The use of dynamic basis functions in proper orthogonal decomposition

Currently, the usefulness of proper orthogonal decomposition (POD) is limited to computational domains with fixed meshes and fixed boundaries. This paper presents a new POD method that enables the modeling of flow through computational domains with deforming meshes and/or moving boundaries. To achieve this goal, the solution is approximated using basis functions which, although not explicitly functions of time, depend on parameters associated with flow unsteadiness. Results are shown for transonic flow through the Tenth Standard Configuration. Comparisons are made between this method and the standard approach for on- and off-reference flow conditions. This method properly captured flow nonlinearities and shock motion for cases in which the classical POD method failed.     

Efficient identification of uncertain parameters in a large‐scale tidal model of the European continental shelf by proper orthogonal decomposition

The adjoint method can be used to identify uncertain parameters in large‐scale shallow water flow models. This requires the implementation of the adjoint model, which is a large programming effort. The work presented here is inverse modeling based on model reduction using proper orthogonal decomposition (POD). An ensemble of forward model simulations is used to determine the approximation of the covariance matrix of the model variability and the dominant eigenvectors of this matrix are used to define a model subspace. An approximate linear reduced model is obtained by projecting the original model onto this reduced subspace. Compared with the classical variational method, the adjoint of the tangent linear model is replaced by the adjoint of a linear reduced forward model. The minimization process is carried out in reduced subspace and hence reduces the computational costs. In this study, the POD‐based calibration approach has been implemented for the estimation of the depth values and the bottom friction coefficient in a large‐scale shallow sea model of the entire European continental shelf with approximately 10⁶ operational grid points. A number of calibration experiments is performed. The effectiveness of the algorithm is evaluated in terms of the accuracy of the final results as well as the computational costs required to produce these results. The results demonstrate that the POD calibration method with little computational effort and without the implementation of the adjoint code can be used to solve large‐scale inverse shallow water flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A proper orthogonal decomposition of a simulated supersonic shear layer

The Karhunen–Loève procedure is applied to the analysis of an ensemble of snapshots obtained from a conditionally sampled localized shear layer simulation. The computed set of optimal basis functions is used to economically characterize sampled flow realizations. Pictorially it is seen that the essential features (and roughly 80% of the energy) of typical flows are captured by retaining roughly 10–20 parameters in the expansion. Smaller-scale features are resolved by retaining more terms in the series.     

An optimizing implicit difference scheme based on proper orthogonal decomposition for the two‐dimensional unsaturated soil water flow equation

An optimizing reduced implicit difference scheme (IDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for the two‐dimensional unsaturated soil water flow equation is presented. An ensemble of snapshots is compiled from the transient solutions derived from the usual IDS for a two‐dimensional unsaturated flow equation. Then, optimal orthogonal bases are reconstructed by implementing SVD and POD techniques for the ensemble of snapshots. Combining POD with a Galerkin projection approach, a new lower dimensional and highly accurate IDS for the two‐dimensional unsaturated flow equation is obtained. Error estimates between the true solution, the usual IDS solution, and the reduced IDS solution based on POD basis are derived. Finally, it is shown by means of a numerical example using the technology of local refined grids that the computational load is greatly diminished by using the reduced IDS. Also, the error between the POD approximate solution and the usual IDS solution is proved to be consistent with the derived theoretical results. Thus, both feasibility and efficiency of the POD method are validated. Copyright © 2011 John Wiley & Sons, Ltd.     

International journal of computational fluid dynamics real-time prediction of unsteady flow based on POD reduced-order model and particle filter

An integrated method consisting of a proper orthogonal decomposition (POD)-based reduced-order model (ROM) and a particle filter (PF) is proposed for real-time prediction of an unsteady flow field. The proposed method is validated using identical twin experiments of an unsteady flow field around a circular cylinder for Reynolds numbers of 100 and 1000. In this study, a PF is employed (ROM-PF) to modify the temporal coefficient of the ROM based on observation data because the prediction capability of the ROM alone is limited due to the stability issue. The proposed method reproduces the unsteady flow field several orders faster than a reference numerical simulation based on Navier–Stokes equations. Furthermore, the effects of parameters, related to observation and simulation, on the prediction accuracy are studied. Most of the energy modes of the unsteady flow field are captured, and it is possible to stably predict the long-term evolution with ROM-PF.     

Time and frequency domain nonlinear system characterization for mechanical fault identification

Mechanical systems are often nonlinear with nonlinear components and nonlinear connections, and mechanical damage frequently causes changes in the nonlinear characteristics of mechanical systems, e.g. loosening of bolts increases Coulomb friction nonlinearity. Consequently, methods which characterize the nonlinear behavior of mechanical systems are well-suited to detect such damage. This paper presents passive time and frequency domain methods that exploit the changes in the nonlinear behavior of a mechanical system to identify damage. In the time domain, fundamental mechanics models are used to generate restoring forces, which characterize the nonlinear nature of internal forces in system components under loading. The onset of nonlinear damage results in changes to the restoring forces, which can be used as indicators of damage. Analogously, in the frequency domain, transmissibility (output-only) versions of auto-regressive exogenous input (ARX) models are used to locate and characterize the degree to which faults change the nonlinear correlations present in the response data. First, it is shown that damage causes changes in the restoring force characteristics, which can be used to detect damage. Second, it is shown that damage also alters the nonlinear correlations in the data that can be used to locate and track the progress of damage. Both restoring forces and auto-regressive transmissibility methods utilize operational response data for damage identification. Mechanical faults in ground vehicle suspension systems, e.g. loosening of bolts, are identified using experimental data.