ten-node shape functions

In this paper, the numerical manifold method (NMM) with a 9-node quadrilateral element and a 10-node triangular element is developed. Furthermore, complex Fourier shape functions are used to improve the 9-node quadrilateral NMM. Also, the two approaches of higher-order NMM construction are compared, increasing the order of weight functions or local approximation ones; for this purpose, six-node triangular and three-node triangular using second-order and third-order NMM is used. For validation of the suggested method, one free vibration and two forced vibration numerical examples are assessed. The results show that the proposed methods are more accurate than conventional NMM. In addition, the superiority of complex Fourier shape functions compared to classical Lagrange ones in improving accuracy is perceived.