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A comprehensive approach for real-time computations using a database of parametric, linear, projection-based reduced-order models (ROMs) based on arbitrary underlying meshes is proposed. In the offline phase of this approach, the parameter space is sampled and linear ROMs defined by linear reduced operators are pre-computed at the sampled parameter points and stored. Then, these operators and associated ROMs are transformed into counterparts that satisfy a certain notion of consistency. In the online phase of this approach, a linear ROM is constructed in real-time at a queried but unsampled parameter point by interpolating the pre-computed linear reduced operators on matrix manifolds and therefore computing an interpolated linear ROM. The proposed overall model reduction framework is illustrated with two applications: a parametric inverse acoustic scattering problem associated with a mockup submarine, and a parametric flutter prediction problem associated with a wing-tank system. The second application is implemented on a mobile device, illustrating the capability of the proposed computational framework to operate in real-time.

Large scale finite element analysis requires model order reduction for computationally expensive applications such as optimization, parametric studies and control design. Although model reduction for nonlinear problems is an active area of research, a major hurdle is modeling and approximating contact problems. This manuscript introduces a projection-based model reduction approach for static and dynamic contact problems. In this approach, non-negative matrix factorization is utilized to optimally compress and strongly enforce positivity of contact forces in training simulation snapshots. Moreover, a greedy algorithm coupled with an error indicator is developed to efficiently construct parametrically robust low-order models. The proposed approach is successfully demonstrated for the model reduction of several two-dimensional elliptic and hyperbolic obstacle and self contact problems. ∗Corresponding author Email address: maciej.balajewicz@stanford.edu (Maciej Balajewicz) 1Postdoctoral Fellow 2Engineering Research Associate 3Vivian Church Hoff Professor of Aircraft Structures

Solving large-scale PDE-constrained optimization problems presents computational challenges due to the large dimensional set of underlying equations that have to be handled by the optimizer. Recently, projection-based nonlinear reduced-order models have been proposed to be used in place of high-dimensional models in a design optimization procedure. The dimensionality of the solution space is reduced using a reduced-order basis constructed by Proper Orthogonal Decomposition. In the case of nonlinear equations, however, this is not sufficient to ensure that the cost associated with the optimization procedure does not scale with the high dimension. To achieve that goal, an additional reduction step, hyper-reduction is applied. Then, solving the resulting reduced set of equations only requires a reduced dimensional domain and large speedups can be achieved. In the case of design optimization, it is shown in this paper that an additional approximation of the objective function is required. This is achieved by the construction of a surrogate objective using radial basis functions. The proposed method is illustrated with two applications: the shape optimization of a simplified nozzle inlet model and the design optimization of a chemical reaction.

Summary Reduced-order models for linear time-invariant dynamical systems are considered, and the error between the full-order model and the reduced-order model solutions is characterized. Based on the analytical representation of the error, an a posteriori error indicator is proposed that combines a Krylov-based exponential integrator and an a posteriori residual-based estimate. Numerical experiments illustrate the quality of the error estimator. Copyright © 2014 John Wiley & Sons, Ltd.

Galerkin reduced-order models for the semi-discrete wave equation, that preserve the second-order structure, are studied. Error bounds for the full state variables are derived in the continuous setting (when the whole trajectory is known) and in the discrete setting when the Newmark average-acceleration scheme is used on the second-order semi-discrete equation. When the approximating subspace is constructed using the proper orthogonal decomposition, the error estimates are proportional to the sums of the neglected singular values. Numerical experiments illustrate the theoretical results.

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

International audience; In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element (RVE). We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model (ROM) can be built.

SUMMARY A new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced-order bases is presented. It is particularly suited for problems characterized by different physical regimes, parameter variations, or moving features such as discontinuities and fronts. Instead of approximating the solution of interest in a fixed lower-dimensional subspace of global basis vectors, the proposed model order reduction method approximates this solution in a lower-dimensional subspace generated by most appropriate local basis vectors. To this effect, the solution space is partitioned into subregions, and a local reduced-order basis is constructed and assigned to each subregion offline. During the incremental solution online of the reduced problem, a local basis is chosen according to the subregion of the solution space where the current high-dimensional solution lies. This is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower-dimensional solution space. Because it is also applicable to the process of hyper reduction, the proposed method for nonlinear model order reduction is computationally efficient. Its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid-structure-electric interaction problems. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY A rigorous method for stabilizing projection-based linear reduced-order models without significantly affecting their accuracy is proposed. Unlike alternative approaches, this method is computationally efficient. It requires primarily the solution of a small-scale convex optimization problem. Furthermore, it is nonintrusive in the sense that it operates directly on readily available reduced-order operators. These can be precomputed using any data compression technique including balanced truncation, balanced proper orthogonal decomposition, proper orthogonal decomposition, or moment matching. The proposed method is illustrated with three applications: the stabilization of the reduction of the Computational Fluid Dynamics-based model of a linearized unsteady supersonic flow, the reduction of a Computational Structural Dynamics system, and the stabilization of the reduction of a coupled Computational Fluid Dynamics–Computational Structural Dynamics model of a linearized aeroelastic system in the transonic flow regime. Copyright © 2012 John Wiley & Sons, Ltd.

This paper describes a computational-fluid-dynamics-based computational methodology for fast on-demand aeroelastic predictions of the behavior of a full aircraft configuration at variable flight conditions and demonstrates its feasibility. The methodology relies on the offline precomputation of a database of reduced-order bases and models associated with a discrete set of flight parameters, and its training for an interpolation method suitable for reduced-order information. The potential of this near-real-time computational methodology for assisting flutter flight testing is highlighted with the aeroelastic identification of an F-16 configuration in the subsonic, transonic, and supersonic regimes.

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to an online real-time implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive-definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are offered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real-time processing speeds. Copyright © 2009 John Wiley & Sons, Ltd.

Reduced-order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real-time analysis. Although most reduced-order models can operate in near real-time, their construction can be computationally expensive, as it requires accumulating a large number of system responses to input excitations. Furthermore, reduced-order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation. Together, these two issues underline the need for a fast and robust method for adapting precomputed reduced-order models to new sets of physical or modeling parameters. To this effect, this paper presents an interpolation method based on the Grassmann manifold and its tangent space at a point that is applicable to structural, aerodynamic, aeroelastic, and many other reduced-order models based on projection schemes. This method is illustrated here with the adaptation of computational-fluid-dynamics-based aeroelastic reduced-order models of complete fighter configurations to new values of the freestream Mach number. Good correlations with results obtained from direct reduced-order model reconstruction, high-fidelity nonlinear and linear simulations are reported, thereby highlighting the potential of the proposed reduced-order model adaptation method for near real-time aeroelastic predictions using precomputed reduced-order model databases.

We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online $$L^1$$ L 1 - norm minimization of the residual. It is shown why this is a natural framework for hyperbolic problems and tested on nonlinear problems such as Burgers’ equation and the one-dimensional Euler equations involving shocks and discontinuities. Efficient algorithms are presented for the computation of the $$L^1$$ L 1 -norm minimizer, both in the cases of linear and nonlinear residuals. Results indicate that the method has the potential of being accurate when involving only very few modes, generating physically acceptable, oscillation-free, solutions.

The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reduced-order basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups.

A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints.

The Galerkin projection method based on modes generated by the Proper Orthogonal Decomposition (POD) technique is very popular for the dimensional reduction of linearized Computational Fluid Dynamics (CFD) models, among many other typically high-dimensional models in computational engineering. This, despite the fact that it cannot guarantee neither the optimality nor the stability of the Reduced- Order Models (ROMs) it constructs. Short of proposing any variant of this model order reduction method that guarantees the stability of its outcome, this paper contributes a best practice to its application to the construction of linearized CFD ROMs. It begins by establishing that whereas the solution snapshots computed using the descriptor and non-descriptor forms of the discretized Euler or Navier-Stokes equations are identical, the ROMs obtained by reducing these two alternative forms of the governing equations of interest are different. Focusing next on compressible fluid-structure interaction problems associated with computational aeroelasticity, this paper shows numerically that the POD-based fluid ROMs originating from the non-descriptor form of the governing linearized CFD equations tend to be unstable, but their counterparts originating from the descriptor form of these equations are typically stable and reliable for aeroelastic applications. Therefore, this paper argues that whereas many computations are performed in CFD codes using the non-descriptor form of discretized Euler and/or Navier-Stokes equations, POD-based model reduction in these codes should be performed using the descriptor form of these equations.

A real-time approach for model predictive control of an aircraft system is proposed. CFD-based predictions cannot, in general, be used in such a setting as they incur intensive computations. To address this issue, the approach proposed in this paper relies on a database of CFD-based parameterized reduced-order models. More specifically, this database is used for fast computations of the aeroelastic behavior of the system at a variety of operating conditions. Fluid-structure constraints can then be included in the optimal control procedure. The framework is applied to the minimum-time path computation of an aircraft system by dynamic programming under fuel consumption and aeroelastic constraints. The implementation of the proposed procedure on an Android tablet highlights the amenability of the method to mobile computing and predictions on the field.

A method for simultaneously constructing a reduced-order model and using it as a surrogate model to solve a PDE-constrained optimization problem is introduced. A reducedorder model is built for the parameters corresponding to the initial guess of the optimization problem. Since the resulting reduced-order model can be expected to be accurate only in the vicinity of this point in the parameter space, emphasis is placed on constructing this model by searching for regions of high error. These are determined by solving a small, nonlinear program with the objective defined as a linear combination of a residual error indicator and the objective function of the original PDE-constrained optimization problem. The reduced-order model is updated with information from the high-dimensional model in the regions of large error, and the process is repeated with more emphasis placed on solving the PDE-constrained optimization problem. The iteration terminates when the optimality conditions of the surrogate PDE-constrained optimization problem are satisfied. Application to a standard, nonlinear CFD shape optimization problem shows that the proposed method effectively solves a PDE-constrained optimization problem with few full CFD simulation queries.

The sloshing effects of an internal fluid on the flutter envelope of an aeroelastic system have received little attention in the open literature. This issue is nevertheless relevant for many aircraft, especially high-performance fighter jets carrying stores. This paper addresses some aspects of this problem as well as related modeling and analysis issues. These include the importance or insignificance of accounting for the hydroelastic effect when modeling an internal fluid and its container as well as accounting for that container when modeling the aerodynamics of the overall aeroelastic system. The paper also reports on the findings of four independent sets of flutter analyses performed for a wing–store test configuration and various fuel fill levels in the subsonic, transonic, and early supersonic regimes. Two of these sets of numerica l experiments relied on a computational-fluid-dynamics-based computational technology, and two of them on the doublet-lattice method or a supersonic lifting-surface theo...

A model reduction framework based on the concept of local reduced-order bases is presented. The offline phase of the method builds the local reduced-order bases using an unsupervised learning algorithm. In the online phase of the method, the choice of the local basis is based on the current state of the system. Inexpensive rank-one updates to the local bases are performed during the online phase for increased accuracy. Applications to nonlinear CFD simulations show that the method is effective in producing small and accurate reduced order models.

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart.

A rigorous method for stabilizing linear projection-based reduced-order models without signicantly aecting their accuracy is proposed. Unlike alternative approaches, this method is computationally ecient as it requires primarily the solution of a small-scale convex optimization problem. Furthermore, it is non-intrusive in the sense that it operates directly on readily available reduced-order operators. These can be precomputed using any technique such as Balanced Truncation, Balanced Proper Orthogonal Decomposition, Proper Orthogonal Decomposition, Moment Matching, or other. The proposed method is illustrated with two applications: The reduction with a stability guarantee of the CFDbased model of a linearized unsteady supersonic ow, and that of a coupled CFD-CSD model of a linearized aeroelastic wing-store system in the transonic ow regime.

nite-volume-based uid dynamics models at low computational cost. To accomplish this objective, this work employs the Gauss{ Newton with approximated tensors (GNAT) nonlinear model reduction method originally presented in Ref. 1. This technique decreases the system dimension by a least-squares Petrov{Galerkin projection, and decreases computational complexity by approximating the residual and Jacobian using the \Gappy POD" method; the latter requires computing only a few rows of the approximated quantities. This work introduces an ecient implementation of the GNAT method based on a novel \sample mesh" concept tailored for the nite volume formulation. When the reduced-order model is evaluated, this approach loads into memory only the subset of the mesh needed to sample the residual and Jacobian. This minimizes required computational resources, communication overhead, and computational complexity. A post-processing step that employs only the subset of the mesh needed for computing outputs is also proposed. Results obtained for a one-dimensional shock propagation problem highlight the method’s capability to decrease solution times by orders of magnitude while retaining high levels of accuracy, even in predictive scenarios. The application of GNAT to a large-scale, compressible, turbulent ow problem with over 17 million unknowns illustrates the method’s favorable performance compared with other nonlinear model reduction techniques (including collocation and discrete empirical interpolation approaches), and speedups exceeding 350 with errors less than 1% are observed. Finally, results show that the sample mesh enables the GNAT model to use many fewer processors compared with the full-order simulation.

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to a real-time and on-line implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are oered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real-time processing speeds.

This paper demonstrates the feasibility of a CFD-based computational strategy aimed at on-demand predictions of aeroelastic responses of full aircraft configurations for variable flight conditions. The strategy relies on the pre-computation of a database of reduced-order bases and models for discrete flight parameters, and an interpolation method suitable for adapting in real-time the stored reduced-order information to parameter values not populated in the database. It also features a database training and reduction scheme based on concepts from machine learning to maximize both the robustness and performance of local interpolations. The application of this computational strategy to the broad aeroelastic identification of a complete F-16 fighter configuration highlights its near-real-time processing capability and demonstrates its potential for assisting flutter flight testing.

Reduced-order models (ROMs) are usually thought of as computationally inexpensive mathematical representations that oer the potential for near real-time analysis. Indeed, most ROMs can operate in near real-time. However, their construction can be computationally intensive as it requires accumulating a large number of system responses to input excitations. Furthermore, ROMs usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation. Together, these two issues underline the need for a fast and robust method for adapting pre-computed ROMs to new sets of physical or modeling parameters. To this eect, this paper reports on recent advances in this topic. In particular, it describes a recently developed interpolation method based on the Grassmann manifold and its tangent space at a point that is applicable to structural, aerodynamic, aeroelastic and many other ROMs based on projection schemes. This method is illustrated here with the adaptation of CFD-based aeroelastic ROMs of complete fighter configurations to new values of the free-stream Mach number. Good correlations with results obtained from direct ROM reconstruction and high-fidelity linear and nonlinear simulations are reported, thereby highlighting the potential of the described ROM adaptation method for near real-time aeroelastic predictions using pre-computed ROM databases.

A fast and robust method for constructing aeroelastic reduced-order models (ROMs) at new Mach numbers and angles of attack from a discrete set of aeroelastic ROMs available at dierent Mach numbers and angles of attack is presented. This method is based on interpolation on a tangent space to a Grassman manifold. Its accuracy is assessed with the parameteric aeroelastic identification of two F-16 and F-18/A configurations in subsonic, transonic, and supersonic air streams. In particular, the aeroelastic frequencies and damping coecients predicted by this new ROM interpolation method are compared to counterpart results obtained from full-order nonlinear aeroelastic simulations, alternative ROM adaptation methods, and flight test data. Good correlations are observed, including in the transonic regime. Concluding remarks about the potential of the proposed ROM interpolation method for assisting the flutter test and store clearance of fighter aircraft are also oered.

Projection-based model order reduction (MOR) using local subspaces is becoming an increasingly important topic in the context of the fast simulation of complex nonlinear models. Most approaches rely on multiple local spaces constructed using parameter, time or state-space partitioning. State-space partitioning is usually based on Euclidean distances. This work highlights the fact that the Euclidean distance is suboptimal and that local MOR procedures can be improved by the use of a metric directly related to the projections underlying the reduction. More specifically, scale-invariances of the underlying model can be captured by the use of a true projection error as a dissimilarity criterion instead of the Euclidean distance. The capability of the proposed approach to construct local and compact reduced subspaces is illustrated by approximation experiments of several data sets and by the model reduction of two nonlinear systems.