应用数学和力学

2013 Vol. 34, No. 3

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Effect of Poisson’s Ratio on Flexural Wave Propagation in Step-Beam

ZHU Zhi-wei, DENG Zi-chen

2013, 34(3): 217-225. doi: 10.3879/j.issn.1000-0887.2013.03.001

Abstract(2373) PDF(1475)

Abstract:
The purpose of this study was to investigate the effect of Poisson’s ratio on the flexural wave propagation in stepbeam. The transmission and reflection matrices, based on Timoshenko theory, were derived for the junction of two beams. The results show that the negative Poisson’s ratio has a significant influence on amplitudes and energies of tansmission and reflection propagating waves. Due to the analysis of energy of decaying waves, EulerBernoulli theory sometimes is not valid even in low frequencys.

Stress Fields of Creep Damage Notch Tip in Steady Propagation

MENG Qing-hua, LIANG Wen-yan, WANG Zhen-qing

2013, 34(3): 226-234. doi: 10.3879/j.issn.1000-0887.2013.03.002

Abstract(2069) PDF(1504)

Abstract:
The stress fields of the steady propagation notch tip in creep damage materials under tensile loading and antiplane shear loading were investigated. The stress potential function and the displacement potential function were adopted. The governing equations of the steady propagation notch tip were obtained under small scale creep conditions. The governing equations were solved numerically considering the effect of notch tip blunting and the boundary condition of problems. The stress fields of the notch tip were obtained. Variations of the stress fields near the notch tip as the parameter were discussed. The results show that the stress has an r^1/(1-n) singularity and the rates of stress posses the r^n/(1-n) singularity near the notch tip, where n is the creep exponent.

An Improved Precise Integration Method for Ill-Conditioned Algebraic Equations

ZHANG Wen-zhi, HUANG Pei-yan

2013, 34(3): 235-239. doi: 10.3879/j.issn.1000-0887.2013.03.003

Abstract(2012) PDF(1121)

Abstract:
An improved process of iterative algorithm was added to the precise integration method for ill-conditioned algebraic equations and an improved precise integration method for ill-conditioned algebraic equations was therefore constructed. The method further improves the accuracy and efficiency of the precise integration method for ill-conditioned algebraic equations and is of good application prospect. Numerical examples show the validity of the present method.

A Parallel Preconditioned Modified Conjugate Gradient Method for a Kind of Matrix Equation

CAO Fang-ying, Lü Quan-yi, XIE Gong-nan

2013, 34(3): 240-251. doi: 10.3879/j.issn.1000-0887.2013.03.004

Abstract(2223) PDF(1282)

Abstract:
In view of a parallel algorithm of preconditioned modified conjugate gradient method for solving a kind of matrix equationAXB=C，a preconditioned model was proposed. Based on this thought, firstly the preconditioned matrix was constructed, which was strictly diagonally dominant matrix, secondly the parallel algorithm for preprocessing matrix equation iterative format was formed, and finally the modified conjugate gradient method was used for parallel solving the preconditioned matrix equation. Through numerical experiments, comparing our algorithm with the modified conjugate gradient method, ours has higher parallel efficiency.

Identification and Parameter Estimation for Classical SIR Model

HE Yan-hui, TANG San-yi

2013, 34(3): 252-258. doi: 10.3879/j.issn.1000-0887.2013.03.005

Abstract(4262) PDF(2541)

Abstract:
Whether a model can be identified is a basic characteristic of the model before studying parameter estimation. Until recently, the classical susceptibleinfectiousrecovered (SIR) model is still one of the most commonly used models. In present work the algebraic identifiability of the SIR model by using highorder derivative method (HODM) and multiple time points method (MTPM) was studied. The results indicatet that the SIR model can be identified if only the infectious was reported, and MTPM is much beter than HODM. Using the data of the flu, the least square method was adopted to estimate the parameters of the SIR model. The result further confirmed that the SIR model was identifiable. The methods developed here could be applied to investigate other type models and left those for future studies.

Adaptive Fast Multipole Regularized Meshless Method for Large-Scale Three Dimensional Potential Problems

LIU Cong-jian, CHEN Wen, WANG Hai-tao, GU Yan

2013, 34(3): 259-271. doi: 10.3879/j.issn.1000-0887.2013.03.006

Abstract(2186) PDF(1493)

Abstract:
The regularized meshless method (RMM) is a new meshless boundary collocation method. This method overcame the perplexing fictitious boundary associated in the method of fundamental solutions (MFS), while inherited all its merits being truly meshless, integration-free, and easy toprogram. Like the MFS, the RMM also produced dense and nonsymmetric coefficient interpolation matrix, which required O(N2) multiplication operations and memory requirements in an iterative solution procedure. Since the calculation operations would dramatically increase with the number of DOF, the RMM was computationally too expensive to solve largescale problems. In order to overcome this bottleneck, this study combined the RMM with the popular diagonal form fast multipole method (FMM) to develop the fast multipole regularized meshless method (FM-RMM). The proposed scheme was integrationfree and meshfree and significantly reduced

A Hybrid Generalized Element Method Based on H-R Variational Principle

YANG Sen-sen, MA Yong-qi, FENG Wei

2013, 34(3): 272-281. doi: 10.3879/j.issn.1000-0887.2013.03.007

Abstract(1810) PDF(1129)

Abstract:
Combining HellingerReissner variational principle and the way of constructing displacement interpolation function of generalized finite element method to construct stress field and displacement field independently, through the suitable stress field could get a more precise stress value of node conveniently, and in the same time to increase the order of displacement function without increasing the number of element’s nodes, in this way a more accurate result was got. This method combines the above two methods of flexibility of constructing the stress field and displacement field, meanwhile, using less memory and equations on the same condition compared with some other methods, and the results also show that of efficiency and higher presicion.

Analysis Between Earing and Drawing for Orthotropic Sheet of Cubic in Deep Drawing

YUE Wen-xia, WU Lang, GUAN Ming-wen, TANG Meng-jian

2013, 34(3): 282-296. doi: 10.3879/j.issn.1000-0887.2013.03.008

Abstract(1841) PDF(1214)

Abstract:
In deep drawing (or the swift cupping test) of sheet metals, a circular blank of radius R_b is placed symmetrically under a blankholder over a die with a circular aperture of radius r_ab. A flatheaded punch of circular crosssection is forced down onto the blank, which is allowed to slip under the blankholder and formed into a cylindrical cup, open at the top and closed at the base.Because of plastic anisotropy of the sheet metal, it is commonly observed that “ears” form in positions roughly symmetrical with respect to the direction of rolling in the original sheet. Under the assumption that the blank be rigidplastic and obey Hill’s quadratic yield function, an explicit formula expressing the earing value of a deepdrawn cup in terms of the drawing ratio ν=R_b/r_a was derived. The validity of this formula(including texture coefficient) is examined against results of cupping tests conducted on batches of commercially pure (L2Y2) aluminum blanks that had undergone various heat treatments.

Trigonometric Series Approach for Forced Parametric Vibration Response

HUANG Di-shan

2013, 34(3): 297-305. doi: 10.3879/j.issn.1000-0887.2013.03.009

Abstract(2186) PDF(1325)

Abstract:
Modulation feedback method was used to predict the forced response of a linear system that was governed by an ordinary differential equation with periodic coefficients. The system was excited by both periodic coefficients and external force terms that had different periods. In the method, the forced response is expressed as a special trigonometric series. By applying harmonic balance and limitation operation, all coefficients of the harmonic components in the forced response solution are fully approached. The investigation result shows that the new approach has an advantage in the complete and analytical solution of forced response and in the expression of nonlinear dynamic characteristics, and it is very significant for the theoretical research and engineering application in dealing with the problem of forced parametric vibration.

Centrosymmetric Solutions of Constrained Matrix Equation and Its Application to Inverse Problem of Vibration Theory

ZHOU Shuo, WANG Lin, HAN Ming-hua

2013, 34(3): 306-317. doi: 10.3879/j.issn.1000-0887.2013.03.010

Abstract(1896) PDF(1346)

Abstract:
The centrosymmetric solutions of constrained matrix equation under a central principal submatrix constraint were studied.By using matrix to quantify, Kronecker product and singular value decomposition (SVD) method, the necessary and sufficient conditions for solvability and the general expression of solutions were obtained. Then, the expression of solution to the related optimal approximation problem was considered. Moreover, the applications in the inverse problem of vibration theory were given, by using the reduction principal mass matrix (or principal stiffness matrix), reduction modal matrix and central principal submatrix of the mass matrix (or stiffness matrix), the mass matrix (or stiffness matrix) of the system was obtained.Finally, the proposed method was demonstrated by two examples.

Seven-Mode Truncation and Chaotic Characteristics of Kolmogorov Flow Model

LI Zhen, LIAN Xin-yu

2013, 34(3): 318-326. doi: 10.3879/j.issn.1000-0887.2013.03.011

Abstract(2125) PDF(1091)

Abstract:
To provide a mathematical description of the chaotic behavior in Kolmogorov flow model,with k=3was researched,NavierStokes equation was truncated by seven basic modes and a new sevendimensional chaotic system described by ordinary differential equations was obtained. The basic dynamical behaviors and chaotic behaviors were simulated numerically according to control parameter changes and the chaotic characteristics were analyzed. The result verifies that the mathematical object which accounts for turbulence is attributed to lowdimensional chaotic attractors and this is helpful to understand turbulent flow.