Richard P. Dwight

Effect of Approximations of the Discrete Adjoint on Gradient-Based Optimization

An exact discrete adjoint of an unstructured nite-volume solver for the RANS equations has been developed. The adjoint is exact in the sense of being based on the full linearization of all terms in the solver, including all turbulence model contributions. From this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method; considered are many approximations already seen in the literature. The eect of these approximations on the accuracy of the resulting design gradients, and the convergence and nal solution of optimizations is studied, as it applies to a two-dimensional high-lift conguration.

Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation

A heuristic method is proposed to estimate a posteriori that part of the total discretization error which is attributable to the smoothing effect of added dissipation, for finite volume discretizations of the Euler equations. This is achieved by observing variation in a functional of the solution as the level of dissipation is varied, and it is deduced for certain test-cases that the dissipation alone accounts for the majority of the functional error. Based on this result an error estimator and mesh adaptation indicator is proposed for functionals, relying on the solution of an adjoint problem. The scheme is considerably implementationally simpler and computationally cheaper than other recently proposed a posteriori error estimators for finite volume schemes, but does not account for all sources of error. In mind of this, emphasis is placed on numerical evaluation of the performance of the indicator, and it is shown to be extremely effective in both estimating and reducing error for a range of 2d and 3d flows.

Robust Mesh Deformation using the Linear Elasticity Equations

A modification is proposed to the equations of linear elasticity as used to deform Euler and Navier-Stokes meshes. In particular it is seen that the equations do not admit rigid body rotations as solutions, and it is shown how these solutions may be recovered by modifying the constitutive law. The result is significantly more robust to general deformations, and combined with incremental application generates valid meshes well beyond the point at which remeshing is required.

Efficient Uncertainty Quantification using Gradient-Enhanced Kriging

A fexible non-intrusive approach to parametric uncertainty quantifcation problems is developed, aimed at problems with many uncertain parameters, and for applications with a high cost of functional evaluations. It employs a Kriging response surface in the parameter space, augmented with gradients obtained from the adjoint of the deterministic equations. The Kriging correlation parameter optimization problem is solved using the Subplex algorithm, which is robust for noisy functionals, and whose effort typically increases only linearly with problem dimension. Integration over the resulting response surface to obtain statistical moments is performed using sparse grid techniques, which are designed to scale well with dimensionality. The efficiency and accuracy of the proposed method is compared with probabilistic collocation, direct application of sparse grid methods, and Monte-Carlo initially for model problems, and finally for a 2d compressible Navier-Stokes problem with a random geometry parameterized by 4 variables.

Eect of Various Approximations of the Discrete Adjoint on Gradient-Based Optimization

An exact discrete adjoint of an unstructured nite-v olume solver for the RANS equations has been developed. The adjoint is exact in the sense of being based on the full linearization of all terms in the solver, including all turbulence model contributions. From this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method; considered are many approximations already seen in the literature. The eect of these approximations on the accuracy of the resulting design gradients, and the convergence and nal solution of optimizations is studied, as it applies to a two-dimensional high-lift conguration.

Predictive RANS simulations via Bayesian Model-Scenario Averaging

The turbulence closure model is the dominant source of error in most Reynolds-Averaged Navier-Stokes simulations, yet no reliable estimators for this error component currently exist. Here we develop a stochastic, a posteriori error estimate, calibrated to specific classes of flow. It is based on variability in model closure coefficients across multiple flow scenarios, for multiple closure models. The variability is estimated using Bayesian calibration against experimental data for each scenario, and Bayesian Model-Scenario Averaging (BMSA) is used to collate the resulting posteriors, to obtain a stochastic estimate of a Quantity of Interest (QoI) in an unmeasured (prediction) scenario. The scenario probabilities in BMSA are chosen using a sensor which automatically weights those scenarios in the calibration set which are similar to the prediction scenario. The methodology is applied to the class of turbulent boundary-layers subject to various pressure gradients. For all considered prediction scenarios the standard-deviation of the stochastic estimate is consistent with the measurement ground truth. Furthermore, the mean of the estimate is more consistently accurate than the individual model predictions.

Goal-Oriented Mesh Adaptation using a Dissipation-Based Error Indicator

A method is proposed to estimate a posteriori that part of the total discretization error which is attributable to the smoothing effect of added dissipation, for finite volume discretizations of the Euler equations. This is achieved by observing variation in a functional of the solution as the level of dissipation is varied, and it is deduced for certain test-cases that the dissipation alone accounts for the majority of the functional error. Based on this result an error estimator and mesh adaptation indicator is proposed for functionals, relying on the solution of an adjoint problem. The scheme is considerably implementationally simpler and computationally cheaper than other recently proposed a posteriori error estimators for finite volume schemes, but does not account for all sources of error. In mind of this, emphasis is placed on numerical evaluation of the performance of the indicator, and it is shown to be extremely effective in both estimating and reducing error for a range of 2d and 3d flows.

Speeding up Kriging through fast estimation of the hyperparameters in the frequency-domain

Kriging is a widely applied data assimilation technique. The computational cost of a conventional Kriging analysis of N data points is dominated by the m iterations of the maximum likelihood estimate (MLE) optimization, resulting in a computational cost of O(mN^3). We propose two fast methods for estimating the hyperparameters in the frequency domain: frequency-domain maximum likelihood estimate (FMLE) and frequency-domain sample variogram (FSV), both of which reduce the cost of the optimization to O(N^2+mN) in the case of a regular Fourier transform (FT), and to O(NlnN+mN) in the case of a fast Fourier transform (FFT). In addition to this speed up, problems concerning positive definiteness of the gain matrix - which limit the robustness of the conventional approach - vanish in the proposed methods.

Bayesian inference for data assimilation using least-squares finite element methods

It has recently been observed that Least-Squares Finite Element methods (LS-FEMs) can be used to assimilate experimental data into approximations of PDEs in a natural way, as shown by Heyes et al. in the case of incompressible Navier Stokes ow [1]. The approach was shown to be effective without regularization terms, and can handle substantial noise in the experimental data without filtering. Of great practical importance is that { unlike other data assimilation techniques { it is not signifcantly more expensive than a single physical simulation. However the method as presented so far in the literature is not set in the context of an inverse problem framework, so that for example the meaning of the final result is unclear. In this paper it is shown that the method can be interpreted as finding a maximum a posteriori (MAP) estimator in a Bayesian approach to data assimilation, with normally distributed observational noise, and a Bayesian prior based on an appropriate norm of the governing equations. In this setting the method may be seen to have several desirable properties: most importantly discretization and modelling error in the simulation code does not affect the solution in limit of complete experimental information, so these errors do not have to be modelled statistically. Also the Bayesian interpretation better justifies the choice of the method, and some useful generalizations become apparent. The technique is applied to incompressible Navier-Stokes flow in a pipe with added velocity data, where its effectiveness, robustness to noise, and application to inverse problems is demonstrated.

Algebraic multigrid within defect correction for the linearized Euler equations

Given the continued difficulty of developing geometric multigrid methods that provide robust convergence for unstructured discretizations of compressible flow problems in aerodynamics, we turn to algebraic multigrid (AMG) as an alternative with the potential to automatically deal with arbitrary sources of stiffness on unstructured grids. We show here that AMG methods are able to solve linear problems associated with first-order discretizations of the compressible Euler equations extremely rapidly. In order to solve the linear problems resulting from second-order discretizations that are of practical interest, we employ AMG applied to the first-order system within a defect correction iteration. It is demonstrated on two- and three-dimensional test cases in a range of flow regimes (sub-, trans- and supersonic) that the described method converges rapidly and robustly. Copyright