A methodology for non-Intrusive projection-based model reduction of expensive black-box PDE-based systems and application in the many-query context

Abstract

The projection-based reduced order modeling, typically requires access to the discrete form of governing equations of the high-fidelity model. The projection is commonly done on a subspace determined via POD. However when commercial codes are used as the high-fidelity model, such an approach is not possible in general. Usually in such circumstances, a ‘POD+Interpolation’ approach is taken where the reduced state variable is directly interpolated to adapt for change in time/parameters. This thesis devices a method to develop projection-based ROM with commercial codes, specifically CFD codes. The novelty of the work is that it converts the original non-linear PDE system into a linear PDE system with auxiliary non-linear algebraic equations which are then projected onto the POD subspace. By such a linearization, it is shown that the governing equations can be extracted by directly discretizing the linear terms (which is easier compared to non-linear terms) at a computational cost that scales linearly with grid size ($N$). Other methods that exist to ‘discover’ governing equations from data, are known to also involve a similar or higher cost, while being tailored towards time-dependent systems. Finally, the ROM is posed as a constrained optimization problem that can be solved cheaply. Since the thesis specifically addresses static parametric systems, a database of such ROMs are generated for a pre-determined set of parameter snapshots which are then interpolated by mapping them to the tangent space of the manifold they are embedded in (manifold of symmetric positive definite matrices in this case) to adapt for parametric changes. The method is tested on canonical PDEs and flow past airfoils at subsonic and transonic flow regimes. A prediction error of < 5% was achieved in subsonic cases in terms of the state, pressure distributions, lift and drag. Under transonic conditions with moving shocks, the approach incurs higher error unless a sufficiently dense snapshot distribution is used. Model parameters are identified and experiments are conducted to determine settings that improve accuracy. The usefulness of the method is also demonstrated on application problems in the many-query context - design optimization and uncertainty quantification. Overall, the strength and weaknesses of the approach are identified, demonstrated and explained.