应用数学和力学

2017 Vol. 38, No. 8

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ZHONG Wan-xie, WU Feng, SUN Yan

2017, 38(8): 845-852. doi: 10.21656/1000-0887.380078

Abstract(1539) PDF(1211)

Abstract:
The hydraulic jump problem was studied by means of the displacement method and the Lagrangian coordinates. The discussion and the numerical example show that, under the fundamental assumption that the horizontal displacement is independent of the vertical coordinate, the hydraulic jump is not a type of strong discontinuous solution because of the kinetic energy of the vertical motion, but a continuous solution which fluctuates near the discontinuity. The strong discontinuity can be seen as the limit of the continuous solution.

Instability and Transition Prediction of Stationary Crossflow Vortices Over Supersonic Swept Elliptic Cylinders

YANG Zhi-yang, ZHAO Lei, LUO Ji-sheng

2017, 38(8): 853-862. doi: 10.21656/1000-0887.370303

Abstract(1281) PDF(970)

Abstract:
Some experiments had shown that stationary crossflow vortices may cause transition over the 1st halves of supersonic swept cylinders. The swept elliptic cylinders with infinite spanwise lengths were used to simulate the swept wings of supersonic aircrafts at high altitudes. Based on the e^N method and the N factor, the influence of changing the parameters including the upwind axis length, the Reynolds number, the swept angle and the Mach number on the instability of the stationary crossflow vortices over supersonic infinite swept elliptic cylinders was studied. The results show that, the instability of stationary crossflow vortices is stronger with longer upwind axes or larger Reynolds numbers. Meanwhile, the relationship between the instability of stationary crossflow vortices and the Reynolds number is almost linear. The results also show that the stationary crossflow vortices are more stable in the conditions with greater Mach numbers. The change of the swept angle in a certain range has a small effect on the instability of stationary crossflow vortices. These results would be helpful to improve the understanding of the transition mechanism over the leading edges of supersonic aircraft wings and provide theoretical guidance for crossflow transition prediction.

A Conjugate Boundary Element Method for Complex Analysis of Analytic Function

LI Guo-qing

2017, 38(8): 863-876. doi: 10.21656/1000-0887.370315

Abstract(1311) PDF(1423)

Abstract:
An analytic function is composed of 2 real conjugate harmonic functions, of which the complex analysis plays an important role in the fields of applied mathematics and mechanics. A set of weighted residual equations were proposed and proved to be equivalent to the approximate solution to the original problem involving 2 governing equations in the domain, the boundary condition and the CauchyRiemann equation at the boundary. 2 conventional direct boundary integral equations at the boundary collocation points were deduced from 2 of the weighted residual equations, and 1 finite difference equation was deduced from the rest one. The mathematical problem arising from the illconditioned linear equations was solved and the Cauchy integral equation was adopted for numerical calculation of the fields at the internal points inside the domain. Finally, the proposed conjugate boundary element method with constant elements was completely established. 3 examples demonstrate that, the proposed method is valid for analytic functions in terms of the power function, the exponential function and the logarithmic function in interior or exterior domains, and the error estimation of the proposed method is at the same order as that of the boundary element method for 2D potential problems.

Some Stable and Unstable Critical States of a Compression Rod With a Flexible Support

ZHAO Yan-ping, LI Lin, JIN Ming

2017, 38(8): 877-887. doi: 10.21656/1000-0887.370299

Abstract(1134) PDF(765)

Abstract:
Under Euler’s critical load, the stability of a slender compression rod with one end fixed and the other clamped in rotation but translationally restrained by a spring was studied. The potential energy of the system was expressed with the functional of the rod deflection angle; the disturbance was expanded into the Fourier series; the 2nd-order variation of the potential energy was expressed with a quadratic form. The 2nd-order positive semidefinite variation in the critical state was derived with the buckling mode and the critical load obtained. A further study of the positive definiteness of higher-order variations, including the 4th and 6th variations, indicates that the stability of the compression rod with a flexible support is related to the stiffness of the flexible constraint and may be stable or unstable, which is different from the case of a rigid constraint. In the stable and unstable critical states the ranges for the relative stiffness of the flexible support were also given.

A Simplified Model for Coupled Fire-Structure Analysis and Its Application

CHEN Shi-cai, XU Shan, ZHANG Lei, YAN Wei-ming

2017, 38(8): 888-898. doi: 10.21656/1000-0887.370351

Abstract(810) PDF(1489)

Abstract:
To consider the actual fire characteristics in the fire response analysis of building structures and simplify the complex relationship between the fire analysis model and the structural finite element analysis model, a spatio-temporal model of fire temperature and heat flux boundary for heat conduction analysis was developed. The model adopted the 2-way orthogonal polynomial approach to fit the discrete data from fire simulation and obtained the continuous spatial polynomial equations, which was proved to be accurate for capturing the distribution of temperature and heat flux required in heat conduction analysis and thermo-mechanical coupling analysis. Finally, the model was applied through user subroutines UTEMP and DFLUX in ABAQUS to the fire resistance analysis of a new archive building in Beijing. The results show that this method can be used to combine fire simulation and structural analysis for safety evaluation of structures under fire.

Waterproofness Optimization for Elastic Rubber Gaskets in Shield Tunnels With Random Parameters

LEI Zhen-yu, LIU Ming

2017, 38(8): 899-910. doi: 10.21656/1000-0887.370289

Abstract(1208) PDF(1030)

Abstract:
Geometric parameters of elastic rubber gaskets in shield tunnels often present stochasticity influenced by the manufacturing process. Accordingly, the waterproof performance of elastic rubber gaskets will be affected. The coordinates of the hole center, the hole diameter, the section width and height were selected as the input random variables, and the sensitivity values with respect to these random parameters of the gasket’s waterproof performance were obtained by means of the ANSYS PDS module. The results show that the hole diameter has larger effect on the closure pressure and the contact stress than other geometric parameters. At the same time, the vertical position of the hole has greater influence than the horizontal one. On this basis, the closure pressure and the contact stress on the lower surface were selected as state variables, and the maximum contact stress was deemed as the objective function with the closure pressure not higher than a set value. The ANSYS design optimization module was used to conduct the parameter optimization for rubber gaskets, and give a new optimal gasket section geometry for one shield tunnel. The full-size test results verify the reliability of the optimal analysis.

Limit Analysis of Surrounding Rock Pressure for Shallow Tunnels Considering Floor Heave

ZHANG Xiang, LI Lin-an, WANG Shi-bin, LIU Xi-jun, GENG Wen-bin

2017, 38(8): 911-921. doi: 10.21656/1000-0887.370346

Abstract(1090) PDF(717)

Abstract:
For the floor heave phenomena after excavation of shallow tunnels, the limit analysis upper bound method was applied to build a surrounding rock pressure calculation model of rigid bodies in view of tunnel floor heave effects. Then, according to the linear MohrCoulomb criterion and the associated flow rule, the theoretical expression of the surrounding rock limit pressure was deduced. Through constraints, the calculation of surrounding rock pressure was transformed into an optimization problem in mathematics so that a computer program was compiled to conduct the optimal computation. The calculated results were compared with the measured data and the reference results, to verify the reliability of the proposed method. The results show that, in the application of the limit analysis upper bound method to deal with the surrounding rock pressure problems of shallow tunnels, the tunnel floor heave effects shall be fully considered; the supporting of the tunnel floor has significant influence on the surrounding rock pressure. The research provides a theoretical reference for the excavation and supporting of shallow tunnels.

A Numerical Method for Calculation of Structural Jerk Responses

LI Yan-ting, XU Ji-qing, XU Xi-bin, PU Yan-ru

2017, 38(8): 922-931. doi: 10.21656/1000-0887.370181

Abstract(1535) PDF(1032)

Abstract:
Jerk is of great significance in engineering practice. A numerical method for solving jerk responses was constructed through combination of the radial basis function (RBF) approximation and the collocation method. The proposed method was used to calculate the jerk and the 3rd-order jerk equations, and the RBF interpolation was adopted to approximate the real motion rule, which made good the defect that the traditional methods can’t be used to calculate the jerk. Aimed at the numerical characteristics of the dynamic differential equations, an improved RBF expression of multivariable joint interpolation combining the all-order derivatives of the variable was presented. The initial-value condition of the same order with the differential equation was added to obviously decrease the numerical oscillation. The results of the numerical examples indicate that the proposed method has the advantages of a simple calculation process, high accuracy and high applicability to jerk equations.

Block-Sparse Signal Recovery Based on l₁-l₂Norm Minimization

CHEN Peng-qing, HUANG Wei

2017, 38(8): 932-942. doi: 10.21656/1000-0887.370230

Abstract(1343) PDF(923)

Abstract:
Compressed sensing (CS) is a newly developed theoretical framework for information acquisition and processing. Through the solution of non-linear optimization problems, sparse and compressible signals can be recovered from small-scale linear and non-adaptive measurements. Block-sparse signals as typical sparse ones exhibit additional block structures where the non-zero elements occur in blocks (or clusters). Based on the previous l_1-2 norm minimization method given by YIN Peng-hang, LOU Yi-fei, HE Qi, et al. for common sparse signal recovery, the l₁-l₂ minimization recovery algorithm was extended to the block-sparse model, the properties of thel₁-l₂norm were proved and the sufficient condition for block-sparse signal recovery was established. Meanwhile, an iterative method for block-sparsel₁-l₂minimization was presented by means of the DCA (difference of convex functions algorithm) and the ADMM (alternating direction method of multipliers). The numerical simulation results demonstrate that the signal recovery success rate of the proposed algorithm is higher than those of the existing algorithms.

A Numerical Approximation Method for Solutions to Nonlinear Dynamic Systems Based on Multiquadric Quasi-Interpolation Functions

DU Shan, LI Feng-jun

2017, 38(8): 943-955. doi: 10.21656/1000-0887.370368

Abstract(1374) PDF(682)

Abstract:
The multiquadric quasi-interpolation function has advantages of high accuracy and good stability. A new numerical method for resolving the initial value problems of nonlinear dynamic systems was proposed via combination of the multiquadric quasi-interpolation function and the 4th-order Runge-Kutta method. The advantages and disadvantages were analyzed between this new method and the existing numerical methods for nonlinear dynamic systems, according to the numerical example and error estimation. The results show that the proposed method needs less computation cost and enables fine approximation to the analytical solutions to nonlinear dynamic systems.

A New Coupled Common Fixed Point Theorem for Square Contractive Mappings in 2-Metric Spaces

LIU Li-ya, GU Feng

2017, 38(8): 956-966. doi: 10.21656/1000-0887.370261

Abstract(952) PDF(504)

Abstract:
In the framework of 2-metric spaces, a new class of square contractive conditions were established and the existence and uniqueness of the coupled common fixed point in given conditions were discussed, by means of the ω-compatible condition for the pair of mappings. Then a new coupled common fixed point theorem was proved. Several illustrative examples were also given in support of the new theorem. The work improves the relative results in the recent literatures.