The Phase-Field Approach to Fracture

This workshop presents the theoretical foundations and numerical implementation of the phase-field approach to fracture using the FEniCS computing platform. The goal is to introduce participants to modern variational fracture mechanics and demonstrate how fracture nucleation and propagation can be modeled computationally in brittle and soft materials.

The course combines mathematical foundations, finite element implementation, and live coding demonstrations in Python using FEniCS. Topics include Griffith fracture theory, variational formulations, phase-field regularization, fracture nucleation, adaptive mesh refinement, and large-scale simulations.

The workshop materials, including lecture notes, tutorials, and simulation examples, are presented in Jupyter Notebooks, allowing participants to interactively explore both the theory and implementation of phase-field fracture models.

What is FEniCS?

FEniCS is a high-performance computing (HPC) capable tool that efficiently utilizes supercomputers and high-performance clusters to solve complex scientific problems. It supports parallel computing, JIT compilation, and integrates with PETSc and MPI for scalability and performance. Its HPC capabilities enable researchers to perform large-scale simulations and analyses effectively.

FEniCS is an acronym

FEniCS is an acronym that stands for "Finite Element Computational Software." The inclusion of "ni" in the name is to create a balanced and appealing composition. The FEniCS software package was compiled at the University of Chicago, whose Phoenix mascot likely influenced the choice of the name.

What is the Phase-Field Approach to Fracture?

The phase-field approach to fracture is a variational and PDE-based framework for modeling crack nucleation and propagation without explicitly tracking crack surfaces. Instead of representing cracks as sharp discontinuities, the method regularizes cracks using a smooth scalar field called the phase-field variable.

This framework enables:

  • Automatic crack nucleation and propagation
  • Complex crack topologies and branching
  • Coupled multi-physics fracture simulations
  • Large-scale finite element simulations
  • Parallel and adaptive computations

The methodology combines ideas from:

  • Griffith fracture mechanics
  • Variational calculus
  • Finite element methods
  • Regularization theory
  • Nonlinear PDEs

Contents

Instructions for Installing FEniCS

Software Framework

The numerical implementation is based on:

  • FEniCS
  • Python
  • PETSc
  • MPI
  • Gmsh
  • PyVista

The workshop examples demonstrate scalable finite element implementations suitable for both educational and research applications.

Workshop Features

  • Variational fracture mechanics
  • Phase-field PDE derivation
  • Finite element implementation
  • Live coding sessions
  • Adaptive mesh refinement
  • Hyperelastic fracture
  • Dynamic crack branching
  • Thermo-mechanical fracture
  • Large-scale simulations in FEniCS

Github Repository

Instructors