Peridynamics for Solving Linear/Nonlinear Ordinary and Partial Differential Equations

Abstract

This study concerns the construction of numerical solutions to linear/nonlinear ordinary and partial differential equations with certain challenges by using the Peridynamic Differential Operator (PDDO). These challenges may be due to the presence of discontinuities arising from a crack, dissimilar material interface or a moving interface due to phase changes or due to the presence of a strong degree of coupling among the coupled field equations. The peridynamic (PD) discretization can be both in time and space. Therefore, their numerical solutions can be achieved by employing either implicit or explicit methods. The PDDO enables differentiation through integration over a domain of interaction. The association among the points within the range of this domain defines the degree of nonlocality. Since the PDDO permits differentiation through integration, it is very robust for determining higher order derivatives of spatial and temporal functions that present singular and discontinuous behavior. The solution process involves neither a derivative reduction process nor a special treatment to remove a jump discontinuity or a singularity. Comparison of the solutions from the PDDO with those of the existing analytical or numerical methods proves that the PD modeling provides accurate solutions to rather complex linear and nonlinear partial differential equations of parabolic, hyperbolic and elliptic type. In order to demonstrate the crack growth analysis capability, the PDDO is applied to model a diagonal plate with a pre-existing crack under tension. In the case of coupled field equations, the PDDO is applied to model thermoelasticity, thermoelectricity, thermo-oxidation, and lithium diffusion and stress evolution.