应用数学和力学

2005 Vol. 26, No. 7

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Dynamicl Character for a Perturbed Coupled Nonlinear Schr^L dinger System

YU Pei, GAO Ping, GUO Bo-ling

2005, 26(7): 757-762.

Abstract(2313) PDF(627)

Abstract:
The dynamics for a perturbed coupled nonlinear Schr^L dinger system with periodic boundary condition was studiesd. First, the dynamics of perturbed and unperturbed systems on the invariant plane was analyzed by the spectrum of the linear operator. Then the existence of the locally invariant manifolds was proved by the sigular perturbation theory and the fixed-point argument.

Method of Green’s Function of Corrugated Shells

YUAN Hong, ZHANG Xiang-wei

2005, 26(7): 763-769.

Abstract(2636) PDF(551)

Abstract:
By using the fundamental equations of axisymmetric shallow shells of revolution, the nonlinear bending of a shallow corrugated shell with taper under arbitrary load has been investigated. The nonlinear boundary value problem of the corrugated shell was reduced to the nonlinear integral equations by using the method of Green's function. To solve the integral equations, expansion method was used to obtain Green's function. Then the integral equations were reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral equations become nonlinear algebraic equations. Newton's iterative method was utilized to solve the nonlinear algebraic equations. To guarantee the convergence of the iterative method, deflection at center was taken as control parameter. Corresponding loads were obtained by increasing deflection one by one. As a numerical example, elastic characteristic of shallow corrugated shells with spherical taper was studied. Calculation results show that characteristic of corrugated shells changes remarkably. The snapping instability which is analogous to shallow spherical shells occurs with increasing load if the taper is relatively large. The solution is close to the experimental results.

Bifurcations of Travelling Wave Solutions for a Coupled Nonlinear Wave System

ZHANG Ji-xiang, LI Ji-bin

2005, 26(7): 770-778.

Abstract(3189) PDF(750)

Abstract:
By using the bifurcation theory of dynamical systems to the coupled nonlinear wave equations, the existence and stability of periodic wave solutions by Hopf bifurcations are obtained. Theory of travelling wave was applied to transforma kind of the coupled nolinear wave equations into three- dimension dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence and stability of the above so lutions are given.

Solving 2-D Nonlinear Coupled Heat and Moisture Transfer Problems via a Self-Adaptive Precise Algorithm in Time Domain

YANG Hai-tian, LIU Yan, WU Rui-feng

2005, 26(7): 779-784.

Abstract(2136) PDF(627)

Abstract:
A self-adaptive precise algorithm in the time domain was employed to solve 2-D nonlinear coupled heat and moisture transfer problems. By expanding variables at a discretized time interval, the variations of variables can be described more precisely, and a nonlinear coupled initial and boundary value problem was converted into a series of recurrent linear boundary value problems which are solved by FE technique. In the computation, no additional assumption and the nonlinear iteration are required, and a criterion for selfadaptive computation is proposed to maintain sufficient computing accuracy for the change sizes of time steps. In the numerical comparison, the variations of material properties with temperature, moisture content, and both temperature and moisture content are taken into account, respectively. Satisfactory results have been obtained, indicating that the proposed approach is capable of dealing with complex nonlinear problems.

Time Domain Parameters Identification of Foundation-Structure Interaction System

HUANG Yi, LIU Zeng-rong

2005, 26(7): 785-793.

Abstract(2433) PDF(657)

Abstract:
The time domain parameter identification method of the foundation-structure interaction system is presented. On the basis of building the computation mode and the motion equation of the foundation-structure interaction system, the system parameter identification method was established by using the Extended Kalman Filter technique and taking the unknown parameters in the system as the augment state variables. And the time parameter identification process of the foundation-structure interaction system was implemented by using the data of the layer foundation-storehouse interaction system model test on the large vibration platform. The computation result shows that the established parameter identification method can induce good parameter estimation.

Well Posedness of Initial Value Problem for Euler Equations of Inviscid Compressible Adiabatic Fluid

WANG Yue-peng

2005, 26(7): 794-800.

Abstract(2731) PDF(783)

Abstract:
The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local (exact) solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.

Small-Stencil Pad Schemes to Solve Nonlinear Evolution Equations

LIU Ru-xun, WU Ling-ling

2005, 26(7): 801-809.

Abstract(2732) PDF(676)

Abstract:
A set of small-stencil new Pad schemes with the same denominator are presented to solve high-order non-linear evoltuion equations. Using this scheme, the fourth-order precision cannot only be kept, but also the final three-diagonal discrete systems are solved by simple Doolittle methods, or ODE systems by Runge-Kutta technique. Numerical samples show that the schemes are very satisfactory. And the advantage of the schemes is very clear compared to other finite difference schemes.

Dynamic Analysis of Two-Degree-of-Freedom Oblique Impact System With Non-Fixed Impact Positions

JIN Li, LU Qi-shao, WANG Qi

2005, 26(7): 810-818.

Abstract(2536) PDF(602)

Abstract:
The dynamic behavior of a two-degree-of-freedom oblique impact system consisted of two pendulums with non-fixed impact positions is investigeated. The relations between the restitution coefficient, the friction coefficient, as well as other parameters of the system and the states before or after impact, are clarified in this oblique impact process. The existence criterion of single impact periodic-n subharmonic motions was deduced based on the Poincar map method and the oblique impact relations with non-fixed impact positions. The stability of these subharmonic periodic motions was analyzed by the Floquet theory, and the formulas to calculate the Floquet multipliers were given. The validity of this method is shown through numerical simulation. At the same time, the probability distribution of impact positions in this oblique system with non-fixed impact positions is analyzed.

Galerkin Meshless Methods Based on Partition of Unity Quadrature

ZENG Qing-hong, LU De-tang

2005, 26(7): 819-825.

Abstract(2871) PDF(660)

Abstract:
Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature (PUQ), for Galerkin meshless methods was presented. The technique was based on finite covering and partition of unity. There was no need to decompose the physical domain into small cell. If possessed remarkable integration accuracy. Using Element-free Galerkin methods as example, Galerkin meshless methods based on PUQ were studied in detail. Meshing is always not required in the procedure of constitution of approximate function or numerical quadrature, so Galerkin meshless methods based on PUQ are "truly" meshless methods.

Configuration Controllability for Non-Zero Potential Mechanical Control Systems With Dissipation

KANG Jian-ling, WANG Hong, YE Hua-wen

2005, 26(7): 826-832.

Abstract(2561) PDF(704)

Abstract:
Within the affine connection framework of Lagrangian control systems, based on the results of Sussmann on controllability of general affine control systems defined on a finite-dimensional manifold, a computable sufficient condition of configuration controllability for the simple mechanical control systems was extended to the case of systems with strictly dissipative energy terms of linear isotropic nature, and a sufficient conditon of equilibrium controllability for the systems was also given, where Lagrangian is kinetic energy minus potential energy. Lie bracketting of vector fields in controllable Lie algebra, and the symmetric product associated with Levi-Civita connection show virtues in the discussion. Liouville vector field simplified the computation of controllable Lie algebra for the systems, although the terms of potential energy complicated the study of configuration controllability.

Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations

GAN Zai-Hui, ZHANG Jian

2005, 26(7): 833-839.

Abstract(2685) PDF(894)

Abstract:
The asymptotic theory of initial value problems for a class of nonlinear perturbed Klein- Gordon equations in two space dimensions is considered. Firstly, using the contraction mapping principle, combining some priori estimates and the convergence of Bessel function, the well-posedness of solutions of the initial value problem in twice continuous differentiable space was obtained according to the equivalent integral equation of initial value problem for the Klein-Gordon equations. Next, formal approximations of initial value problem was constructed by perturbation method and the asymptotic validity of the formal approximation is got. Finally, an application of the asymptotic theory was given, the asymptotic approximation degree of solutions for the initial value problem of a specific nonlinear Klein-Gordon equation was analyzed by using the asymptotic approximation theorem.

Analysis on the Stability of an Autonomous Dynamics System for SARS Epidemic

ZHANG Shuang-de, HAO Hai

2005, 26(7): 840-846.

Abstract(2873) PDF(735)

Abstract:
An extended dynamic model for SARS epidemic was deduced on the basis of the K-M infection model with taking the density constraint of susceptible population and the cure and death rate of patients into consideration. It is shown that the infection-free equilibrium is global asymptotic stability for under given conditions, and endemic equilibrium is not asymptotic stability. It comes to the conclusion that the epidemic system is permanent persistence existence under appropriate conditions.

New Method for Geometric Nonlinear Analysis of Large Displacement Drill Strings

TAN Mei-lan, WANG Xin-wei

2005, 26(7): 847-853.

Abstract(2603) PDF(586)

Abstract:
Based on the actual measured well depth, azimuth and oblique angles, a novel interpolation method to obtain the well axis is developed. The initial stress of drill string at the reference state being consistent with well axis can be obtained from the curvatures and the tortuosity of well axis. By using the principle of virtual work, formula to compute the equivalent load vector of the initial stress was derived. In the derivation, natural (curvilinear) coordinate system was adopted since both the curvature and the tortuosity were generally not zero. A set of displacement functions fully reflecting the rigid body modes was used. Some basic concepts in the finite element analysis of drill string have been clarified. It is hoped that the proposed method would offer a theoretical basis for handling the geometry nonlinear problem of the drill string in a 3-D large displacement wellbore.

Curvature Relation of Wave Front and Wave Changing in External Field

LIU Shen-quan, SONG Le

2005, 26(7): 854-860.

Abstract(2778) PDF(980)

Abstract:
The changing of wave structure in excitable media in external field is studied and the curvature relation of wave front is analyzed. Under external stimulus the normal velocity of wave front has linear relation with mean curvature of wave front, plane velocity and external field. The simulation methods have been used to analyze Bar-Eiswirth model with external field and obtain the wave pattern of excitable media contained external stimulus. These theoretical analysis and simulation results are identical with experiments of BZ reaction. So the results here theoretically explain the BZ phenomenon under external field and the simulation results here have rich wave patterns.

Error Estimate for the Influence of Model Reduction of Nonlinear Dissipative Autonomous Dynamical System on the Long-Term Behaviours

ZHANG Jia-zhong, LIU Yan, CHEN Dang-min

2005, 26(7): 861-866.

Abstract(2868) PDF(735)

Abstract:
From viewpoint of nonlinear dynamics, the model reduction and its influence on the long- term behaviours of a class of nonlinear dissipative autonomous dynamical system with higher dimen sion are investigated theoretically under some assumptions. The system was analyzed in the state space with an introduction of a distance definition which can be used to describe the distance between the full system and the reduced system, and the solution of the full system was then projected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. As are sult, the influence of mode series truncation on the long-term behaviours and the error estimate are derived, showing that the error is dependent on the first products of frequencies and damping ratios in the subspace spanned by the eigenvectors with higher modal damping. Furthermore, the fundamental understanding for the topological change of the solution due to the application of different model reduction is interpreted in a mathematically precise way, using the qualitative theory of nonlinear dy namics.

Region-Wise Variational Principles and Generalized Variational Principles on Large Strain for Consolidation Theory

LUO Xiao-hui, LI Yong-le, LUO Xin

2005, 26(7): 867-874.

Abstract(2765) PDF(881)

Abstract:
The difference of constitutive character and large deformation as to soil mass are basic questions to analyze deformational feature. According to the description method of limited deformation, the large deformation consolidation equations of soil mass were created and its variational principles were rigorous testimony. The region-wise variational principles of consolidation theory were deduced using sub-structure continuous condition of region-wise. Quoting the method of Lagrangian multiplier operator, generalized variational principles of region-wise of large deformation consolidation in the constrained condition were created and approved.

Multi-Monte Carlo Method for General Dynamic Equation Considering Particle Coagulation

ZHAO Hai-bo, ZHENG Chu-guang, XU Ming-hou

2005, 26(7): 875-882.

Abstract(2553) PDF(1372)

Abstract:
Monte Carlo (MC) method is widely adopted to take into account general dynamic equation (GDE) for particle coagulation, however popular MC method has high computation cost and statistical fatigue. A new Multi-Monte Carlo (MMC) method, which has characteristics of time-driven MC method, constant number method and constant volume method, was promoted to solve GDE for coagulation. Firstly MMC method was described in details, including the introduction of weighted fictitious particle, the scheme of MMC method, the setting of time step, the judgment of the occurrence of coagulation event, the choice of coagulation partner and the consequential treatment of coagulation event. Secondly MMC method was validated by five special coagulation cases in which analytical solutions exist. The good agreement between the simulation results of MMC method and analytical solutions shows MMC method conserves high computation precision and has low computation cost. Lastly the different influence of different kinds of coagulation kernel on the process of coagulation was analyzed: constant coagulation kernel and Brownian coagulation kernel in continuum regime affect small particles much more than linear and quadratic coagulation kernel, whereas affect big particles much less than linear and quadratic coagulation kernel.