Cokriging is a flexible tool for constructing surrogate models on the outputs of computer models. It can readily incorporate gradient information, in which form it is named gradient-enhanced Kriging (GEK), and promises accurate surrogate models in >10 dimensions with a moderate number of sample locations for sufficiently smooth responses. However, GEK suffers from several problems: poor robustness and ill-conditionedness of the surface. Furthermore it is unclear how to account for errors in gradients, which are typically larger than errors in values. In this work we derive GEK using Bayes' Theorem, which gives an useful interpretation of the method, allowing construction of a gradienterror contribution. The Bayesian interpretation suggests the "observation error" as a proxy for errors in the output of the computer model. From this point we derive analytic estimates of robustness of the method, which can easily be used to compute upper bounds on the correlation range and lower bounds on the observation error. We thus see that by including the observation error, treatment of errors and robustness go hand in hand. The resulting GEK method is applied to uncertainty quantification for two test problems.

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After a jet engine is shut down, hot air rising inside the compressor disc cavities, secondary air systems, and the gas path annulus will result in a vertical temperature gradient. As the compressor rotor cools and contracts in the presence of this thermal gradient, it will bend, in a phenomenon known as thermal rotor bow. Starting an engine under bowed conditions can result in rubbing of the rotor and stator seals, adding heat to the rotor, exacerbating the rotor bow. This causal sequence of rubbing and bending is called the Newkirk Effect. In this study, 30 simulations of simplified compressor geometries have been run in three-dimensional unsteady conjugate heat transfer computational fluid dynamics coupled with finite element modelling, using a Sobol’ quasi-random sequence coupled with kriging interpolation to study the effects of three important geometric parameters on the thermal bow response of an engine. The three parameters, rotor length, rotor diameter, and compressor case wall thickness, were selected based on a similar screening test analysis performed by authors in a previous study. The results include response maps of each parameter with respect to rotor bow and clearance reduction onset time, duration, and severity, and show that length and case wall thickness exhibit linear responses due to their effect on stiffness, whereas diameter exhibits a non-linear response, due to the conflicting and competing effects on stiffness and vertical temperature difference.

3D Numerical simulations of a biplane flapping wing MAV have been performed using an immersed boundary method Navier-Stokes finite volume solver. To obtain a realistic simulation, the wing deformation has been captured using a stereo-vision system. The raw data obtained is further post-processed using Kriging interpolation and the results with and without the interpolation are compared. Results show that Kriging interpolation gives smoother force variation and is able to give reasonable converged solution using only ten wing positions (frames) over one period. The simulation results managed to capture the first peak of the experimental force results both in terms of approximate location and magnitude. However, the simulation only managed to capture the second peak in term of location; its magnitude is smaller than the experimental force. Various reasons for the discrepancies have been discussed. Nevertheless, the simulations reveal strong leading edge and tip vortices, which will enable us to get a better understanding of the underlying flapping wing aerodynamics. Nomenclaturec = mean chord length of wing ct = thrust coefficient dx = minimum grid size f = reduced frequency fc = external body force p = pressure Q = Q criterion Re = Reynolds number S = wing surface area t = time, nondimensionalized with the flapping period T = flapping period Uref = reference velocity

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We apply Ordinary Kriging to predict 75,000 terrain survey data from a randomly sampled subset of < 2500 observations. Since such a Kriging prediction requires a considerable amount of CPU time, we aim to reduce its computational cost. In a conventional approach, the cost of the Kriging analysis would be dominated by the optimization routine required to find the maximum likelihood, which provides an estimate of the correlation ranges. We propose to transform the optimization problem to the frequency domain, such that the cost of the optimization is now dominated by that of a single Fourier transform required to find the power spectrum of the observations, as a result of which the computational cost is now virtually independent of the number of optimization steps. For the present application, we find that the proposed approach is as accurate as the conventional approach for a sample size of 100 or more. The CPU time increases with the number of optimization steps for the conventional approach, while it is virtually constant for the proposed approach.