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2010年 第31卷 第10期    刊出日期：2010-10-01

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Accuracy analysis of immersed boundary method using method of manufactured solutions

宫兆新 鲁传敬 黄华雄

2010, 31(10):  1197-1208.  doi:10.1007/s10483-010-1353-x

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The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. This paper analyzes the accuracy of the immersed boundary method. The procedure contains two parts, i.e., the code verification and the accuracy analysis. The code verification provides the confidence that the code used is free of mistakes, and the accuracy analysis gives the order of accuracy of the immersed boundary method. The method of manufactured solutions is taken as a means for both parts. In the first part, the numerical code employs a second-order discretization scheme, i.e., it has second-order accuracy in theory. It matches the calculated order of accuracy obtained in the numerical calculation for all variables. This means that the code contains no mistake, which is a premise of the subsequent work. The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations. By analyzing the discretization errors, the accuracy of the immersed boundary method is proven to be first order even though the discretization scheme is second order. It has been found that the coarser mesh may not be sensitive enough to capture the influence of the immersed boundary, and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.

Attenuation analysis of hydraulic transients with laminar-turbulent flow alternations

万五一 朱嵩 胡云进

2010, 31(10):  1209-1216.  doi:10.1007/s10483-010-1354-9

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

J.ZUECO S.AHMED

2010, 31(10):  1217-1230.  doi:10.1007/s10483-010-1355-6

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An exact and a numerical solutions 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 are presented. A uniform magnetic field is assumed to be applied transversely to the direction of the flow with the consideration of the induced magnetic field with viscous and magnetic dissipations of energy. The porous plate is subjected to a constant suction velocity as well as a uniform mixed stream velocity. The governing equations are solved by the perturbation technique and a numerical method. The analytical expressions for the velocity field, the temperature field, the induced magnetic field, the skin-friction, and the rate of heat transfer at the plate are obtained. The numerical results are demonstrated graphically for 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 in the velocity field, the temperature field, the concentration field, and the induced magnetic field from the plate to the fluid are discussed. An increase in the heat source/sink or the Eckert number is found to strongly enhance the fluid velocity values. The induced magnetic field along the x-direction increases with the increase in the Hartmann number, the magnetic Prandtl number, the heat source/sink, and the viscous dissipation. It is found that the flow velocity, the fluid temperature, and the induced magnetic field decrease with the increase in the destructive chemical reaction. Applications of the study arise in the thermal plasma reactor modelling, the electromagnetic induction, the magnetohydrodynamic transport phenomena in chromatographic systems, and the magnetic field control of materials processing.

Exact solution to peristaltic transport of power-law fluid in asymmetric channel with compliant walls

T.HAYAT M.JAVED

2010, 31(10):  1231-1240.  doi:10.1007/s10483-010-1356-7

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

Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

侯国林 阿拉坦仓

2010, 31(10):  1241-1250.  doi:10.1007/s10483-010-1357-9

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The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated. The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region. The completeness of the eigenfunctions is then proved, providing the feasibility of using separation of variables to solve the problems. A general solution is obtained with the symplectic eigenfunction expansion theorem.

Symplectic multi-level method for solving nonlinear optimal control problem

彭海军 高强 吴志刚 钟万勰

2010, 31(10):  1251-1260.  doi:10.1007/s10483-010-1358-6

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By converting an optimal control problem for nonlinear systems to a Hamiltonian system, a symplecitc-preserving method is proposed. The state and costate variables are approximated by the Lagrange polynomial. The state variables at two ends of the time interval are taken as independent variables. Based on the dual variable principle, nonlinear optimal control problems are replaced with nonlinear equations. Furthermore, in the implementation of the symplectic algorithm, based on the 2^N algorithm, a multilevel method is proposed. When the time grid is refined from low level to high level, the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve e±ciency. Numerical simulations show the precision and the e±ciency of the proposed algorithm in this paper.

Three dimensional large deformation analysis of phase transformation in shape memory alloys

夏开明 潘同燕 刘山洪

2010, 31(10):  1261-1272.  doi:10.1007/s10483-010-1359-7

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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. This paper presents 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. The Martensite fraction of SMAs is assumed to be dependent on deviatoric stress tensor. Therefore, phase transformation of SMAs is volume preserving during the phase transformation. The model is implemented in large deformation finite element code and cast in the updated Lagrangian scheme. In order to use the Cauchy stress and the linear strain in constitutive laws, a frame indifferent stress objective rate has to be used. In this paper, the Jaumann stress rate is used. Results of the numerical experiments conducted in this study 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

张石生 李向荣 陈志坚 柳京爱

2010, 31(10):  1273-1282.  doi:10.1007/s10483-010-1360-x

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This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction. The purpose is to find 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 that is also a solution to a variational inequality. Under suitable conditions, some strong convergence theorems are established in the framework of Hilbert spaces. The results presented in the paper improve and extend the corresponding results of Colao et al. (Colao, V., Acedo, G. L., and Marino, G. An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 71, 2708–2715 (2009)), Plubtieng and Punpaeng (Plubtieng, S. and Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007)), Colao et al. (Colao, V., Marino, G., and Xu, H. K. An iterative method for finding common solutions of equilibrium problem and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)), Yao et al. (Yao, Y., Liou, Y. C., and Yao, J. C. Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Application 2007, Article ID 64363 (2007) DOI 10.1155/2007/64363), and others.

Three solutions to inequalities of Dirichlet problem driven by p(x)-Laplacian

葛斌 薛小平 郭梦舒

2010, 31(10):  1283-1292.  doi:10.1007/s10483-010-1361-9

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A class of nonlinear elliptic problems driven by p(x)-Laplacian with a non-smooth locally Lipschitz potential is considered. By applying the version of the non-smooth three-critical-point theorem, the existence of three solutions to the problems is proved.

Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

原培新 李永强

2010, 31(10):  1293-1304.  doi:10.1007/s10483-010-1362-6

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A homotopy analysis method (HAM) is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces. The validity of the HAM is independent of the existence of small parameters in the considered equation. The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter. Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar´e method and the incremental harmonic balance method.

Nonlinear and chaotic analysis of a financial complex system

林勇新 陈予恕 曹庆杰

2010, 31(10):  1305-1316.  doi:10.1007/s10483-010-1363-7

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In this paper, determination of the characteristics of futures market in China is presented by the method of the phase-randomized surrogate data. There is a significant difference in the obtained critical values when this method is used for random timeseries and for nonlinear chaotic timeseries. The singular value decomposition is used to reduce noise in the chaotic timeseries. The phase space of chaotic timeseries is decomposed into range space and null noise space. The original chaotic timeseries in range space is restructured. The method of strong disturbance based on the improved general constrained randomized method is further adopted to re-deternination. With the calculated results, an analysis on the trend of futures market of commodity is made in this paper. The results indicate that China’s futures 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. S. ARTEM L. AYDIN

2010, 31(10):  1317-1324.  doi:10.1007/s10483-010-1364-8

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This paper presents a closed form solution to 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, i.e., a₁cn²(t, k²) + a₂sn²(t, k²) + a₃dn²(t, k²). The solution to the governing equation is obtained in the form of Fourier sine series. The resulting ordinary differential equation is solved analytically. Finding the exact analytical solutions to the dynamic buckling problems is difficult. However, the availability of exact solutions can provide adequate understanding for the physical characteristics of the system. In this study, the frequency-response characteristics of the system, the effects of the static load, the driving forces, and the frequency ratio on the critical buckling load are also investigated.

Advanced method to estimate reliability-based sensitivity of mechanical components with strongly nonlinear performance function

张义民 朱丽莎 王新刚

2010, 31(10):  1325-1336.  doi:10.1007/s10483-010-1365-x

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Based on the random perturbation technique for reliability sensitivity design, some realistic reliability-based sensitivity issues are discussed, some of which have a structure of high nonlinear performance functions. Combining the related theories of the moment method of the reliability analysis, the matrix differential, and the Kronecker algebra, the reliability-based sensitivity method based on the perturbation method is modified if the first four moments of random variables are given. Meanwhile, a reliability-based sensitivity computation method is proposed. Some examples are used to show that using this method can effectively improve the accuracy of the reliability-based sensitivity computation and offer a reliable theoretic basis in engineering.