# The Objective Space and the Formulation of Design Requirement in Natural Laminar Flow Optimization
## 1 Introduction
This paper proposes to formulate design requirement of NLF optimization from the perspective of objective space via data manipulation approach. For an aerodynamic design with given algorithm and geometric representation method (i.e., a given design space), the objective space is the projection from design space via aerodynamics or CFD (Computational Fluid Dynamics) models. This paper believes that objective space contains the intrinsic information of the 'optimization potentials' of samples in terms of aerodynamic performance: thus, it can be injected in the design requirement formulation as an aide to capture the nature of the problem at hand. In the two proof-of-concept studies in this paper, the improved formulation of concerned design requirement injected with objective space information is compared with traditional design requirement formulation with evenly distributed weight distribution in terms of design efficiency and comprehensive aerodynamic performance, respectively. The solution proposed in this paper is done via Principal Component Analysis (PCA). PCA is used to strengthen preprocessing of data to improve manipulation efficiency [21; 22], dimension reduction [23], and in the sample selection of initialization [24], etc. The methods for data manipulation have been utilized in aerodynamic designs, but mostly in design space than in objective space. In fact, apart from extending the subjects of aerodynamic design to complex aircraft components [25; 26; 27] and strengthening the link between geometry and aerodynamic performance [21; 28; 29; 30], aerodynamic designers deployed data preparation in design space for preprocessing, e.g., dimensional reduction by parameter filtering according to physical intuitiveness(e.g., cutting out geometry parameters with less impact on ultimate aerodynamic design goal [31]) and feature extracting from flowfield for economical use of data [32; 33], etc. It is noted that the intention of PCA in this paper is not simply for variable reduction but for extracting the main modes from objective space considered to be the essential information that may help establish a more reasonable formulation of design requirement in aerodynamic designs. The rest of the paper is organized as follows: Section 2 studies the CFD solver used in the paper, which is validated by wind-tunnel experiment. First, grid sampling is applied to prepare for the creation of objective space, which offers data to be processed by PCA so that a more problem-oriented formulation of design requirement can be obtained. The eigenvectors of the result are projected onto the principal axis generated via PCA and thus forms the formulation of design requirement. Secondly, the approach of objective space analysis is applied to an NLF optimization of transonic airfoil under fixed lift via Differential Evolution algorithm, the result of which shows that the information of objective space help improve the optimization route in aerodynamic design (cf. Section 3). Thirdly, a reformulated design requirement is also applied to an NLF aero engine compressor blade design, showing that a third aerodynamic performance objective's concern can be accessorily reflected in a two-objective problem if an objective space of three aerodynamic performances (i.e., the pressure drag and laminar length related drag as well as pressure loss in the case of blade design) is created and influence design requirement formulation (cf. Section 4).