应用数学和力学

2006 Vol. 27, No. 10

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Support Vector Machine for Structural Reliability Analysis

LI Hong-shuang, LÜ Zhen-zhou, YUE Zhu-feng

2006, 27(10): 1135-1143.

Abstract(2562) PDF(681)

Abstract:
Support vector machine (SVM) was introduced to analyze the reliability of the implicit performance function, which is difficult to implement by the classical methods such as the first order reliability method (FORM) and the Monte Carlo simulation (MCS). As a classification method where the underlying structural risk minimization inference rule is employed, SVM possesses excellent learning capacity with a small amount of information and good capability of generalization over the complete data. Hence, two approaches, i. e. SVM-based FORM and SVM-based MCS, were presented for the structural reliability analysis of the implicit limit state function. Compared to the conventional response surface method (RSM) and the artificial neural network (ANN), which are widely used to replace the implicit state function for alleviating the computation cost, the more important advantages of SVM are that it can approximate the implicit function with higher precision and better generalization under the small amount of information and avoid the "curse of dimensionality". The SVM-based reliability approaches can approximate the actual performance function over the complete sampling data with the decreased number of the implicit performance function analysis (usually finite element analysis), and the computational precision can satisfy the engineering requirement, which are demonstrated by illustrations.

Analytical Solution for Fixed-Fixed Anisotropic Beam Subjected to Uniform Load

DING Hao-jiang, HUANG De-jin, WANG Hui-ming

2006, 27(10): 1144-1149.

Abstract(2754) PDF(712)

Abstract:
The analytical solutions of the stresses and displacements were obtained for fixed-fixed anisotropic beams subjected to uniform load. A stress function involving unknown coefficients was constructed, and the general expressions of stress and displacement were obtained by means of Airy stress function method. Two types of the description for the fixed end boundary condition were considered. The introduced unknown coefficients in stress function were determined by using the boundary conditions. The analytical solutions for stresses and displacements were finally obtained. Numerical tests show that the analytical solutions agree with the FEM results. The analytical solution supplies a classical example for the elasticity theory.

Oscillation Behavior of Solutions for Even Order Neutral Functional Differential Equations

T. Cadan

2006, 27(10): 1150-1158.

Abstract(2089) PDF(599)

Abstract:
Even order neutral functional differential equations are considered. Sufficient conditions for the oscillation behavior of solutions for this differential equation are presented. The new results are presented and the some examples also given.

Stochastic Algorithm and Numerical Simulation for Drop Scavenging of Aerosols

ZHAO Hai-bo, ZHENG Chu-guang

2006, 27(10): 1159-1168.

Abstract(2593) PDF(713)

Abstract:
The time evolution of aerosol size distribution during precipitation, which is founded mathematically by general dynamic equation (GDE) for wet removal, describes quantitatively the process of aerosol wet scavenging. The equation depends on aerosol size distribution, raindrop size distribution and the complicated model of scavenging coefficient that takes account of the important wet removal mechanisms such as Brownian diffusion, interception and inertial impaction. Normal numerical methods can hardly solve GDE, which is a typical partially integro-differential equation. A new multi-Monte Carlo method was introduced to solve GDE for wet removal, and then was used to simulate the wet scavenging of aerosols in the real atmospheric environment. The results of numerical simulation show that, the smaller the lognormal raindrop size distribution and lognormal initial aerosol size distribution, the smaller geometric mean diameter or geometric standard deviation of raindrops can help scavenge small aerosols and intermediate size aerosols better, though large aerosols are prevented from being collected in some ways.

Exact Solution of Large Deformation Basic Equations of Circular Membrane Under Central Force

HAO Ji-ping, YAN Xin-li

2006, 27(10): 1169-1172.

Abstract(2976) PDF(564)

Abstract:
Exact solution of the nonlinear boundary value problem for the basic equation and boundary condition of circular membrane under central force were obtained by using new simple methods. The existence and uniqueness of the solution were discussed by making use of modern immovable point theorems. Although specific problem is treated, the basic principles of the methods can be applied to a considerable variety of nonlinear problems.

High Performance Sparse Solver for Unsymmetrical Linear Equations With Out-of-Core Strategies and Its Application on Meshless Methods

YUAN Wei-ran, CHEN Pu, LIU Kai-xin

2006, 27(10): 1173-1181.

Abstract(2648) PDF(775)

Abstract:
A new direct method for solving unsymmetrical sparse linear systems(USLS) arising from meshless methods was introduced. Computation of certain meshless methods such as meshless local Petrov-Galerkin (MLPG) method need to solve large USLS. The proposed solution method for unsymmetrical case performs factorization processes symmetrically on the upper and lower portion of matrix, which differs from previous work based on general unsymmetrical process, and attains higher performance. It is shown that the solution algorithm for USLS can be simply derived from the existing approaches for the symmetrical case. The new matrix factorization algorithm in the method can be implemented easily by modifying a standard JKI symmetrical matrix factorization code. Multi-blocked out-of-core strategies were also developed to expand the solution scale. The approach convincingly increases the speed of the solution process, as is demonstrated with the numerical tests.

Directional Derivative of the Vector Field and Regular Curves on Time Scales

Emin Özyilmaz

2006, 27(10): 1182-1192.

Abstract(2630) PDF(736)

Abstract:
The general idea is to study curves where in the parametric equations the parameter varies in a so-called time scale, which may be an arbitrary closed subset of the set of all real numbers. The directional derivative according to the vector fields was introduced.

Double-Medium Constitutive Model of Geological Material in Uniaxial Tension and Compression

LIU Xiao-li, WANG Si-jing, WANG En-zhi, XUE Qiang

2006, 27(10): 1193-1201.

Abstract(2529) PDF(603)

Abstract:
Based on elasto-plasticity and damage mechanics, a double-medium constitutive model of geological material under uniaxial tension and compression was presented, on the assumption that rock and soil materials being pore-fracture double-medium, and porous medium occurring no damage, while fracture medium occurring damage with load. To the implicit equation of the model, iterative method was adopted to obtain the complete stress-strain curve of the material. The result shows that many different distributions (uniform distribution, concentrated distribution and random distribution) of fractures in rock and soil material are the essential reasons of the daedal constitutive relations. By the reason that the double-medium constitutive model separating the material to be porous medium part, which is the main body of elasticity, and fracture medium part, which is the main body of damage, it is of important practical values and theoretical meanings to the study on failure of rock and soil or materials containing damage.

Second-Order Moment Model for Dense Two-Phase Turbulent Flow of Bingham Fluid With Particles

ZENG Zhuo-xiong, ZHOU Li-xing, LIU Zhi-he

2006, 27(10): 1202-1210.

Abstract(2792) PDF(500)

Abstract:
The USM-theta model of Bingham fluid for dense two-phase tur bulent flow is developed, which combines the unified second-order moment model for two-phase tur bulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extraterm of Bingham fluid yield stress were taken into account. An algorithm for second-order moment model in dense two-phase flow was proposed, in which the influence of particle volume fraction was accounted for. This model was used to simulate turbulent flow of single-phase and dense two-phase in pipe, it is shown the USM-theta model has better prediction result than five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while tur bulence of both phases is simulated by the two-equation tur bulence model. The USM-theta model was also used to simulate the dense two-phase turbulent flow of Bingham fluid with particles. With the incre asing of the yield stress, the velocities of Bingham and particle decre ase near the pipe centre, comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.

Efficient Numerical Integrators for Highly Oscillatory Dynamic Systems Based on Modified Magnus Integrator Method

LI Wen-cheng, DENG Zi-chen, HUANG Yong-an

2006, 27(10): 1211-1218.

Abstract(2585) PDF(599)

Abstract:
Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method is proposed for the second-order dynamic systems with time-dependent high frequencies. Firstly, the second-order dynamic system was reformulated as a system of the first-order and transfered the frame of reference by introducing new variables so that highly oscillatory behaviour is inherited from the entries in the meantime. Then the modified Magnus integrator method based on local linearization was appropriately-designed for solving the above new form and some improved ones are also presented. Finally, numerical examples are presented and analyzed to show that the proposed methods appear to be quite adequate for integration for highly oscillatory dynamic systems including Hamiltonian systems problem with long time and effectiveness.

Resistance Effect of Electric Double Layer on Liquid Flow in Microchannel

GONG Lei, WU Jian-kang

2006, 27(10): 1219-1225.

Abstract(2640) PDF(732)

Abstract:
Poisson-Boltzmann equation for electric double layer and Navier-Stokes equation for liquid flows to investigate resistance effect of electric double layer on liquid flow in microchannel were nu merically solved. The dimension analysis indicates that the resistance effect of electric double layer can be estimated by an electric resistance number, which is proportional to the square of the liquid dielec tric constant and the solid surface zeta potential, and inverse-proportional to the liquid dynamic vis cosity, electric conductivity and the square of the channel width. An electric current density balancing (ECDB) condition was proposed to evaluate the flow-induced streaming potential and electric resis tance, instead of conventional electric current balancing(ECB) condition which may induce spurious local backflow in neighborhood of solid wall of the microchannel. The numerical results of the flow rate loss ratio and velocity profile are also given to demonstrate the resistance effect of electric double layer in microchannel.

Application of Wu Elimination Method to Constrained Dynamics

JIA Yi-feng, CHEN Yu-fu, XU Zhi-qiang

2006, 27(10): 1226-1234.

Abstract(2787) PDF(736)

Abstract:
The polynomial type Lagrange equation and Hamilton equation of finite dimensional constrained dynamics are considered. A new algorithm was presented for solving constraints based on Wu elimination method. The new algorithm does not need to calculate the rank of Hessian matrix and determine the linear dependence of equations, so the steps of calculation decrease greatly. In addition, the expanding of expression occurring in the computing process is smaller. Using the symbolic computation software platform, the new algorithm can be executed in computers.

On the Cauchy Problem of One Type of Atmosphere Evolution Equations

HE Juan-xiong, HE You-hua

2006, 27(10): 1235-1242.

Abstract(2364) PDF(563)

Abstract:
One type of evolution atmosphere equations was discussed. It is found that according to the stratification theory, 1) the inertial force has no influence on the criterion of the well-posed Cauchy problem; 2) the compressibility plays no role on the well posed condition of the Cauchy problem of the viscid atmosphere equations, but changes the well posed condition of the viscid atmosphere equations; 3) this type of atmosphere evolution equations is ill-posed on the hyperplane t=0 in spite of its compressibility and viscosity; 4) the Cauchy problem of compressible viscosity atmosphere with still initial motion is ill-posed.

Nonlinear Vibration of Circular Sandwich Plates Under Circumjacent Load

DU Guo-jun, MA Jian-qing

2006, 27(10): 1243-1249.

Abstract(3008) PDF(526)

Abstract:
Based on von Karman plate theory, the issue about nonlinear vibration for circular sandwich plates under circumjacent load with the loosely clamped boundary condition is researched. Nonlinear differential eigenvalue equations and boundary conditions of the problem were formulated by variational method and then their exact static solution can be got. The solution was derived by modified iteration method, so the analytic relations between amplitude and nonlinear oscillating frequency for circular sandwich plates were obtained. When circumjacent load makes the lowest natural frequency zero, critical load is obtained.

Static Analysis of Cable Structure

HUANG Yan, LAN Wei-ren

2006, 27(10): 1250-1254.

Abstract(2671) PDF(534)

Abstract:
Based on the nonlinear geometric relation between strain and displacement for flexible cable, the equilibrium equation under self-weight and influence of temperature were established and an analytical solution of displacement and tension distribution defined in Eulerian coordinate system was accurately obtained. The nonlinear algebraic equations caused by cable structure were solved directly using the modified Powell hybrid algorithm with high preision routine DNEQNE of Fortran. For example, a cable structure consisting of three cables jointly supported by a vertical spring and all the other ends fixed was calculated and compared with various methods by other scholars.

Solitary Waves in Finite Deformation Elastic Circular Rod

LIU Zhi-fang, ZHANG Shan-yuan

2006, 27(10): 1255-1260.

Abstract(2825) PDF(817)

Abstract:
A new nonlinear wave equation of a finite deformation elastic circular rod simultaneously introducing transverse inertia and shearing strain was derived by means of Hamilton principle. The nonlinear equation includes two nonlinear terms caused by finite deformation and double geometric dispersion effects caused by transverse inertia and transverse shearing strain. Nonlinear wave equation and corresponding truncated nonlinear wave equation were solved by the hyperbolic secant function finite expansion method. The solitary wave solutions of these nonlinear equations are obtained. The necessary condition of these solutions existence is given also.