تجزیه و تحلیل حساسیت تغییرات و بهینه سازی طراحی

Variational method (VM) is employed to derive the co-state equations, boundary (transversality) conditions, and functional sensitivity derivatives. The converged solutions of the state equations together with the steady state solution of the co-state equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational method to aerodynamic shape optimization problems is demonstrated on internal flow problems at supersonic Mach number range of 1.5. Optimization results for flows with and without shock phenomena are presented. The study shows that while maintaining the accuracy of aerodynamical objective function and constraint within the reasonable range for engineering prediction purposes, variational method provides a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference computations.

The new emerging sensitivity analysis technique for gradient-based optimization method is the continuous or variational sensitivity analysis. From a modified functional, variational method derives a set of partial differential equations, i.e., the co-state equations with their boundary conditions and the sensitivity derivatives. In computing the sensitivity derivatives with respect to the control points or design variables, this methodology utilizes the converged solutions of the state and co-state equations. In recent years, variational sensitivity analysis has significantly contributed to the progress of aerodynamic design optimization. Pironneau [1] showed the usefulness of variational approach in fluid mechanical problems by illustrating how to compute the minimum drag profile in two-dimensional viscous and laminar flows. Chen and Seinfeld [2] developed a methodology to compute the performance sensitivity derivatives using optimal control theory. Koda et al. [3] used this procedure to solve atmospheric diffusion problems. Koda [4], [5] and [6] further developed this approach and outlined a numerical algorithm for the computation of functional derivatives. Meric [7] and [8] treated optimal control problems governed by parabolic and elliptic partial differential equations and solved them numerically using variational method. In their effort to compare the gradients obtained by “implicit” and “variational” approaches, Shubin and Frank [9] implemented VM to optimize the shape of a nozzle of a variable cross-sectional area for steady one-dimensional Euler equations. Jameson [10] regarded the boundary of the flow domain as a control parameter and then designed airfoils using the potential as well as the two- and three-dimensional compressible inviscid flows. Cabuk and Modi [11] implemented a perturbation method to compute the optimum profile of a diffuser for a maximum static pressure in a two-dimensional steady viscous incompressible flow. Ta’asan et al. [12] have successfully implemented variational method and optimized an airfoil in the potential flow field. Ibrahim and Baysal [13] demonstrated the versatility of the variational method to solve aerodynamical design problems for internal flows in different Mach number regimes including shock flows. Following the same approach as Jameson [10], Reuther and Jameson [14] optimized airfoils in potential flows. Iollo and Salas [15] used variational method to solve a two-dimensional internal flow optimization problem with embedded shock to match a pressure distribution. References [17], [28], [29], [30], [33] and [34] applied variants of discrete sensitivity approaches to optimize engineering problems. Epstein and Peigin [26] used Genetic Algorithm to optimize three-dimensional lifting surfaces for wing-body aircraft configuration. In all these classes of optimization, the functional sensitivity derivatives are directly coupled to the solution of a set of linear partial differential equations, i.e., the co-state equations and their boundary conditions that result from the variation of the augmented Lagrangian function [27], [31] and [32]. The success of any optimization by this approach is, therefore, destined to a stable and converged solution of the state and co-state equations.

A nozzle optimization problem is considered, and the application of variational method to compute the optimal shape for the maximum thrust is presented. During the design process, the supersonic nozzle remained supersonic while improving the performance index or thrust (Table 2). Design optimization with strong flow discontinuity was also performed which shows a slight gain in pressure attenuation and much better mass-error minimization at the shock location. While the computational accuracy of variational method (Table 2) is comparable with the finite difference, its computational efficiency and memory savings (Table 3) are found to be substantial. As memory and computational efficiency are the bottle-neck issues for large two-dimensional and three-dimensional problems in general, variational method is one of the most viable candidates in solving design optimization problems.