Zhijuan Meng’s research while affiliated with Taiyuan University of Science and Technology and other places

This study proposes the efficient element-free Galerkin (EEFG) method, which is based on the improved moving least-squares (IMLS) method, for addressing the three-dimensional (3D) Schrödinger equations. The dimension splitting method (DSM) is vital to this approach, since it splits the 3D Schrödinger equation in a specific direction and transforms it into a sequence of two-dimensional (2D) and one-dimensional (1D) issues. The improved element-free Galerkin (IEFG) method is subsequently applied to solve the corresponding 2D problems, using the finite difference method (FDM) employed for both the splitting direction and the time domain. Finally, the EEFG method for addressing the 3D Schrödinger equations is established. Compared to the IEFG approach, the EEFG method offers greater computational accuracy. Furthermore, this method outperforms the IEFG method in terms of computational efficiency for the same precision. To demonstrate the problem, numerical instances are solved using the EEFG approach, demonstrating its advantages in terms of computing correctness and efficiency.

In this study, a wheeled mobile manipulator was modeled while considering the case of wheel-ground slippage. Considering the whole system, the number of states in the dynamic system equation was reduced by the kinematic solution, which facilitates the control of the system. For the system state equation, the improved sliding mode control method was applied, a flush continuous control term and a super-twisting algorithm were used to achieve the improved sliding mode control, and the gain of the super-twisting algorithm was adaptively controlled. The modeling error in the subsystem and the uncertain slippage were approximated by a neural network. The middle layer weights of the neural network were updated by the adaptive control law, and the stability of the system was demonstrated by a quadratic Lyapunov function. The simulation results show the superiority of the control method proposed in this paper by comparing them with results for the traditional sliding-mode control method and the improved sliding-mode control method.

A fast interpolating meshless (FIM) method for three-dimensional (3D) heat conduction equations is presented. Transforming a 3D problem into the relevant two-dimensional (2D) problems using the dimension splitting method (DSM) is the main idea of FIM method. The improved interpolating moving least-squares (IIMLS) method is applied in 2D problems to obtain required approximation function with interpolation property. Finite difference method (FDM) is utilized in time domain and the direction of splitting. Take the improved element-free Galerkin (IEFG) method as a comparison, difficulties created by the singularity of weight functions, such as truncation error and calculation inconvenience, are overcome by the FIM method. And it can directly implement the Dirichlet boundary conditions. To prove the advantages of the new method, three examples are selected and solved by the FIM method. Comparing and analyzing the calculation results, it can be shown that the FIM method effectively improves computation speed and precision.

A hybrid interpolating meshless (HIM) method is established for dealing with three-dimensional (3D) advection–diffusion equations. To improve computational efficiency, a 3D equation is changed into correlative two-dimensional (2D) equations. The improved interpolating moving least-squares (IIMLS) method is applied in 2D subdomains to obtain the required approximation function with interpolation property. The finite difference method (FDM) is utilized in time domain and the splitting direction. Setting diagonal elements to one in the coefficient matrix is chosen to directly impose Dirichlet boundary conditions. Using the HIM method, difficulties created by the singularity of the weight functions, such as truncation error and calculation inconvenience, are overcome. To prove the advantages of the new method, some advection–diffusion equations are selected and solved by HIM, dimension splitting element-free Galerkin (DSEFG), and improved element-free Galerkin (IEFG) methods. Comparing and analyzing the calculation results of the three methods, it can be shown that the HIM method effectively improves computation speed and precision. In addition, the effectiveness of the HIM method in the nonlinear problem is verified by solving a 3D Richards’ equation.

In this paper, a fast element-free Galerkin (FEFG) method for three-dimensional (3D) elasticity problems is established. The FEFG method is a combination of the improved element-free Galerkin (IEFG) method and the dimension splitting method (DSM). By using the DSM, a 3D problem is converted to a series of 2D ones, and the IEFG method with a weighted orthogonal function as the basis function and the cubic spline function as the weight function is applied to simulate these 2D problems. The essential boundary conditions are treated by the penalty method. The splitting direction uses the finite difference method (FDM), which can combine these 2D problems into a discrete system. Finally, the system equation of the 3D elasticity problem is obtained. Some specific numerical problems are provided to illustrate the effectiveness and advantages of the FEFG method for 3D elasticity by comparing the results of the FEFG method with those of the IEFG method. The convergence and relative error norm of the FEFG method for elasticity are also studied.