The use of Fourier analysis for frequency domain representation or high-order trigonometric interpolation can be found in a plethora of applications throughout science and engineering, including those related to surface representation, scattered data approximation, signal processing, and differential equations. In particular, Fourier-based methods for solving high-resolution discretized problems (e.g., those governed by PDEs) in solid and fluid mechanics have enjoyed particular interest due to the relatively low computational complexity and memory requirements involved in computing fast Fourier transforms (FFTs). Indeed, by reducing convolution and differentiation operations to simple products in the frequency domain (which can be easily obtained by means of the FFT), fast and accurate numerical methods (including recently-introduced neural network operators) can be developed to treat a variety of initial and/or boundary value problems for the mathematical modeling of mechanical behavior (e.g., wave propagation and dynamics, diffusion, material homogenization, etc.). However, widespread use for such applications has been hindered in no small part by the well-known difficulties of Fourier analysis in handling complex physical boundary conditions, complicated domains, discontinuities/irregularities, non-uniformly spaced data, and, more commonly, non-periodicity (i.e., Gibb’s phenomenon).
In the spirit of addressing such challenges and facilitating broader applicability, this special issue aims to highlight recent contributions from applied mathematicians and computational mechanicians on the general development of Fourier-based approaches toward solving problems in acoustics, solids, or fluids.
Topics of particular interest include, but are not limited to:
• Modified/non-standard Fourier series representations
• FFT-based numerical methods for PDEs
• Fourier-based machine learning techniques
Faisal Amlani (Lead Guest Editor), LMPS, Université Paris-Saclay, France)
Niema M. Pahlevan, University of Southern California, USA firstname.lastname@example.org
Carlos Pérez-Arancibia, University of Twente, Netherlands email@example.com
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