The complex-step approximation for the calculation of derivative information has two significant advantages: the formulation does not suffer from subtractive cancellation errors and it can provide exact derivatives without the need to search for an optimal step size. However, when used for the calculation of second derivatives that may be required for approximation and optimization methods, these advantages vanish. In this work, we develop a novel calculation method that can be used to obtain first (gradient) and second (Hessian) derivatives and that retains all the advantages of the complex-step method (accuracy and step-size independence). In order to accomplish this task, a new number system which we have named hyper-dual numbers and the corresponding arithmetic have been developed. The properties of this number system are derived and explored, and the formulation for derivative calculations is presented. Hyper-dual number arithmetic can be applied to arbitrarily complex software and allows the derivative calculations to be free from both truncation and subtractive cancellation errors. A numerical implementation on an unstructured, parallel, unsteady Reynolds-Averaged Navier Stokes (URANS) solver, Joe, is created using C++ operator overloading. Results are presented demonstrating the potential of this method in computations of relevance to the analysis and design of multidisciplinary aerospace systems.

This report describes work performed from June 2012 through May 2014 as a part of a Sandia Early Career Laboratory Directed Research and Development (LDRD) project led by the first author. The objective of the project is to investigate methods for building stable and efficient proper orthogonal decomposition (POD)/Galerkin reduced order models (ROMs): models derived from a sequence of high-fidelity simulations but having a much lower computational cost. Since they are, by construction, small and fast, ROMs can enable real-time simulations of complex systems for onthe-spot analysis, control and decision-making in the presence of uncertainty. Of particular interest to Sandia is the use of ROMs for the quantification of the compressible captive-carry environment, simulated for the design and qualification of nuclear weapons systems. It is an unfortunate reality that many ROM techniques are computationally intractable or lack an a priori stability guarantee for compressible flows. For this reason, this LDRD project focuses on the development of techniques for building provably stable projection-based ROMs. Model reduction approaches based on continuous as well as discrete projection are considered.

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