Title

A unified, multifidelity quasi-Newton optimization method with application to aero-structural design

Date of Award

2017

Degree Name

Ph.D. in Engineering

Department

Department of Mechanical and Aerospace Engineering

Advisor/Chair

Advisor: Markus P. Rumpfkeil

Abstract

A model's level of fidelity may be defined as its accuracy in faithfully reproducing a quantity or behavior of interest of a real system. Increasing the fidelity of a model often goes hand in hand with increasing its cost in terms of time, money, or computing resources. The traditional aircraft design process relies upon low-fidelity models for expedience and resource savings. However, the reduced accuracy and reliability of low-fidelity tools often lead to the discovery of design defects or inadequacies late in the design process. These deficiencies result either in costly changes or the acceptance of a configuration that does not meet expectations. The unknown opportunity cost is the discovery of superior vehicles that leverage phenomena unknown to the designer and not illuminated by low-fidelity tools. Multifidelity methods attempt to blend the increased accuracy and reliability of high-fidelity models with the reduced cost of low-fidelity models. In building surrogate models, where mathematical expressions are used to cheaply approximate the behavior of costly data, low-fidelity models may be sampled extensively to resolve the underlying trend, while high-fidelity data are reserved to correct inaccuracies at key locations. Similarly, in design optimization a low-fidelity model may be queried many times in the search for new, better designs, with a high-fidelity model being exercised only once per iteration to evaluate the candidate design. In this dissertation, a new multifidelity, gradient-based optimization algorithm is proposed. It differs from the standard trust region approach in several ways, stemming from the new method maintaining an approximation of the inverse Hessian, that is the underlying curvature of the design problem. Whereas the typical trust region approach performs a full sub-optimization using the low-fidelity model at every iteration, the new technique finds a suitable descent direction and focuses the search along it, reducing the number of low-fidelity evaluations required. This narrowing of the search domain also alleviates the burden on the surrogate model corrections between the low- and high-fidelity data. Rather than requiring the surrogate to be accurate in a hyper-volume bounded by the trust region, the model needs only to be accurate along the forward-looking search direction. Maintaining the approximate inverse Hessian also allows the multifidelity algorithm to revert to high-fidelity optimization at any time. In contrast, the standard approach has no memory of the previously-computed high-fidelity data. The primary disadvantage of the proposed algorithm is that it may require modifications to the optimization software, whereas standard optimizers may be used as black-box drivers in the typical trust region method. A multifidelity, multidisciplinary simulation of aeroelastic vehicle performance is developed to demonstrate the optimization method. The numerical physics models include body-fitted Euler computational fluid dynamics; linear, panel aerodynamics; linear, finite-element computational structural mechanics; and reduced, modal structural bases. A central element of the multifidelity, multidisciplinary framework is a shared parametric, attributed geometric representation that ensures the analysis inputs are consistent between disciplines and fidelities. The attributed geometry also enables the transfer of data between disciplines. The new optimization algorithm, a standard trust region approach, and a single-fidelity quasi-Newton method are compared for a series of analytic test functions, using both polynomial chaos expansions and kriging to correct discrepancies between fidelity levels of data. In the aggregate, the new method requires fewer high-fidelity evaluations than the trust region approach in 51% of cases, and the same number of evaluations in 18%. The new approach also requires fewer low-fidelity evaluations, by up to an order of magnitude, in almost all cases. The efficacy of both multifidelity methods compared to single-fidelity optimization depends significantly on the behavior of the high-fidelity model and the quality of the low-fidelity approximation, though savings are realized in a large number of cases.The multifidelity algorithm is also compared to the single-fidelity quasi-Newton method for complex aeroelastic simulations. The vehicle design problem includes variables for planform shape, structural sizing, and cruise condition with constraints on trim and structural stresses. Considering the objective function reduction versus computational expenditure, the multifidelity process performs better in three of four cases in early iterations. However, the enforcement of a contracting trust region slows the multifidelity progress. Even so, leveraging the approximate inverse Hessian, the optimization can be seamlessly continued using high-fidelity data alone. Ultimately, the proposed new algorithm produced better designs in all four cases. Investigating the return on investment in terms of design improvement per computational hour confirms that the multifidelity advantage is greatest in early iterations, and managing the transition to high-fidelity optimization is critical.

Keywords

Multidisciplinary design optimization, Structural optimization Mathematics, Engineering models Mathematics, Algorithms, Aerospace Engineering, multifidelity, multidisciplinary design optimization, trust region model management, variable-fidelity, polynomial chaos

Rights Statement

Copyright 2017, author

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