A Fast Cell-Smoothed Finite Element Method for Solving Static–Dynamic Problems Using a Hybrid Quadtree Mesh

We present a fast cell-smoothed finite element method (CSFEM) that integrates hybrid quadtree mesh and polygonal techniques to solve static and dynamic problems, addressing the hanging node problem by constructing CSFEM polygons. In our method, we introduce acceleration technology based on the parent element to enhance computational efficiency. The computational efficiency and accuracy of the proposed approach are validated by solving several benchmark static and dynamic problems. Numerical results show that the hybrid quadtree mesh achieves high-precision surface representation while reducing the computational cost through mesh slicing. Compared with conventional finite element methods, our proposed CSFEM achieved superior efficiency and precision, offering a robust and automated solution for static–dynamic engineering challenges by using elements of arbitrary shapes, which grants CSFEM greater flexibility. The results demonstrated that this method was particularly advantageous for applications that require detailed geometric approximation and efficient computation, which could make it valuable for static–dynamic analysis.