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Recent advances in nonlinear model reduction for design and associated uncertainty quantification

Abstract : This lecture will be organized in two complementary parts focused on advances in the construction of parametric, nonlinear, projection-based, reduced-order models that are suitable for design and design optimization, and a corresponding nonparametric probabilistic method for quantifying their uncertainties. In the first arena, this talk will present a scalable approach for nonlinear model order reduction that is based on: a clustering scheme for precomputed parametric snapshots that promotes simultaneously locality on the solution manifold, and low dimensionality; a stable, Petrov-Galerkin projection method based on local reduced-order bases constructed by compressing the clustered snapshots [1, 2, 3]; a family of hyper reduction methods tailored to the mathematical underpinnings of both of the constructed reduced-order models and their underlying high-dimensional counterparts [4] ; and an online solution strategy that blends appropriate interpolation schemes and Newton-like methods for delivering real-time performance [6]. Most importantly, this part of the lecture will present for the first time the results of two realistic case studies on the performance of this framework for parametric nonlinear model reduction for “what-if?” aerodynamic design scenarios. The first case study focuses on NASA’s Common Research Model (CRM), which is representative of modern jet airliners. The second one focuses on the 2009 Volkswagen Passat, a midsize car. It will also contrast two approaches for training nonlinear reduced-order models and discuss their strengths and weaknesses for practical computations: the celebrated greedy method (for example, see [5]), and simple uniform sampling (for example, see [6]). In the second arena, this lecture will present a nonparametric probabilistic method for modeling various uncertainties associated with nonlinear reduced-order models, including model form uncertainties [7]. When experimental data is available, this approach can also quantify uncertainties in the underlying high-dimensional parametric model. The main idea here is two-fold: to substitute the deterministic reduced-order basis with a stochastic counterpart; and to construct the probability measure of the stochastic reduced-order basis on a subset of a compact Stiefel manifold in order to preserve some fundamental properties of a reduced-order basis. The potential of this nonparametric probabilistic method for uncertainty quantification in general, and updating nonlinear reduced-order models in particular, will be demonstrated through several sample computational mechanics problems. Finally, the lecture will conclude with some perspectives on the synergy between model order reduction and uncertainty quantification, and their likelihood for ecoming a game changer in computational mechanics, along the lines anticipated by Pierre Ladeveze and co-workers in their series of workshops on model reduction. References [1] K. Carlberg, C. Bou-Mosleh and C. Farhat, “Efficient Nonlinear Model Reduction via a Least-Squares Petrov-Galerkin Projection and Compressive Tensor Approximations,” International Journal for Numerical Methods in Engineering, Vol. 86, pp. 155-181 (2011). [2] K. Carlberg, C. Farhat, J. Cortial and D. Amsallem, “The GNAT Method for Nonlinear Model Reduction: Effective Implementation and Application to Computational Fluid Dynamics and Turbulent Flows,” Journal of Computational Physics, Vol. 242, pp. 623-647 (2013). [3] D. Amsallem, M. Zahr and C. Farhat, “Nonlinear Model Order Reduction Based on Local Reduced-Order Bases,” International Journal for Numerical Methods in Engineering, Vol. 92, pp. 891-916 (2012). [4] C. Farhat, P. Avery, T. Chapman and J. Cortial, “Dimensional Reduction of Nonlinear Finite Element Dynamic Models with Finite Rotations and Energy-Conserving Mesh Sampling and Weighting for Computational Efficiency,” International Journal for Numerical Methods in Engineering, Vol. 98, pp. 625-662 (2014). [5] T. Bui-Thanh, K. Willcox and O. Ghattas, “Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space,” SIAM Journal on Scientific Computing, Vol. 30, pp. 3270-3288 (2008). [6] K.Washabaugh, M. Zahr and C. Farhat, “On the Use of Discrete Nonlinear Reduced-Order Models for the Prediction of Steady-State Flows Past Parametrically Deformed Complex Geometries,” AIAA-2016-1814, AIAA SciTech 2016, San Diego, CA, January 4-8 (2016). [7] C. Soize and C. Farhat, “A Nonparametric Probabilistic Approach for Quantifying Uncertainties in Low- and High-Dimensional Nonlinear Models,” International Journal for Numerical Methods in Engineering, (in review).
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Contributor : Christian Soize Connect in order to contact the contributor
Submitted on : Monday, September 18, 2017 - 8:34:23 AM
Last modification on : Thursday, September 29, 2022 - 2:21:15 PM


  • HAL Id : hal-01588910, version 1



Charbel Farhat, K Washabaugh, T. Chapman, Christian Soize. Recent advances in nonlinear model reduction for design and associated uncertainty quantification . New Challenges in Computational Mechanics, May 2016, Cachan, France. pp.1-2. ⟨hal-01588910⟩



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