Charbel Farhat

Kevin Carlberg, Charbel Farhat, Julien Cortial et al.

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.

Jad Mounayer, Sebastian Rodriguez, Chady Ghnatios et al.

The choice of an appropriate bottleneck dimension and the application of effective regularization are both essential for Autoencoders to learn meaningful representations from unlabeled data. In this paper, we introduce a new class of deterministic autoencoders, Rank Reduction Autoencoders (RRAEs), which regularize their latent spaces by employing a truncated singular value decomposition (SVD) during training. In RRAEs, the bottleneck is defined by the rank of the latent matrix, thereby alleviating the dependence of the encoder/decoder architecture on the bottleneck size. This approach enabled us to propose an adaptive algorithm (aRRAEs) that efficiently determines the optimal bottleneck size during training. We empirically demonstrate that both RRAEs and aRRAEs are stable, scalable, and reliable, as they do not introduce any additional training hyperparameters. We evaluate our proposed architecture on a synthetic data set, as well as on MNIST, Fashion MNIST, and CelebA. Our results show that RRAEs offer several advantages over Vanilla AEs with both large and small latent spaces, and outperform other regularizing AE architectures.

Sebastian Grimberg, Charbel Farhat, Radek Tezaur et al.

The energy-conserving sampling and weighting (ECSW) method is a hyperreduction method originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when the underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, this hyperreduction method is extended to Petrov-Galerkin PROMs where the underlying high-dimensional models can be associated with arbitrary finite element, finite volume, and finite difference semi-discretization methods. Its scope is also extended to cover local PROMs based on piecewise-affine approximation subspaces, such as those designed for mitigating the Kolmogorov $n$-width barrier issue associated with convection-dominated flow problems. The resulting ECSW method is shown in this paper to be robust and accurate. In particular, its offline phase is shown to be fast and parallelizable, and the potential of its online phase for large-scale applications of industrial relevance is demonstrated for turbulent flow problems with $O(10^7)$ and $O(10^8)$ degrees of freedom. For such problems, the online part of the ECSW method proposed in this paper for Petrov-Galerkin PROMs is shown to enable wall-clock time and CPU time speedup factors of several orders of magnitude while delivering exceptional accuracy.

Philip Avery, Daniel Z. Huang, Wanli He et al.

A general-purpose computational homogenization framework is proposed for the nonlinear dynamic analysis of membranes exhibiting complex microscale and/or mesoscale heterogeneity characterized by in-plane periodicity that cannot be effectively treated by a conventional method, such as woven fabrics. The framework is a generalization of the "finite element squared" (or FE2) method in which a localized portion of the periodic subscale structure is modeled using finite elements. The numerical solution of displacement driven problems involving this model can be adapted to the context of membranes by a variant of the Klinkel-Govindjee method[1] originally proposed for using finite strain, three-dimensional material models in beam and shell elements. This approach relies on numerical enforcement of the plane stress constraint and is enabled by the principle of frame invariance. Computational tractability is achieved by introducing a regression-based surrogate model informed by a physics-inspired training regimen in which FE$^2$ is utilized to simulate a variety of numerical experiments including uniaxial, biaxial and shear straining of a material coupon. Several alternative surrogate models are evaluated including an artificial neural network. The framework is demonstrated and validated for a realistic Mars landing application involving supersonic inflation of a parachute canopy made of woven fabric.

Sebastian Grimberg, Charbel Farhat, Noah Youkilis

In the literature on projection-based nonlinear model order reduction for fluid dynamics problems, it is often claimed that due to modal truncation, a projection-based reduced-order model (PROM) does not resolve the dissipative regime of the turbulent energy cascade and therefore is numerically unstable. Efforts at addressing this claim have ranged from attempting to model the effects of the truncated modes to enriching the classical subspace of approximation in order to account for the truncated phenomena. This paper challenges this claim. Exploring the relationship between projection-based model order reduction and semi-discretization and using numerical evidence from three relevant flow problems, it argues in an orderly manner that the real culprit behind most if not all reported numerical instabilities of PROMs for turbulence and convection-dominated turbulent flow problems is the Galerkin framework that has been used for constructing the PROMs. The paper also shows that alternatively, a Petrov-Galerkin framework can be used to construct numerically stable PROMs for convection-dominated laminar as well as turbulent flow problems that are numerically stable and accurate, without resorting to additional closure models or tailoring of the subspace of approximation. It also shows that such alternative PROMs deliver significant speedup factors.

Cristina White, Daniela Ushizima, Charbel Farhat

Incorporating computational fluid dynamics in the design process of jets, spacecraft, or gas turbine engines is often challenged by the required computational resources and simulation time, which depend on the chosen physics-based computational models and grid resolutions. An ongoing problem in the field is how to simulate these systems faster but with sufficient accuracy. While many approaches involve simplified models of the underlying physics, others are model-free and make predictions based only on existing simulation data. We present a novel model-free approach in which we reformulate the simulation problem to effectively increase the size of constrained pre-computed datasets and introduce a novel neural network architecture (called a cluster network) with an inductive bias well-suited to highly nonlinear computational fluid dynamics solutions. Compared to the state-of-the-art in model-based approximations, we show that our approach is nearly as accurate, an order of magnitude faster, and easier to apply. Furthermore, we show that our method outperforms other model-free approaches.

David Amsallem, Radek Tezaur, Charbel Farhat

A comprehensive approach for real-time computations using a database of parameterized linear reduced-order models (ROMs) is proposed. The method proceeds by sampling offline ROMs for specific values of the parameters and interpolating online the associated reduced operators. In the offline phase, a pre-processing step transforms the reduced operators into consistent sets of generalized coordinates prior to their interpolation. The present paper also introduces a consistency enforcement approach for models defined on arbitrary underlying meshes. In the online phase, the operators are interpolated on matrix manifolds. The proposed framework is illustrated on two realistic multi-physics problems: an inverse acoustic scattering problem around a submarine and flutter predictions for a wing-tank system. The second application is implemented on a mobile device, illustrating the capability of the proposed framework to operate in real-time.

David Amsallem, Daniel Neumann, Youngsoo Choi et al.

An iterative, CFD-based approach for aeroelastic computations in the frequency domain is presented. The method relies on a linearized formulation of the aeroelastic problem and a fixed-point iteration approach and enables the computation of the eigenproperties of each of the wet aeroelastic eigenmodes. Numerical experiments on the aeroelastic analysis and design optimization of two wing configurations illustrate the capability of the method for the fast and accurate aeroelastic analysis of aircraft configurations and its advantage over classical time-domain approaches.

Maciej Balajewicz, David Amsallem, Charbel Farhat

To be feasible for computationally intensive applications such as parametric studies, optimization and control design, large-scale finite element analysis requires model order reduction. This is particularly true in nonlinear settings that tend to dramatically increase computational complexity. Although significant progress has been achieved in the development of computational approaches for the reduction of nonlinear computational mechanics models, addressing the issue of contact remains a major hurdle. To this effect, this paper introduces a projection-based model reduction approach for both static and dynamic contact problems. It features the application of a non-negative matrix factorization scheme to the construction of a positive reduced-order basis for the contact forces, and a greedy sampling algorithm coupled with an error indicator for achieving robustness with respect to model parameter variations. The proposed approach is successfully demonstrated for the reduction of several two-dimensional, simple, but representative contact and self contact computational models.