPAD computer algebra system had been retained because it was freely available for non commercial purposes and showed very good technical features. Unfortunately, MuPAD is nowadays only available through commercial channels and needs to be replaced to keep EasyDyn publicly available. This paper presents why Xcas/Giac is finally selected, among other long-term promising projects like Axiom, Maxima, Sage or Yacas. Among the choice criteria, the accessibility, the portability, the ease of use, the automatic export to C language, and the similarity with the MuPAD language are all considered. The performances of the MuPAD and Xcas/Giac implementations are also compared on some examples.     

A Note on Analysis of the Natural Vibrations of Shallow Shells with Cuts on the Surface

A new approach is proposed to study the dynamic behavior of shells with an arbitrary planform weakened by surface cuts (cracks.) The approach is based on the method of R-functions. The computer simulation is carried out using the POLE-SHELL problem-oriented system, which implements the method of R-functions and variational methods. Numerical results are presented     

Full-field speckle pattern image correlation with B-Spline deformation function

A full-field speckle pattern image correlation method is presented that will determine directly the complete, two-dimensional deformation field during the image correlation process on digital images obtained using computer vision systems. In this work, a B-Spline function is used to represent the object deformation field throughout the entire image area. This is an improvement over subset-based image correlation methods by implicitly maintaining position and derivative continuity constraints among subsets up to a specified order. The control point variables within the B-Spline deformation function are optimized iteratively with the Levenberg-Marquardt method to achieve minimum disparity between the predicted and actual deformed images. Results have shown that the proposed method is computationally efficient, accurate and robust. The general framework of this method can be applied ton-dimensional image correlation systems that solve for multi-dimension vector fields.