A dual reciprocity BEM approach using new Fourier radial basis functions applied to 2D elastodynamic transient analysis

In this paper, a new boundary element analysis for two-dimensional (2D) transient elastodynamic problems is proposed. The dual reciprocity method (DRM) is reconsidered by employing new radial basis functions (RBFs) to approximate the domain inertia terms. These new RBFs, which are in the form of ζ+κ sin (ωr+α), are called Fourier RBFs hereafter. Using the method of variation of parameters, the particular solution kernels of Fourier RBFs corresponding to displacement and traction, whose a few terms are singular, has been explicitly derived. Therefore, a new simple smoothing trick has been employed to resolve the singularity problem. Moreover, the limiting values of the particular solution kernels have been evaluated. In order to find the unknown parameters of Fourier RBFs, an optimization problem seeking for the optimum value of the Houbolt scheme parameter β that minimizes the mean squared error (MSE) function of the problem is established. Since the MSE function of the proposed RBFs is a function of five unknown parameters (i.e., ζ, κ, ω, α, and β), the genetic algorithm (GA) has been used to solve the necessary optimization problem. In order to illustrate the validity, accuracy, and superiority of the present study, several numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. Compared to other RBFs, Fourier RBFs show more accurate and stable results. Moreover, these results are obtained using less degree of freedom without any additional internal points that are commonly used to improve the accuracy of the results.