INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2000; 34: 685–700 Optimal design of paper machine headboxes J. Ha ¨ma ¨la ¨inen a,1 , R. A. E. Ma ¨kinen b,2 and P. Tarvainen c, * ,3 a Valmet Corporation, PO Box 587, FIN-40101 Jya ¨skyla ¨ , Finland b Department of Mathematical Information Technology, Uniersity of Jya ¨skyla ¨ , PO Box 35, FIN-40351 Jya ¨skyla ¨ , Finland c Numerola Oy (Limited), Va ¨ino ¨nkatu 11A, FIN-40100 Jya ¨skyla ¨ , Finland SUMMARY A shape optimization problem for incompressible flows within a stabilized finite element framework is studied. The goal is to develop and test numerical realizations of optimal shape design problems that could be applied to non-trivial industrial problems. The resulting algorithm is applied to the optimization of the geometry of a tapered header in a paper machine headbox. Copyright © 2000 John Wiley & Sons, Ltd. KEY WORDS: optimal shape design; paper machine headbox; shape optimization 1. INTRODUCTION The headbox is the first component in the paper making process on a paper machine, located at the wet end (Figure 1). Fluid flow phenomena taking place in the headbox largely determine the quality of paper produced, e.g., the basis weight and the fiber orientation variations. The first flow passage in the headbox is a tapered header (Figure 2). The header is designed to distribute fiber suspension (wood fibers, filler clays and chemicals mixed in water) so that the produced paper will have an optimal basis weight (thickness) and fiber orientation across the width of a paper machine. Considerable progress has been made in the mathematical modeling and numerical compu- tations related to paper machine technology; see, for instance, References [1 – 3]. However, until now, very little attention has been given to the question of controlling physical flow situations, for example, shape optimization problems governed by flow equations. The shape optimization problem considered in this paper is to find optimal tapering of the header such that the outlet flow rate distribution from the headbox will result in optimal paper * Correspondence to: Numerola Oy (Limited), Va ¨ino ¨ nkatu 11A, FIN-40100 Jyva ¨skyla ¨, Finland. 1 E-mail: jari.p.hamalainen@valmet.com 2 E-mail: rainom@mit.jyu.fi 3 E-mail: pasi.tarvainen@numerola.fi Copyright © 2000 John Wiley & Sons, Ltd. Receied July 1999 Reised December 1999