Higher Order Dynamic Mode Decomposition

Abstract

Abstract. This paper deals with an extension of dynamic mode decomposition (DMD), which is appropriate to treat general periodic and quasi-periodic dynamics, and transients decaying to periodic and quasiperiodic attractors, including cases (not accessible to standard DMD) that show limited spatial complexity but a very large number of involved frequencies. The extension, labeled as higher order dynamic mode decomposition, uses time-lagged snapshots and can be seen as superimposed DMD in a sliding window. The new method is illustrated and clarified using some toy model dynamics, the Stuart-Landau equation, and the Lorenz system. In addition, the new method is applied to (and its robustness is tested in) some permanent and transient dynamics resulting from the complex Ginzburg-Landau equation (a paradigm of pattern forming systems), for which standard DMD is seen to only uncover trivial dynamics, and the thermal convection in a rotating spherical shell subject to a radial gravity field. Schmid [48], and has become a useful tool for postprocessing massive spatiotemporal data in numerical and experimental fluid mechanics [49,50,51,52,32] and other fields [28,40]. The ability of DMD to extract relevant patterns makes this method potentially useful in identifying nonlinear dynamics in many physical systems, but this application requires some care since the standard DMD may give completely spurious results. The main goal of this paper is to analyze the application of DMD to general (both low-dimensional and infinite-dimensional) dynamical systems, which will require a nontrivial extension of the method.In order to fix ideas, DMD applies to spatio-temporal data organized in K equispaced J-dimensional snapshots as

A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines

“…Higher Order Dynamic Mode Decomposition (HODMD) [11] is an extension of the Dynamic Mode Decomposition (DMD) approach [23], which has been recently introduced as a tool to analyze complex and noisy flows [24] or data whose spatial dimension is restricted to external conditions (i.e., hot-wire measurements or flight flutter test experiments [25]). The DMD approach extracts modes (or flow structures) from complex flows, which are ranked by frequency.…”

“…The HODMD algorithm is only presented briefly in this article, but the interested reader may find details in [11]. As a prior step, it is necessary to construct a snapshot matrix V K 1 (of dimension J × K) containing a set of K time equispaced snapshots as…”

“…The benefits of our ROM is that it is based on Higher Order Dynamic Mode Decomposition (HODMD) [11], a technique that extracts information of the flow dynamics of any dynamical system, thus enabling the characterisation of the system through its dominant frequencies. By selecting the most relevant frequencies of the flow system at hand (in our case, the complex flow around a Vertical Axis Turbine (VAT)), it is possible to generate a cheap ROM that can mimic the flow evolution in space and time.…”

Experimental Analysis of a Bubble Wake Influenced by a Vortex Street

“…For example, there is only a given amount of frequencies that can be calculated during the FFT. When the flow is non-linear and the data are noisy, higher order dynamic mode decomposition can be used to calculate dominant frequencies [28]. However, with the given temporal resolution (see Table 1), the data basis in this study is considered to be large enough.…”

Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines

“…The shape of this mode suggests that the wake may be divided in two regions: a middle-distance-wake, x/D < 10 and a very-far-wake , x/D > 10, as observed in the previous section: vorticity curves in Figure 6 and the shape of the direct mode in Figure 8a. A spatial stability analysis performed using spatial-DMD [8,[52][53][54]] reveals a spatial growth of 0.4 for the middle-distance-wake and of 0.9 for the very-far-wake showing that the convective instability of the wake is enhanced aft 10 turbine diameters. The two distinct wake regions will lead to varying sensitivities (see next section).…”