Global Proper Orthogonal Decomposition Research Articles

A new method to interpolate POD reduced bases–Application to the parametric model order reduction of a gas bearings supported rotor

SummaryThe proper orthogonal decomposition (POD) is successfully employed in a variety of projection‐based methods for parametric model order reduction (pMOR) of large dynamical systems. It extracts the most energetic modes describing the dynamics of a system from time snapshots of the solution. In practice, these snapshots are computed at user‐defined parameters of the system with a high fidelity model. Then either all the snapshots are concatenated to extract a global POD basis, or a different POD basis is extracted for each parameter value. In the latter case, for unseen target parameters, the POD bases need to be adapted with a dedicated technique. An established method addressing this task is the interpolation on the tangent space to the Grassmann manifold (ITSGM). It interpolates the linear subspaces spanned by the known POD bases, discarding by construction the knowledge on the modes ordering by energy levels. In this paper, we formalize an ordered reduced basis interpolation (ORBI) technique which preserves such ordering. This approach improves the adaptation accuracy of the resulting POD basis as more information is taken into account in the construction of the interpolation operator. Numerical investigations are conducted on the parametric reduced order model of a dynamical system representing a small‐scale gas‐bearings supported rotor. Results show that the proposed method is more accurate than the ITSGM method at a similar computation cost. Trained at only three points of the parameters space, the developed hyper reduced order model (h‐ROM) performs to faster simulations with satisfactory accuracy, even far from the training points.

Fast Vibration Reduction Optimization Approach for Complex Thin-Walled Shells Accelerated by Global Proper Orthogonal Decomposition Reduced-Order Model

A fast vibration reduction optimization approach accelerated by the global proper orthogonal decomposition (POD) reduced-order model (ROM) is proposed, aiming at increasing the efficiency of frequency response analysis and vibration reduction optimization of complex thin-walled shells. At the offline stage, the global POD ROM is adaptively updated using the sample configurations generated by the CV (cross validation)–Voronoi sequence sampling method. In comparison to the traditional direct sampling method, the proposed approach achieves higher global prediction accuracy. At the online stage, the fast vibration reduction optimization is performed by combining the surrogate-based efficient global optimization (EGO) method and the proposed ROM. Two representative examples are carried out to verify the effectiveness and efficiency of the proposed approach, including examples of an aerospace S-shaped curved stiffened shell and a Payload Attach Fitting. The results indicate that the proposed approach achieves high prediction accuracy and efficiency through the verification by FOM and obtains better optimization ability over the direct optimization method based on FOM.

Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models

Low-dimensional and computationally less-expensive reduced-order models (ROMs) have been widely used to capture the dominant behaviors of high-4dimensional systems. An ROM can be obtained, using the well-known proper orthogonal decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes that are learned from experimental, simulated, or observational data, i.e., training data. However, the optimal basis can change with the parameter settings. When an ROM, constructed using the POD basis obtained from training data, is applied to new parameter settings, the model often lacks robustness against the change of parameters in design, control, and other real-time operation problems. This paper proposes to use regression trees on Grassmann manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected. Motivated by the observation that a subspace spanned by a POD basis can be viewed as a point in the Grassmann manifold, we propose to grow a tree by repeatedly splitting the tree node to maximize the Riemannian distance between the two subspaces spanned by the predicted POD bases on the left and right daughter nodes. Five numerical examples are presented to comprehensively demonstrate the performance of the proposed method, and compare the proposed tree-based method to the existing interpolation method for POD basis and the use of global POD basis. The results show that the proposed tree-based method is capable of establishing the mapping between parameters and POD bases, and thus adapt ROMs for new parameters.

A POD-Galerkin reduced order model for the Navier–Stokes equations in stream function-vorticity formulation

We develop a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for the efficient numerical simulation of the parametric Navier–Stokes equations in the stream function-vorticity formulation. Unlike previous works, we choose different reduced coefficients for the vorticity and stream function fields. In addition, for parametric studies we use a global POD basis space obtained from a database of time dependent full order snapshots related to sample points in the parameter space. We test the performance of our ROM strategy with the well-known vortex merger benchmark and a more complex case study featuring the geometry of the North Atlantic Ocean. Accuracy and efficiency are assessed for both time reconstruction and physical parameterization.

Data-driven sparse reconstruction of flow over a stalled aerofoil using experimental data

Abstract Recent work has demonstrated the use of sparse sensors in combination with the proper orthogonal decomposition (POD) to produce data-driven reconstructions of the full velocity fields in a variety of flows. The present work investigates the fidelity of such techniques applied to a stalled NACA 0012 aerofoil at $ {Re}_c=75,000 $ at an angle of attack $ \alpha ={12}^{\circ } $ as measured experimentally using planar time-resolved particle image velocimetry. In contrast to many previous studies, the flow is absent of any dominant shedding frequency and exhibits a broad range of singular values due to the turbulence in the separated region. Several reconstruction methodologies for linear state estimation based on classical compressed sensing and extended POD methodologies are presented as well as nonlinear refinement through the use of a shallow neural network (SNN). It is found that the linear reconstructions inspired by the extended POD are inferior to the compressed sensing approach provided that the sparse sensors avoid regions of the flow with small variance across the global POD basis. Regardless of the linear method used, the nonlinear SNN gives strikingly similar performance in its refinement of the reconstructions. The capability of sparse sensors to reconstruct separated turbulent flow measurements is further discussed and directions for future work suggested.

Efficient representation of turbulent flows using data‐enriched finite elements

AbstractEfficient modal decomposition of high‐dimensional turbulent flow data is an important first step for data reduction, analysis, and low‐dimensional predictive modeling. The conventional modal decomposition techniques, such as proper orthogonal and dynamic mode decompositions, aim to represent the system response using spatially global basis vectors that span a broad spatial domain. A significant challenge facing approaches based on global domain decomposition is the rapid increase in both the amount of training data and the number of modes that must be retained for an accurate representation of convection dominated turbulent flows. An alternative generalized finite element (GFEM) based approach is explored for efficient representation of high‐dimensional fluid flow data. Here, the standard finite element interpolation method is enriched with numerical functions that are learned from a small amount of high‐fidelity training data over spatially localized subdomains. The GFEM approach is demonstrated on a 3D flow past a cylinder at Reynolds number of 100 000 and flows inside a 2D lid‐driven cavity over a range of Reynolds numbers. Compared with a global proper orthogonal decomposition, the GFEM‐based approach increases efficiency in reconstructing the datasets while also substantially reducing the amounts of training data.

Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load

This paper presents an efficient structural design optimization strategy that combines the reduced basis method (RBM) with the equivalent static load (ESL). In dynamic response optimization using ESL, the computation of a static system optimization is repeatedly executed under multiple static loads. In this process, we propose parametrizing the static system and employing the RBM with global proper orthogonal decomposition (POD). In general, the snapshots for the sampling procedure under multiple loads increase proportionally to the number of loads, which results in an inefficient computational procedure. Thus, we propose taking snapshots with a proper orthogonal mode (POM) of the multiple loads rather than for the multiple loads themselves. The number of snapshots then decreases, and the original system can be efficiently reduced. We directly employ the framework of the proposed RBM with the POM of multiple loads to ESL-based design optimization, and the results indicate that the proposed method is more efficient than conventional ESL-based optimization as well as a full order model. Various numerical examples, including comparisons of relative errors and the dynamic response optimizations, support the strength of the proposed strategy.

Fast Multiscale Reservoir Simulations With POD-DEIM Model Reduction

Summary We present a global/local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media. Our approach identifies a low-dimensional structure in the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that achieves a high compression of the model. This compression is achieved because the velocity field is conservative for any low-order reduced model in our framework, whereas a typical global model reduction that is based on proper-orthogonal-decomposition (POD) Galerkin projection cannot guarantee local mass conservation. The lack of mass conservation can be observed in numerical simulations that use finite-volume-based approaches. The discrete empirical interpolation method (DEIM) approximates fine-grid nonlinear functions in Newton iterations. This approach delivers an online computational cost that is independent of the fine-grid dimension. POD snapshots are inexpensively computed with local model-reduction techniques that are based on the generalized multiscale finite-element method (GMsFEM) that provides (1) a hierarchical approximation of the snapshot vectors, (2) adaptive computations with coarse grids, and (3) inexpensive global POD operations in small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology provides an error bound in simulations. Our numerical results, by use of a two-phase immiscible flow, show a substantial speedup, and we compare our results with the standard POD-DEIM in a finite-volume setup.

Control of Nonlinear PDEs based on Space Vectors Clustering reduced order systems

Nonlinear PDEs domain discretization yields finite but high dimensional nonlinear systems. Proper Orthogonal Decomposition (POD) is widely used to reduce the order of such systems but it assumes that data belongs to a linear space and therefore fails to capture the nonlinear degrees of freedom. To overcome this problem, we develop a Space Vector Clustering (SVC) POD and use the reduced order model to design the controller which will then be applied to the full order system. A space vector is the solution at a particular space location over all times. The solution space is grouped into clusters where the behavior has significantly different features, then local POD modes will be constructed based on these clusters. We apply our method to reduce and control a nonlinear convective PDE system governed by the Burgers' equation over 1D and 2D domains and show a significant improvement over global POD. We also design a reference tracking controller and compare the controlled systems. We show that the controller based on our SVC local POD reduced system yields more accurate tracking results over the one based on global POD.

Improved extrapolation of steady turbulent aerodynamics using a non-linear POD-based reduced order model

AbstractA reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.

Non-linear reduced order models for steady aerodynamics

Abstract A reduced order modelling approach for predicting steady aerodynamic flows and loads data based on Computational Fluid Dynamics (CFD) and global Proper Orthogonal Decomposition (POD), that is, POD for multiple different variables of interest simultaneously, is presented. A suitable data transformation for obtaining problemadapted global basis modes is introduced. Model order reduction is achieved by parameter space sampling, reduced solution space representation via global POD and restriction of a CFD flow solver to the reduced POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimization problem. Methods for obtaining feasible starting solutions for the optimization procedure are discussed. The method is demonstrated by computing reduced-order solutions to the compressible Euler equations for the NACA 0012 airfoil based on two different snapshot sets; one in the subsonic and one in the transonic flow regime, where shocks occur. Results are compared with those obtained by POD-based interpolation using Kriging and the Thin Plate Spline method (TPS).