## 1 Introduction  
Pumps play an important role in industrial production, urban drainage, sewage treatment, and other fields, and they consume about 15% of the total energy consumption [1]. Mixed flow pumps have a smaller volume with a lighter weight compared to centrifugal pumps. They also have lower shaft power compared to axial-flow pumps [2]. Due to these characteristics, mixed flow pumps have increasingly gained recognition and have been widely used in recent years. Like most pumps, widening the high-efficiency region has been challenging, and therefore some in-depth research is required to attempt to meet the challenge. In recent years, with the improvement of science and technology, computational fluid dynamics (CFD) has been widely used in the field of mixed flow pump optimization [3]. Compared with experiment-based pump optimization, CFD-based optimization has the advantages of less cost involvement and more convenience. The shape of the impeller and vane diffuser can be easily modified by varying the design parameters in CFD-based optimization [4]. However, CFD analysis cannot directly give the ideal results. To obtain the desired performance, a trial and error process is needed, which requires a lot of time and extensive experience. Over the past years, researchers have introduced various algorithms in CFD-based optimization systems to solve this problem, including the use of the design of experiment (DOE) to discretize the design space [5, 6], surrogate models to construct the approximate model between the design parameters and objective function [7, 8], and optimization algorithms to search the global optimization solution in the design space [9, 10]. One of the challenges faced by the above optimization system is how to accurately describe the three-dimensional shape of the blade with as few design parameters as possible. With the improvement of computer technology and design methods, the use of inverse design method (IDM) to parameterize the blade has become an ideal method [11], which uses circulation, stacking, and blade loading to parameterize the blade [12]. Compared with the use of geometric parameters for impeller parameterization, the number of design parameters used in IDM is fewer, and the design parameters are more closely related to hydraulic performance. More importantly, compared with the traditional design, which makes use of the conformal transformation method to obtain the blade angle, in IDM, the design parameters directly control the blade angle and are therefore much likely to achieve better results [13]. The effectiveness of IDM has been verified by [PERSON] and [PERSON] [14; 15] and has been extensively applied in mixed flow pump [16; 17], centrifugal pump [18; 19], and axial-flow pump [20; 21] optimization designs. Although the abovementioned optimizations have achieved satisfactory results, there are still some limitations. For example, most studies only use the spanwise distribution of impeller exit circulation (SDIEC) to control the impeller Euler's head, while the influence of its distribution form on pump efficiency, cavitation, and other performances of the impeller have been ignored. As a result, during the optimization process, constant distribution of SDIEC is adopted and only the stacking condition and blade loading are used as design variables. However, the theoretical derivation of [PERSON] [22] and [PERSON] et al. [23] shows that the distribution form of SDIEC has an important influence on impeller performance and should be considered in the impeller design. The correctness of the above theory has been confirmed through a series of experimental studies [24; 25]. The superiority of nonlinear SDIEC in mixed flow pump optimization has also been verified in the author's previous work [26]. In this paper, to quantitatively study the influence of SDIEC on the optimization results, the mixed flow pump impeller was optimized in two different cases by using a comprehensive optimization system.