Improvement of adjoint based methods for efficient computation of response surfaces

In computational fluid dynamics, the solution of the dual (or adjoint) equations in conjunction with the primal (or flow equations) offers valuable information that can be used in design, sensitivity analysis, error estimation, mesh adaptation, uncertainty quantification, etc. From the view point of implementation and application of the adjoint technique, one of the following two approaches may be pursued: the continuous method and the discrete one. In the continuous approach (Jameson 1988; Anderson & Venkatakrishnan 1997; Kim et al. 2002; Castro et al. 2007), the adjoint equations are derived from the continuous flow equations and then subsequently discretized using a numerical method of choice, whereas in the discrete approach (Griewank 2000; Nadarajah & Jameson 2000; Giles et al. 2003; Mavriplis 2006; Mader et al. 2008), the adjoint equations are directly derived from the discretized governing equations. These approaches offer distinct advantages in different scenarios and have both been implemented for purposes of Verification and Validation (V&V) and Uncertainty Quantification (UQ) in hypersonic flow applications. In this work, we outline preliminary work in two areas of research that involve the use of adjoints in uncertainty quantification: • Uncertainty quantification using non-intrusive methods such as stochastic collocation (Eldred 2009) typically involves the construction of a response surface of the quantity of interest as a function of the uncertain parameters. To achieve the level of desired accuracy, a large number of Computational Fluid Dynamics (CFD) simulations is typically required. To improve the efficiency of this process, we propose a new “robust” grid adaptation technique that is aimed at minimizing the numerical error over a small variation of the parameters around a nominal state. In this approach, computational grids are generated with the knowledge that small variations of the uncertain parameters can locally change the solution (and hence, the error distribution). This is in contrast to classical adjoint techniques (Pierce & Giles 2002; Fidkowski & Darmofal 2009), which seek to adapt the grid with the aim of minimizing numerical errors for a nominal flow condition. • In the discrete sensitivity framework (such as discrete adjoints, complex step, and finite differences), the grid convergence of nonlinear flow calculations does not guarantee the convergence of linear sensitivities (Giles et al. 2003). This attribute has a special relevance in problems involving strong shocks, where the gradient of an output functional of interest may exhibit noise, and hence, cannot be directly utilized. To counter this difficulty, we utilize a bi-grid filtering strategy (Glowinski 1992; Zuazua 2005) that offers the promise of converged linear sensitivities using a discrete computation technique. In summary, the pursuit of UQ science in hypersonic flow applications, together with sensitivity computations using adjoints, has brought to the table some interesting limitations of the state-of-the-art in adjoint methodologies. In this report, our early work