Recent advances in parametric nonlinear model order reduction: treatment of shocks, contact and interfaces, structure-preserving hyper reduction, acceleration of multiscale formulations, and application to design optimization
Résumé
Parametric, projection-based, Model Order Reduction (MOR) is a mathematical tool for constructing a parametric Reduced-Order Model (ROM) by projecting a given parametric High Dimensional Model (HDM) onto a Reduced-Order Basis (ROB). It is rapidly becoming indispensable for a large number of applications including, among others, computational-based design and optimization, multiscale analysis, statistical analysis, uncertainty quantification, and model predictive control. It is also essential for scenarios where real-time simulation responses are desired. During the last two decades, linear, projection-based, parametric MOR has matured and made a major impact in many fields of engineering including electrical engineering, acoustics, and structural acoustics, to name only a few. By comparison, nonlinear, projection-based, parametric MOR remains somehow in its infancy. Nevertheless, giant strides have been recently achieved in many of its theoretical, algorithmic, and offline/online organizational aspects. The main purpose of this lecture is to highlight some of these advances, discuss their mathematical and computer science underpinnings, and report on their impact for an important class of problems in aerodynamics, fluid mechanics, nonlinear solid mechanics and structural dynamics, failure analysis, multiscale analysis, uncertainty quantification, and design optimization. To this effect, nonlinear, projection-based, parametric MOR will be first interpreted as a constrained semidiscretization on a subset of a compact Stiefel manifold, using a low-dimensional basis of global shape functions constructed a posteriori — that is, after some knowledge about the response of the system of interest has been developed. Usually, such a knowledge is gathered using the given parametric HDM and an offline training procedure where the model parameters are sampled with a greedy strategy based on a cost-effective ROM error indicator. Specifically, a set of problems related to the parametric problem of interest are solved at the sampled parameter points using the given HDM, and the computed solution snapshots are compressed to obtain the desired global ROB. Depending on the mathematical type of the governing equations underlying the given HDM, a dual ROB is also constructed and the sought-after nonlinear parametric ROM is constructed by Galerkin (or Petrov-Galerkin) projection of the HDM onto the global ROB (and its dual counterpart).
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