Data-driven reduced-order modeling for nonlinear aerodynamics using an autoencoder neural network

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.

The Wright brothers were credited with having flown the first manned, heavier-than-air, powered aircraft in December 1903. During the first few years of the century, however, the U.S. was relatively slow in accepting the newly introduced airplane. Pre-World War I advances in aviation in Europe created some concern that the U.S. was lagging behind other nations in the development of aircraft. This lead to the establishment in the U.S. in 1915 of a government-funded organization that was charged to direct and conduct research and experiments in aeronautics. The initial organization was the National Advisory Committee for Aeronautics (NACA). The organization was renamed the National Aeronautics and Space Administration (NASA) in 1958 in order to include space research. The research in aeronautics has been continuous since the early 1920’s and has resulted in many advances in aviation. The purpose of this paper is to relate some of the aerodynamic research conducted at NACA/NASA–Langley that has contributed to the growth of aviation during the first century of flight.

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.

The proper orthogonal decomposition (POD) is an efficient model order reduction method, which is frequently coupled with the discrete empirical interpolation method (DEIM) to solve nonlinear electromagnetic problems. A drawback of this method is that instabilities can occur related to the reduction operator of the nonlinear part. In this contribution, different DEIMs and the Gappy POD are employed and analyzed. Consecutively, the methods are employed to efficiently estimate the behavior of a permanent magnet synchronous machine in terms of global quantities, such as torque and iron losses.

From its creation, the U.S. Air Force (USAF) had supported advanced aircraft development, typically in conjunction with or for the National Advisory Committee for Aeronautics (NACA) and then the National Aeronautics and Space Administration (NASA). This support, which included personnel, facilities, and development expertise, started before the X-1 program and has continued to the present. The USAF assisted Litton’s 1950s’ development of the Mark I pressure suit in the form of funding and test personnel. In the X-15 program, the USAF provided pressure suit selection and development to NACA.

During lamination, residual thermo-mechanical stresses are induced in the encapsulated solar cells composing photovoltaic (PV) modules. Depending on the material and geometrical configuration of the layers of the laminate, this residual stress field can be beneficial since it may lead to a compressive stress state in Silicon and therefore crack closure effects in the presence of cracks, with a recovery of electrical conductivity in cracked solar cells. It is therefore important to investigate the distribution of thermo-mechanical stresses within the PV laminate with a view to optimizing the coupling between the electrical response and elastic deformation in the operation of PV modules. A promising approach proposed in the present thesis regards the prediction of residual stresses in composite laminates by using a shear-lag theory to model the epoxy-vinil-acetate polymeric layers, accounting for their thermo-visco-elastic response. Moreover, it will be shown that thermomechanical formulations for stress analysis of a PV laminate lead to a system of higher order ordinary differential equations or partial differential equations in which the exact solutions may be impossible to be determined in closed form and hence numerical schemes become desirable. However, the computational cost associated with the implementation of the numerical scheme may be significantly expensive. Therefore, a method to reduce the computational complexity is expected to be very important. To this aim, Model Order Reduction (MOR) techniques are applied hierarchically, first to the thermal system of a PV module in service, and then extended to coupled thermo-mechanical problems. A combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) with a modified formulation is proposed for the first-order thermal equations of photovoltaic system during service and a new coupled second-order Krylov based formulation is developed for model order reduction of the coupled thermo-mechanical model of the photovoltaic module. The results of these reduction schemes show a huge computational gain in the reduced system solutions and a high accuracy of the reduced system outputs

This study aims to develop a linear nonintrusive reduced-order model (NIROM) for facilitating combined fast parametric analysis and reconstruction of unsteady aerodynamic flow variables by using a novel signal interpolation approach, which is combined with a subspace-identification-based method. The NIROM is then applied to airfoil flutter and gust interaction problems where the assumption of linearized dynamics gives reasonably accurate estimates. In the traditional subspace-identification method, the training datasets of force coefficient signals corresponding to each flow parameter (like the Mach number and angle of attack) are required to generate the linear time-invariant system. The generation of the training signal for each flow parameter is the most computationally expensive process in any parametric reduced-order model (ROM). In the present work, the discrete empirical interpolation method (DEIM) combined with the radial basis function (RBF) method is used for the interpolation of training signals at a specific flow parameter to reduce the overall computational cost. The conventional nonintrusive approaches are generally constrained by the number of output channels, and hence not suitable for the reconstruction of unsteady surface force distribution or flowfield variables. A few suitable sensor locations using the DEIM are selected as output channels of the subspace-identification method to realize the full datasets of surface force distribution over the airfoil. Finally, a suitable algorithmic structure is proposed to achieve unsteady distributive flow-variable reconstruction and parametric variation simultaneously. The current signal interpolation-based NIROM is applied to a wing section with two- and four-degree-of-freedom structural motions.

Nonlinear Model Order Reduction for Catalytic Tubular Reactors

The multiphase fluid flow in porous media is one of the most fundamental phenomena in various physical processes, such as oil/gas flow in reservoir, subsurface contamination dispersion, chemical separation, etc. Due to its importance, the efficient and accurate solution and prediction of multiphase flow in porous media is highly required in engineering applications and mechanism studies, which has been a research hot spot with increasing interest in recent years. However, the strong nonlinearity implicated in the multiphase flow model has brought great challenges for the computation and analysis. In addition, the permeability in Darcy-type pressure equation is always represented as a high contrast coefficient, which further complicates the simulation difficulty especially for high fidelity prediction and decision-making process. In this study, we attempt to apply one popular global model reduction method, namely Proper Orthogonal Decomposition (POD), to accelerate the simulation of incompressible multiphase flow in porous media. The cornerstone of POD is processing information from a sequence of snapshots (or instantaneous solutions) to identify a set of basis functions to construct a low-dimensional space, the most energetic part of these basis functions is then selected to represent the solution of the original system. In our work, one key point is how to establish the accurate POD reduced order model (ROM) for incompressible mul-tiphase porous flow problems. According to Li et al. [1], the traditional Galerkin projection method (directly project the original governing equation into the low-dimensional space spanned by basis functions) is incapable of introducing the fluid saturation and permeability located on the cell face in POD-ROM for incompressible multiphase flow whether in homogeneous or heterogeneous porous media, thus we adopt the matrix operation method (project the discrete governing equation instead of the original governing equation into the low-dimensional space spanned by basis functions) to build the POD-ROM in this study. Based on the proposed POD-ROM framework, we use the implicit method to solve the (non-) wetting-phase pressure and saturation equations. However, in the implicit solution of the pressure and saturation equations, the standard POD-ROM alone sometimes cannot yield an expected acceleration owing to two main reasons: (1) the nonlinearity between pressure and saturation is sensitive, thus the saturation would change largely even with slight pressure change. And the small error of pressure results would lead to a large deviation of saturation, which in turn exerts negative impacts on the computation of pressure; (2) the complexity for computing a projected nonlinear term in POD-ROM still depends on the dimension of the original full-order system, it is unavoidable in conventional POD-ROM. Therefore, more POD modes are needed to be used in order to achieve the prescribed numerical accuracy, which could worsen the speed up efficiency of proposed POD-ROM. To further reduce the computational complexity of POD-ROM to speed up the implicit solution of pressure and saturation equations, another model reduction method called Discrete Empirical Interpolation Method (DEIM) [2] is applied to treat the projected nonlinear term in the proposed incompressible multiphase porous flow POD-ROM. Different from previous studies, we develop an adaptive DEIM (ADEIM) in this work to achieve a more flexible selection of the interpolation points. Combined the POD-ROM with ADEIM, a POD-ADEIM-ROM for incompressible multiphase flow in porous media is established. The computational efficiency and numerical accuracy of the proposed model is validated through several representative numerical examples. Numerical results indicate that an outstanding acceleration (2~3 orders of magnitude faster) without sacrificing numerical accuracy significantly is obtained from the proposed model when comparing to the traditional solution method that without any acceleration technique. In addition, the effects of POD mode number, the distribution of permeability field, and the mesh density on the overall performances of the proposed model are investigated in detail.

From the Editor’s Desk Steven N. Handel (bio) As I am writing this letter, the United States has just begun its celebration of Black History Month, the aim of which is to highlight the “the too-often neglected accomplishments of black Americans in every area of endeavor throughout our history.” In previous years, the history of computing community has not had much to contribute to this celebration. Until recently, our discipline has paid little attention to questions of race and ethnicity. This year, I am happy to say, things are different. The triple Oscar-nominated film Hidden Figures, based on the book of the same name by Margot Shetterly, has brought welcome and widespread attention to the stories of Katherine Johnson, Dorothy Vaughn, Mary Jackson, and the many other African American women who worked as human computers and computer programmers at the National Advisory Committee for Aeronautics (NACA) and the National Aeronautics and Space Administration (NASA). The contributions of these women were essential to both the Space Race and the development of the computing disciplines, and have been shamefully neglected. But as the article by Arvid Nelsen on “Race & Computing: the Problem of Sources, the Potential of Prosopography, and the Lesson of Ebony Magazine” so clearly reveals, the story of these NASA computers is only the tip of the iceberg in terms of African-American contributions to computing. In his close study of both archival and popular sources Nelsen demonstrates that, despite their almost complete invisibility in most of the history of computing literature, people of color were omnipresent in the actual history of computing. In the years between 1959 and 1996, for example, Ebony Magazine alone published 57 profiles of African Americans working in the electronics and computer industry in a broad range of occupations from computer operator to systems analyst to manager of sales for time sharing computing at General Electric. For each of these named individuals, there are certainly many, many more whose mere identities, much less their larger histories, remain to be established. To give just one example, we know almost nothing about the two African American men featured on the cover of this issue of the Annals, other than that they worked (most likely as engineers) at the Burroughs Corporation in the early 1960s. Their anonymity is a poignant reminder of both their vital presence in this history and the enormous amount of work that remains to be done. And as Nelsen’s article reveals, there are formidable barriers—practical and methodological as well as theoretical and conceptual—associated with doing this research. Nelsen, who for many years served as the archivist of the Charles Babbage Institute for the History of Computing, is a skilled researcher with intimate familiarity with the available primary sources, but even for him the work was time-consuming and painstaking. But if our discipline is to come to terms with the critical category of race—beginning with the hidden histories of people of color, but including as well the role of “whiteness” in shaping the dominant narratives in much of our histories, then we are going to need to do pursue the difficult path that Nelsen has described for us. Think of the similar challenges that historians of computing faced while contemplating the incorporation of gender as an analytical category. This too was an effort that began slowly and took decades to realize, but which has culminated in a dramatic and productive reshaping of our very conception of what are the fundamental questions in the history of computing. Consider, for example, the recently published research by Annals associate editor Marie Hicks, which demonstrates the central role of gender in the promising origins of but ultimately disappointing ending to the British computer industry in the mid-20th century.1 For Hicks, gender is central to her analysis, not a secondary curiosity. Similarly, we can expect that as we begin to incorporate race and ethnicity into our scholarship, we will discover new insights, methods, and perspectives that will radically reshape the focus of our discipline. In his article “MCM on Personal Software,” Zbigniew Stachniak argues that the origins of the personal computer industry ought to be pushed back before 1975 (the...

Reduced-order modeling techniques enable a remarkable speed up in the solution of the parametrized electromechanical model for heart dynamics. Being able to rapidly approximate the solution of this problem allows to investigate the impact of significant model parameters querying the parameter-to-solution map in a very inexpensive way. The construction of reduced-order approximations for cardiac electromechanics faces several challenges from both modeling and computational viewpoints, because of the multiscale nature of the problem, the need of coupling different physics, and the nonlinearities involved. Our approach relies on the reduced basis method for parametrized PDEs. This technique performs a Galerkin projection onto low-dimensional spaces built from a set of snapshots of the high-fidelity problem by the Proper Orthogonal Decomposition technique. Snapshots are obtained for different values of the parameters and computed, e.g., by the finite element method. Then, suitable hyper-reduction techniques, in particular the Discrete Empirical Interpolation Method and its matrix version, are called into play to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of both the electrical and the mechanical model can be achieved by developing two separate reduced order models where the interaction of the cardiac electrophysiology system with the contractile muscle tissue, as well as the sub-cellular activation-contraction mechanism, are included. Open challenges and possible perspectives are finally outlined.

In this work, a model-order reduction methodology based on proper orthogonal decomposition (POD) and Galërkin projection is presented and applied to the simulation of the self-ignition of a stockpile of solid fuel. Self-ignition is a phenomenon associated with steep changes in space and time, yielding high gradients of state variables which demand grid refinement and, thus, increase of the computational burden. To cope with this difficulty, first, a full order model (FOM), generated by finite-difference discretisation of the PDEs constituting the differential model, is employed to generate reference solutions. Two different POD-based formulations are proposed: the classical POD-Galërkin is employed to generate reduced order models (ROM), then discrete empirical interpolation method (DEIM) is employed to deal with nonlinearities in a more efficient manner. These reduction techniques are further supplemented with an innovative sampling approach based on k-means clustering. The resulting agile ROM is validated against the FOM. Both model-order reduction strategies, particularly the POD-DEIM model, reproduce the FOM solutions with high accuracy and much lower computational cost: The results of the application of a combination of the DEIM algorithm and k-means clustering show that the computational time for the calculation of one solution reduces up to 1020 times, while remaining able to reproduce all bifurcation points found with the FOM, thus demonstrating quantitative and qualitative agreement.

Preface

The objective of this paper is to reduce the computational effort in reservoir flooding optimization problems by a combination of different optimization parametrization methods and model order reduction techniques. We compare three different parametrization methods that reduce the cardinality of the original infinite set of control-decision variables to a finite set. The three methods include a traditional piece-wise constant (PWC) approximation, a polynomial approximation by Chebyshev orthogonal polynomials and a piece-wise polynomial approximation by cubic Spline interpolation. The Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD-DEIM) accomplishes the reduced order modeling (ROM).. We compare a gradient-free global stochastic search approach and a gradient-based local search approach. We used Particle Swarm Optimization (PSO) as a gradient-free algorithm and Interior-Point Optimization (IPOPT) with L-BFGS method as a gradient-based algorithm. First, we compare the performances of the three parametrization methods solved by each optimizer, using fine scale simulations for an increasing level of parametrization refinement. Then, in the second part of this paper, we combine the parametrization methods with the reduced modeling workflow. For a given level of parametrization refinement, we compare the performance of each parametrization method coupled with POD-DEIM, and solved by each optimizer. In this part, we introduce an online training procedure, where the first optimization iteration is used to construct the snapshot matrix. The results demonstrate how refining the control approximation with more decision variables per well lead to better NPV values, but with a higher computational cost. The best NPV was achieved using the highest refining level with Chebyshev polynomial approximation. Both polynomial and piece-wise polynomial approximations served as better training sets for POD-DEIM leading to a more accurate and fast reduced model. With the strategy proposed, POD-DEIM showed the best optimization accuracy for Chebyshev polynomial with the gradient-free optimizer, thus permitting the use of the model reduction methodology for global-stochastic search methods. However, the gradient-based approach seems to consistently outperform the gradient-free approach in terms of NPV and number of iterations for the cases shown.

Model order reduction applied to heat conduction in photovoltaic modules