The accurate and efficient analysis of laminated composite structures under vibrational loading remains a critical challenge in modern engineering design. Due to their layered configuration and anisotropic mechanical properties, such materials exhibit complex dynamic behaviors that cannot be effectively captured by classical discretization techniques alone. In this work, we present a novel approach to the free vibration analysis of laminated composites using the Haar wavelet discretization method, which offers distinct advantages in terms of localization, sparsity, and computational efficiency [1]. The Haar wavelet, as the simplest member of the family of orthonormal wavelets, provides a compact and efficient basis for representing localized features in structural deformation fields. Its stepwise nature makes it particularly suited for problems where abrupt changes or discontinuities in material properties or geometry are present, such as those often encountered in composite laminates [2, 3].
In the proposed method, the displacement field is approximated using Haar wavelet scaling functions in the spatial domain. The governing partial differential equations, derived based on first-order shear deformation theory (FSDT) or higher-order laminate theories, are transformed into a system of algebraic equations through Galerkin projection using Haar wavelet bases. The resulting system exhibits a sparse and structured matrix form, significantly reducing computational costs compared to conventional finite element or finite difference methods.
To validate the accuracy and robustness of the method, benchmark problems involving simply supported, clamped, and free boundary conditions are examined. Results are compared with analytical solutions, where available, as well as with high-fidelity numerical solutions from traditional approaches. The method demonstrates excellent convergence characteristics and maintains high accuracy even with relatively low discretization levels. Notably, it captures both the global vibrational modes and local variations near layer interfaces or geometrical discontinuities without the need for mesh refinement or artificial smoothing. Furthermore, the influence of various laminate parameters on natural frequencies and mode shapes is investigated. Parameters such as the number of layers, fiber orientation angles, stiffness ratios, and stacking sequences are varied systematically. The Haar wavelet method proves effective in revealing sensitivity trends and providing physical insights into the dynamic response of the structure. In particular, the method efficiently captures the coupling effects between bending and shear deformations, which are often underestimated in classical theories when applied to thick or anisotropic laminates.
A key strength of the method is its low computational footprint. Since the Haar basis functions are localized in space and possess compact support, the stiffness and mass matrices generated in the discretized system are highly sparse. This not only reduces the memory requirements but also allows for the use of fast solvers optimized for sparse systems. Compared to traditional finite element models with dense or banded matrices, the Haar wavelet model achieves similar or better accuracy at a fraction of the computational cost.
Another advantage of the proposed approach is its flexibility in handling nonuniform and adaptive discretization schemes. By selectively increasing the resolution in regions of interest, such as near cutouts, delaminations, or supports, the method achieves local refinement without a significant increase in global computational effort. This makes the method suitable for large-scale applications or integration into design optimization frameworks where rapid yet reliable dynamic analysis is essential.
The developed framework is also expandable to more complex cases, including laminated plates with variable thickness, curvilinear fibers, and functionally graded layers. Since the Haar wavelet method does not rely on geometric regularity or isotropy, it naturally extends to such configurations with minimal modification to the computational pipeline. Furthermore, it can be adapted for use in hybrid models that incorporate experimental data or machine learning components to refine boundary conditions or material properties in situ.
In conclusion, the Haar wavelet discretization method presents a compelling alternative to conventional numerical approaches for the vibration analysis of laminated composites. Its strengths lie in its ability to combine accuracy, computational efficiency, and adaptability to structural complexity. The method’s localized nature, sparse representation, and suitability for multi-scale modeling offer unique advantages in capturing the intricate vibrational behavior of composite laminates. Ongoing work includes the extension of this method to nonlinear and time-dependent vibration problems, as well as its integration with experimental data for inverse analysis and material characterization.
References:
1. Majak, J., Shvartsman, B., Karjust, K., Mikola, M., Haavajõe, A., & Pohlak, M. (2015). On the accuracy of the Haar wavelet discretization method. Composites Part B: Engineering, 80, 321-327. https://doi.org/ 10.1016/j.compositesb.2015.06.008
2. Jena, S. K., Chakraverty, S., Mahesh, V., & Harursampath, D. (2022). Application of Haar wavelet discretization and differential quadrature methods for free vibration of functionally graded micro-beam with porosity using modified couple stress theory. Engineering Analysis with Boundary Elements, 140, 167-185. https://doi.org/ 10.1016/j.enganabound.2022.04.009
3. Zhu, B., & Hiraishi, T. (2024). A decomposed Karhunen–Loève expansion scheme for the discretization of multidimensional random fields in geotechnical variability analysis. Stochastic Environmental Research and Risk Assessment, 38(4), 1215-1233. https://doi.org/ 10.1007/s00477-023-02625-8
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