Simulations of Systems with Spatio-Temporal Structure: A Model Reduction Approach for Complex Geometrics

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A systematic procedure for computer- aided studies of largescale engineering systems combining complex spatial structure with nonlinear dynamics behavior will be developed. A general methodology is developed, interfacing direct simulations, model reduction, and modern stability and bifurcation algorithms. This procedure will be validated on a number of representative engineering problems in complex geometries, involving and reaction phenomena. More specifically, high-order direct simulations based on spectral element discretizations will be combined with the Proper Orthogonal Decomposition method, in order to extract a small number of global modes governing the behavior of spatially extended systems. A Galerkin weighted residual formulation employing these global modes yields low-dimensional accurate dynamic models, which can subsequently be used to analyze spatial and temporal system behavior, including stability and transition to time-periodic and chaotic states. The proposed methodology can be expected to become a standard powerful computational tool for studying the dynamic behavior of realistic engineering systems. Furthermore, the low-dimensional engineering models resulting from this methodology will be useful in a wide variety of applications, from modeling and analysis, to design, prediction and control of spatially extended systems.