应用数学和力学

2010 Vol. 31, No. 10

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Accuracy Analysis of the Immersed Boundary Method Using the Method of Manufactured Solutions

GONG Zhao-xin, LU Chuan-jing, HUANG Hua-xiong

2010, 31(10): 1141-1151. doi: 10.3879/j.issn.1000-0887.2010.10.001

Abstract(1885) PDF(903)

Abstract:
The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics.The principle work was to analyze the accuracy of the immersed boundary method.The whole procedure contained two parts:code verification and accuracy analysis.The code verification part provided the confidence that the code used here was mistake free and the accuracy analysis part gave the order of accuracy of the immersed boundary method.The method of manufactured solutions was taken as the research means for both parts.In the first part, the numerical code employed a second order discretization scheme, i.e., the theoretical order of accuracy was second order.It was matched by the calculated order of accuracy obtained by numerical calculation for all variables.This meant that the code contained no mistake, which was the premise of the following work.The second part introduced a jump in the manufactured solution for pressure and added corresponding singular forcing terms in the momentum equations.By analyzing the discretization errors, the accuracy of the immersed boundary method was proved to be first order, even though the discretization scheme was second order.In the meantime, it was found that the coarser mesh might be not sensitive enough to capture the influence of the immersed boundary and that refinement on the Lagrangian markers barely had any effects on the numerical calculation.

Attenuation Analysis of Hydraulic Transients With Laminar-Turbulent Flow Alternations

WAN Wu-yi, ZHU Song, HU Yun-jin

2010, 31(10): 1152-1159. doi: 10.3879/j.issn.1000-0887.2010.10.002

Abstract(1544) PDF(1031)

Abstract:
An improved compound mathematical model was established to simulate attenuation of hydraulic transients with laminar-turbulent alternations, usually occured when pipeline flow velocity fluctuates near the critical velocity.Laminar friction resistance and turbulent friction resistance were considered respectively in this model by applying different resistance schemes to the characteristics method of fluid transient analysis.The hydraulic transients of valve closing process were simulated using the model.More reasonable attenuation of hydraulic transients was obtained.Accurate attenuation is more distinct than that obtained from traditional models.The research shows that hydraulic transient is a type of energy waves, and its attenuation is caused by friction resistance.Laminar friction resistance is more important than turbulent friction resistance if the flow velocity is smaller than the critical velocity.Otherwise turbulent friction resistance is more important.Laminar friction resistance is important in the attenuation of hydraulic transients for valve closing process. Therefore it is significant to consider different resistances separately in order to obtain more accurate attenuation of hydraulic transients.

Combined Heat and Mass Transfer by Mixed Convection MHD Flow Along a Porous Plate With Chemical Reaction in Presence of Heat Source

Joaqu韓 Zueco, Sahin Ahmed

2010, 31(10): 1160-1171. doi: 10.3879/j.issn.1000-0887.2010.10.003

Abstract(1850) PDF(825)

Abstract:
An exact and numerical solution to the problem of a steady mixed convective MHD flow of an incompressible viscous electrically conducting fluid past an infinite vertical porous plate with combined heat and mass transfer was presented.A uniform magnetic field was assumed to be applied transversely to the direction of the flow, taking into account the induced magnetic field with viscous and magnetic dissipations of energy.The porous plate was subjected to a constant suction velocity as well as uniform mixed stream velocity.The governing equations were solved by perturbations technique and numerical method.The analytical expressions for the velocity field, temperature field, induced magnetic field, skin-friction and the rate of heat transfer at the plate were obtained.The numerical results were demonstrated graphically for the various values of the parameters involved in the problem.The effects of the Hartmann number, the chemical reaction parameter, the magnetic Prandtl number, and the other parameters involved on the velocity field, temperature field, concentration field and induced magnetic field from the plate to the fluid were discussed.An increase in heat source/sink or Eckert number was found to strongly enhance fluid velocity values.The induced magnetic field along x-direction increases with the increase in Hartmann number, magnetic Prandtl number, heat source/sink and the viscous dissipation.However, it is found that the flow velocity, fluid temperature, and induced magnetic field decrease with the increase in destructive chemical reaction(K0).Applications of the study arise in thermal plasma reactor modelling, electromagnetic induction, magnetohydrodynamic transport phenomena in chromatographic systems and magnetic field control of materials processing.

Exact Solution for Peristaltic Transport of Power-Law Fluid in an Asymmetric Channel With Compliant Walls

T. Hayat, Maryiam Javed

2010, 31(10): 1172-1180. doi: 10.3879/j.issn.1000-0887.2010.10.004

Abstract(1630) PDF(759)

Abstract:
Effects of compliant wall properties on the peristaltic flow of a non-Newtonian fluid in an asymmetric channel were investigated.The rheological characteristics were characterized by the constitutive equations of a power-law fluid.Long wavelength and low Reynolds number approximations were adopted in the presentation of mathematical developments.Exact solutions were established for the stream function and velocity.The streamlines pattern and trapping are given due attention.The salient features of the key parameters entering into the present flow are displayed and important conclusions have been pointed out.

Symplectic Eigenfunction Expansion Theorem for the Rectangular Plane Elasticity Problems With Two Opposite Simply Supported

HOU Guo-lin, Alatancang

2010, 31(10): 1181-1190. doi: 10.3879/j.issn.1000-0887.2010.10.005

Abstract(2046) PDF(1008)

Abstract:
The eigenvalue problem of the Hamiltonian operator associated with the plane elasticity problems was investigated.First, the eigenfunctions of the operator with the mixed boundary conditions for the displacement and stress in the rectangular region was solved directly.Then, the completeness of the eigenfunctions was proved, thereby demonstrating the feasibility of using separation of variables to solve the problems.Finally, the general solution was obtained by using the symplectic eigenfunction expansion theorem.

Symplectic Multi-Level Method for Solving Nonlinear Optimal Control Problem

PENG Hai-jun, GAO Qiang, WU Zhi-gang, ZHONG Wan-xie

2010, 31(10): 1191-1200. doi: 10.3879/j.issn.1000-0887.2010.10.006

Abstract(2238) PDF(820)

Abstract:
The optimal control problem for nonlinear system was transformed into Hamiltonian system and a symplectic-preserving method was proposed.The state and costate variables were approximated by Lagrange polynomial and state variables at two ends of the time interval were taken as the independent variables, and then based on the dual variable principle, nonlinear optimal control problems were replaced by nonlinear equations.In the implement of symplectic algorithm, based on the 2^N algorithm, a multi-level method was proposed.When the time grid was refined from the low level to the high level, the initial state variables and costate variables of nonlinear equations could be obtained from Lagrange interpolation at the low level grid, which could improve the efficiency.Numerical simulations show the precision and efficiency of the proposed algorithm.

Three Dimensional Large Deformation Analysis of Phase Transformation in Shape Memory Alloys

XIA Kai-ming, PAN Tong-yan, LIU Shan-hong

2010, 31(10): 1201-1210. doi: 10.3879/j.issn.1000-0887.2010.10.007

Abstract(2117) PDF(1142)

Abstract:
Shape memory alloys (SMAs) have been explored as smart materials and used as dampers, actuator elements and smart sensors.An important character of SMAs is its ability to recover all of its large deformations in mechanical loading-unloading cycles, without showing permanent deformation.A stress-induced phenomenological constitutive equation for SMAs, which can be used to describe the superelastic hysteresis loops and phase transformation between martensite and austenite was presented.The martensite fraction of SMAs was assumed to be dependent on deviatoric stress tensor.Therefore phase transformation of shape memory alloys was volume preserving during the phase transformation.The model was implemented in large deformation finite element code and cast in the updated Lagrangian scheme.In order to use Cauchy stress and the linear strain in constitutive laws, a frame indifferent stress objective rate has to be used and the Jaumann stress rate was used. The results of the numerical experiments conducted show that the superelastic hysteresis loops arising with the phase transformation can be effectively captured.

Viscosity Approximation With Weak Contractions for Fixed Point Problem Equilibrium Problem and Variational Inequality Problem

ZHANG Shi-sheng, LEE Heung-wing Joseph, CHAN Chi-kin, LIU Jing-ai

2010, 31(10): 1211-1219. doi: 10.3879/j.issn.1000-0887.2010.10.008

Abstract(1838) PDF(909)

Abstract:
The purpose was by using the viscosity approximation method with a weak contraction to propose a modified iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems which was also a solution of a variational inequality.Under suitable conditions,some strong convergence theorems were established in the framework of Hilbert spaces.The results presented improve and extend the corresponding results in Colao, et al [Nonlinear Anal,2009, 71:27082715], Plubtieng, et al [J Math Anal Appl,2007, 336:455-469],Colao, et al [J Math Anal Appl, 2008, 344:340-352],Yao, et al [Fixed Point Theory Appl,2007, Art ID64363] and others.

Three Solutions for Inequalities Dirichlet Problem Driven by p(x)-Laplacian

GE Bin, XUE Xiao-ping, GUO Meng-shu

2010, 31(10): 1220-1228. doi: 10.3879/j.issn.1000-0887.2010.10.009

Abstract(2072) PDF(859)

Abstract:
A class of nonlinear elliptic proplem driven by p(x)-Laplacian with a nonsmooth locally Lipschitz potential was considered.Applying the version of non-smooth three-critical-point theorem, existence of three solutions of the problem in W₀^{1, p(x)}(Ω)was proved.

Study on Primary Resonance of Multi-Degree-of-Freedom Dynamic Systems With Strongly Non-Linearity Using the Homotopy Analysis Method

YUAN Pei-xin, LI Yong-qiang

2010, 31(10): 1229-1238. doi: 10.3879/j.issn.1000-0887.2010.10.010

Abstract(1609) PDF(277)

Abstract:
The homotopy analysis method (HAM) was presented for the primary resonance of multidegree-of-freedom system with strongly non-linearity excited by harmonic forces.The validity of the HAM is independent of whether or not there are small parameters in the considered equation.The HAM has provided a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter h.Two examples were presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincarémethod and the incremental harmonic balance method.

Nonlinear and Chaotic Analysis of a Financial Complex System

LIN Yong-xin, CHEN Yu-shu, CAO Qing-jie

2010, 31(10): 1239-1248. doi: 10.3879/j.issn.1000-0887.2010.10.011

Abstract(1803) PDF(956)

Abstract:
A determination on the characteristics of futures market of commodity of China was presented by the method of the phase-randomized surrogate data.There was a significant difference in critical values obtained when this method was used in random timeseries and nonlinear chaotic timeseries.The technology of the singular value decomposition was used to reduce noise of chaotic timeseries and then the phase space of chaotic timeseries was decomposed to range space and null noise space, and the original chaotic timeseries in range space was restructured.The method of strong disturbance on the basis of the improved general constrained randomized method was further adopted to re-deternine.According to the calculated result an analysis on the trend of futures's market of commodity is made.The results indicate that the Chinese futures's market of commodity is a complicated nonlinear system with obvious nonlinear chaotic characteristic.

Exact Solution and Dynamic Buckling Analysis of a Beam-Column System Having the Elliptic Type Loading

H. Secil Artem, Levent Aydin

2010, 31(10): 1249-1255. doi: 10.3879/j.issn.1000-0887.2010.10.012

Abstract(1766) PDF(723)

Abstract:
A closed form solution of the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time varying compressive force of an elliptic type was presented.Solution of the governing equation was obtained in the form of Fourier sine series and the resulting ordinary differential equation was solved analytically.Finding the exact analytical solutions of the dynamic buckling problems were difficult.However, availability of exact solutions provided adequate understanding about the physical characteristics of the system.The frequency-response characteristics of the system, the effects of static load, driving forces and frequency ratio on the critical buckling load were also investigated.

Advanced Method to Estimate the Reliability-Based Sensitivity of Mechanical Components With Strongly Nonlinear Performance Function

ZHANG Yi-min, ZHU Li-sha, WANG Xin-gang

2010, 31(10): 1256-1266. doi: 10.3879/j.issn.1000-0887.2010.10.013

Abstract(1617) PDF(961)

Abstract:
Based on the random perturbation technique of reliability sensitivity design, some realistic reliability-based sensitivity issues were discussed, with some of them having a structure of high nonlinear performance function.Combining the relating theories of moment method of reliability analysis, matrix differential and Kronecker algebra, the reliability-based sensitivity method based on perturbation method was modified, given that the first four moments of random variables.Meanwhile, a reliability-based sensitivity computing methodology also was put forward.Some examples were adopted to prove that using this methodology could highly improve the accuracy of reliabilitybased sensitivity computation and this methodology also offers some more dependable theoretic basis for engineering.