Planar contraction design with specified fibre orientation distribution at the outlet

In the present study, an optimal shape design problem of a papermachine headbox is considered to control the fibre orientation distribution at the outlet of contraction. First, by the aid of a bijective transformation, we transform the optimization problem governed by weak formulation on a fixed domain. Then, by means of a process of embedding, the class of admissible shapes is replaced by a class of non-negative Radon measures. The modified problem consists of the minimization of a linear functional over a set of pairs of positive Radon measures satisfying linear constraints. Each measure of optimal pair is approximated by a finite combination of unitary atomic measures and furthermore the approximate solution of the first problem is found by the optimal solution of a finite-dimensional linear programming problem. The solution of this problem is used to construct an optimal piecewise constant control. Finally, using the approximate control signals, we obtain the approximate optimal shapes.