(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
In this study, we combine orthonormal basis functions with the traditional moving least-squares (MLS) approximation to establish a new approximation function via the improved moving least-squares (IMLS) approximation, and the corresponding formula derivation is provided in Section 2. Afterwards, the equilibrium, geometrical, and physical equations for bending problems of plates on elastic foundations are presented respectively, and the equivalent functional of such problems is established by imposing the essential boundary conditions via the penalty method. Subsequently, the calculation formulas for the numerical solution are derived using the improved element-free Galerkin (IEFG) method based on the IMLS approximation. In the numerical examples, we verify the convergence of the IEFG method by increasing the number of nodes. Compared with the EFG method, the IEFG method exhibits faster convergence. Furthermore, by adopting the IEFG method for solving three numerical examples, smaller errors and higher computational speed are achieved.