It is found that the classic finite element method (FEM) requires much time for adequate meshes to acquire satisfactory numerical solutions, and is restricted to acoustic problems with low and middle frequencies. In this work, a coupled overlapping finite element method (OFEM) is employed by combining the overlapping finite element and the modified Dirichlet-to-Neumann (mDtN) boundary condition to solve underwater acoustic scattering problems. The main difference between the OFEM and the FEM lies in the construction of the local field approximation. In the OFEM, virtual nodes are utilized to form the partition of unity functions while no degree of freedom is assigned to these virtual nodes, which suppresses the linear dependence issue in other generalized finite element methods. Moreover, the user-defined enrichment functions can be flexibly utilized in the local field, and thus the numerical dispersions can be significantly mitigated. To truncate the infinite problem domain and satisfy the Sommerfeld radiation condition, an artificial boundary is constructed by incorporating the mDtN technique. Several numerical examples are studied and it is shown that the proposed method can greatly diminish the numerical error and is insensitive to distorted meshes, indicating that the proposed method is promising in predicting underwater acoustic scattering.

The traditional finite element method (FEM) usually exhibits significant numerical dispersion error for solving the Helmholtz equation in relatively high-frequency range, resulting in insufficiently accurate solutions. To address this problem, this paper proposes a novel enriched finite element method (EFEM) based on radial basis functions (RBFs) which are frequently used in meshless numerical techniques. In the proposed method, the partition of unity (PU) framework is retained, and nodal interpolation functions are formed using the RBFs. Furthermore, the linear dependence (LD) problem commonly encountered in many of the PU-based methods using polynomial basis functions (PBFs) is effectively avoided by using the present RBFs. To enrich the approximation space generated by the RBFs, the PBFs are introduced to construct the local enrichment functions. Several typical numerical experiments are conducted in this work. The results indicate that the proposed method can significantly reduce the dispersion error and yield accurate solutions even for relatively high-frequency Helmholtz problems. More importantly, the proposed method can be directly implemented with standard quadrilateral meshes as in FEM. Therefore, the proposed method represents a promising numerical scheme for solving relatively high-frequency Helmholtz problems.