A generalized finite difference method based on the Peridynamic differential operator for the solution of problems in bounded and unbounded domains

In this paper, a generalized finite difference method (GFDM) based on the Peridynamic differential operator (PDDO) is investigated. The weighted moving least square (MLS) procedure involved in the GFDM is replaced by the PDDO. PDDO is capable to recast differentiation operators through an integration procedure which has the following advantages: the method is free of using any particular treatment near sharp gradients and where the solution is governed by a steep variation of field variables; it facilitates the imposition of boundary conditions compared to certain meshless methods as its formulation does not need a symmetric kernel function; it is free of any particular correction function and parameter tunings; the method is easy to implement and we shall show that it generates a well-conditioned stiffness matrix for both structured and unstructured discretization. In this paper, the method is applied to 2D Helmholtz-type problems including time-harmonic acoustic problems, though the extension to 3D is straightforward. We also present a simple and efficient way to equip the method in the solution of unbounded domains. Moreover, a comprehensive study on the accuracy, convergence, and behavior of the method through a patch test is conducted. The test may be considered as a way to find the optimal region of the required integrations. We shall benefit from the advantage of the method in terms of accurate calculation of derivatives to determine the flow of acoustic energy, in a frequency problem domain, without employing any particular singular basis functions.