# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
###### Abstract
[12] presented a new element-based finite-volume discretization approach for the solution of incompressible flow problems on co-located grids. The proposed method, which was called the method of proper closure equations (MPCE), employed a proper set of physically relevant equations to the constraints of velocity and pressure at an integration point. The decoupled algorithms are mainly different from coupled ones due to their shape modification strategy. In mathematical-based decoupled shape design algorithms, which are commonly called optimization methods, the difference between the target and current pressure distribution is defined as an objective function, and the minimum of this function is sought using ideas rooted in calculus. In physical-based decoupled shape design algorithms, assumptions are made, and the physical analogies are employed to guide the shape evolution during the iterations. Gradient-based optimization methods [13] generate information regarding the objective function gradients and use it to specify the search direction and its step size in the design space. Evolutionary methods find the optimum solution based on repeated function evaluations. [PERSON] et al. [14] developed an efficient algorithm for the design optimization of the compressible fluid flow problem through a flexible structure. They coupled a supersonic flow solver, an aeroelastic solver, and genetic algorithm to calculate the optimum shape of a supersonic diffuser with flexible wall for a prescribed pressure distribution. [PERSON] et al. [15] optimized a three-dimensional S-shaped diffuser using multi-objective optimization strategy and a modified shear stress transport (SST) turbulent model. Optimization methods are generally computationally expensive and rather mathematically complex. However, they can be used to minimize different objective functions while satisfying various constraints on the design variables. [PERSON] et al. [16] coupled the free form deformation method with a multi-objective genetic algorithm to optimize the shape of a diffusing S-duct. They controlled the number of geometrical variables to limit the complexity of the flow simulation methods and obtain acceptable optimization times. A fast multi-objective optimization method for S-duct scoop inlets considering both inflow and outflow was developed and validated by [PERSON] et al. [17]. They developed an automated optimization system integrating the computational fluid dynamics analysis, non-uniform rational B-spline geometric representation technique, and non-dominated sorting genetic algorithm to minimize the total pressure loss and distortion at the exit of diffuser. [PERSON] et al. [18] performed the multi-objective optimization of an S-shaped duct based on a Tabu search algorithm. They used a free-form deformation approach with 36\({}^{\circ}\) of freedom. A substantial improvement in the values of both objective functions was reported, but 1300 simulations were needed to obtain the results. Physical-based decoupled shape design algorithms are rather mathematically simple, and a linear relation is often established between the boundary node displacement \((\Delta r_{i})\) and the local surface pressure mismatch \((\Delta p_{i})\) to update the shape. Residual-correction methods have also been developed to solve shape design problems. [PERSON] and [PERSON] [19] related the boundary curvature to the tangential velocity mismatch on a wall for an inviscid low. [PERSON] and [PERSON] [20] proposed a residual-correction method called the GM method to design a part of a transonic airfoil geometry. [PERSON] et al. [21] developed a modified version called the MGM method to overcome some of the limitations of the GM method. [PERSON] and [PERSON] [22] proposed a method using elastic membranes, which considerably increased the convergence rate of the MGM method. [PERSON] et al. [23] presented the flexible string algorithm (FSA) as a residual-correction method for duct design.