High-Order Interior Penalty Finite Element Methods for Fourth-Order Phase-Field Models in Fracture Analysis

TL;DR

This work tackles the numerical challenges of simulating brittle fracture with fourth-order phase-field models by adopting a high-order Interior Penalty Finite Element Method (IP-FEM). It combines a fourth-order phase-field formulation with a hybrid strain-energy decomposition and a history field to enforce irreversibility, discretized via a FEALPy-enabled, high-order IP-FEM framework. The authors derive a comprehensive nonlinear residual formulation and a Newton-Raphson solution strategy, including a staggered scheme that decouples displacement and phase-field updates. Numerical experiments show that increasing the polynomial degree reduces mesh dependence and improves stability and convergence, enabling accurate fracture predictions on coarser meshes. The results guide efficient setup for practical engineering fracture analyses and point to future extensions to more complex geometries, dynamics, and adaptive refinement.

Abstract

This paper presents a novel approach for solving fourth-order phase-field models in brittle fracture mechanics using the Interior Penalty Finite Element Method (IP-FEM). The fourth-order model improves numerical stability and accuracy compared to traditional second-order phase-field models, particularly when simulating complex crack paths. The IP-FEM provides an efficient framework for discretizing these models, effectively handling nonconforming trial functions and complex boundary conditions. In this study, we leverage the FEALPy framework to implement a flexible computational tool that supports high-order IP-FEM discretizations. Our results show that as the polynomial order increases, the mesh dependence of the phase-field model decreases, offering improved accuracy and faster convergence. Additionally, we explore the trade-offs between computational cost and accuracy with varying polynomial orders and mesh sizes. The findings offer valuable insights for optimizing numerical simulations of brittle fracture in practical engineering applications.

Figures (2)

• Figure 1: The square model with a circular hole (unit:cm)
• Figure 2: The final results (left) and the residual force for different degrees and mesh sizes in the square model with a circular hole (right).