Summary. Surrogate-based optimization has proven very useful for novel or exploratory design tasks because it offers a global view of the characteristics of the design space, and it enables one to refine the design of experiments, conduct sensitivity analyses, characterize tradeoffs between multiple objectives, and, if necessary, help modify the design space. In this article, a framework is presented for design optimization on problems that involve two or more objectives which may be conflicting in nature. The applicability of the framework is demonstrated using a case study in space propulsion: a response surface-based multi-objective optimization of a radial turbine for an expander cycle-type liquid rocket engine. The surrogate model is combined with a genetic algorithm-based Pareto front construction and can be effective in supporting global sensitivity evaluations. In this case study, due to the lack of established experiences in adopting radial turbines for space propulsion, much of the original design space, generated based on intuitive ideas from the designer, violated established design constraints. Response surfaces were successfully used to define previously unknown feasible design space boundaries. Once a feasible design space was identified, the optimization framework was followed, which led to the construction of the Pareto front using genetic algorithms. The optimization framework was effectively utilized to achieve a substantial performance improvement and to reveal important physics in the design. IntroductionWith continuing progress in computational simulations, computational-based optimization has proven to be a useful tool in reducing the design process duration and expense. Numerous methods exist for conducting design optimizations. Popular methods include gradient-based methods [4,14], adjoint methods [6,10], and surrogate model-based optimization methods such as the response surface approximation (RSA) [9]. Gradient-based methods rely on a

A multiple surrogate-based optimization strategy in conjunction with an evolutionary algorithm has been employed to optimize the shape of a simplified hydraulic turbine diffuser utilizing three-dimensional Reynolds-averaged Navier–Stokes computational fluid dynamics solutions. Specifically, the diffuser performance is optimized by changing five geometric design variables to maximize the average pressure recovery factor for two inlet boundary conditions with different swirl, corresponding to different operating modes of the hydraulic turbine. Polynomial response surfaces and radial basis neural networks are used as surrogates, while a hybrid formulation of the NSGA-IIa evolutionary algorithm and a ϵ-constraint strategy is applied to construct the Pareto front from the two surrogates. The proposed optimization framework drastically reduces the computational load of the problem, compared to solely utilizing an evolutionary algorithm. For the present problem, the radial basis neural networks are more accurate near the Pareto front while the response surface performs better in regions away from it. By using a local resampling updating scheme the fidelity of both surrogates is improved, especially near the Pareto front. The optimal design yields larger wall angles, nonaxisymmetrical shapes, and delay in wall separation, resulting in 14.4% and 8.9% improvement, respectively, for the two inlet boundary conditions.

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A response surface-based dual-objective design optimization was conducted in the preliminary design of a compact radial turbine for an expander cycle rocket engine. The optimization objective was to increase the efficiency of the turbine while maintaining low turbine weight. Polynomial response surface approximations were used as surrogates, and the accuracies of such approximations improve by limiting the size of the domain and the number of variables for each response of interest. The optimization was accomplished in three stages using an approximate, one-dimensional model. In the first stage, a relatively small number of points were used to identify approximate constraint boundaries of the feasible domain and to reduce the number of variables used to approximate each one of the constraints. In the second stage, a moderate number of points in this approximate feasible domain were used to identify the region where both objectives had reasonable values. The last stage focused on obtaining high accuracy approximation in the region of interest with large number of points. The approximations were used to identify the Pareto front and to perform a global sensitivity analysis. Significant improvement was achieved compared to a baseline design.

A three-dimensional computational model of an experimental rectangular combustion chamber was developed to explore the wall heat transfer of a GO 2 /GH 2 shear coaxial single element injector. The CFD model allowed for the direct analysis of heat transfer effects due to flow dynamics-an analysis that would be very difficult using experimental studies alone. The use of a 3-D CFD model revealed heat transfer effects due to flow streamlines and eddy conductivity, and provided insight into the two-dimensional nature of the wall heat flux. A grid sensitivity study was conducted to determine the effects of grid resolution on the combustion chamber length and heat flux. The results of a grid sensitivity study were inconclusive, as a grid-independent solution could not be reached. However, it was found that the predicted heat flux was largely independent of the grid resolution, as long as the near-wall region was well resolved. Finally, a single-element injector model was constructed to explore the sensitivity of the peak heat flux and combustion chamber length to the circumferential and radial spacing of injector elements in the outer row of a multi-element injector. Many cases, including the baseline case, had a recirculation region that was oriented such that the outer shear layer was directed at the combustion chamber wall, resulting in a large peak heat flux near the injector face. It was found that by increasing the spacing between injector elements of the outer row while reducing the distance of the outer row to the wall, a relatively flat heat flux profile could be obtained.