The wavelet theory shows very unique time-frequency localization and multi-resolution analysis ability in signal processing and function approximation. The wavelet basis function has excellent mathematical properties such as orthogonality, compactness, low-pass filtering and interpolation, which endows the wavelet analysis theory with great application potential in the fields of computational mathematics and computational mechanics, and creates new opportunities for breakthrough development in these fields. Since the 1990s, a large number of studies have proved that the numerical method based on the wavelet theory has very obvious advantages in solving differential equations, but at the same time, have exposed some limitations of numerical calculation application caused by the wavelet basis function itself and its unique approximation method. In order to promote the innovative application of the wavelet theory in the fields of computational mathematics and mechanics and provide researchers with a new research perspective, the development background of the wavelet analysis and the research history of methods based on the wavelet theory were reviewed, and the numerical method problems were emphasized and the research progresses made in recent years discussed. The conclusions and comments may provide a meaningful reference for the further development and improvement of quantitative mathematical solution methods based on the wavelet theory and applications in mechanics as well as solutions of a wide range of engineering problems.

The active control on band gap properties of elastic wave metamaterial beams was studied by means of the negative capacitance circuits attached to piezoelectric sheets periodically. External circuits were used to change the material constants of the connected piezoelectric materials, which could tune the equivalent parameters of the structure and control the band gap characteristics. Through controlling the unit cell, the generation and disappearance of band gaps can be observed with the active control system. Then, an elastic wave metamaterial beam with the interface was constructed to discuss the effects of the active control system on the interface transmission.

The stability and the accuracy of the precision devices installed in spacecraft depend on the local vibration characteristics of the spacecraft. In return, the local vibration characteristics of the spacecraft are influenced by the layout of the precision devices, which implies that, a rational layout of the precision devices in the spacecraft is the precondition for the stable and efficient work of the precision devices. The dynamic model for a flexible panel bearing several precision devices was presented and the structure-preserving method was developed to investigate the local vibration characteristics of the panel. In view of the sizes of the precision devices and the heat dissipation clearances between the precision devices, the layout optimization for the precision devices was performed to minimize the weighted values of the out-of-plane vibration accelerations of the precision devices. The optimization results show that, benefiting from the excellent structure-preserving properties of the numerical method employed during the vibration analysis, the weighted values of the out-of-plane vibration accelerations of the precision devices decrease by about 88.05% through the layout optimization, which provides a useful guide for the layout scheme of the precision devices in the spacecraft and improves the stability of the precision devices.

The classical fiber beam model based on the Euler-Bernoulli beam theory ignores the influence of shear deformation on the section. To get a more accurate beam element model, based on the fiber beam element with shear effects and the Timoshenko beam theory, the stiffness matrix of the fiber beam element was deduced, and the dual effects of geometric nonlinearity and material nonlinearity were considered at the same time, combined with the elastoplastic incremental theory. Then, the nonlinear finite element analysis theory for the structure under the complex stress state of compression, bending and shear was established. Finally, a program was coded on MATLAB to conduct finite element numerical simulation of the typical compression-bending-shear members of reinforced concrete and rectangular concrete-filled steel tube, and the nonlinear full-process load-displacement curves were obtained. The analysis of the numerical examples show that, the established nonlinear finite element analysis theory is universal, feasible and correct.

基于流体体积(VOF)法追踪自由液面，研究了壁面结构对三维可压缩气泡群流动的影响。通过在待测壁面上设置不同形状的壁面结构（长方体、椭球体和圆锥体）并改变它们各自的几何参数（位置和长度），来研究壁面结构对壁面附近的气泡群流动的影响，该影响表现为气泡群对壁面的空间平均压力。研究发现，壁面结构对气泡群的拓扑结构的影响会造成壁面压力的变化，其中长方体壁面结构降低壁面平均压力的效果最好，且通过适当调整该结构的位置和长度，能使壁面的压力脉动现象消失。

Based on the lattice Boltzmann method, the numerical simulation of droplet impacting on orifice plates with different wettabilities was carried out. The effects of the Weber number (We), the wettability of the orifice surface and the orifice size on different states of droplets passing through orifice plates during impacts were studied. The numerical simulation results show that, different phenomena will occur in the processes of droplets impacting on the orifice plates. If the orifice plate is hydrophilic, the droplet will not detach from the orifice plate surface, but adhere to the lower surface of the orifice plate for a relatively low We number, and then the droplet will rise for a certain distance in the orifice channel under the action of capillarity, forming the liquid plugging phenomenon. For relatively high We numbers, droplets will pass through the orifice plates and rupture will occur. If the orifice plate is hydrophobic, the droplet will not pass through the orifice plate and migrate to the lower surface for a relatively low We number, and will finally stabilize in the orifice channel. For higher We numbers, droplets will pass through the orifice plates, and then break up, leaving droplet remains on the surfaces of the orifice plates. For various orifice sizes, the droplet will be more difficult to pass through the plate for a smaller orifice size or a lager orifice plate thickness.

To solve the nonlinear problem of airfoil shape optimization induced by nonlinear large perturbation, an optimization method was proposed based on the convolutional neural network (CNN) aerodynamic reduced order model (ROM). In the method, the aerodynamic forces on different airfoils were used as the training data for the proposed ROM. For the sake of the maximum lift-drag ratios, the ROM was applied to optimize the airfoil shape. The results show the method applies well to the prediction and optimization of airfoil shape dynamics under large perturbation. The improving effects of the parameter pooling and the radial basis function method based training data method on the accuracy of the dimensionality reduction model, were discussed. The reason for the improvement is that, the training data dimensionality reduction can cut down the number of undetermined parameters in the CNN model and make the CNN model converge better under the same data volume.

The Darboux polynomials and invariant algebraic surfaces of the Vallis system were investigated. In the proofs, the weighted homogeneous polynomials and the characteristic curve method were used to solve linear partial differential equations. Finally, 3 types of Darboux polynomials for the Vallis system were obtained under suitable conditions of parameters.

This paper is concerned with the finite time synchronization of quaternion-value neural networks with time delays. Based on finite time control technique, the protocol of adaptive control is first proposed. Then by utilizing Lyapunov function method and inequalities skills, some sufficient conditions are derived to ensuring master systems and slave systems achieved synchronization in finite time, the settling time can also be theoretically given. Finally, the numerical simulation on quaternion-value neural networks with time delays is included to show the effectiveness of the theorem.

Aimed at the optimal leader-following consensus problem of fractional-order multi-agent systems, an reinforcement learning strategy was designed based on the intermittent event trigger. With the periodic intermittent strategy as the basic mechanism, the event trigger and the actor-critic algorithm in reinforcement learning were organically integrated. According to the 1st-order approximation of the fractional differential, the reinforcement learning algorithm structure based on the periodic intermittent event trigger strategy was proposed. Finally, the feasibility and effectiveness of the algorithm was proved through numerical simulation experiments.

The spatial properties of solutions for a class of thermoelastic plates with biharmonic operators were studied in a semi-infinite strip in R². Firstly, an energy expression was constructed. Then, by means of the differential inequality technique, a differential inequality whose energy expression can be controlled with its 1st derivative was derived. Finally, the spatial decay estimates of the solution were obtained. The result can be regarded as an application of the Saint-Venant principle to hyperbolic parabolic coupled biharmonic equations.