PDBlend, a 2D Peridynamic Solver

This repository contains a Peridynamic solver for simulating failure and fracture processes in two-dimensional solid domains. The code implements a blended-Peridynamics approach that efficiently captures crack initiation and propagation mechanisms. As a demonstration, the provided input files are configured to simulate a three-point bending test of a concrete beam under monotonic loading conditions.

Prerequisites

• C++11 compatible compiler (g++ is preferred)
• Linux/macOS environment

Installation

Clone this repository to your local machine:

bash git clone https://github.com/SemsiCoskun/PDBlend.git cd PDBlend

How to Run

1. Make the script executable: bash g++ -std=c++11 -Wall *.cpp -o PDBlend_Solver -fopenmp

   • g++:The C++ compiler.
   • -std=c++11: Ensures compilation with C++11 standard features.
   • -Wall: Enables all warning messages (recommended for robust code).
   • *.cpp: Compiles all .cpp source files in the current directory.
   • -o PDBlend_Solver: Specifies the output executable name as PDBlend_Solver.
   • -fopenmp: Crucial for enabling OpenMP parallelization in the solver.

2. Run the script: bash ./PDBlend_Solver

The simulation will execute, process the inputs, and generate results.

Understanding the Code

The simulation workflow in PDBlend conceptually involves setting up the problem (mesh, boundaries, loading) and then executing the peridynamic time-stepping solver. All necessary functionalities for both are co-located in the single codebase.

The solver's core functionality is encapsulated in several key C++ header files, each responsible for a specific part of the peridynamic simulation. This includes functions for problem setup and simulation execution:

• geometricalFeatures.h: Provides utility functions and definitions related to geometrical calculations, used by other modules like createMesh.h and createBoundary.h.
• ZonesBoundary.h: Defines structures or classes for specifying boundary zones.
• ZonesCrackLines.h: Defines structures or classes for specifying pre-existing crack lines.
• ZonesExclusion.h: Defines structures or classes for specifying regions to be excluded from the mesh or neighbor searches.
• createMesh.h: Generates the discretization (mesh) of the simulation domain, defining nodal coordinates based on geometricalFeatures.h and potentially considering ZonesExclusion.h for excluded regions.
• createNeighbors.h: Establishes the neighbor list for each node within its peridynamic horizon, crucial for bond definition and interaction. This includes handling ZonesCrackLines.h and ZonesExclusion.h.
• createBoundary.h: Defines and applies boundary conditions (fixed or assigned displacements, velocities, and forces) to the simulation domain, utilizing ZonesBoundary.h and geometricalFeatures.h for defining boundary regions.
• createLoading.h: Manages the application of external loads over the simulation's loading steps.
• calculateDamage.h: Manages the calculation of local damage accumulation and identifies newly broken bonds within the material.
• calculateDisplacement.h: Computes nodal displacements, velocities, and accelerations based on internal and external forces, incorporating bond health degradation. The half-step time integration algorithm is utilized.
• calculateDt.h: Determines the stable time step (dt) for the dynamic simulation based on material properties and discretization.
• calculateEffectiveHorizonArea.h: Calculates the effective area of interaction for each node within its peridynamic horizon.
• calculatePartialAreas.h: Computes partial areas, likely used in peridynamic force calculations or influence functions.
• degradate.h: Implements the bond degradation law, determining the health of bonds based on critical bond strains (linear and bilinear models).

Simulation Workflow The simulation proceeds by first setting up the geometric domain, mesh, and boundary conditions, then iteratively calculating the dynamic response:

1. Mesh Generation: The domain is discretized into a set of nodes and their initial coordinates are defined (createMesh). This can include defining zones for exclusion.
2. Neighbor Determination: For each node, its neighbors within a defined horizon are identified, forming the "bonds" of the peridynamic model (createNeighbors). Pre-existing crack lines can influence bond creation.
3. Boundary and Loading Application: Boundary conditions (fixed nodes, applied displacements/velocities/forces) are established (createBoundary), and the external loading schedule is set up (createLoading).
4. Iterative Solver: The core simulation loop then begins, where at each time step:
   • Internal forces are calculated based on bond deformations and material properties.
   • Damage is assessed and bonds can degrade or break (calculateDamage, degradate).
   • Nodal accelerations, velocities, and displacements are updated (calculateDisplacement).
5. Output: Relevant simulation data (e.g., displacements, damage) is periodically written to output files.

Contributing

If you'd like to contribute to this project:

1. Fork the repository
2. Create a feature branch (git checkout -b feature/your-feature)
3. Commit your changes (git commit -m 'Add some feature')
4. Push to the branch (git push origin feature/your-feature)
5. Open a Pull Request

License

This project is licensed under the GNU General Public License v3.0 (GPL-3.0) - see the LICENSE file for details.

Copyright (C) 2025

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.

Contact

For questions, feedback, or collaboration inquiries, please contact: - Email: semsicoskun@gmail.com - GitHub: [SemsiCoskun](https://github.com/SemsiCoskun

References

• Semsi Coskun and Davood Damircheli and Robert Lipton. "PDBlend: A Blended Peridynamics Solver for Fracture Simulation." arXiv preprint arXiv:2503.20109, 2025. https://arxiv.org/abs/2503.20109
• Silling, S., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids 48, 175–209. doi:https://doi.org/10.1016/S0022-5096(99)00029-0.
• Underwood, P., 1983. Dynamic relaxation, in: Computational Methods in Mechanics, Vol 1: Computational methods for transient analysis. Elsevier Science Publishers B.V., pp. 245–265.
• D. Littlewood. Roadmap for software implementation. In Handbook of peridynamic modeling, pages 147–178. Chapman and Hall/CRC, 2016.
• Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Computers & structures 83, 1526–1535. doi:https://doi.org/10.1016/j.compstruc.2004.11.026.