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各向异性管道超声导波频散特性弱形式偏微分方程求解方法研究

Study on the Solution Method of Weak Form Partial Differential Equation for Dispersion Characteristics of Ultrasonic Guided Waves in Anisotropic Pipelines

  • 摘要: 在管道超声导波检测中,高效获取导波频散曲线是优化检测效果、实现模态合理选择的核心问题。本研究旨在开发一种高精度、低计算复杂度的频散特性求解方法,解决传统三维有限元分析在复杂管道结构中计算效率低的难题。研究提出一种弱形式偏微分方程与半解析有限元法(PDE-SAFE)耦合技术,通过在柱坐标系下引入谐波解,将三维波动控制方程降维为一维半解析模型。利用弱形式理论将强形式方程转化为积分形式,结合COMSOL有限元软件构建管道结构的一维模型,系统推导了纵波、扭转波、弯曲波及周向导波的传播方程与速度公式。通过定义检验函数提升数值稳定性,完整求解各向异性及各向同性管道的导波模态。研究结果表明,该方法成功获取了管道中不同模态的频散曲线与波结构图,计算效率较传统三维模型显著提升。与全局矩阵法(global matrix method, GMM)对比验证显示,其频散特性计算结果吻合良好,尤其在多模态识别中表现出更高的完整性和精度。研究表明,PDE-SAFE耦合技术通过降维建模与弱形式处理,有效平衡了计算复杂度与求解精度,为管道超声导波检测的模态优化和缺陷精准识别提供了可靠的理论与方法支撑,在管道结构无损检测与监测领域具有重要应用价值。

     

    Abstract: In ultrasonic guided wave testing of pipelines, efficiently obtaining guided wave dispersion curves is critical for optimizing inspection performance and selecting appropriate modes, as these curves directly characterize modal behavior and influence wave propagation and detection outcomes. This study presents a high-precision, low-complexity method to address the computational inefficiency of traditional 3D finite element analysis in complex pipeline structures. A coupled technique combining weak-form partial differential equations (PDEs) and the semi-analytical finite element method (SAFE)—termed PDE-SAFE—is developed. By introducing harmonic solutions in cylindrical coordinates, the 3D wave governing equations are reduced to 1D semi-analytical models. Using weak-form theory, the strong-form equations are transformed into integral formulations, and a 1D finite element model of pipelines is constructed in COMSOL to systematically derive propagation equations and phase velocity formulas for longitudinal, torsional, flexural, and circumferential guided wave modes. Test functions are employed to enhance numerical stability, enabling comprehensive computation of guided wave modes in both isotropic and anisotropic pipelines. Numerical results show that PDE-SAFE successfully generates dispersion curves and wave structure diagrams for various modes in pipelines, achieving significantly improved computational efficiency compared to traditional 3D models. Validation against the global matrix method confirms good agreement in dispersion characteristics—particularly superior completeness and accuracy in multi-mode identification. By combining dimensionality reduction with weak-form formulation, the PDE-SAFE technique balances computational complexity and solution accuracy, providing a reliable framework for modal optimization and precise defect detection in pipeline ultrasonic guided wave testing. This method holds substantial practical value for structural health monitoring of industrial pipelines.

     

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