Wavelet-based numerical methods and their applications in computational mechanics
Mechanics research advances towards interdisciplinary research,cross-scale correlations,and extreme environmental impacts.Strong nonlinearity,strong discontinuity,significant multi-phys-ics coupling,multiscale,and complex geometries have become common characteristics in solving vari-ous 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 accur-ately 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-fre-quency localization characteristics of wavelet theory,as well as the various selectivity of basis func-tions,align precisely with the demands of this mathematical feature.Therefore,they can provide fun-damental theory and diverse approaches for developing efficient quantitative methods to address vari-ous 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 refer-ences 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.
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