Peridynamics
Peridynamics: Theoretical Foundations
What is Peridynamics?
Professor, what is peridynamics?
Peridynamics (PD) is a nonlocal continuum mechanics theory proposed by Silling (2000, Sandia National Laboratories). Unlike conventional FEM (based on differential equations), it describes motion using integral equations. The initiation, branching, and merging of cracks can be represented naturally in a mesh-free manner.
Fundamental Equation
$H_\mathbf{x}$ is a sphere (horizon) centered at point $\mathbf{x}$. $\mathbf{f}$ is the pairwise force function.
No derivatives! It's an integral equation, not a partial differential equation.
Because there are no derivatives, discontinuities (cracks) can be handled naturally. When a bond (connection between points) stretches beyond a critical value, it breaks โ crack. Cracks propagate without the need for mesh regeneration.
Summary
Dr. Silling's Invention at Sandia Laboratories
Peridynamics is a nonlocal continuum mechanics theory published in 2000 by Dr. Stewart Silling of Sandia National Laboratories. It circumvents the problem of undefined derivatives at cracks in conventional partial differential equations by using integral equations (interactions between material points called "bonds"). It was developed with the motivation of simulating the fracturing of nuclear waste glass.
Computational Methods for Peridynamics
Peridynamics Implementation
Comparison with FEM
| Characteristic | FEM | Peridynamics |
|---|---|---|
| Equation | Partial Differential Equation | Integral Equation |
| Crack | Mesh-dependent (requires remeshing or XFEM) | Mesh-independent |
| Branching/Merging | Difficult | Natural |
| Computational Cost | Low | High (nonlocal interactions) |
| Commercial Implementation | Mature | Limited |
Summary
Difference Between Bond-based and State-based
Peridynamics has two types: Bond-based (BPD) and State-based (SPD). BPD is simpler to compute but has the constraint that Poisson's ratio is fixed at 1/4 (2D) or 1/3 (3D). SPD removes this constraint, allowing arbitrary Poisson's ratios, making it superior for application to real materials. However, SPD's computational cost is 2-3 times higher than BPD, and parallel implementation for large-scale models is a challenge.
Peridynamics in Practice
Peridynamics in Practice
Still in the research stage, but research papers are increasing for impact fracture of glass, rock fragmentation, and composite material damage.
Practical Checklist
Peridynamics Analysis of Composite Material Impact Damage
For CFRP impact damage (combination of delamination, matrix cracking, and fiber breakage), peridynamics, which can naturally handle multiple crack propagations, is more promising than FEM. Since 2016, NASA Marshall Space Flight Center has analyzed high-speed particle impact damage on CFRP launch vehicle panels using peridynamics, achieving 80% agreement in damage area with CT scan experiments.
Peridynamics: Software & Solver Comparison
Peridynamics Tools
Peridigm: Open-Source Peridynamics
Peridigm is an open-source peridynamics code developed and released by Sandia National Laboratories, supporting MPI parallelization for large-scale computation. It uses the Trilinos numerical computation library and can scale to over 1000 cores in MPICH/OpenMPI environments. Used by NASA, DOE, and military-related agencies, the code is publicly available on GitHub (last updated in the 2020s).
Advanced Technology
Advanced Peridynamics
Frontline of Peridynamics GPU Parallelization
Peridynamics involves massive "bond" loops, making computation...
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