Optimal Shape Design in a Fibre Orientation Model

We study a 2D model of the orientation distribution of fibres in a paper machine headbox. The goal is to control the orientation of fibres at the outlet by shape variations. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Existence of solutions both to the state and the optimization problem is analyzed and sensitivity analysis is performed. Introduction This paper deals with a model describing the orientation of fibres moving through a paper machine headbox. The goal is to achieve an optimal orientation of fibres at the outlet, realized by changing the shape of the headbox. The mathematical model is justified in Olson et al. [2004] and its numerical solution is presented in Mäkinen and Hämäläinen. More results on this topic can be found e.g. in Hämäläinen [1993] and Haslinger, Málek, Stebel. The geometry of a planar contraction (one-dimensional) headbox is described by a Lipschitz function α (see Figure 1). For simplicity, one-dimensional steady flow u = (u1(x1), 0) is assumed. The model further considers only the distribution ψ = ψ(x1, φ) of the projected angle φ of the fibre along the central streamline.