Parallel Solution of Optimal Shape Design Problem Governed by Helmholtz/Potential Flow Equations

In this paper the implementation of shape optimization techniques based on genetic algorithms for airfoil design is presented. The shape of an airfoil is sought such that a cost function depending on the scattered electromagnetic wave and the pressure distribution on the airfoil is minimized. The combined electromagnet-ics//ow problem is modeled by Helmholtz equation and potential ow. The state equations are discretized using the nite element method and the resulting linear systems of equations are solved using the ctitious domain method. The optimizer is parallelized for distributed memory multiprocessor by using PVM message passing library. Finally, numerical examples are given. 1 Setting of the problem Numerical shape optimization has been under extensive study during the past twenty years 1], 6], 12]. Traditionally shape optimization has been restricted into one discipline only. Recently, there has been much interest in shape optimization of systems governed by equations of both electromagnetics and uid mechanics. Numerical solutions based on nonlinear optimization methods have been presented in papers 5], 15]. Accurate numerical computation of a radar wave scattered by a ying obstacle and the aerodynamical properties of the obstacle is a challenging problem as the solutions of three-dimensional Maxwell and Navier-Stokes equations are needed. Optimization of shape via nonlinear programming is an iterative process where solutions of several analysis problems are needed. To reduce the huge computational burden several simpliications must be done. Assuming two-dimensional design and the case of T. M. polarization for the incoming wave the Maxwell's equations reduce to Helmholtz equation. Moreover, it is often assumed that the ow is irrotational and even incompressible. In this paper we consider the case of a two-dimensional lifting airfoil. The computational domain is shown in Figure 1. The interior of the ow eld is denoted by and the surface of the airfoil by ?. The far eld boundary, boundary of a large rectangle , is denoted by ? 1. A mathematical model describing the scattering of a two-dimensional