Minimizing the reflection of waves by surface impedance using boundary elements and global optimization (Q1098669)

In this paper we consider the problem of minimizing the scattered field intensity with respect to the boundary impedance. Usual optimization procedures based on gradient type local minimization algorithms will not be effective for this problem because the scattered field intensity is not a convex function of the impedance, and it has many local extrema. We approach the problem here by first discretizing the Helmholtz partial differential equation using the boundary element method. Then we apply some recently developed global optimization algorithms [e.g. \textit{Q. Zheng}, Numer. Math., Nanking 7, 31-43 (1985; Zbl 0586.65048)] to find approximate distributions of the boundary impedance for particular shapes which minimize the reflected field intensity. The boundary element method effects a reduction of dimensionality resulting in much greater computational efficiency. The global optimization algorithm allows us to pick out nearly global minimum solutions among many local minima. Numerical solutions are represented graphically and discussed. Our results show that a variable boundary impedance is much more effective for minimizing the scattered field than a constant boundary impedance.