ABSTRACT
Introduction Shape optimization is quite indispensable for designing and constructing industrial components. Many problems that arise in application, particularly in structural mechanics and in the optimal control of distributed parameter systems, can be formulated as the minimization of functionals defined over a class of admissible domains. Therefore, such problems have been intensively studied in the literature throughout the last 25-30 years (see [16-17], [21, 23], and the references therein). From
the practical as well as from the theoretical point of view, to prevent optimal shapes from degeneration, several constraints have to be taken into account. More recently, the computation of the related dual variables, i.e. the Lagrange multipliers, becomes of increasing interest due to applicational and theoretical reasons. On the one hand, there are several applications where these multipliers have an important physical meaning like in the electromagnetic shaping of liquid metals (see, e.g., [19-20]). On the other hand, their computation is important by itself for the investigation of sufficient optimality conditions in shape optimization [6]. In [9, 11]we developed first and second order optimization algorithms for a class of elliptic shape optimization problems, where a powerful wavelet BEM is proposed for the computation of the state and related quantities. Additional equality and/or inequality constraints of functional type are treated by an augmented Lagrangian technique, which turns out to be more stable and efficient than a penalty method. In particular, by the standard update procedure, a linear convergence is provided for the Lagrange multipliers. However, due to the efficiency of the Newton method with respect to the primal variables, that is, the shape respective to its finite dimensional approximation, this slow convergence of the Lagrange multipliers becomes, in our opinion, the bottleneck of the overall algorithm. Furthermore, the question of a faster approximation of the dual variables is important as we explained above. Following an idea of Ma˚rtensson [18], second order convergence is realizable if only active equality constraints are present. Due to known degeneration tendencies in shape problems, exactly this situation occurs very often in practical and theoretical shape optimization problems. Consequently, the goal of the present paper is to demonstrate this approach for the class of shape problems considered in [9]. Of course, both the standard method and its modification are independent of the underlying shape calculus and of the numerical method for solving the state equation.
