应用数学和力学

2007 Vol. 28, No. 8

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Verification of the Parabolized Stability Equationsfor Its Application to Compressible Boundary Layers

ZHANG Yong-ming, ZHOU Heng

2007, 28(8): 883-893.

Abstract(3060) PDF(887)

Abstract:
Parabolized stability equations(PSE)were used to study the evolution of disturbances in compressible boundary layers.The results were compared with those obtained by direct numerical simulations(DNs),to check if the results from PSE method were reliable or not.The results of comparison showed that no matter for subsonic or supersonic boundary layers,results from both the PSE method and DNS method agreed with each other reasonably well.And the agreement between temperatures is better than those between velocities.In addition,linear PSE was used to calculate the neutral curve for small amplitude disturbances in a supersonic boundary layer.Compared with those obtained by linear stability theory(LST),the situation is similar to those for incompressible boundary layer.

Bifurcations of Travelling Wave Solutions for Jaulent-Miodek Equations

FENG Da-he, LI Ji-bin

2007, 28(8): 894-900.

Abstract(2923) PDF(760)

Abstract:
By using the theory of bifurcations of planar dynamical systems to the coupled Jaulent-Miodek equations,the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown.Under the given parametric conditions,all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.

Multidimensional Modal Analysis of Liquid Nonlinear Sloshing in Right Circular Cylindrical Tank

YU Yan-sheng, MA Xing-rui, WANG Ben-li

2007, 28(8): 901-911.

Abstract(2242) PDF(842)

Abstract:
The multidimensional modal theory proposed by Faltinsen,et al(2000)was applied to solve liquid nonlinear free sloshing in right circular cylindrical tank.After selecting the leading modes and fixing the order of magnitudes based on the Narimanov-Moiseiev third order asymptotic hypothesis, the general infinite dimensional modal system was reduced to a five dimensional asymptotic modal system(the system of second order nonlinear ordinary differential equations coupling the generalized time dependent coordinates of free surface wave elevation).The numerical integrations of this modal system discover most important nonlinear phenomena,which agree well with both pervious analytic theories and experimental observations.The results indicate that the multidimensional modal method is a very good tool for solving liquid nonlinear sloshing dynamics and will be developed to investigate more complex sloshing problem in our following work.

Mechanism of Transition in a Hypersonic Sharp Cone Boundary Layer With Zero Angle of Attack

DONG Ming, LUO Ji-sheng

2007, 28(8): 912-920.

Abstract(2630) PDF(746)

Abstract:
Firstly,the steady laminar flow field of a hypersonic sharp cone boundary layer with zero angle of attack was computed.Then two groups of finite amplitude T-S wave disturbances were introduced at the entrance of the computational field,and the spatial mode transition process was studied by direct numerical simulation(DNS).The mechanism of the transition process was analyzed.It was found that the change of the stability characteristics of the mean flow profile was the key issue.Moreover,the characteristics of evolution for the disturbances of different modes in the hypersonic sharp cone boundary layer was discussed.

Three-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities

DING Xie-ping, LIN Yen-cherng, YAO Jen-chih

2007, 28(8): 921-928.

Abstract(2878) PDF(931)

Abstract:
The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space was studied.A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities was introduced.Strong convergence of this method was established under suitable assumptions imposed on the algorithm parameters.

Vibration Reduction of Floating Elastic Plates in Surface Waves

ZHAO Cun-bao, ZHANG Jia-zhong, HUANG Wen-hu

2007, 28(8): 929-938.

Abstract(2632) PDF(587)

Abstract:
Based on the dynamical theories of water waves and dynamics of Mindlin thick plates,the investigation of the wave-induced responses and the vibration reduction of an elastic floating plate were presented by using the Wiener-Hopf technique.Firstly,regardless of without the case of elastic connector,the calculated results obtained by the present method were in good agreement with the results from the literature and the experiment.So it can be shown that the present method is valid.Finally,the relation between the spring stiffness to be used to connect the sea bottom and the floating plate,and the parameters of wave-induced responses of floating plates was analyzed by using the present method.Therefore,these results can be used as theoretical bases at the design stage of the super floating platform systems.

Elastodynamic Analysis of an Anisotropic Liquid-Saturated Porous Medium Due to Mechanical Sources

Rajneesh Kumar, Aseem Miglani, N. R. Garg

2007, 28(8): 939-948.

Abstract(3135) PDF(683)

Abstract:
Elastodynamic analysis of an anisotropic liquid-saturated porous medium has been made to study a deformation problem of a transversely isotropic liquid-saturated porous medium due to mechanical sources.Certain physical problems are of the nature,in which the deformation takes place only in one direction,e.g.,the problem relating to deformed structures and columns.In soil mechanics,assumption of only vertical subsidence is often invoked and this leads to the one dimensional model of poroelasticity.By considering a model of one-dimensional deformation of anisotropic liquid-saturated porous medium,the variations in disturbances were observed with reference to time and distance.The distribution of displacements and stresses are affected due to anisotropy of the medium, and also due to the type of sources causing the disturbances.

Solvability on Boundary-Value Problems of Elasyicity of Three-Dimensional Quasicrystals

GUO Li-hui, FAN Tian-you

2007, 28(8): 949-957.

Abstract(2701) PDF(871)

Abstract:
Weak solution(or generalized solution)for the boundary-value problems of partial differential equations of elasticity of 3D(three-dimensional)quasicrystals was given,in which the matrix expression was used.In terms of Korn inequality and theory of function space,the uniqueness of the weak solution was proved.This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals,and develops the weak solution theory of elasticity of 2D quasicrystals.

Continuous Finite Element Methods of Hamilton Systems

TANG Qiong, CHEN Chuan-miao

2007, 28(8): 958-966.

Abstract(2895) PDF(735)

Abstract:
By applying the continuous finite element methods of ordinary differential equations,the linear element methods are proved have pseudo-symplectic scheme of order 2 and the quadratic element methods have pseudo-symplectic scheme of order 3 respectively for general Hamiltonian systems,as well as energy conservative.The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems.The numerical results are in agreement with theory.

Nonlinear Vibration and Buckling of Circular Sandwich Plate Under Complex Load

DU Guo-jun, MA Jian-qing

2007, 28(8): 967-975.

Abstract(2737) PDF(735)

Abstract:
The nonlinear vibration fundamental equation of circular sandwich plate under uniformed load and circumjacent load and the loosely clamped boundary condition were established by von K鄏m鄋 plate theory,and then accordingly exact solution of static load and its numerical results is given.Based on time mode hypothesis and the variational method,the control equation of the space mode was derived,and then the amplitude frequency-load character relation of circular sandwich plate was obtained by the modified iteration method.Consequently the rule which the two kinds of load affected the vibration character of the circular sandwich plate was investigated.When circumjacent load makes the lowest natural frequency zero,critical load is obtained.

Improved Non-Singular Local Boundary Integral Equation Method

FU Dong-jie, CHEN Hai-bo, ZHANG Pei-qiang

2007, 28(8): 976-982.

Abstract(2900) PDF(659)

Abstract:
When the source nodes are on the global boundary in the implementation of local boundary integral equation method(LBIEM),singularities in the local boundary integrals need to be treated specially.Local integral equations were adopted for the nodes inside the domain and moving least square approximation(MLSA)for the nodes on the global boundary,thus singularities will not occur in the new algorithm.At the same time,approximation errors of boundary integrals reduce significantly.As applications and numerical tests,Laplace equation and Helmholtz equation problems were considered and excellent numerical results were obtained.Furthermore,when solving the Helmholtz problems, the modified basis functions with wave solutions were adopted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.

Reduced Projection Augmented Lagrange Bi-Conjugate Gradient Method for Contact and Impact Problems

LI Nan-sheng, REN Kui-sheng, SHA De-song

2007, 28(8): 983-990.

Abstract(2896) PDF(769)

Abstract:
Based on the numerical governing formulation and non-linear complementary conditions of contact and impact problems,a reduced projection augmented Lagrange bi-conjugate gradient method was proposed for contact and impact problems by translating non-linear complementary conditions into-equivalent formulation of non-linear programming.For contact-impact problems,a larger time-step can be adopted arriving at numerical convergence compared with penalty method.By establishment of the impact-contact formulations which are equivalent with original non-linear complementary conditions,a reduced projection augmented Lagrange bi-conjugate gradient method is deduced to improve precision and efficiency of numerical solutions.A numerical example shows that the algorithm suggested is valid and exact.

Symmetry, Cusp Bifurcation and Chaos of an Impact Oscillator Between Two Rigid Sides

YUE Yuan, XIE Jian-hua

2007, 28(8): 991-998.

Abstract(3362) PDF(827)

Abstract:
Both the symmetric period n-2 motion and asymmetric one of a one-degree-of-freedom impact oscillator were considered.The theory of bifurcations of the fixed point was applied to such model,and it was proved that the symmetric periodic motion only has pitchfork bifurcation by the analysis of the symmetry of the Poincar map.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmetric ones via pitchfork bifurcation.While the control parameter changes continuously,the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subsequently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp,and the pitch-fork changes into one unbifurcated branch and one fold branch.

Superconvergence Analysis of a Lower Order Anisotropic Finite Element

ZHU Guo-qing, SHI Dong-yang, CHEN Shao-chun

2007, 28(8): 999-1008.

Abstract(2972) PDF(615)

Abstract:
The convergence analysis of the lower order nonconforming element proposed by Park and Sheen was applied to the second order elliptic problem under anisotropic meshes.And the corresponding error estimation was obtained.Moreover,by using the interpolation postprocessing technique,a global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself was derived.Numerical results were also given to verify the theoretical analysis.