A Fast Reduced-Order Method for High-Order FPM Equations and Its Applications

Finite Particle Method (FPM) has emerged as a significant meshless method, which innovatively realizes the simultaneous solution of function value and its derivative values by solving a linear system. Accordingly, the computational accuracy in the whole computational domain is enhanced. However, when facing problems with higher-order partial differential governing equations, the original FPM usually needs to solve higher-order linear equations for each particle, whose computational complexity will be unacceptable for large-scale engineering problems. In this paper, a fast reduced-order method for high-order FPM equations is proposed. First, the original FPM equations are rewritten to reduce the order of the linear system. Second, matrix decoupling is further performed on the reduced-order system and a fast computation scheme is proposed. Finally, the particle consistency and computational efficiency of the proposed method are analyzed, and its superiorities are verified based on several typical engineering cases. Compared with the traditional FPM, the proposed fast reduced-order FPM could significantly improve the computational efficiency while maintaining its second-order accuracy.