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中文核心期刊

结构断裂损伤高效分析的隔离扩展有限元法

Inelasticity-Separated Extended Finite Element Method for Efficient Structural Fracture Damage Analysis

  • 摘要: 疲劳与灾害荷载作用下,工程结构内部裂纹的萌生与扩展引起的断裂损伤是威胁其安全性的关键因素。然而,由于断裂损伤问题的非线性特征和微观尺度特征,其分析效率通常较低。本研究结合隔离非线性理论与扩展有限元法(XFEM),提出结构断裂损伤高效分析的隔离扩展有限元法。本研究首先基于XFEM方法基本原理建立断裂区单元模型,并引入广义协调原理、自由度凝聚、矩阵广义逆等对该方法进行改进,将其中用于模拟断裂损伤的单元间共享附加扩展自由度转换为单元内部自由度,避免传统XFEM方法中结构整体刚度的维数变化,随后,通过将断裂区各单元刚度改写为无损单元刚度的摄动展开形式,进一步考虑断裂损伤损伤的局部化特征,提出了断裂损伤结构的隔离非线性控制方程,该方程避免了结构整体刚度中的元素时变,断裂损伤效应通过对初始无损结构刚度的低秩摄动进行描述,引入Woodbury公式进行控制方程求解,在每个计算步中仅需对代表局部断裂损伤的小规模矩阵进行更新,在保证计算精度的同时显著提升效率。典型算例验证表明,该方法在确保裂纹计算准确性的同时,大幅降低了计算成本。

     

    Abstract: Fatigue and catastrophic loads can induce crack initiation and propagation in engineering structures, leading to fracture damage that threatens structural safety. However, the nonlinear and microscale nature of fracture problems often results in low computational efficiency. This study integrates the Inelasticity-Separated Finite Element Method (IS-FEM) and the Extended Finite Element Method (XFEM) to propose an Inelasticity-Separated Extended Finite Element Method (IS-XFEM). First, a fracture-zone element model is established based on XFEM principles, incorporating generalized compatibility conditions, degree-of-freedom condensation, and matrix generalized inverse to convert shared enrichment degrees of freedom (DOFs) into internal element DOFs. This avoids dimension changes in the global stiffness matrix typical of conventional XFEM. Next, by reformulating the stiffness of fracture-zone elements as a perturbation of undamaged element stiffness, the localized nature of fracture damage is captured. The isolated nonlinear governing equations for fracture-damaged structures are derived, eliminating time-varying terms in the global stiffness matrix. Instead, fracture effects are described via low-rank perturbations of the initial undamaged stiffness. The Woodbury formula is employed for equation solving, updating only small-scale matrices representing localized damage at each step, ensuring accuracy while significantly improving efficiency. Numerical examples demonstrate that the proposed method maintains crack prediction accuracy while drastically reducing computational costs.

     

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