Koopman analysis by the dynamic mode decomposition in wind engineering

Abstract

The Koopman theory, a concept to globally model nonlinear signals by a linear Hamiltonian, has been at the frontier of fluid mechanics research for the last decade. Wind engineering research may well benefit from the new opportunities and insights into turbulence and fluid-structure interactions (FSI), but the principal Koopman algorithm, the Dynamic Mode Decomposition (DMD), has only been preliminarily applied in the field. This review aims to promote the understanding and practice of the DMD and Koopman analysis through a wind engineering-oriented perspective. First, a thorough Koopman literature review has been conducted in the Journal of Wind Engineering and Industrial Aerodynamics, the field's prime journal, to assess the current research status. Second, the DMD's inseparable connection to four fundamental mathematical principles, namely the Koopman theory, the Fourier and Laplace transform, the Proper Orthogonal Decomposition (POD), and machine learning, has been elucidated. Third, the core DMD algorithm has been presented and dissected, sparking a user guide and some discussions on its spectral implications. Last, several key topics in wind tunnel experimentation and numerical simulations have been discussed with practice-oriented recommendations and suggested DMD variants; the topics include noise-contamination, non-uniform sample domain, data sparsity, observable choice, input sample range and resolution, FSI decoupling, mean-subtraction, and truncation. Some discussions on the continuity assumption, coefficient of weight, reduced-order modeling, moving boundaries, compressed sensing, and fluid phenomenology have also been appended.