ترکیب تئوری بازی ها و الگوریتم ژنتیک با کاربرد برای مسایل بهینه سازی DDM ـ نازل

The goal of this paper is to discuss a new evolutionary strategy for the multiple objective design optimization of internal aerodynamic shape operating with transonic flow. The distributed optimization strategy discussed here and inspired from Lions’ new distributed control approach (J.L. Lions, Distributed active control approach for pde systems, Fourth WCCM CD-ROM, Buenos Aires, Argentina, 1998) relies on genetic algorithms (GAs). GAs are different from traditional optimization tools and based on digital imitation of biological evolution. Game theory replaces here a global optimization problem by a non-cooperative game based on Nash equilibrium with several players solving local constrained sub-optimization tasks. The transonic flow simulator uses a full potential solver taking advantage of domain decomposition methods and GAs for the matching of non-linear local solutions. The main idea developed here is to combine domain decomposition methods for the flow solver with the geometrical optimization procedure using local shape parameterization. Numerical results are presented for a simple nozzle inverse problem with subsonic and transonic shocked flows. A comparison of the nozzle reconstruction using domain decomposition method (DDM) or not for the simulation of the flow is then presented through evolutionary computations and convergence of the two surface parts of the throat is discussed. The above results illustrate the robustness and primising inherent parallelism of GAs for mastering the complexity of 3D optimizations.

From the experiments described in this paper, it is clear that GAs and DDMs may provide powerful tools to solve complex distributed optimization problems. The multiple objective techniques discussed here demonstrate that combining ideas from economics or game theory with GAs is worth considering for solving engineering problems. The main interest of Nash-GAs is that they are more robust than cooperative optimizations such as Pareto-GAs and easily parallelizable, even if the solutions found here are near-optimal rather than optimal. A forthcoming development consists in combining different optimization strategies (Pareto, Nash or Stackelberg) depending on the environment. The DDM associated to the optimization procedure will play a major role in decreasing the CPU needed for the PDE based evaluation of the fitness function via the use of several small genotypes instead of a large one. Namely, the incomplete information-based paradigm introduced recently by Bank [17] for parallel adaptive meshing algorithms using variable domain decomposition techniques with a posteriori error estimates should undoubtedly be an important companion of GAs working in a noisy environment. In the future, more complex problems will be investigated, such as multi-element configuration with different design variables and 3-D multi-criteria optimization.