Purpose

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions.

Design/methodology/approach

This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error.

Findings

Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance.

Originality/value

The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

中文翻译：

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用于二阶 ODE 系统中特征问题的 h 版自适应 FEM：向量 Sturm-Liouville 问题和弯曲梁的自由振动

目的

本研究旨在克服所涉及的具有挑战性的问题并提供高精度的特征解。常微分方程 (ODE) 系统中的一般特征问题用作向量 Sturm-Liouville (SL) 和自由振动问题的数学模型。高精度的特征值和特征函数解是结构可靠动力分析的重要基础。然而，对于变量矩阵的系数、重合和相邻的近似特征值、特征对的连续阶数和变化的边界条件等问题，很难获得满足指定误差容限的解。

设计/方法/方法

本研究针对二阶 ODE 系统中的特征问题，提出了一种基于超收敛补丁恢复位移法的h 版自适应有限元方法。进一步引入高阶形函数插值技术得到本征函数的超收敛解，通过计算瑞利商得到本征值的超收敛解。特征函数的超收敛解用于估计有限元解在能量范数中的误差。然后根据误差对网格进行细分以生成改进的网格。

发现

选择具有代表性的特征问题示例，包含典型向量 SL 和梁的自由振动问题，涉及上述具有挑战性的问题，以评估所提出方法的准确性和可靠性。建立非均匀细化网格以适应特征函数的变化，并且数值解满足预先指定的误差容限。

原创性/价值

论文中描述的方法的建议组合，导致了一个强大的h 版网格细化算法，用于二阶 ODE 系统中的特征问题，可以扩展到其他类别的应用程序，用于基于高精度本征解。