当采用低维动力系统模型研究Navier-Stokes方程的动力学性质时，保持低维模型的吸引域与Navier-Stokes方程的吸引域相同是非常重要的。然而，到目前为止，还没有一种普适的方法能确保对于一般问题都能达到这一目的。该文发现任何基于空间基的低维模型，如本征正交分解基、优化空间基和其他经典空间基，都不具有可预测性，即低维动力系统的误差随着流场的时间演化而增大。在构造优化动力系统的理论框架和时空耦合谱展开的新概念下，该文构造了可压缩Navier-Stokes方程的低维模型来逼近大涡模拟方程的数值解，给出了高精度的流场数值模拟结果和全新的时空耦合基时空演化数值结果。全场误差在10⁻²%以下，而每个网格点的平均误差在10⁻⁸%以下。时空耦合化化低维动力系统可以保证低维模型的吸引域与Navier-Stokes方程的吸引域相同。因此，保证了时空耦合优化低维动力系统的特征动力学性质与真实流场的特征动力学性质是一致的。

The soap bubble heated at the bottom was introduced as a novel quasi-2D turbulence system. The curved geometry of the bubble brings challenges for the direct numerical simulations (DNS) of the turbulence on the bubble. In order to overcome the difficulties due to the curved geometry, a numerical method based on the stereographic projection was implemented for the DNS of the soap bubble. The numerical methods to compute the spectrum, the flux and the structure functions of the flows on the bubble were described in detail. Three different Rayleigh numbers Ra=3×10⁷，3×10⁹，3×10¹¹ were used in the simulation based on the present numerical methods. Then, the related spectrum, flux and structure functions were calculated. The results indicate that, both the inversed energy cascade and forward enstrophy cascade can be observed in all the calculation cases. The Bo59 scaling law fits the small-scale fluctuations on the soap bubble. With the increase of the Rayleigh number, the turbulent energy decreases for the large-scale plumes, and the kinetic energy increases for the higher wave number structures.

A finite-difference unsteady coupled heat transfer solver was developed. This solver was utilized to simulate the hypersonic flow over a backward-facing step with a transverse gap, and the unsteady thermal conduction in an infinite circular pipe. The backward-facing step leads to local dramatically changing distributions of aerodynamic parameters and wall heat fluxes. The gas flow in the gap decelerates rapidly along with the increase of the gap depth, and there is rather weak convective heat transfer at the bottom of the gap. In the case of hypersonic flow around the infinitely long circular pipe, there exists large temperature gradient in the boundary layer, and the wall temperature increases with time, otherwise the aerodynamic parameters outside the boundary layer change quite slightly. The predicted results are in good agreement with the tested data. The comparison between numerical simulation results and tested data verifies the calculation ability of the developed solver.

The computation consumption of finite element analysis for structural optimization design of holding poles is large, and it is difficult to determine the feasible region. The response surface method (RSM) was used to simulate the real response of the holding pole, and an improved arithmetic optimization algorithm (IAOA) was proposed to optimize the holding pole. The fractional-order calculus was introduced into the arithmetic optimization algorithm (AOA) to improve the exploitation ability of the AOA. The Latin hypercube sampling was applied to select the test samples of each member of the holding pole, and the least square method was employed to analyze the sample points. Then, the 2nd-order response surface surrogate model for the stress and displacement of the holding pole on the cross-sectional sizes of each member was established. An optimization model was constructed with the minimum mass as the optimization objective and the allowable stress and displacement as constraints, and the IAOA was implemented to solve the model. The results show that, the 2nd-order response surface model can accurately predict the response value of the holding pole. The solution accuracy of the IAOA is significantly improved. The surrogate model can greatly decrease the calculation cost of the finite element analysis. The mass of the holding pole is reduced by 8.2% after optimization. The RSM and the IAOA can be combined to solve the optimization design problem of large spatial truss structures effectively.

To avoid transverse shear locking and improve the accuracy, the RES-DSG3 method was proposed through the incorporation of the non-isoparametric DSG method with a novel edge-based smoothing technique based on global coordinates, and all the integration of smoothed matrices can be calculated along the boundary segments of smoothed cells without coordinate mapping. Based on the 1st-order shear theory, the RES-DSG3 method was used to analyze free vibration natural frequencies of laminated composite plates with different material parameters, edge-thickness ratios and boundary conditions. The free vibrations of the composite plates were analyzed numerically. The calculation of the example verifies the feasibility and effectiveness of the proposed method.

Based on the modified couple stress theory and the Timoshenko beam theory, the free vibration and buckling mechanics model for 2D functionally graded microbeams with variable cross sections was established by means of the variational principle. The model contains the intrinsic material length scale parameters of the metal and ceramic components, which can predict the size effects of microbeams. The Ritz method was used to obtain the numerical solution of the vibration frequencies and critical buckling loads of the microbeams under arbitrary boundary conditions. Numerical examples reveal that, when the thickness of the microbeam decreases, the dimensionless 1st-order frequency and the dimensionless critical buckling load will increase, and the scale effect will grow larger. The effect of the taper ratio on the dimensionless 1st-order frequency of the microbeam is closely related to the boundary conditions. At the same time, the effects of the taper ratios of the thickness and the width are also significantly different. The dimensionless 1st-order frequencies of microbeams increase with the material length scale parameter ratios of ceramic and metal, and the degrees of increase are different under different boundary conditions. The thick-direction and axial material gradient indexes also have significant influences on the free vibration and buckling behavior of the microbeam.

The elastic problem of anisotropic functionally graded rotating disks with non-uniform thicknesses was studied. The material properties and thicknesses of the disk change with an arbitrary gradient along the radial direction, and the disk displacement is constrained at the central axis of rotation and the edge is free. According to the equilibrium differential equations for the anisotropic rotating disk, the Fredholm integral equation of radial stresses was obtained, and then numerically solved to calculate the displacement and stress fields. For any specific situation of gradient change, it is only needed to substitute the corresponding gradient parameters into the equation for solution. In the numerical examples, firstly, the parameters such as the thickness and the elastic modulus were assumed to be of special power functions. The numerical solutions obtained from the Fredholm integral equation were compared with the corresponding exact solutions and the FEM solutions by the ANSYS software for the common Voigt model, to validate the method. Then, the effects of the thickness change, the gradient parameters and different degrees of anisotropy on the stress and displacement fields, were numerically analyzed with the common Voigt model. The proposed elastic analysis method for disk structures with arbitrary gradient changes along the radial direction, is promising in the application of structural optimization. The analysis results can guide the engineers to design safer and more economical functionally graded structures.

The propagation dynamics was studied for a class of spatially discrete multi-type SIS epidemic model with time periodicity. Firstly, the theory of spreading speeds and travelling waves for monotonic periodic semiflows was applied to prove the existence of asymptotic spreading speed c_*. Secondly, by means of the comparison principle, the asymptotic spreading speed was proved to coincide with the minimal wave speed of monotonic periodic traveling waves.

The existence and non-existence of periodic traveling wave solutions of a class of time-periodic SIR epidemic models with external supplies were considered. Firstly, the appropriate upper and lower solutions of the auxiliary system were built and a closed convex cone was defined, the existence of periodic traveling waves was transformed into a fixed-point problem of the non-monotonic operator defined on the closed convex cone. The existence of periodic solutions of the auxiliary system was established under the Schauder fixed-point theorem, and the Arzela-Ascoli theorem was used to prove the existence of periodic traveling waves for the original model. Secondly, the non-existence of periodic traveling waves was obtained by analytic techniques.

The blow-up problems of the solutions are considered for reaction-diffusion equations with nonlocal sources and variable exponents. Firstly, the local existence and uniqueness of solutions to the problem were proved under the fixed-point theorem. Secondly, by means of the super- and sub-solution method, some sufficient conditions for the occurrence of finite-time blow-up were determined under the homogeneous Dirichlet boundary conditions, i.e., the variable exponent is positive and the initial value is large enough. Moreover, the estimates of upper and lower bounds of the blow-up time were given.

The fractional-order modified unstable Schrödinger equation (FMUSE) was studied, which describes the dispersion, nonlinearity, gain or absorption variation of optical pulses propagating in nonuniform fiber systems. First, the generalized fractional wave transform was appropriately used to convert the FMUSE into an ordinary differential equation, and the real and imaginary parts were separated and set as zero respectively, and the dispersion relation was obtained. By means of the modified (G'/G)-expansion method, a series of new exact analytical solutions with parameters were obtained, including trigonometric solutions, hyperbolic solutions and rational solutions, and the constraints ensuring the existence of solutions were given. Finally, the solutions of the dark solitary wave and the periodic wave were obtained with the parameters of special values.