The paper presents a review of Haar wavelet methods and an application of the higher-order Haar wavelet method to study the behavior of multilayered composite beams under static and buckling loads. The Refined Zigzag Theory (RZT) is used to formulate the corresponding governing differential equations (equilibrium/stability equations and boundary conditions). To solve these equations numerically, the recently developed Higher-Order Haar Wavelet Method (HOHWM) is used. The results found are compared with those obtained by the widely used Haar Wavelet Method (HWM) and the Generalized Differential Quadrature Method (GDQM). The relative numerical performances of these numerical methods are assessed and validated by comparing them with exact analytical solutions. Furthermore, a detailed convergence study is conducted to analyze the convergence characteristics (absolute errors and the order of convergence) of the method presented. It is concluded that the HOHWM, when applied to RZT beam equilibrium equations in static and linear buckling problems, is capable of predicting, with a good accuracy, the unknown kinematic variables and their derivatives. The HOHWM is also found to be computationally competitive with the other numerical methods considered.

本文对Haar小波方法进行了综述，并应用了高阶Haar小波方法研究了多层复合梁在静载荷和屈曲载荷下的性能。精确的曲折理论（RZT）用于制定相应的控制微分方程（平衡/稳定性方程和边界条件）。为了用数值方法求解这些方程，使用了最新开发的高阶Haar小波方法（HOHWM）。将找到的结果与通过广泛使用的Haar小波方法（HWM）和广义差分正交方法（GDQM）获得的结果进行比较。通过将它们与精确的解析解进行比较，可以评估和验证这些数值方法的相对数值性能。此外，进行了详细的收敛研究，以分析所提出方法的收敛特性（绝对误差和收敛阶数）。结论是，将HOHWM应用于静态和线性屈曲问题中的RZT梁平衡方程时，能够很好地预测未知的运动学变量及其导数。还发现，HOHWM与其他考虑的数值方法在计算上具有竞争力。