An iterative algorithm for POD basis adaptation in solving parametric convection–diffusion equations

A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations

An aspect of recent developments in computational methods for control of fluids is the design of reduced‐order controllers, for reducing the CPU costs of flow solves. In this paper a reduced order approach will be introduced for the optimal boundary control problem governed by the unsteady Navier Stokes equations with the help of the Galerkin proper orthogonal decomposition(POD) method. The paper summarizes the results of the diploma‐thesis [1].The Galerkin Proper orthogonal decomposition ([2], [3], [4], [5], [6]) provides a method for deriving reduced order models of dynamical systems. It’s based on projecting the dynamical system onto subspaces of snapshots ensemble, which is composed of the solutions for the physical system at pre‐specified time instances or experimental measurements. These snapshots are not suitable as the basis for the ensemble spanned by themselves by reason of the possible linear dependence. We will find the orthogonal basis for the ensemble by solving an eigenvalue problem, and these basis will be denoted in the thesis as POD basis, and number of these POD basis can be very small in comparison with the number of the snapshots. The POD basis spanned subspace is just that, which the dynamical system will be projected onto.The goals of the paper are at first applying the POD basis to simulate the original flow in two dimensional rectangle. We hope that in this manner the POD method can reduce the computational cost of the nonlinear flow solutions. Moreover, a boundary condition should be found, so that the unsteady Navier Stokes equations is solved with this boundary condition and meanwhile its solution minimized dynamically an extra cost functional, which contains the state and the control variable. In this optimization procedure we continue with the help of POD basis to derive an optimal control system for the reduced order models. POD‐based optimization is going to work, only if the snapshots contains enough information to model all the dynamical behavior of the flows that are encountered throughout the optimization process.

The proper orthogonal decomposition method has been shown to produce accurate reduced-order models for the aeroelastic analysis of complete aircraft configurations at fixed flight conditions. However, changes in the Mach number or angle of attack often necessitate the reconstruction of the reduced-order model to maintain accuracy, which destroys the sought-after computational efficiency. Straightforward approaches for reduced-order model adaptation-such as the global proper orthogonal decomposition method and the direct interpolation of the proper orthogonal decomposition basis vectors-that have been attempted in the past have been shown to lead to inaccurate proper orthogonal decomposition bases in the transonic flight regime. Alternatively, a new reduced-order model adaptation scheme is described in this paper and evaluated for changes in the freestream Mach number and angle of attack. This scheme interpolates the subspace angles between two proper orthogonal decomposition subspaces, then generates a new proper orthogonal decomposition basis through an orthogonal transformation based on the interpolated subspace angles. The resulting computational methodology is applied to a complete F-16 configuration in various airstreams. The predicted aeroelastic frequencies and damping coefficients are compared with counterparts obtained from full-order nonlinear aeroelastic simulations and flight test data. Good correlations are observed, including in the transonic regime. The obtained computational results reveal a significant potential of the adapted reduced-order model computational technology for accurate, near-real-time, aeroelastic predictions.

Impact of Geometric Forms on the Effectiveness and Physical Features of POD-based Geometric Parameterization

In this thesis we analyze and develop methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs). The methods combine a direct solution approach with model reduction via proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). The reduced-order models (ROMs) are used to approximate the high-dimensional dynamic systems originating from a spatial discretization of a PDE. However, when used in an optimization algorithm, conventional POD/DEIM ROMs often lack the ability to give adequate approximations of the gradient. We propose methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces. We distinguish two types of error between quantities evaluated with the high-dimensional model and its ROM approximation in dependency on the optimization variable q: The reconstruction error which is evaluated with the same q0 which is used constructing the ROM and the prediction error which assesses approximations at q with a ROM constructed at q0 different to q. The novel reconstruction results we present include estimates for solutions of the adjoint equation and the sensitivity equations as well as for the gradient of the objective function. Based on the estimates we explain how the POD and DEIM bases should be extended with either adjoint or sensitivity information. The enhanced ROMs allow control of the reconstruction error for the objective and its gradient up to machine precision. Moreover, we propose a POD prediction estimate for the objective of the optimization problem in a neighborhood of q where the ROM is constructed. In case of sensitivity-extended POD and DEIM bases we give an analogous result for solutions of the states. The derivative-extended ROMs are then used to develop adaptive algorithms for the solution of optimal control and parameter estimation problems which results in great runtime improvements for the optimization while ensuring high approximation quality of the solution of the original problem. For the parameter estimation case a novel a posteriori error estimate is proposed which assesses the quality of suboptimal solutions obtained with the ROM. A further fundamental contribution is a discussion of discretize-then-optimize (DTO) vs. optimize-then-discretize (OTD) approaches in the context of MOR for optimization. We analyze advantages and disadvantages of both approaches and discuss to which extent our methods exhibit properties of either strategy. We also give examples of representative optimization problems in which standard POD/DEIM ROMs show an inacceptable behavior and can be successfully solved by derivative-extended ROMs. We have further implemented the developed methods emphasizing an efficient realization which is important for the investigation of the MOR potential. We showcase the practical performance of the proposed algorithms and the superiority of derivative-extended over conventional ROMs on two academic and one industry-relevant application which exhibit a variety of challenges for the model reduction approach in optimization.

Inverse viscoelastic material characterization using POD reduced-order modeling in acoustic–structure interaction

Using traditional high‐fidelity numerical simulation to simulate fluid flow in fractured porous media in a real field remains challenging. It involves a large number of degrees of freedom when matrix and fracture equations are solved. To address this challenge, we propose a Galerkin‐free framework to construct a reduced‐order model (ROM) based on the proper orthogonal decomposition (POD). Compared with the typical POD‐based modeling process commonly used in previous studies, the POD‐ROM can be built without performing the Galerkin projection of flow equations onto the low‐dimensional space spanned by the POD basis functions. The numerical integration method was incorporated to obtain the POD time coefficients based on the flow equations solved by the conventional finite volume method. Two complex fracture cases reflecting high‐contrast porous media in a two‐dimensional domain were designed to verify the accuracy and efficiency of the established Galerkin‐free POD‐ROM. Sensitivity analysis of parameters was conducted to examine the adaptability of the ROM. The results illustrate that, compared with the fine‐scale model, the ROM can significantly reduce the CPU time without compromising the quality of the numerical solutions.

A POD-based reduced-order model for free surface shallow water flows over real bathymetries for Monte-Carlo-type applications

A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier–Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier–Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

In aerospace engineering, various problems such as restoring impaired experimental flow data can be handled by gappy proper orthogonal decomposition (POD). Similarly to gappy POD, probabilistic principal component analysis (PPCA) can approximate missing data with the help of an expectation-maximization (EM) algorithm, yielding an EM algorithm applied to PPCA (EM-PCA). Although both gappy POD and EM-PCA address the same missing date estimation problem, their antithetical formulation perspectives hinder their direct comparison; the development of the former is deterministic whereas that of the latter is probabilistic. In order to effectively differentiate both methods, this research provides a unifying least squares perspective to qualitatively dissect them within a unified least squares framework. By virtue of the unifying least squares perspective, gappy POD and the EM-PCA turn out to be similar in that they are twofold: basis and least squares coefficient evaluations. On the other hand, they are dissimilar because the EM-PCA, unlike gappy POD, dispenses with either a gappy norm or a POD basis. To illustrate the theoretical analysis of both methods, numerical experiments using simple and complex data sets quantitatively examine their performance in terms of convergence rates and computational cost. Finally, comprehensive comparisons, including theoretical and numerical aspects, establish that the EM-PCA is simpler and thereby more efficient than gappy POD.

An approach has been developed to use the strain sensed in a membrane wing to estimate aerodynamic loads. Through experimental and theoretical correlation of membrane strain state due to a normally distributed load, the pressure over the surface of the wing was estimated. Elastic deformations and strains of the membrane wing were measured in a low-speed wind tunnel using digital image correlation. Simultaneously, aerodynamic loads were measured using a six-component load cell, inclinometer, and pitot tube. Digital image correlation displacement measurements were used to form a reduced-order model of the membrane displacements via a snapshot proper orthogonal decomposition method. Digital image correlation strain measurements were used to form virtual strain sensors, which served as the sensory input to the estimation. The Poisson equation for a two-dimensional linear-elastic membrane with out-of-plane deformation was used to calculate the normal pressure distribution from virtual strain sensors using proper orthogonal decomposition basis functions and a recursive least-squares minimization. Estimated pressure distributions were compared with a high-fidelity three-dimensional computational fluid dynamic model and pressures were calculated from digital image correlation deformations. Coefficients of lift and pitching moment for steady-state flow conditions were estimated and compared with measured wind-tunnel loads. Results show promise toward the application of a low-fidelity estimation approach for real-time load-estimation applications.

Numerical model reduction (NMR) is used to solve the microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. The reduction technique and an associated error estimator for the NMR error have been presented in prior work, where both spectral decomposition (SD) and proper orthogonal decomposition (POD) were used to construct the reduced basis. It was shown that the POD basis performs better w.r.t. minimizing the residual, but the SD basis has some advantageous properties for the estimator. Since it is the estimated error that will govern the error control, the most efficient procedure is the one that results in the lowest error bound. The main contribution of this paper is further development of the previous work with a proposed combined basis constructed using both SD and POD modes together with an adaptive mode selection strategy. The performance of the combined basis is compared to (i) the pure SD basis and (ii) the pure POD basis via numerical examples. The examples show that it is possible to find a combination of SD/POD modes which is improved, i.e. it yields a smaller estimate, compared to the cases of pure SD or pure POD.

This work proposes a new method for using reduced-order models in lieu of high-fidelity analysis during the sensitivity analysis step. Reduced-order models are developed using a combination of proper orthogonal decomposition and radial basis functions. Interpolation using reduced-order models based on proper orthogonal decomposition is compared against normal optimization on a general airfoil shape optimization problem. The interpolation procedure does not require additional high-fidelity evaluations to construct new reduced-order models. The errors associated with the reduced-order models themselves as well as the gradients calculated from them are compared. The effects of each approach on the overall optimization paths, times, and function counts are also examined.