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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

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.

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.

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.