Abstract
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.
Similar content being viewed by others
References
Advani SG, Tucker CL (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 31:751–784
Cui H, Grace JR (2007) Flow of pulp fibre suspension and slurries: a review. Int J Multiph Flow 33:921–934
Farahi MH, Rubio JE, Wilson DA (1996) The optimal control of the linear wave equation. Int J Control 63:833–848
Farhadinia B, Farahi MH (2005) Optimal shape design of an almost straight nozzle. Int J Appl Math 17(3):319–333
Farhadinia B, Farahi MH, Esfahani JA (2008) Shape optimization of a nozzle with specified flow field including viscosity effect. Acta Appl Math 104:243–256
Hyensjö M, Dahlkild A (2008) Study of the rotational diffusivity coefficient of fibres in planar contracting flows with varying turbulence levels. Int J Multiph Flow 34:894-903
Hyensjo M, Dahlkild A, Krochak P, Olson J, Hamalainen J (2007) Modelling the effect of shear flow on fibre orientation anisotropy in a planar contraction. Nord Pulp Pap Res J 22:376–382
Kamyad AV, Rubio JE, Wilson DA (1992) An optimal control problem for the multidimensional diffusion equation with a generalized control variable. J Optim Theory Appl 75:101–132
Makinen RAE, Hamalainen J (2006) Optimal control of a turbulent fibre suspension flowing in a planar contraction. Commun Numer Methods Eng 22:567–575
Mehne HH, Farahi MH, Esfahani JA (2005) Slot nozzle design with specified pressure in a given subregion by embedding method. Appl Math Comput 168:1258–1272
Nazemi AR, Farahi MH (2009) Control the fibre orientation distribution at the outlet of contraction. Acta Appl Math 106:279–292
Olson JA, Kerekes RJ (1998) The motion of fibres in turbulent flow. J Fluid Mech 377:47–64
Olson JA, Frigaard I, Chan C, Hamalainen JP (2004) Modeling a turbulent fibre suspension flowing in a planar contraction: the one-dimensional headbox. Int J Multiph Flow 30:51–66
Parsheh M, Brown ML, Aidun CK (2006) Variation of fiber orientation in turbulent flow inside a planar contraction with different shapes. Int J Multiph Flow 32:1354–1369
Rubio JE (1986) Control and optimization: the linear treatment of nonlinear problems. Manchester University Press, Manchester, and John Wiley, New York and London
Rudin W (1987) Real and complex analysis, 3rd edn. New York, McGraw-Hill
Shaqfeh ESG, Koch DL (1990) Orientation dispersion of fibers in extensional flows. Phys Fluids A 2:1077–1093
Shin M, Koch DL (2005) Rotational and translational dispersion of fibers in isotropic turbulent flows. J Fluid Mech 540:143–173
Stebel J, Makinen RAE, Toivanen JI (2007) Optimal shape design in a fibre orientation model. Appl Math 52:391–405
Wilson DA, Rubio JE (1977) Existence of optimal controls for the diffusion equation. J Optim Theory Appl 22:91–101
Young LC (1969) Calculus of variations and optimal control theory. Sunders, Philadelphia
Acknowledgments
The author would like to thank to the editor and the anonymous referees for comments which improved this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Farhadinia, B. Planar contraction design with specified fibre orientation distribution at the outlet. Struct Multidisc Optim 46, 503–511 (2012). https://doi.org/10.1007/s00158-012-0775-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-012-0775-7