By

Sudharshan Ashwin Renganathan

(Advisor: Prof. Dimitri N. Mavris)

1:00 PM, Tuesday, March 20, 2018

*Weber Space Science and Technology Building (SST-II)*

*Collaborative Visualization Environment (CoVE)*

**A Methodology for Non-Intrusive projection-based model reduction of expensive black-box PDE-BASED SYSTEMS and application in the many-query context**

**ABSTRACT:**

In the design of next-generation aerospace systems, the need for high-fidelity models that accurately capture the physics is inevitable. High-fidelity models are in general computationally expensive and are not per se suitable to be used in many-query contexts frequently encountered in aerospace design such as design optimization and probabilistic analysis, where the model is evaluated several thousand times. In such situations, use of a surrogate model in lieu of the high-fidelity model is popular. However, when the underlying physics is nonlinear and the design space is high-dimensional, such models are known to (i) be not robust to parameter changes (ii) suffer from the curse of dimensionality and (iii) provide no guarantees for accuracy. Reduced Order Models are a class of surrogate models that have the potential to alleviate these problems by ensuring that they satisfy some approximate form of the governing equations of the high-fidelity model. Such a *physics-based* approach to surrogate modeling is the focus of this thesis.

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 Proper Orthogonal Decomposition. 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 is 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. Overall, the method is tested on canonical PDEs and flow past airfoils at subsonic and transonic flow regimes. Overall, 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 design optimization and uncertainty quantification. Overall, the strength and weaknesses of the approach are identified, demonstrated and explained.

**Committee Members: **Professor Dimitri N. Mavris, Professor Graeme J. Kennedy, Professor Daniel Schrage, Professor Juan J. Alonso (Stanford), Dr Steven Berguin

]]>