# The Objective Space and the Formulation of Design Requirement in Natural Laminar Flow Optimization
## 1 Introduction  
The formulation of design requirements is assumed critical for effective optimization in all design disciplines [1], since they characterize the design process through design space exploration for the best solution via evaluation and analysis [2; 3; 4; 5]. This paper is focused on providing new perspective from objective space via formulating problem-oriented design requirement in the field of aerodynamic design such as Natural Laminar Flow optimization of a transonic airfoil and aero engine compressor blade. The destructive impact of carbon emissions and the enduring bad economic situation of airlines makes it necessary for aircraft manufacturers to minimize fuel burn in aircraft design, one solution to which has been drag reduction [6; 7]. The skin friction is important part of drag composition (it occupies nearly 50% of total aircraft drag in some cases [6]). Therefore, Natural Laminar Flow (NLF) optimization has been worked on for skin friction drag reduction by extending laminar area over aircraft surface, on account of lower skin friction drag magnitude in laminar than in turbulence [6; 8; 9]. In many aerodynamic designs, numerical optimization algorithms based on geometrical parameterization are used for the samples in design space to be iterated towards a pre-defined design requirement. The classification of skin friction drag and pressure drag comes from their differentgeneration mechanisms and the fact that they can be easily calculated without involving the other's information. The formulation of design requirement in NLF optimization usually reduces to the summation of pressure drag \(C_{p}\) and laminar-related performance (i.e., skin friction drag \(C_{f}\)) as in Equation (1)  
\[\min(w_{1}\cdot C_{p}+w_{2}\cdot C_{f}),\text{s.t. geometrical constraints, etc.} \tag{1}\]  
where \(w_{1}\) and \(w_{2}\) are the assigned weights that represents the importance attached to the aerodynamic performance behind variables given by designers. [PERSON] et al. and [PERSON] et al. put \(w_{2}\) under major consideration (solely minimize \(C_{f}\)) in design requirement formulation with \(C_{p}\) being kept under a passive constant constraint [8; 10]. [PERSON] and [PERSON] and [PERSON] et al. added wave drag (a subset of pressure drag) in design requirement formulation where they put much consideration on shock wave at transonic regime [11; 12]. [PERSON] et al. and [PERSON] and [PERSON] simply put total drag as the design requirement [13; 14; 15]. [PERSON] et al. set 80% to \(w_{2}\) in Equation (1) for increasing the effect of laminarization \(C_{f}\)[16]. The above settings represent different perspectives in design process. From industrial view, making total drag of a geometry alone as the design requirement is ultimate, but this would hinder the effort of methodological manipulation in emphasizing NLF's influence [17; 18; 19; 20]. Usually, it is not how much importance should be adequately given to laminarization that matters in such cases, but whether laminarization is considered in drag reduction design. In fact, these studies are more of verification of transition models in the prediction of laminar transition with coupling of ordinary drag force computation. Adjusting the corresponding weights in Equation (1) is a solution to the challenge of reasonable design requirement formulation. The problem, however, is that this is usually done according to experience. For example, simply singling out laminarization in the design requirement formulation ignores the fact that the change of geometry is also linked to the change of pressure drag force as well. Simple collocation of skin friction drag and pressure drag in multi-objective problem of \(\min C_{p},C_{f}\) is also not a perfect solution as well in that it ignores the factor of impact of each objective. However, it can be used as a good measurement of optimization progress, and it will be used in the first case in this paper.