Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hybrid-High Order method for fracture modelling

Alessandra Crippa, Julien Coatléven, Daniele A. Di Pietro, Nicolas Guy, Yousef Soleiman

TL;DR

The paper introduces a novel Hybrid High-Order (HHO) discretization for phase-field fracture models, enabling fracture simulations on general polygonal/polyhedral meshes with fully discontinuous spaces. It combines a staggered time-stepping scheme with static condensation to efficiently solve the coupled mechanical and phase-field subproblems, incorporating a history field to enforce fracture irreversibility. The method uses a higher-order strain reconstruction and stabilization in the mechanical block, and a diffusion–reaction discretization with local reconstructions for the phase-field block, including both isotropic and hybrid (tensile–compressive split) formulations. Numerical tests on mode I and II fracture demonstrate accurate, mesh-flexible behavior and favorable convergence compared to standard FEM, highlighting the approach's robustness and computational efficiency for complex fracture geometries.

Abstract

In this work, we introduce a new Hybrid High-Order method for the numerical simulation of fracture propagation based on phase-field models. The proposed method supports general meshes made of polygonal/polyhedral elements, which provides great flexibility in mesh design and adaptation, and can accommodate large variations of both the displacement and damage variables thanks to the use of fully discontinuous spaces. The resolution of the corresponding algebraic problem is based on a staggered time stepping scheme which takes advantage of static condensation for each subproblem. We provide extensive numerical validation of the method on classical two-dimensional fracture propagation problems, including a comparison with a more standard finite element scheme.

A Hybrid-High Order method for fracture modelling

TL;DR

The paper introduces a novel Hybrid High-Order (HHO) discretization for phase-field fracture models, enabling fracture simulations on general polygonal/polyhedral meshes with fully discontinuous spaces. It combines a staggered time-stepping scheme with static condensation to efficiently solve the coupled mechanical and phase-field subproblems, incorporating a history field to enforce fracture irreversibility. The method uses a higher-order strain reconstruction and stabilization in the mechanical block, and a diffusion–reaction discretization with local reconstructions for the phase-field block, including both isotropic and hybrid (tensile–compressive split) formulations. Numerical tests on mode I and II fracture demonstrate accurate, mesh-flexible behavior and favorable convergence compared to standard FEM, highlighting the approach's robustness and computational efficiency for complex fracture geometries.

Abstract

In this work, we introduce a new Hybrid High-Order method for the numerical simulation of fracture propagation based on phase-field models. The proposed method supports general meshes made of polygonal/polyhedral elements, which provides great flexibility in mesh design and adaptation, and can accommodate large variations of both the displacement and damage variables thanks to the use of fully discontinuous spaces. The resolution of the corresponding algebraic problem is based on a staggered time stepping scheme which takes advantage of static condensation for each subproblem. We provide extensive numerical validation of the method on classical two-dimensional fracture propagation problems, including a comparison with a more standard finite element scheme.

Paper Structure

This paper contains 18 sections, 38 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Continuous setting
  3. HHO Scheme
  4. Discrete spaces
  5. Discretization of the mechanical equilibrium equation
  6. Strain reconstruction operator
  7. Global discrete function $a_h$
  8. Discretization of the crack phase-field evolution equation
  9. Local reconstructions
  10. Global discrete function $b_h$
  11. Discrete history field variable
  12. Space semi-discrete problem
  13. Staggered time stepping scheme
  14. Time discretization and non-linear iterations
  15. History field update
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Domain and boundary conditions for the traction and shear test.
  • Figure 2: Evolution of the crack phase-field for different formulations (from top to bottom: isotropic, hybrid–VD, hybrid–SP) at increasing time steps, using a uniform mesh with element size $h = 0.01mm$
  • Figure 3: Load-displacement curves for the traction test with different mesh sizes using the hybrid–VD formulation.
  • Figure 4: Load-displacement curves for the traction test with different mesh sizes using HHO (order $k=1$, $l=0$) and P2-P1 finite elements.
  • Figure 5: Crack phase-field in mode I with HHO scheme on hexagonal meshes
  • ...and 5 more figures

Theorems & Definitions (7)

  • Remark : The role of time
  • Remark : Two-dimensional case
  • Remark : Choice of the polynomial degrees
  • Remark : The case of volumetric-deviatoric energy split
  • Remark : Static condensation
  • Remark : Update at quadrature nodes
  • Remark : Initial notch