Abstract
The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.
Highlights
As an important elliptic differential equation, the Helmholtz equation has been widely applied in many different fields, such as mechanics, acoustics, physics, electromagnetics, engineering, and so on
Many meshless methods have been used for researching Helmholtz equations, such as the element-free Galerkin (EFG) method [1], meshless Galerkin least-square method [2], meshless hybrid boundary-node method [3], boundary element-free method [4], and complex variable boundary element-free method [5,6]
In order to overcome the disadvantage of the lower efficiency of the EFG method, this paper presents the improved element-free Galerkin (IEFG) method for solving
Summary
As an important elliptic differential equation, the Helmholtz equation has been widely applied in many different fields, such as mechanics, acoustics, physics, electromagnetics, engineering, and so on. The improved CVEFG method has higher computational accuracy and efficiency than the EFG method, but it cannot be applied to 3D problems directly because the complex theory is used. By combining meshless methods and the finite difference method, the hybrid CVEFG method [52,53,54,55,56], dimension-splitting EFG method [57,58,59,60], dimension-splitting reproducing kernel particle method [61,62,63,64], interpolating dimension-splitting EFG method [65] and hybrid generalized interpolated EFG method [66] were proposed These methods can greatly improve the computational efficiency of the traditional meshless method for solving multi-dimensional problems.
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