Peridynamics is a nonlocal reformulation of classical continuum mechanics suitable for material failure and damage simulation. Governing equations in peridynamics are based on spatial integration rather than spatial differentiation, allowing natural representation of material discontinuities, such as cracks. Applications in peridynamics include failure and damage in composite laminates, crack propagation and branching in glass, crack nucleation, impact damage, fracture in polycrystalline materials, structural health monitoring, and damage in concrete, among many others. 

There are two main distinctive discretization methods employed by the peridynamic community. These are the finite element method and the meshfree method of Silling and Askari \[1\]. The former discretization method, in spite of robust supportive mathematical analysis, is not amenable in terms of computational cost and software implementation, particularly for simulations involving complex geometry in three dimensions. Two specific challenges of finite element discretizations of peridynamic models are proper quadratures for peridynamic weak formulations, which double the number of spatial dimensions to be discretized relative to strong formulations, and the necessity to adapt finite element meshes to conform to evolving crack surfaces. In contrast, the latter discretization method, i.e., the meshfree method, has been rapidly adopted by the engineering community, due to its implementation simplicity and its ability to handle material separation in a relatively simple manner. A downside of the meshfree approach is its poor convergence behavior. Specifically, it has been shown by Seleson and Littlewood \[2\] that standard meshfree discretizations of peridynamic models exhibit irregular (non-smooth) convergence behavior and a low (average) first-order convergence. Nevertheless, numerical tools have been proposed and studied by Seleson \[3\] and Seleson and Littlewood \[2\] to correct such irregular convergence behavior, resulting in smoother convergence curves. However, a robust quantitative assessment of the performance of this meshfree method, particularly in fracture scenarios, is lacking. 

In \[2\], the convergence of meshfree numerical solutions of static peridynamic problems has been investigated. Related convergence studies for peridynamic wave propagation problems appeared in \[3\]. In this talk, we will report recent convergence studies of meshfree peridynamic fracture simulations. We will begin by reviewing the meshfree peridynamic discretization, related issues present in peridynamic convergence studies, and possible remedies proposed in the literature. In particular, we will discuss two numerical tools, partial-volume algorithms and influence functions, to improve the convergence behavior of numerical solutions in peridynamics. Then, we will report our finding concerning convergence studies of dynamic crack propagation and branching in meshfree peridynamic simulations. We will conclude this talk by mentioning some attempts to increase the order of convergence of peridynamic computations. 

REFERENCES

\[1\] S. A. Silling and E. Askari, A meshfree method based on the peridynamic model of solid mechanics, *Computers & Structures* 83 (2005): 1526–1535.

\[2\] P. Seleson and D. J. Littlewood, Convergence studies in meshfree peridynamic simulations, *Computers and Mathematics with Applications* 71 (2016): 2432–2448.

\[3\] P. Seleson, Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations, *Computer Methods in Applied Mechanics and Engineering* 282 (2014): 184–217. 

\[4\] P. Seleson and D. J. Littlewood, Numerical tools for improved convergence of meshfree peridynamic discretizations, in  Handbook of Nonlocal Continuum Mechanics for Materials and Structures, G.Voyiadjis (ed.), Springer, Cham, 2018.   
 


Convergence Studies in Meshfree Peridynamic Crack Propagation

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

IMECE2020 - Virtual Conference

Supporting the need among researchers worldwide for additional online and multimedia content, ASME now offers the ASME Conference Video Collection. This platform will collect and provide long-term access to conference presentation videos, encompassing all technology areas within ASME's technical conference and events program. About 30,000-40,000 engineers across all mechanical engineering and related disciplines participate in ASME conferences each year. This includes authors of approximately 6,000 to 8,000 technical papers published by ASME, the majority of whom also submit video presentations for our virtual conferences. The video collection will be available as an add-on to existing product packages or as a separate subscription.