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

ABSTRACTIn this work, with elasticity governed by the Giesekus constitutive equation, a proper orthogonal decomposition (POD) reduced-order model of viscoelastic turbulent channel flow is established for the first time. The established reduced-order model is based on small sets of basis functions from the POD of the sampling data obtained by direct numerical simulation (DNS) for the studied flow. The POD reduced-order model is tested on cases with that are different from the samplings for viscoelastic turbulent channel flow. The results show that the errors for root-mean-square (rms) velocity fluctuations are significant at the top and bottom walls. It is found that each basis function plays an important role in describing the studied turbulence which makes it unmanageable to obtain accurate velocity field (including mean velocity and velocity fluctuations) through solving the reduced-order model. It is of necessity to take all the basis functions into consideration to depict the flows more accurately. However, the mean velocity obtained from the reduced-order model is of high precision, which states that the POD-based reduced-order model is a potential approach to obtain an accurate mean velocity field for viscoelastic turbulent flow, which has great significance in academic study as well as engineering. The calculation speed of the established reduced-order model is much faster than that of DNS, which indicates that the POD is a highly efficient way of obtaining the statistic characteristics, such as mean velocity in turbulent channel flow.

Fast calculation of latent heat storage process in the direct steam generation solar thermal power system using a POD reduced-order model

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

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.

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.

The proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) are applied to coupled Burgers equations to develop its reduced-order model (ROM) by the Galerkin projection. A calibrated POD ROM is developed in the current study through adding and multiplying a set of time-dependent random parameters to recover the loss of accuracy due to the truncation of the POD modes. Calibrating the ROM becomes essentially a high-dimensional statistical inverse inference problem. To reduce the computational effort, the polynomial chaos based ensemble Kalman filter (PC-EnKF) is adopted in this work. By using a sparse optimization algorithm, a sparse PC expansion is obtained to facilitate further calculation of statistical moments used in ensemble Kalman filter. We apply the well-defined calibrated POD ROM for the coupled Burgers equations with the Reynolds numberRe= 10 000. The numerical results show that the PC-EnKF method is efficient in reducing the uncertainty included in the initial guess of input parameters and feasible in correcting the behavior of the POD ROM. The study suggests that the PC-EnKF is quite general as a calibration tool for calibrating the POD ROM.

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

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.

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

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

A data-driven online prediction method for surface-deformed liquid level in vessels under ocean conditions

建立二维Burgers方程基于特征投影分解(POD)方法的时间二阶精度的Crank-Nicolson (CN)有限元降维外推算法，给出这种算法的误差估计，并用误差估计作为算法的POD基数目选取及POD更新的准则。最后用数值实验说明该算法的优越性，这表明了该算法对于求解二维Burgers方程的数值解是有效可行的。 A Crank-Nicolson (CN) finite element reduced-order extrapolating algorithm with second-order accuracy based on proper orthogonal decomposition (POD) technique is established for two-dimensional Burgers equation, its error estimates are provided for criterions of the CN finite element reduced-order extrapolating algorithm to choose the number of POD basis and to renew POD basis. Some numerical experiments are used to show that the advantage of the CN finite element reduced-order extrapolating algorithm. It is shown that the CN finite element reduced-order extrapolating algorithm based on POD technique is feasible and efficient for finding the numerical solutions for two-dimensional Burgers equation.

A time-domain POD approach based on numerical implicit and explicit Green’s functions for 3D elastodynamic analysis

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.

An object-oriented framework for reduced-order models using proper orthogonal decomposition (POD)