Discussion on the form of construction function in the peridynamic differential operator based on relative function

Abstract

The peridynamic differential operator (PDDO) adopts nonlocal idea, realizes the solution of any order derivative of a function through the integration of a constructed PD function, and expands the motion equation based on function derivative into the nonlocal peridynamic integral equation. Even at the discontinuity, the peridynamic integral equation still has a definition and the form remains unchanged. When calculating any order derivative of a function f(x), two PDDO formulas can be used, one is the PDDO formula based on proximal function f(x + ξ) and the other is PDDO formula based on relative function f(x + ξ) − f(x). The key to using these two formulas is to construct the PD function. This paper discusses the construction method of the PD function in PDDO formula based on relative function. It is found that it can be constructed by supplementing an equation as well as by reducing a construction term of PD polynomial. After derivation, it is found that it is necessary to reduce a construction term that is even power to all variables, and it is invalid to reduce a construction term with an odd power to a certain variable. Numerical examples show that the two construction methods can lead to high-precision calculation results.