应用数学和力学

Nonlinear Model Reduction Based on the Mori-Zwanzig Scheme and Partial Least Squares

LAI Xuefang, WANG Xiaolong, NIE Yufeng

2021, 42(6): 551-561. doi: 10.21656/1000-0887.410230

Abstract(1116) PDF(115)

Abstract:
The proper orthogonal decomposition and the Galerkin projection are widely used methods for solving the model reduction problems of complex nonlinear systems. However, only a part of basis function modes are extracted with these methods to construct the reduced systems, which usually makes the reduced systems inaccurate. For this issue a method was proposed to efficiently correct the errors of the reduced systems. First, the Mori-Zwanzig scheme was employed to analyze the errors of the reduced systems, with the theoretical form of the error model and the effective predictive variables obtained. Then, the error prediction model was built by means of the partial least square method to construct the multiple regression model between the predictive variables and the system errors. The constructed error prediction model was directly embedded into the original reduced system, to get a modified reduced system formally equivalent to the model obtained with the Petrov-Galerkin projection on the right side of the original model. The error estimation of the modified reduced system was given. Numerical results illustrate that, the proposed method can improve the stability and accuracy of the reduced systems effectively, and has high computation efficiency.

Numerical Analysis of a Class of Fractional Langevin Equations With the Block-by-Block Method

ZHANG Man, CAO Yanhua, YANG Xiaozhong

2021, 42(6): 562-574. doi: 10.21656/1000-0887.410337

Abstract(848) PDF(83)

Abstract:
The fractional Langevin equation is of great scientific significance and engineering application value. Based on the classical block-by-block method, the numerical solution of a class of fractional Langevin equations with Caputo derivatives was obtained. Through introduction of the quadratic Lagrange basis function interpolation, the block-by-block convergent nonlinear equations were constructed, and the numerical solution of the Langevin equation was obtained by coupling in each block. Under the condition of 0<α<1, the stochastic Taylor expansion was used to prove that the block-by-block method is (3+α)-order convergent. Numerical experiments show that, the block-by-block method is stable and convergent under different values of α and time step h,and overcomes the existing methods’ disadvantages of slow speed and poor accuracy for solving fractional Langevin equations.

The Minimal Wave Speed of a Lotka-Volterra Competition Model

ZHANG Yafei, ZHOU Yinbo

2021, 42(6): 575-585. doi: 10.21656/1000-0887.410279

Abstract(746) PDF(44)

Abstract:
The minimal wave speed selection for the traveling waves of a 3-species competition system with monostable nonlinearity was considered. First, a comparison method was used to establish minimal-speed selection mechanisms through construction of subtle upper and lower solutions. Then, the speed selection about the threshold values of competition coefficients and the estimations were obtained. Finally, some numerical simulations were performed to show that the obtained results extend those in the previous literatures.

Observer-Based Adaptive Neural Network Control for Nonstrict-Feedback Nonlinear Systems With Time Delays

LIU Xiang, TONG Dongbing, CHEN Qiaoyu

2021, 42(6): 586-594. doi: 10.21656/1000-0887.410325

Abstract(1026) PDF(73)

Abstract:
An observer-based adaptive neural network control problem was investigated for a class of nonstrict-feedback nonlinear systems with time delays. A state observer was constructed to estimate unknown variables in nonlinear systems. With the approximation ability of RBF NNs and the backstepping technique, an adaptive neural network output feedback control approach was proposed. The designed controller ensures the semi-global uniform boundedness of all signals in the closed-loop system. Finally, the simulation example shows the effectiveness of the proposed control approach.

Mixed Type Duality for Nonsmooth Multiobjective Semi-Infinite Programming Problems

LIU Juan, LONG Xianjun

2021, 42(6): 595-601. doi: 10.21656/1000-0887.410342

Abstract(754) PDF(42)

Abstract:
The mixed type duality for nonsmooth multiobjective semi-infinite programming problems was studied. Firstly, by the Lagrange function, the definitions of weakly efficient solutions and efficient solutions to the mixed type duality were introduced. Secondly, by means of the Dini-pseudoconvexity, the weak duality theorems, the strong duality theorems and the converse duality theorems were obtained. The results generalize the main results in previous literatures.

Equivalent Characterization of McRow Optimal Solutions to Multiobjective Optimization Problems

ZHAO Chunjie, GAO Ying, LIU Fuping

2021, 42(6): 602-610. doi: 10.21656/1000-0887.410338

Abstract(602) PDF(40)

Abstract:
Based on the McRow model for multiobjective optimization problems, the relationships between the W-robust efficient solution (also known as the McRow optimal solution) and the weakly efficient solution, the efficient solution and the properly efficient solution were established. Firstly, the relationship between the W-robust efficient solution and various exact solutions to multiobjective optimization problems was studied. Then, the concept of the McRow optimal solution to stochastic multiobjective optimization problems was introduced, and the relationship between the McRow optimal solution and other solutions was given. The examples show that, the solutions obtained with the McRow model are of better robustness.

Analysis of Deformation and Bearing Capacity of Flexible Beams Under Gravitational Loads

WANG Dan, WANG Jian

2021, 42(6): 611-622. doi: 10.21656/1000-0887.410169

Abstract(634) PDF(81)

Abstract:
The large deformation of flexible structure can decrease the load. The relation between the large deformation of the flexible beam and the gravitational load was studied. Based on the experiments, the distribution mode for the gravitational load was built. With the large deflection constitutive model for Timoshenko beams, the governing equation for the large-deformation beam under the gravitational load was derived. Two dimensionless parameters were defined, i.e. the Cauchy number and the deformation coefficient. Through numerical calculation of the governing equation, the quantitative relation between the Cauchy number and the deformation coefficient was discussed. The theoretical results are compared with the experimental data to confirm the reliability of the theoretical model. The model was used to analyze the snow load data in previous literatures to verify the applicability in reality. The research indicates that, the proposed method applies to the design of deflection and bearing capacity of flexible beams in engineering systems, as well as the prediction of the lodging resistance of vegetations in sand storm or snow storm.

Microbeam Model and Related Differential Quadrature Finite Elements

LIU Songzheng, ZHANG Bo, SHEN Huoming, ZHANG Xu

2021, 42(6): 623-636. doi: 10.21656/1000-0887.410260

Abstract(721) PDF(35)

Abstract:
A size-dependent quasi-3D functionally graded (FG) microbeam model was presented within the combined framework of the modified couple stress theory and a 4-unknown higher-order shear and normal deformation theory. Then the model was applied to analyze the static bending and free vibration of FG microbeams. With the 2nd Lagrange equation, the corresponding motion equations and the appropriate boundary conditions were obtained. A 2-node 16DOF differential quadrature finite element combining the Gauss-Lobatto quadrature rule with the differential quadrature rule was constructed to handle the general static/dynamic boundary value problems of FG microbeams. A comparison study was performed to show the efficacy of the proposed theoretical model and solution method. Finally, the effects of the gradient index, the intrinsic length scale parameter, the geometrical parameters and the boundary conditions on the static and dynamic characteristics of FG microbeams were examined. Numerical results reveal that the developed beam model and element are applicable to the analysis of mechanical behaviors of FG microbeams with various slenderness ratios. Besides, introduction of the couple stress effect can significantly change the static and dynamic characteristics of FG microbeams.

Research on the Whole Creep Process of the Generalized Kelvin Model Based on Damaged Body Elements

SONG Yanqi, LI Xiaolong, MA Hongfa, FU Hang

2021, 42(6): 637-644. doi: 10.21656/1000-0887.410255

Abstract(690) PDF(29)

Abstract:
The creep properties of rocks often have important control effects on the stability of tunnels and underground works. According to the stage characteristics of rock creep, the whole process of rock creep can be divided into 4 stages. The generalized Kelvin model can better reflect the rock creep characteristics of the 1st 3 stages, but cannot ideally reflect the characteristics of the accelerated creep stage. Through introduction of damaged body elements and Kachanov’s damage factor evolution formula, a generalized Kelvin model with damaged body elements was constructed, thereby a creep model was established to reflect the whole process of rock creep, and a relatively simple combined model parameter calculation method was proposed. The model well describes the whole process of rock creep, and the model parameters are easy to determine. The model works well in fitting and analysis of the experimental curves of sandy mudstone uniaxial compression creep, and the research results provides a reference for similar projects.

Fracture Mechanism of Unstable Rock With Double-Crack Control Discontinuity Subjected to High Earthquake Intensities

TANG Hongmei, ZHOU Fuchuan, CHEN Song, WANG Linfeng

2021, 42(6): 645-655. doi: 10.21656/1000-0887.410187

Abstract(705) PDF(23)

Abstract:
Unstable rock in the limestone area has typical characteristics of quasimasonry structure and control discontinuity, where the failure essence is the fracture propagation under loads. The unstable rock control discontinuity in the limestone zone was analyzed, and the geological model for a doublecrack unstable rock mass was obtained according to the theory of geomorphologic evolution. The mechanical model and fracture mechanical model for the complex control discontinuity were constructed. Based on the rock weight, the fissure water pressure and the earthquake load, the formula of the fracture stability coefficient was obtained under the maximum circumferential stress criterion. The fracture stability coefficient expression for the unstable rock is rational according to the verifying case analysis. The fracture stability coefficient corresponding to each intensity decreases with the crack length ratio. Under earthquake intensities Ⅷ and Ⅸ, the unstable rock will collapse; the stability of the unstable rock is liable to the coupling effects of the crack length ratio of the main control discontinuity and the earthquake intensity. The critical crack length ratio decreases with the earthquake intensity. Under earthquake intensity Ⅶ, the theoretical value of the critical crack length ratio is 25.8%, which is slightly less than the actual value of 27.7%, and on the slightly safer side. The fracture stability coefficient expression has good applicability. The research results provide an important theoretical support for the treatment of such unstable rock masses.

Fracture Mechanics Analysis of Thermoelectric Materials With Equilateral Triangle Holes

ZHU Mingming, LI Lianhe

2021, 42(6): 656-664. doi: 10.21656/1000-0887.410232

Abstract(556) PDF(22)

Abstract:
The fracture mechanics for thermoelectric materials with equilateral triangle holes subjected to uniform electric current densities and uniform energy fluxes at infinity was studied by means of the complex variable method. The analytic expressions of temperature fields and stress fields were obtained under the boundary conditions of electric insulation and thermal insulation. Effects of the triangle size, the applied electric current density and the energy flux on the thermoelectric material were analyzed. The results show that, the variations of the current density and the triangle size have obvious influences on the annular energy flux, the annular stress and the annular heat flux.

2021 Vol. 42, No. 6