Construction of Bifurcation Diagrams Using POD on the Fly

Terragni, Filippo; Vega, José M.

doi:10.1137/130927267

Cited by 29 publications

(36 citation statements)

References 43 publications

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“…The bifurcation diagram for varying µ is given in Figure 5. As further explained in [56], this bifurcation diagram is obtained by plotting |u(3/4, t)| for the intersections of an orbit in the attractor with the Poincaré hypersurface…”

Section: Analysis Of the Attractors Using All Spatial Datamentioning

confidence: 99%

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Higher Order Dynamic Mode Decomposition

Clainche¹,

Vega²

2017

SIAM J. Appl. Dyn. Syst.

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296

172

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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

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Section: Analysis Of the Attractors Using All Spatial Datamentioning

confidence: 99%

“…This case was considered in [56] to illustrate the construction of bifurcation diagrams for varying µ using an adaptive reduced order model.…”

Section: Analysis Of the Attractors Using All Spatial Datamentioning

confidence: 99%

Higher Order Dynamic Mode Decomposition

Clainche¹,

Vega²

2017

SIAM J. Appl. Dyn. Syst.

Self Cite

296

172

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“…Here, we focus on an approach that has been called local POD (POD modes adapt to the local dynamics) or POD on the fly (POD modes are calculated along the simulation). This idea [21] has been used to both calculating transients [22,23] and approximating very complex bifurcation diagrams [24]. In addition to an updating method to recalculate the POD modes using snapshots calculated in short runs of the FM, some ingredients have been added to the method to further increase the computational efficiency, including the use of a limited number of mesh points to project the governing equations onto the POD modes and monitoring both the approximation and possible mode truncation instabilities using appropriate estimates.…”

Section: Introductionmentioning

confidence: 99%

“…An algorithm will be designed to update the set of POD modes and the effect of the parameters involved in the adaptive technique studied. The benefit of using unconverged Newton iterations as snapshots (as performed in [24] with unconverged transient behavior) will be analyzed.…”

Section: Introductionmentioning

confidence: 99%

Accelerating oil reservoir simulations using POD on the fly

Clainche

Varas

Vega

2016

Numerical Meth Engineering

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A reduced order model (ROM) is presented for the long-term calculation of subsurface oil/water flows. As in several previous ROMs in the field, the Newton iterations in the full model (FM) equations, which are implicit in time, are projected onto a set of modes obtained by applying proper orthogonal decomposition (POD) to a set of snapshots computed by the FM itself. The novelty of the present ROM is that the POD modes are (i) first calculated from snapshots computed by the FM in a short initial stage, and then (ii) updated on the fly along the simulation itself, using new sets of snapshots computed by the FM in even shorter additional runs. Thus, the POD modes adapt themselves to the local dynamics along the simulation, instead of being completely calculated at the outset, which requires a computationally expensive preprocess. This strategy is robust and computationally efficient, which is tested in 10-and 30-year simulations for a realistic reservoir model taken from the SAIGUP project.ACCELERATING OIL RESERVOIR SIMULATIONS USING POD ON THE FLY Here, t is the time variable, is the porosity (void fraction of the rock) and, for j D o (oil) and w (water), m j accounts for sources and sinks associated with the injection and production wells, ¶ www.nr.no/en/SAIGUP obtained with the full model (blue) and the reduced-order model (ROM) (green) in the 30-year simulation.

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“…The method consists in an adaptive strategy in which POD modes are calculated on demand, as the simulation proceeds. This strategy has proven to be very robust in both calculating particular solutions and constructing bifurcation diagrams (Terragni and Vega 2014), which require a huge number of particular timedependent runs. The method uses both a CFD solver and a low dimensional system in interspersed time intervals and can thus be called a POD on the Fly method.…”

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confidence: 99%

Computationally efficient simulation of unsteady aerodynamics using POD on the fly

2016

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Modern industrial aircraft design requires a large amount of sufficiently accurate aerodynamic and aeroelastic simulations. Current computational fluid dynamics (CFD) solvers with aeroelastic capabilities, such as the NASA URANS unstructured solver FUN3D, require very large computational resources. Since a very large amount of simulation is necessary, the CFD cost is just unaffordable in an industrial production environment and must be significantly reduced. Thus, a more inexpensive, yet sufficiently precise solver is strongly needed. An opportunity to approach this goal could follow some recent results (Terragni and Vega 2014 SIAM J. Appl. Dyn. Syst. 13 330-65; Rapun et al 2015 Int. J. Numer. Meth. Eng. 104 844-68) on an adaptive reduced order model that combines 'on the fly' a standard numerical solver (to compute some representative snapshots), proper orthogonal decomposition (POD) (to extract modes from the snapshots), Galerkin projection (onto the set of POD modes), and several additional ingredients such as projecting the equations using a limited amount of points and fairly generic mode libraries. When applied to the complex Ginzburg-Landau equation, the method produces acceleration factors (comparing with standard numerical solvers) of the order of 20 and 300 in one and two space dimensions, respectively. Unfortunately, the extension of the method to unsteady, compressible flows around deformable geometries requires new approaches to deal with deformable meshes, high-Reynolds numbers, and compressibility. A first step in this direction is presented considering the unsteady compressible, two-dimensional flow around an oscillating airfoil using a CFD solver in a rigidly moving mesh. POD on the Fly gives results whose accuracy is comparable to that of the CFD solver used to compute the snapshots.

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Construction of Bifurcation Diagrams Using POD on the Fly

Cited by 29 publications

References 43 publications

Higher Order Dynamic Mode Decomposition

Higher Order Dynamic Mode Decomposition

Accelerating oil reservoir simulations using POD on the fly

Computationally efficient simulation of unsteady aerodynamics using POD on the fly

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