Peridynamics
Theory and Physics
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 → 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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A typical mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — deliberately absorbing vibration energy for a smoother ride. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic inhomogeneities.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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.
Linear Elements (1st-order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix at initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$.
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper" — reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer" — starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler" — represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve" — can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
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.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy the ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question — which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize across at least three levels of mesh density — neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is the same as "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?" — Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software 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).
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for peridynamics? For example, presence of LES support for fluids, contact/large deformation capability for structures makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far will you expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Peridynamics
Frontline of Peridynamics GPU Parallelization
Peridynamics involves massive "bond" loops, making computation...
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