Multiple Surrogates for the Shape Optimization of Bluff Body-facilitated Mixing
Yolanda Mack, Tushar Goel, Wei Shyy, Raphael Haftka, Nestor Queipo
2005
43rd AIAA Aerospace Sciences Meeting and Exhibit
unpublished
Plausible alternative surrogate models can lead to different results in surrogate-based optimization. Since the cost of constructing surrogates is small compared to the cost of the simulations, using multiple surrogates may offer advantages compared to the use of a single surrogate. This idea is explored for a complex design space encountered when shape optimization of a bluff body is performed to facilitate mixing while minimizing the total pressure loss. It is shown that the design space has
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... mall islands where mixing is very effective compared to the rest of the design space. It is difficult to use a single surrogate model to capture such local but critical features. Both polynomial response surfaces and radial basis neural networks are used as surrogates. The former are more accurate away from the high-mixing regions while the latter are more accurate near these regions. A combined use of both models is beneficial. The surrogates are also used to perform global sensitivity analysis and bi-objective optimization. The former help rank the design variables in terms of their influence on the objectives, while the latter elucidates the tradeoffs between mixing efficiency and total pressure loss. I. Introduction HE surrogate-based optimization (SBO) approach has been shown to be an effective approach for the design of computationally expensive models such as those found in aerospace systems, aerodynamics, structures, and propulsion, among other disciplines. The surrogates provide fast approximations of the system response making optimization and sensitivity studies possible. However, the application of the approach in the context of CFD-based shape optimization with complex flow models presents significant challenges: i) the data required to construct the surrogates is severely limited due to time/computational constraints, and ii) there is potential variability in the SBO performance when using alternative surrogate modeling schemes 1 . Response surface methodology 2 , neural network techniques, spline, and kriging are examples of methods used to generate surrogates for simulations in the optimization of complex flows involving applications such as engine diffusers 3 , rocket injectors 4 , and supersonic turbines 5 . The type of surrogate used is highly problem dependent, and it is usually not know beforehand which model will perform better in a given analysis. For example, Alexandrov et al. 6 compared the performances of cubic polynomial, kriging, and spline approximations for use as the low fidelity model in the sequential optimization of a three-dimensional aerodynamic wing. They found that a cubic polynomial provided a poor approximation of the high fidelity model, while spline and kriging approximations performed well. Shyy et al. 7 compared quadratic and cubic polynomial approximations to a radial basis neural network approximation in the multiobjective optimization of a rocket engine injector. In this case, the radial basis neural network performed better for one objective, whereas the cubic polynomial performed better for the other. Therefore, it is often advantageous to try more than one surrogate when performing an optimization. The use of multiple surrogates is particularly attractive for very expensive simulations, because the cost of using surrogates for 2 additional analysis is negligible compared to the cost running the simulations. This approach is tested for the shape optimization of a trapezoidal bluff body model-a computationally expensive simulation from the design optimization viewpoint. Specifically, the optimization task focuses on the mixing and total pressure loss characteristics of time dependent flows over a 2-D bluff body. Bluff body devices are often used as flameholding devices 8 such as in afterburner and ramjet systems. To attain satisfactory design of the bluff body devices, the mixing capability can be maximized by increasing the size and the strength of the recirculation zone induced by shape changes in the bluff body geometry 9 . However, to prevent the deterioration of the efficiency of the device, the total pressure loss on the body needs to be minimized. This leads to a shape optimization problem involving two competing objectives-the total pressure loss coefficient and the mixing capability-that must be simultaneously optimized. There are challenges in optimizing the bluff body shape. The flow is unsteady and difficult to predict. There is recirculating flow in the near-wake region that decays to form a well-developed vortex street in the wake region. The instantaneous loss and degree of mixing changes over time and must be well resolved for accurate solutions. An earlier investigation of the optimization of a trapezoidal bluff body was conducted by Burman et al. 10 This study focused on the response surface optimization of Navier-Stokes flow over the bluff body. The optimization objectives were to minimize the mean drag coefficient and to maximize the measure of mixing, which was the timeaveraged maximum negative velocity. These conflicting objectives were handled by constructing a desirability function. However, because the optimum points obtained are based on the settings in the desirability function, the information in terms of trade-off characteristics is somewhat limited. At the designated Reynolds number, which is 120 based on the freestream velocity and the bluff body dimension, the response surface turned out to be very smooth with little variation in the response as design variables were changed. The measure of mixing is also limited, providing little information on the amount of mixing that occurs in the wake flow. The study uses a relatively coarse mesh (9272 computational cells) that was selected by performing a grid sensitivity study on a single case based on the drag coefficient. The effects of grid resolution on the measure of mixing were not investigated, so it is possible that not all of the flow effects were captured. For higher Reynolds number flow, as observed by, Morton 11 , grid resolution can have a marked effect on the prediction of unsteady flows and can substantially affect the fidelity of the surrogate model. Obtaining the necessary number of CFD solutions needed to construct the surrogate model can be time consuming and computationally expensive. To construct the surrogate model, many sample points are required to gain an accurate representation of the system behavior. The response of the system differs depending on the values of the input variables. In the current study, the optimization process is facilitated by using CFD solutions that allow for quick changes to the input variables. The problem becomes further complicated when the optimization is conducted on models with complex flow situations that may include unsteady dynamics. For these time-dependent solutions, adequate temporal resolution is needed in addition to the adequate spatial resolution needed to resolve flow features such as flow gradients and vortex structures. Because many sample points are needed to determine the system response, there are practical time and computational limits to the degree of resolution obtainable for each simulation. Surrogate models are often used in the CFD optimization of computationally expensive models. In gradient-based optimization, for example, these methods require a number of function evaluations to estimate the direction of the optimum point. Mujumdar 12 used a surrogate model to optimize duct flow based on prior knowledge of the duct physics. This method is not applicable when prior knowledge of the flow physics is unknown. Otto et al. 13 used a response surface model in place of the function evaluation in the gradient-based optimization of trapezoidal ducts and axisymmetric bodies. The general problem in using a low fidelity model in place of actual function evaluations is that it introduces error into the optimization. Alexandrov et al. 6 attempted to reduce this error somewhat by using a variable fidelity model. They began with the same technique as Otto et al., but recalibrated the low fidelity model with occasional high fidelity model data. This helped reduce the error while simultaneously relaxing the necessary computer effort. A companion paper by Goel et al. 14 details the effects of using a low fidelity model for the trapezoidal bluff body problem. Goel found that the low fidelity model failed to capture key flow phenomena. An investigation of bluff body simulations of varying fidelity identified the lowest grid resolution necessary to adequately capture the majority of the flow effects. This grid resolution would provide data in the most efficient way while still maintaining accuracy. Surrogate models will be used to approximate the effect of bluff body geometry changes on total pressure loss and mixing effectiveness. The time-averaged flow field solutions will be compared by looking for common trends and correlations in the flow structures. Optimal designs will be identified. The previous bluff body study by Burman et al. 10 will be improved upon in the following ways: 1. Increase the Reynolds number to explore higher Reynolds number effects.
doi:10.2514/6.2005-333
fatcat:lnn24pc3zzg7zij3ugpozqamja