On long-term boundedness of Galerkin models

Schlegel, Michael; Noack, Bernd R.

doi:10.1017/jfm.2014.736

Cited by 62 publications

(74 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Global converge can strictly be ensured by enforcing the energy preservation on the quadratic term. This energy preservation is derivable from the Navier-Stokes equation (Kraichnan & Chen 1989;Schlegel 2013). In addition, Galerkin systems with nonlinear subscale turbulence representations are shown to be much more robust with respect to changes of the eddy viscosity parameters and the dimension of the model.…”

Section: Discussionmentioning

confidence: 95%

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

et al. 2014

View full text Add to dashboard Cite

We investigate a hierarchy of eddy-viscosity terms in POD Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to Large Eddy Simulation data for a flow around the vehicle-like bluff body called Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (1994). Robustness of the model solution with respect to initial conditions, eddy viscosity values and model order is only achieved for state-dependent eddy viscosities as proposed by Noack, Morzyński & Tadmor (2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier-Stokes based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.

show abstract

Section: Discussionmentioning

confidence: 95%

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

et al. 2014

View full text Add to dashboard Cite

show abstract

“…In the future, we will apply our model to more complicated time-dependent non-linear PDEs and explore the stability of long-term parameteric non-linear dynamical systems. The generalised Lyapunovs direct method [43] can be used to guarantee the long-term boundedness if there is a monotonically attracting trapping region. The concept of longterm boundedness is linked to the stability analysis of parameteric nonlinear PDE systems with respect to the parameters e.g.…”

Section: Resultsmentioning

confidence: 99%

“…initial and boundary values using the energy method. By analysing the spectrum of eigenvalues and Lyapunov exponents, a sufficient criterion for long-term boundedness of Galerkin systems can be used to exclude infinite blow-ups of the system state solutions in finite or infinite periods of time [43]. In the near future we will explore solution boundedness and methodologies for interpolating the ROM basis functions over parameter ranges.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

Xiao¹,

Fang²,

Pain³

et al. 2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ω p ∈ R P ). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ω p can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique [50,53] where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier-Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

show abstract

“…Reduced order models (ROMs) can be derived by a combination of POD and Galerkin projection methods. However, the use of POD/Galerkin methods raises numerical instability and non-linearity inefficiency problems [11,12,13,14]. Several methods have been presented to improve the numerical stability of ROMs, such as calibration [15,16], Fourier expansion [17], regularisation [18] and Petrov−Galerkin methods [2,19].…”

Section: Introductionmentioning

confidence: 99%

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

Xiao¹,

Yang²,

Fang

et al. 2017

Journal of Computational Physics

View full text Add to dashboard Cite

This work presents the first application of a non-intrusive reduced order method to model solid interacting with compressible fluid flows to simulate crack initiation and propagation. In the high fidelity model, the coupling process is achieved by introducing a source term into the momentum equation, which represents the effects of forces of the solid on the fluid. A combined single and smeared crack model with the MohrCoulomb failure criterion is used to simulate crack initiation and propagation. The non-intrusive reduced order method is then applied to compressible fluid and fractured solid coupled modelling where the computational cost involved in the full high fidelity simulation is high. The non-intrusive reduced order model (NIROM) developed here is constructed through proper orthogonal decomposition (POD) and a radial basis function (RBF) multi-dimensional interpolation method.The performance of the NIROM for solid interacting with compressible fluid flows, in the presence of fracture models, is illustrated by two complex test cases: an immersed wall in a fluid and a blasting test case. The numerical simulation results show that the NIROM is capable of capturing the details of compressible fluids and fractured solids while the CPU time is reduced by several orders of magnitude. In addition, the issue of whether or not to subtract the mean from the snapshots before applying POD is discussed in this paper. It is shown that solutions of the NIROM, without mean subtracted before constructing the POD basis, captured more details than the NIROM with mean subtracted from snapshots.

show abstract

On long-term boundedness of Galerkin models

Cited by 62 publications

References 55 publications

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

Contact Info

Product

Resources

About