标题： 高阶内部惩罚有限元方法在断裂分析中四阶相场模型的应用 显示英文标题

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

摘要： 本文提出了一种新颖的方法，利用内罚有限元方法（IP-FEM）求解断裂力学中的四阶相场模型。 与传统的二阶相场模型相比，四阶模型在数值稳定性和精度方面有所提高，尤其是在模拟复杂裂纹路径时。 IP-FEM 为这些模型的离散化提供了一个高效的框架，有效处理了非协调试函数和复杂的边界条件。 本研究中，我们利用 FEALPy 框架实现了一个灵活的计算工具，支持高阶 IP-FEM 离散化。 我们的结果显示，随着多项式阶数的增加，相场模型的网格依赖性降低，从而提高了精度并加快了收敛速度。 此外，我们探讨了多项式阶数和网格尺寸变化时计算成本与精度之间的权衡。 这些发现为优化实际工程应用中脆性断裂的数值模拟提供了宝贵的见解。 显示英文摘要

摘要： 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.