2019 Vol. 40, No. 6

Display Method:
Free Vibration Analysis of Rectangular Honeycomb-Cored Plates Under Elastically Constrained Boundary Conditions
WANG Yongfu, QI Wenkai, SHEN Cheng
2019, 40(6): 583-594. doi: 10.21656/1000-0887.390348
Abstract(880) HTML (78) PDF(666)
Honeycomb-cored plates are widely used in aircraft, high-speed trains and other fields, with clear scientific value and engineering significance worthy of vibration analysis. Other than classical boundary condition assumptions (such as simple supports), the elastic constraints were considered to analyze the free vibration characteristics of honeycomb-cored plates. Specifically, the sandwich plate problem was transformed into a 3-layer structure with the honeycomb core layer simplified as an equivalent anisotropic layer. Furthermore, the displacement field function for the structure was expressed in the form of the improved 2D Fourier series, and the natural frequencies and modal shapes of the structure were derived with the Rayleigh-Ritz method based on the energy principle. The theoretical results are in good agreement with the numerical ones. The proposed theoretical model can be used to systematically discuss the effects of boundary constraints on the free vibration characteristics of honeycomb-cored plates, and provides a theoretical basis for the design of constraint schemes for this kind of structures.
A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems
LIU Xiaomei, ZHOU Gang, ZHU Shuai
2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249
Abstract(694) HTML (66) PDF(424)
Symplectic methods, including the generating function method, the symplectic Runge-Kutta (RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10-13. For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.
Analysis of Shear Lag Effects in Box Girders With Variable-Thickness Flanges
ZHAO Qingyou, ZHANG Yuanhai, SHAO Jiangyan
2019, 40(6): 609-619. doi: 10.21656/1000-0887.390259
Abstract(809) HTML (88) PDF(671)
For concrete box girders with flanges of linearly variable thicknesses along the cross section widths, shear lag effects were analyzed according to the potential energy variation principle. The additional deflection caused by the shear lag effect was used as the generalized displacement to describe the shear lag deformation state, in view of the influence of the axial force equilibrium condition on the shear lag. The calculation results of several simply supported box girders subjected to uniform loads and concentrated loads were compared. The results show that, the proposed method gives stresses in good agreement with those obtained from the finite element method, which confirms the correctness of this method. Compared with the simplified calculation method for box girders with equivalent uniformthickness flanges, this method has enhanced accuracy for real box girders with variablethickness flanges, and the error reduction is up to 5.65%.
High-Order Analytical Solutions and Convergence Discussions of the 2-Step Perturbation Method for Euler-Bernoulli Beams
ZHANG Daguang
2019, 40(6): 620-629. doi: 10.21656/1000-0887.390272
Abstract(938) HTML (75) PDF(482)
High-order analytical solutions of the 2-step perturbation method were first obtained for post-buckling and nonlinear bending of Euler-Bernoulli beams. The nonlinear model with centerline inextensibility was derived with the exact curvature expression according to the energy variational principle. Based on the comparison with the exact solutions or high-order perturbation solutions, the asymptotic property and the suitable range of 2-step perturbation solutions were also discussed. The results show that, the lower-order perturbation solutions are suitable for the initial post-buckling stage and the initial nonlinear bending stage, and the higher-order perturbation solutions are necessary for the late post-buckling stage and the highly nonlinear bending stage. Therefore, the reason why some previous perturbation solutions are inaccurate lies in the offside beyond suitable ranges, and the 2-step perturbation method is developed and improved herein.
Multi-Scale Structure Optimization Design Based on Eigenvalue Analysis
SUN Guomin, ZHANG Xiaozhong, SUN Yanhua
2019, 40(6): 630-640. doi: 10.21656/1000-0887.390207
Abstract(1006) HTML (99) PDF(908)
A multi-scale structure optimization method was proposed based on eigenvlue analysis, to find the macrostructure and microstructure of maximum macro stiffness under the worst load. The constraint that the Euclidian norm of the uncertain load is 1 was introduced, the structural compliance was calculated according to the Rayleigh-Ritz theorem, and the compliance was transformed to a symmetric matrix with the same dimensions as the local load vector. In this way, the compliance minimization problem under the worst load was transformed to the minimum problem of the maximum eigenvalue of the symmetric matrix. Moreover, the worst load case was determined with the eigenvector corresponding to the maximum eigenvalue of the matrix. Several numerical experiments demonstrated the validity of the proposed method, and illustrated the reasonability of the macro topological structure and the micro material distribution. The proposed multi-scale optimization method has virtues of iterative stability and rapid convergence. The update of the density function in the topological optimization was performed based on sensitivity analysis and the method of moving asymptotes (MMA).
An Improved Algorithm for Solving Dynamic Responses of Vehicle-Track Vertically Coupled Systems
LIU Zhangjun, HE Chenggao, ZHANG Chuanyong
2019, 40(6): 641-649. doi: 10.21656/1000-0887.390202
Abstract(822) HTML (68) PDF(547)
Since the track structure length would increase with the system response duration, an improved algorithm for solving the dynamic responses of the vehicle-track vertically coupled system was proposed. In this algorithm, a certain length of the track structure was selected in advance, and the mass matrix, the damping matrix and the stiffness matrix of the track structure can be obtained. During the solution process, the position of the vehicle subsystem was continually determined, meanwhile, it was judged whether the position of the vehicle subsystem and the response matrix of the track structure shall be adjusted so as to achieve the goal of only increasing the system response duration without increasing the track structure length. The numerical results show that, the improved algorithm is of high accuracy and efficiency in realistic simulation of the forward movement of the train on the track, and ensures that the number of elements in the track subsystem does not increase with the system response duration, making an applicable method for solving the dynamic responses of vehicle-track vertically coupled systems.
A Composite Optimization Method for Anchorage Parameters of Rammed Earth Sites Based on Desirability Functions
LU Wei, ZHAO Dong, LI Dongbo, MAO Xiaofei
2019, 40(6): 650-662. doi: 10.21656/1000-0887.390247
Abstract(654) HTML (69) PDF(552)
Aimed at the needs for anchoring engineering in rammed earth sites, a composite optimization method for anchorage parameters based on desirability functions was proposed. The main purpose of the method is to obtain a good balance between the maximum anchoring force and the minimum site damage through optimization of the combination of anchor lengths and anchor hole diameters. The full factorial design was used in the experiment, and the response surface method was applied to build the analysis model. Then the desirability function in statistics was introduced into the optimization of anchor parameters, and the relationship between the multiobjective response and the anchoring parameter level was established. The results show that, when the anchoring force maximization and the site damage minimization goals are respectively satisfied, there will be a conflict between the corresponding anchoring parameters. The multiobjective response optimization can determine the feasible ranges of the anchoring parameters under the goal of response demand, which is convenient for engineering designers to visualize the anchoring parameters according to the actual engineering conditions.
Unconditionally Optimal Error Estimates of the Semi-Implicit BDF2-FEM for Cubic Schrödinger Equations
DAI Meng, YIN Xiaoyan
2019, 40(6): 663-681. doi: 10.21656/1000-0887.390209
Abstract(883) HTML (77) PDF(518)
The optimal error estimates of the semi-implicit BDF2-FEM were studied for cubic Schrödinger equations. First, an error estimate was divided into 2 parts: the temporal-discretization and the spatial-discretization. Through introduction of a temporal-discretization equation, the uniform boundedness of the solution and the temporal error estimate were obtained. The unconditionally optimal error estimates of the 2nd-order backward difference (BDF2-FEM) semi-implicit scheme for cubic Schrdinger equations were given. Finally, numerical examples verify the theoretical analysis.
A Self-Adaptive Uzawa Block Relaxation Algorithm for Free Boundary Problems
GUO Nanxin, ZHANG Shougui
2019, 40(6): 682-693. doi: 10.21656/1000-0887.390347
Abstract(966) HTML (59) PDF(515)
A self-adaptive Uzawa block relaxation algorithm, based on the augmented Lagrangian multiplier method and the self-adaptive rule, was designed and analyzed for free boundary problems with unilateral obstacle. The problem was discretized as a finite-dimensional linear complementary problem which is equivalent to a saddle-point one with an augmented Lagrangian function and an auxiliary unknown. With the Uzawa block relaxation method for the problem, a 2-step iterative method was got with a linear problem as a main subproblem while the auxiliary unknown was computed explicitly. The convergence speed of the method highly depends on the penalty parameter, and it is difficult to choose a proper parameter for an individual problem. To improve the performance of the method, a self-adaptive rule was proposed to adjust the parameter automatically per iteration. Numerical results confirm the theoretical analysis of the proposed method.
Some Robust Approximate Optimality Conditions for Nonconvex Multi-Objective Optimization Problems
ZHAO Dan, SUN Xiangkai
2019, 40(6): 694-700. doi: 10.21656/1000-0887.390289
Abstract(948) HTML (72) PDF(364)
A class of nonconvex multi-objective optimization problems were introduced with data uncertainty. Then, with the robust optimization approach, the robust counterpart model for the uncertain multi-objective optimization problem was built. Moreover, with the scalarization method and the generalized subdifferential properties, the optimality conditions were characterized for robust quasi approximate efficient solutions to the uncertain multi-objective optimization problem. The work generalizes and improves some results in the recent literatures.