Wavelet-based numerical methods and their applications in computational mechanics

YANG Bing; WANG Jizeng; LIU Xiaojing; ZHOU Youhe; FENG Yonggu

doi:10.6052/1000-0992-24-009

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Yang B, Wang J Z, Liu X J, Zhou Y H, Feng Y G. Wavelet-based numerical methods and their applications in computational mechanics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-009

Citation:

Yang B, Wang J Z, Liu X J, Zhou Y H, Feng Y G. Wavelet-based numerical methods and their applications in computational mechanics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-009

Yang B, Wang J Z, Liu X J, Zhou Y H, Feng Y G. Wavelet-based numerical methods and their applications in computational mechanics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-009

Citation:

Yang B, Wang J Z, Liu X J, Zhou Y H, Feng Y G. Wavelet-based numerical methods and their applications in computational mechanics. Advances in Mechanics, in press doi: 10.6052/1000-0992-24-009

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Wavelet-based numerical methods and their applications in computational mechanics

doi: 10.6052/1000-0992-24-009

Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry ofEducation of China, College of Civil Engineering and Mechanics, Lanzhou University,Lanzhou 730000, China

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Corresponding author: jzwang@lzu.edu.cn
Accepted Date: 2024-05-07

Available Online: 2024-05-16

Abstract

Abstract

Mechanics research advances towards interdisciplinary research, cross-scale correlations, and extreme environmental impacts. Strong nonlinearity, strong discontinuity, significant multi-physics coupling, multiscale, and complex geometries have become common characteristics in solving various mechanics problems quantitatively. Long-term quantitative research indicates that one of the core ingredients for solving such problems effectively lies in constructing numerical methods that can accurately identify, locate, capture, and separate different scale characteristics especially small-scale local characteristics, under multiscale and nonlinearity circumstances. These numerical methods should also possess the capacity to isolate and decouple the large-scale lower-order approximation from the small-scale higher-order truncation error effectively. The intrinsic multiresolution analysis and time-frequency localization characteristics of wavelet theory, as well as the various selectivity of basis functions, align precisely with the demands of this mathematical feature. Therefore, they can provide fundamental theory and diverse approaches for developing efficient quantitative methods to address various complex mechanics problems. Based on this fact, this paper provides a comprehensive discussion of wavelet theory, focusing on the theoretical framework of biorthogonal multi-resolution analysis and the construction method of frequently used wavelet bases. Furthermore, wavelet approximation of the function defined on a finite domain is presented in detail. The fundamental principles, development, merits, and shortcomings of different wavelet-based numerical methods are systematically elucidated. Several novel wavelet-based methods with outstanding performance developed recently are elaborated especially, and their applications in solving typical mechanics problems are reviewed. Meanwhile, the present paper also points out challenges encountered by the existing wavelet-based numerical methods when addressing complex and strongly nonlinear mechanics problems. This may provide valuable references for the development of wavelet-based numerical methods and their applications in complex mechanics and engineering problems, and introduce new perspectives and methods for solving these problems efficiently, accurately, and universally.
- wavelet theory,
- multiresolution analysis,
- numerical methods,
- wavelet adaptive algorithms,
- computational mechanics

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