Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems

- The polynomial-basis-functions-based particular solutions are derived analytically.
- The present method is meshless and free from annoying parameters.
- This method is accurate and fast, which has huge potential for real engineering problems.

The traditional polynomial expansion method is deemed to be not suitable for solving two- and three-dimensional problems. The system matrix is usually singular and highly ill-conditioned due to large powers of polynomial basis functions. And the inverse of the coefficient matrix is not guaranteed for the evaluation of derivatives of polynomial basis functions with respect to the differential operator of governing equations. To avoid these troublesome issues, this paper presents an improved polynomial expansion method for the simulation of plate bending vibration problems. At first, the particular solutions using polynomial basis functions are derived analytically. Then these polynomial particular solutions are employed as basis functions instead of the original polynomial basis functions in the method of particular solutions for the approximated solutions. To alleviate the conditioning of the resultant matrix, we employ the multiple-scale method. Numerical experiments compared with analytical solutions, solutions by the Kansa's method, and reference solutions in references confirm the efficiency and accuracy of the proposed method in the solution of Winkler and thin plate bending problems including irregular shapes.