应用数学和力学

2021 Vol. 42, No. 5

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Optimization Software Development for Offshore Turbine Transition Structures Based on LiToSim

YE Yanpeng, GU Shuitao, LIU Min, FENG Zhiqiang

2021, 42(5): 441-451. doi: 10.21656/1000-0887.410354

Abstract(1178) PDF(236)

Abstract:
The software development of stress-based topology optimization was addressed for the offshore turbine transition structure based on the self-developed LiToSim software platform. Firstly, the multi-scale model for the mixed beam and bulk elements was applied to the structural analysis of the wind turbine transition structure. Secondly, the extreme loads of the wind-wave coupling were taken into account. Thirdly, the stress-based topology optimization method was used for the design of the transition structure. Based on the self-developed LiToSim software platform, a customized software TUR/TOPT for the topology optimization of the transition structures of offshore wind turbines was formed. With TUR/TOPT, the traditional compliance optimization and stress-based optimization results of the transition structure were compared to show the advantages of stress-based optimization in reducing the structural stress during the material reduction design process and effectively avoiding stress concentration. The software TUR/TOPT provides important guiding value for the selection process of wind turbine construction.

Effects of the Gravity Field on 2D Fiber-Reinforced Media Under the Fractional Order Theory of Thermoelasticity

DUAN Xiaoyu, MA Yongbin

2021, 42(5): 452-459. doi: 10.21656/1000-0887.410125

Abstract(1015) PDF(205)

Abstract:
Based on the fractional order generalized thermoelastic coupling theory proposed by Sherief et al, the thermoelastic problem of 2D fiber-reinforced elastomers under thermal shock was studied. In view of the effects of gravity on 2D fiber-reinforced linearly thermoelastic isotropic media, the governing equations were established. Through the normal mode analysis and numerical calculation, the governing equations were solved, and the expressions of the dimensionless temperature, the displacement components and the stress under different fractional order parameters and different gravity fields were obtained. The distributions of variables were illustrated and the results were discussed. The results show that, the gravity field and fractional order parameters have significant impacts on the displacements and stresses of the fiber-reinforced media, but the influence on the temperature is limited.

Analysis of 2D Transient Heat Conduction Problems With the Element-Free Galerkin Method

WANG Hong, LI Xiaolin

2021, 42(5): 460-469. doi: 10.21656/1000-0887.410111

Abstract(1203) PDF(225)

Abstract:
The element-free Galerkin (EFG) method was introduced to solve 2D transient heat conduction problems. Firstly, with the 2nd-order BDF scheme to address the time derivative term, the original problem was transformed into a series of time-independent mixed boundary value problems. Then, the penalty method was adopted to treat the Dirichlet boundary condition, and the element-free Galerkin method was established for 2D transient heat conduction problems. Finally, based on the error results of the moving least squares approximation, the error estimation of the element-free Galerkin method for 2D transient heat conduction problems was derived in detail. Numerical examples show that, the calculation results are in good agreement with the analytical solutions or the existing numerical solutions, and the EFG method has higher calculation accuracy and better convergence.

Research on the Flow Field Distribution of Non-Circular Cross-Section Vessels Based on the Schwarz-Christoffel Transformation

ZHANG Wei, ZHANG Wenpu

2021, 42(5): 470-480. doi: 10.21656/1000-0887.410267

Abstract(1339) PDF(180)

Abstract:
(With the Schwarz-Christoffel transformation method, a conformal mapping from a unit circle to a polygonal domain in the complex plane was obtained. Based on the mapping combined with the Womersley algorithm theory for fully developed pulsating flow in a circular pipe, a Womersley velocity boundary model with a non-circular inlet section was established. For this boundary model, the computational fluid dynamics (CFD) method was used to simulate the blood flow in the human pulmonary artery secondary branch based on physiological reality in a cardiac cycle, and the results were compared with those obtained from the connected circular tube method. The analysis of examples indicates that, the simulation results of the 2 methods are highly consistent, but in the aspects of simulation efficiency and certainty, the Womersley velocity boundary model based on the S-C mapping is better than the connected circular tube method, and has more practical significance for the simulation of vascular hemodynamics.

Uncertainty Quantification of Flow and Heat Transfer Problems With Stochastic Boundary Conditions Based on the Intrusive Polynomial Chaos Expansion Method

JIANG Changwei, LIU Xing, SHI Er, LI Taohai, JIANG Yi

2021, 42(5): 481-491. doi: 10.21656/1000-0887.410217

Abstract(1444) PDF(181)

Abstract:
An intrusive polynomial chaos expansion method and a finite element program framework were proposed to quantify the uncertainty of flow and heat transfer problems under stochastic boundary conditions. In this method, the Karhunen-Loeve expansion was used to express the stochastic boundary condition, and the polynomial chaos expansion method was used to express the output random field. At the same time, the control equation was transformed into a set of deterministic equations with the spectral decomposition technology, and each polynomial chaos was solved to obtain the statistical characteristics of the numerical solution. Compared with the Monte Carlo method, this method can accurately and efficiently predict the uncertainty characteristics of flow and heat transfer problems under stochastic boundary conditions, and can save a lot of computing resources.

Passivity of Fractional-Order Delayed Complex-Valued Neural Networks With Uncertainties

CHEN Yu, ZHOU Bo, SONG Qiankun

2021, 42(5): 492-499. doi: 10.21656/1000-0887.410309

Abstract(987) PDF(220)

Abstract:
The passivity for a class of fractional-order delayed complex-valued neural networks with uncertainties was studied. The complex-valued neural network was not divided into 2 real-valued neural networks, but treated as a whole. Through construction of the appropriate Lyapunov function and application of the inequality technique, the sufficient criterion in the form of the linear matrix inequality was established to ensure the passivity of the considered neural networks. Numerical examples and simulations verify the feasibility and effectiveness of the obtained conclusion.

Exponential Stability of Complex-Valued Neural Networks With Time-Varying Delays and Reaction-Diffusion Terms

SHI Jizhong, XU Xiaohui, JIANG Yonghua, YANG Jibin>, SUN Shulei

2021, 42(5): 500-509. doi: 10.21656/1000-0887.410245

Abstract(1282) PDF(200)

Abstract:
The exponential stability of complex-valued neural networks with time-varying delays and reaction-diffusion terms was studied. Firstly, the addressed systems were separated into their real parts with the complex-valued activation functions assumed to be divided into the real parts and imaginary parts. Secondly, some sufficient conditions for ensuring the exponential stability of the equilibrium states of the systems were established based on the vector Lyapunov function method and the M-matrix theory. The obtained criteria have no free variables and reduced conservatism compared with the existing results. A numerical example proves the correctness of the obtained results.

Optimal Harvesting in a Periodic 3-Species Predator-Prey Model With Size Structure in Predators

LIU Rong, LIU Guirong

2021, 42(5): 510-521. doi: 10.21656/1000-0887.410285

Abstract(1381) PDF(292)

Abstract:
The research on population dynamics and related control problems is not only of theoretical significance, but also closely related to biodiversity protection, pest control, and the development and utilization of renewable resources. The optimal harvesting problem was considered in a periodic 3-species predator-prey system with 1 predator and 2 competing preys, where the predator has size structure and was described with 1st-order partial differential equations. First, the existence of a unique non-negative solution of the controlled system was proven by means of the fixed-point reasoning, and the continuous dependence of the solution on the control variables was discussed. Then, the optimal harvesting conditions were given with the techniques of tangential-normal cones and the adjoint system. Finally, with Ekeland’s variational principle, the existence of the optimal harvesting strategy was derived. Here the objective functional represents the net economic benefit in the harvesting of 3 species. The results obtained would be beneficial for exploration of renewable resources.

A Lattice Boltzmann Method for Spatial Fractional-Order Telegraph Equations

LI Mengjun, DAI Houping, WEI Xuedan, ZHENG Zhoushun

2021, 42(5): 522-530. doi: 10.21656/1000-0887.410311

Abstract(1604) PDF(185)

Abstract:
The lattice Boltzmann method (LBM) was applied to numerically solve Riemann-Liouville spatial fractional-order telegraph equations. Firstly, the integral term of the fractional-order operator was discretized and the order of convergence was analyzed. Then, a 1D and 3-velocity (D1Q3) LBM evolution model with modified functions was established. The expressions of equilibrium distribution functions and correction functions were deduced by means of the Chapman-Enskog multi-scale analysis and the Taylor expansion technique. Therefore, the macroscopic equation was exactly recovered from the established evolution model. Numerical results show the stability and effectiveness of the model.

A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations

GAO Xinghua, LI Hong, LIU Yang

2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149

Abstract(1378) PDF(319)

Abstract:
A numerical scheme with the piecewise linear interpolation polynomial method was established to solve a class of nonlinear fractional ordinary differential equations including the Hadamard finite part integral. In the time direction, the fractional derivative was approximated with the piecewise linear interpolation polynomial method, and the integer order time derivative was discretized by means of the 2ndorder backward difference scheme. Through detailed proof, the error estimates with an accuracy of OO(τ^{min{1+α,1+β}}) were obtained. The comparison between the numerical results and the theoretical solution shows the correctness of the theoretical analysis.

A New Regularization Method for Solving Sideways Heat Equations

BAI Enpeng, XIONG Xiangtuan

2021, 42(5): 541-550. doi: 10.21656/1000-0887.410290

Abstract(1346) PDF(221)

Abstract:
The seriously ill-posed sideways heat equations were considered in the quarter plane. The classical quasi-reversibility method was applied to acquire an approximate but non-regularized solution to the problem. Interestingly, a regularization solution to the sideways heat equation was obtained through modification of the denominator of the solution. Then, a new regularization method was proposed, and the Hlder-type error estimates under a priori and a posteriori parameter choice rules were proved, respectively. Numerical experiments show the feasibility and effectiveness of the proposed method.