Abstract

Proper generalized decomposition (PGD), an advanced numerical simulation method, has been successfully used as a model reduction method in many fields. In this study, functionally graded materials are analyzed using the Airy stress function method, and a compatible equation representing the stress function is obtained. PGD is used to solve a compatible equation, which is a complex high-order partial differential equation, by decomposing it into two one-dimensional problems in space. The solution of the complex high-order partial differential equation is then obtained by solving the two one-dimensional problems using Chebyshev’s method. The accuracy of the proposed method is verified by comparing the PGD solution with the analytical solution. Numerical examples show that highly accurate results can be obtained by using several iterative modes. Therefore, the PGD method is an effective numerical technique for the stress analysis of functionally graded materials. This paper provides a theoretical basis for future research on the stress concentrations of irregular shape and rough surface.

中文翻译：

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用于功能分级材料应力分析的适当广义分解方法

摘要

适当广义分解（PGD）是一种先进的数值模拟方法，已在许多领域成功用作模型简化方法。在这项研究中，使用艾里应力函数方法分析了功能梯度材料，并获得了表示应力函数的兼容方程。PGD​​ 用于求解一个兼容方程，它是一个复杂的高阶偏微分方程，通过将其分解为空间中的两个一维问题。然后通过使用切比雪夫方法求解两个一维问题来获得复杂的高阶偏微分方程的解。通过比较PGD解与解析解，验证了所提出方法的准确性。数值例子表明，通过使用多种迭代模式可以获得高精度的结果。因此，PGD 方法是一种有效的数值技术，用于功能梯度材料的应力分析。本文为今后研究不规则形状和粗糙表面的应力集中问题提供了理论依据。