One of the most widely used and reliable methods is the finite element method, which has proven its effectiveness in various applications, from structural analysis to fluid dynamics. The traditional FEM, however, can face challenges when dealing with specific types of problems, such as those involving sharp gradients, material discontinuities, or local damages. These issues often require a very fine mesh, leading to a significant increase in computational cost and complexity. Consequently, researchers have been exploring alternative approaches to enhance the capabilities of numerical methods. A promising direction has been the integration of wavelet theory into the finite element framework [1]. Unlike conventional polynomial basis functions used in the standard FEM, wavelets can represent functions at different scales and locations, making them particularly well-suited for problems with highly localized features. This characteristic allows for adaptive refinement, where a finer resolution is used only in areas of interest, such as crack tips or stress concentration zones, without the need for a globally dense mesh. Furthermore, the inherent properties of wavelets can lead to better conditioned stiffness matrices, improving the stability and efficiency of the numerical solution process. Among the various types of wavelets, the B-spline wavelets have emerged as a particularly strong candidate for this purpose. B-splines possess desirable smoothness properties and a compact support, which are beneficial for constructing efficient and stable finite element formulations [2, 3]. Their use has been explored in a wide range of applications, including the analysis of beams, plates, and other two-dimensional structures. They have been successfully applied to static and dynamic problems, demonstrating their versatility and robustness. By leveraging the combined strengths of the finite element method and wavelet theory, researchers are developing more efficient and accurate tools for solving challenging engineering problems. This integration is not just a simple modification of an existing method but a fundamental re-formulation that opens up new possibilities for advanced analysis and simulation. The continuous development in this area promises to lead to even more powerful and versatile computational methods in the future. The focus on B-spline wavelets is motivated by their excellent approximation properties and computational convenience. The goal of this work is to provide a comprehensive overview of the current state of research in this field, highlighting the key advancements and potential future directions. We will explore how these methods are formulated, their advantages over conventional approaches, and their practical applications in various engineering disciplines. The discussion will cover both the theoretical foundations and the practical implementation of these advanced numerical techniques, providing a clear picture of their potential to revolutionize computational mechanics.
Structural analysis has shown that B-spline wavelets can provide more accurate results compared to traditional finite element, finite difference, and other discretization methods. This work explores the application of B-spline wavelets for the vibration analysis of composite pipes. For modeling composite pipes, a wavelet-based finite element method was developed using B-spline wavelets on the interval. In this formulation, the composite pipe is discretized by a set of beam elements, utilizing two theories: the Euler-Bernoulli beam theory and the Timoshenko beam theory. The obtained numerical results demonstrate the accuracy and efficiency of the developed method. To verify this, a comparison was made with results obtained using the conventional finite element method.
The computational method implemented in this study is based on the use of orthogonal wavelets with compact support. B-splines, which were previously used as interpolation functions in the finite element analysis of homogeneous structures, were applied here to model the composite pipe structure. In our case, the solving domain for a glass-fiber reinforced polymer (FRP) pipe can be divided into subdomains, where each subdomain is mapped into a standard solving domain. Thus, a composite pipe of a fixed length can be discretized into several elements. Both the Euler-Bernoulli and Timoshenko methods were applied to model the pipe with various boundary conditions, including simply-supported, clamped-pinned, and clamped-clamped end conditions. The computational method utilized modal characteristics that included the first four mode shapes of the composite pipe under various boundary conditions using Euler-Bernoulli elements. The mode shapes were normalized to highlight the vibration pattern. In this work, orthogonal wavelets with compact support were used. Detailed expressions for the mass and stiffness matrices for the composite pipe were obtained. The modal characteristics of the composite pipe were determined by formulating and solving the generalized eigenvalue problem. Comparisons with known results from experimental studies demonstrated the accuracy of the developed finite element method. Additionally, the computational efficiency when modeling composite multilayer pipes using B-spline wavelets is demonstrated by using a significantly smaller number of elements compared to the elements used in the shell analysis method. Ultimately, the results obtained using the developed element can be used as benchmark solutions.
References:
1. Chen, X., Yang, S., Ma, J., & He, Z. (2004). The construction of wavelet finite element and its application. Finite Elements in Analysis and Design, 40(5-6), 541-554. https://doi.org/ 10.1016/S0168-874X(03)00077-5
2. Schillinger, D., & Rank, E. (2011). An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry. Computer Methods in Applied Mechanics and Engineering, 200(47-48), 3358-3380. https://doi.org/ 10.1016/j.cma.2011.08.002
3. Roh, H. Y., & Cho, M. (2005). Integration of geometric design and mechanical analysis using B‐spline functions on surface. International Journal for Numerical Methods in Engineering, 62(14), 1927-1949. https://doi.org/ 10.1002/nme.1254
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