Moving least squares peridynamics: a new approach to the irregular node spacing problem in mesh-free mechanics
A new approach to classical continuum mechanics has been developed in the computational studies community and has gained significant traction. Its primary separation from the classical approach is the use of integral equations rather than traditional partial differential equations to handle crack propagation/initiation. This integral approach to continuum mechanics is called Peridynamics - the mechanics of local force interaction. However, an unforeseen consequence of the development of this mesh-free frame work is the discretized model's inability to achieve machine precision in an irregularly meshed condition. This feature limits the capability of Peridynamics as well as its academic credibility in the mechanics community. The de facto standard of accuracy for any computational model is the Irons Patch Test. It describes a scenario wherein a model is subjected to a known linear displacement and is expected to converge to the analytic solution with machine precision. As the Peridynamic model is currently unable to achieve a solution with machine precision, the author has produced an alternative approach to the irregular node spacing problem: a modified Moving Least Squares interpolator. This approach significantly reduces the error in multiple boundary condition scenarios and is proven through comparison of L1 and L2 norms on force magnitude, displacement, and an analogous term of strain in Peridynamics called damage. These results indicate an improvement on this meshfree model. While machine precision was not achieved, it is the author's opinion that this improvement is promising enough to continue development toward such precision.