PERIDYNAMIC OPERATOR METHOD

A new method based on non-local theories, named as peridynamic operator method (PDOM), for solving ordinary and partial differential equations in physics, is proposed in the present work. By using the peridynamic operator method, the local differentiations of any orders as well as their products can be converted into corresponding nonlocal integral forms without any extra remedies or special treatment in the presence of discontinuities or singularities. It can be proved that both the so-called peridynamic differential operator (PDDO) and the nonlocal operator method (NOM), two nonlocal operators which have gained much concern of researchers in the field of computational mechanics in recent years, can be seen as special cases of this proposed PDOM. As a typical application example, linear elastic PDOM model for static and dynamic elasticity problems is developed by employing the variational principles and Lagrange's equations. Theoretical analysis shows that when the nonlocal interaction domain is defined by position-independent or dependent circles, PDOM elasticity model can be simplified to the classical peridynamic (PD) model or the dual-horizon peridynamic (DH-PD) model in literature correspondingly. The accuracy, convergence, as well as the numerical stability of the presented method are validated by analyzing three typical examples, including tension and wave motion in a bar, Helmholtz equation, and tensile deformation of plates with hole. It is shown that the proposed method can be effectively used with both uniform and non-uniform discretization, and it can naturally avoid the zero-energy modes and numerical oscillations which will occur when the original non-ordinary state-based peridynamic simulations, the peridynamic differential operator or the nonlocal operator method is employed without extra remedies. PDOM can provide a potential alternative to develop the nonlocal models for various physical problems especially involving discontinuities.