应用数学和力学

2014 Vol. 35, No. 8

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Numerical Integration Algorithm of the Symplectic-Conservative and Energy-Preserving Method

SUN Yan, GAO Qiang, ZHONG Wan-xie

2014, 35(8): 831-837. doi: 10.3879/j.issn.1000-0887.2014.08.001

Abstract(1470) PDF(944)

Abstract:
A symplectic-conservative algorithm was proposed for the nonlinear dynamic Hamilton systems with the application of the mixed energy variational principle. Based on this, an iterative algorithm for the nonlinear problem was designed in which a parametric variable was introduced into the Hamilton system, and the goal of energy preservation was realized at the integration grid nodes through parametric adjustments. The numerical examples of the undamped Duffing spring systems show that, compared with the only symplectic-conservative algorithm, the proposed symplectic-conservative and energy-preserving algorithm bears far higher accuracy in the simulation of the nonlinear dynamic Hamilton systems.

Bending Vibration and Power Flow Analysis of Plate Assemblies in the Symplectic Space

MA Yong-bin, ZHANG Ya-hui, ZENG Yao-xiang

2014, 35(8): 838-849. doi: 10.3879/j.issn.1000-0887.2014.08.002

Abstract(1293) PDF(764)

Abstract:
The free wave propagation and forced vibration of thin rectangular plate assemblies were investigated with the symplectic method based on wave propagation theory. The governing equations of bending vibration of the thin plates were introduced into the symplectic duality system firstly, then the wave propagation parameters and wave shapes were determined as analytical solution to the symplectic eigenvalue problem. And responses of the thin plates described in physical domain were transformed into wave coordinates. The amplitudes associated with the mode shapes were obtained through solving of the equations involving excitation, scattering and propagation. Superimposition of the wave amplitudes gave the physical responses. Expressions were derived for the mean power flow through the system and mean energy in the plate components. Compared with the traditional wave methods, the provided method is applicable for any combination of classical boundary conditions. The method was applied to the forced vibration of a built-up structure of 3 directly connected thin plates and the results were compared with those from the ABAQUS finite element software. A significant improvement on accuracy and computational efficiency is achieved. As the derivation of the formulae is rigorously rational, the provided method is also applicable for the dynamic analysis of plate assemblies composed of any other types of plates (such as moderately thick plates, and layered plates, etc.).

A Constitutive Description of Cyclic Pseudoelasticity for the NiTi SMAs Involving Coupled Phase Transformation and Plasticity

ZENG Zhong-min, PENG Xiang-he

2014, 35(8): 850-862. doi: 10.3879/j.issn.1000-0887.2014.08.003

Abstract(1519) PDF(793)

Abstract:
According to the main experimental constitutive characteristics of the NiTi shape memory alloys (SMAs) subjected to cyclic tensile loading, and in line with the constitutive relationship for the SMAs, an analytical approach and the corresponding numerical algorithm as well as the computational program were developed. The constitutive description is based on the mixture theory, employs the Voigt hypothesis, and takes into account the effects of forward and inverse martensitic transformation, reorientation deformation and plastic deformation. The pseudoelastic behavior of a NiTi SMA subjected to cyclic tensile loading and unloading was simulated. The comparison between the computed and experimental results shows that the main characteristics of the material under cyclic loading can be satisfactorily described, which demonstrates the validity of the constitutive model and the developed numerical program.

Shape Control and Optimal Design of Piezoelectric Shell Structures

YAN Hao, YAO Lin-quan, SUN Xiao-jie

2014, 35(8): 863-872. doi: 10.3879/j.issn.1000-0887.2014.08.004

Abstract(1162) PDF(702)

Abstract:
The problem of shape control of curved shell structures with the least piezoelectric actuators in the most reasonable layout at optimal multi-point voltages was addressed. First, a solid shell element for shell structures was constructed with the HS-ANS method, to overcome shear locking, trapezoidal locking and thickness locking. Second, the layout of actuators was designed through the GA combinatorial optimization and the input actuator voltage was decided with the GA numerical optimization. The general computation processes were also built respectively. The following examples show the effectiveness of the proposed solid shell element and optimization method to control deformation of the curved shell structures.

Bending of Sandwich Plates With Hard Cores Under Transverse Loading Based on the HighOrder Deformation Theory

HAO Jia-qiong, LI Ming-cheng, DENG Zong-bai

2014, 35(8): 873-882. doi: 10.3879/j.issn.1000-0887.2014.08.005

Abstract(1205) PDF(743)

Abstract:
Based on the high-order deformation theory, the in-plane stiffness and bending stiffness of both surface layer and core layer of the sandwich plate were considered to derive the transverse shearing stiffnesses of all the layers. The transverse stress function was given according to the transverse strain distribution, and the differential equations for the sandwich plate were deduced with the generalized principle of virtual displacement. The bending deformation of simply supported rectangular sandwich plates with different core-to-surface thickness ratios were detailedly studied under transverse loading, and the calculation results were compared with those from the 1st-order deformation theory to give a bigger relative deformation difference at a smaller thickness ratio. The distribution of transverse strain along the thickness direction makes a half sine curve, and the center-plane normal line distortion culminates at the surface height.

Dynamic Analysis of Circular Thin Plates Under Eccentric Impact Load With the StructurePreserving Method

QIN Yu-yue, DENG Zi-chen, HU Wei-peng

2014, 35(8): 883-892. doi: 10.3879/j.issn.1000-0887.2014.08.006

Abstract(1129) PDF(682)

Abstract:
Focused on the local geometric properties of the dynamic system, the multi-symplectic method was used to analyze the vibration behavior of the circular thin plate under eccentric impact load. Firstly, the governing equation for the vibration problem of the plate was redescribed in the multi-symplectic framework. And then, the multi-symplectic scheme was constructed with the explicit midpoint method to simulate the dynamic process of the thin plate under impact load with different relative eccentric distances. Finally, the numerical results were presented and discussed in detail, which demonstrated the structure-preserving properties of the multi-symplectic algorithm. Generally, the numerical results not only present a reference for the estimation of the dynamic responses resulting from the acting position error of the load on the structure, but also propose a new way for the study of the eccentrically impacted plate problems.

Influences of the Cochlear Structure on the Dispersion of Low-Frequency Signals

LI Te, LIU Shao-bao, LI Meng-meng, WU Ying, LI Yue-ming

2014, 35(8): 893-902. doi: 10.3879/j.issn.1000-0887.2014.08.007

Abstract(1197) PDF(1089)

Abstract:
The cochlea is the most precise mechanical component in a human body. With frequencies from dozens to thousands of Hertz, acoustic signals can be processed by the cochlea and captured by the sensory hair cells on the basilar membrane (BM). Experimental research shows that sound waves of different frequencies are scattered at different positions along the basilar membrane as a natural Fourier filter. In this paper, based on Manoussaki’s 3D fluid-solid coupling model for the spiral cochlear basilar membrane and in addition according to the longitudinal gradients of the cochlear duct height and the BM stiffness, a dispersion equation for the acoustic wave propagation along the basilar membrane was deduced. The influences of the duct height and the BM stiffness on the dispersion characteristics were analyzed. It is found that existence of the cochlear endolymph greatly increases the low frequency signal processing ability, and the capture frequency reduces with the decreases of both the BM stiffness and the duct height. Finally, 3 examples of human, gerbil and guinea pig were empirically studied for verification. 3 frequency-position diagrams corresponding to the 3 animals respectively were obtained to prove the correctness of the proposed dispersion model, and reveal the relationship between the biological adaptability and the function of cochlear dispersion. This study is not only beneficial to understanding of the cochlear function but also promising to lay a theoretical basis for the development and design of sound sensors.

A New Method to Calculate the Wave Height of Deformed Shallow Water Based on the Gauss Global Radial Basis Function

LI Yi, WU Lin-jian, SHU Dan, CHEN Jia-yu

2014, 35(8): 903-912. doi: 10.3879/j.issn.1000-0887.2014.08.008

Abstract(1259) PDF(853)

Abstract:
The Gauss global radial basis function (GRBF) was used to simulate the unknown function of the changing wave height differential equation for shallow water deformation. Through a case analysis a new GRBF method was built to obtain the numerical solution of this equation in the conditions of laminar as well as turbulent boundary layers, and the GRBF calculation results were compared with those from the traditional numerical methods. The average calculation error of the GRBF method is smaller than that of the other methods, which indicates higher calculation accuracy of the proposed method. Meanwhile, based on the GRBF approximation results, a fitting function to the numerical solution of the wave height differential equation for deformed shallow water was formulated and intended to replace the corresponding intangible analytical solution in engineering application. The work provides a new way to study the wave motion in the coastal shallow water areas.

High Order Derivative Rational Interpolation Algorithm With Heredity

JING Ke, LIU Ye-zheng, KANG Ning

2014, 35(8): 913-919. doi: 10.3879/j.issn.1000-0887.2014.08.009

Abstract(1250) PDF(955)

Abstract:
Osculatory rational interpolation was an important theme of function approximation, meanwhile, reducing the degree and solving the existence of the osculatory rational interpolation function made a crucial problem for rational interpolation. The previous algorithms of osculatory rational interpolation functions mostly depended on the continued fraction with conditional feasibility and high computation complexity. Based on heredity of the Newton interpolation and the method of piecewise combination, an osculatory rational interpolation function without real poles was constructed to meet the condition of high order derivative interpolation, and was in turn extended to the vector-valued cases. It not only solved the existence problem for the osculatory rational interpolation function, but reduced the degree of the rational function. Furthermore, the error estimates of the new algorithm was given. Results of the numerical examples illustrate the new algorithm’s heredity, low computation complexity and easy programmability.

Research of the Checkerboard Pattern Suppression Method in Structural Topological Optimization

DOU Lin-long, YIN Yi-hui, LIU Yuan-dong

2014, 35(8): 920-929. doi: 10.3879/j.issn.1000-0887.2014.08.010

Abstract(1320) PDF(839)

Abstract:
Aimed at the calculation of elements’weight coefficients in the checkerboard pattern filtering technique, a previous checkerboard pattern suppression method was improved, and a more generalized weight coefficient formula with more specific physical meanings was proposed. The topological optimization model, in which the material elastoplastic deformation was considered, was established under stress constraint conditions, and the improved checkerboard pattern suppression method was employed in the structural topological optimization process with the ESO method. Two numerical examples were implemented. The results show that the improved checkerboard pattern suppression method has better suppression effects than the previous one, and is applicable to the topological optimization of not only linearmaterial structures, but also nonlinearmaterial structures.

Pattern Formation of Nonconstant Steady-State Solutions to the n-Dimensional Glycolysis Model

WEI Mei-hua, CHANG Jin-yong, QI Lan>, ZHANG Qiao-wei

2014, 35(8): 930-938. doi: 10.3879/j.issn.1000-0887.2014.08.011

Abstract(919) PDF(776)

Abstract:
A glycolysis model under the Neumann boundary condition was investigated in the n-dimensional space. Based on the local bifurcation theory, the local structure of the nonconstant steady-state solution to the model was studied with diffusion coefficient d₁ as the bifurcation parameter. Then, according to the global bifurcation theory and the Leray-Schauder degree theory, global existence of the nonconstant steady-state solution was discussed. Moreover, the theoretical results were confirmed through numerical simulations. It is shown that the spatial pattern can form for the glycolysis model.

Calculation Model for the Permeability Coefficient of Shale Gas in Shale Matrix

YAO Tong-yu, HUANG Yan-zhang, LI Ji-shan

2014, 35(8): 939-948. doi: 10.3879/j.issn.1000-0887.2014.08.012

Abstract(1019) PDF(1167)

Abstract:
According to the seepage behavior of shale gas in shale matrix, a coupling model for both the shale gas release and diffusion was established. With the Laplace transform method, the coupling model was converted to initial and boundary value problems of ordinary differential equations in the Laplace space. Then the analytical solution of pressure was found, and the permeability coefficient was obtained. The calculation of some specific examples prove reliability of the seepage model and correctness of the solution. The results indicate that the shale gas release process is mainly composed of two actions of adsorption and diffusion, in which the adsorption action has substantial influences on both the effective porosity and the permeability coefficient of the shales. This research enriches the unsteady seepage theory about shale gas.