## 2 3-D Optimization Methodology  
In this paper, a flexible 3-D multidisciplinary optimization system is developed and Figure 1 shows its flowchart. The system makes use of an optimization algorithm, a metamodel, a dynamic sampling database and high fidelity simulation codes for CFD and CSM analyses, respectively. Four optimization algorithms are integrated into this system, including genetic algorithm (GA), differential evolution algorithm (DEA) [22], nondominated sorting genetic algorithm (NSGA-II) [23] and simulated annealing algorithm (SAA). Besides, three metamodel techniques including the Kriging method, artificial neural network (ANN), and support vector machine (SVM) are integrated into this system. It is to be noted that NSGA-II used in this paper is from a toolbox called Inspyred, which is a free, open source framework for creating biologically-inspired computational intelligence algorithms in Python, and the algorithms provided in Inspyred are all tested through benchmark test problems with excellent convergence. Also, NSGA-II is relatively popular and widely used in turbomachinery design optimization area. Many papers can be found within ASME (American Society of Mechanical Engineering) proceedings and journals that use this algorithm to drive their optimization. In the present work, NSGA-II and ANN are used to perform the optimization and shorten the computational time. The system can easily switch between the accurate optimization and metamodel-assisted optimization. The initial database is generated using the Latin center hypercube method and then evaluated by high fidelity CFD and CSM codes. The metamodel is then built based  
Figure 1: Flowchart of the optimization system. CFD: computational flow dynamics, CSM: computational structural mechanics, NSGA: nondominated sorting genetic algorithm. on the database and used to substitute the computational tools (CFD and CSM). After a specified number of generations, 10 new individuals will be added in the database and the metamodel will be updated for better performance. The 10 new individuals are selected through a dynamic sampling algorithm, which makes the sampling as uniform as possible for fear that the database only reflects a subzone of the whole design space. The ANN topology (number of hidden layers and the number of neurons per hidden layer) and the initial weight are optimized using GA because they greatly affect the precision of the obtained ANN. The population size for the ANN optimization is 10, the maximum generation is 15, and then the time for constructing metamodel increases, but the performance of the obtained metamodel is improved at the same time. The measurement for the ANN training is the mean relative error. In the present paper, the optimization is conducted for T-100 and T-100 re turbines, respectively, so as to assess the preliminary design influence of the radial inflow turbine according to the final results. Aerodynamic optimization is firstly performed for the nozzle, and multidisciplinary optimization is then performed for the rotor. The nozzle aerodynamic optimization involves two strategies. Accurate optimization (red loop in Figure 1) without metamodel is adopted for T-100 nozzle optimization and metamodel-assisted optimization (blue loop in Figure 1) is adopted for T-100 re nozzle optimization. Both T-100 and T-100 re rotors use the accurate optimization because the GA optimized ANN becomes time-consuming as the database grows. The detailed information about the metamodel is described in the discussion section. The objective for the nozzle optimization is to maximize the total-to-static efficiency while keeping the mass flow rate constant. The main purpose is to lower the nozzle flow loss and improve the match between the nozzle and the rotor. The objective for the rotor optimization is to maximize the total-to-static efficiency and minimize the rotor weight while keeping the mass flow rate constant and the stress at the allowable level.