Towards shape optimization of profile extrusion dies with respect to homogeneous die swell

Abstract Plastics extrusion is a manufacturing process suited for continuous profiles with a fixed cross-section. The function of the extrusion die is to reshape the melt, which originally has a circular cross-section, to the desired profile shape. When constructing new extrusion dies, the key challenge is to design the transition region between outflow and inflow of the die. While in general the design of the transition region is arbitrary, there are influences on the shape accuracy of the product which need to be considered during die design. One of those influence factors is die swell. This paper presents first steps towards numerical die design with the objective of homogeneous die swell. It introduces a shape-optimization framework and an appropriate objective function. Since the accurate computation of die swell is still a topic of ongoing research, the applicability of the Galerkin/Least-Squares stabilization method in a space–time finite element setting and in conjunction with the Oldroyd-B and the Giesekus model is discussed. Furthermore, for three space dimensions, we suggest an interface tracking approach combined with a smoothing based on non-uniform rational B-splines for the definition of the free-surface shape.

[1]  Nhan Phan-Thien,et al.  Understanding Viscoelasticity: An Introduction to Rheology , 2012 .

[2]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[3]  R. Darby,et al.  Viscoelastic fluids: An introduction to their properties and behavior , 1976 .

[4]  Marek Behr,et al.  On the usage of NURBS as interface representation in free‐surface flows , 2012 .

[5]  H. M. Bücker,et al.  Sensitivity of optimal shapes of artificial grafts with respect to flow parameters , 2010 .

[6]  J. M. Rallison,et al.  Creeping flow of dilute polymer solutions past cylinders and spheres , 1988 .

[7]  Marek Behr,et al.  Stabilized space‐time finite element formulations for free‐surface flows , 2001 .

[8]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[9]  R. Rutgers,et al.  On the evaluation of some differential formulations for the pom-pom constitutive model , 2003 .

[10]  Roland Keunings,et al.  An algorithm for the simulation of transient viscoelastic flows with free surfaces , 1986 .

[11]  E. TezduyarT.,et al.  A new strategy for finite element computations involving moving boundaries and interfacesthe deforming-spatial-domain/space-time procedure. II , 1992 .

[12]  H. R. Warner,et al.  Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells , 1972 .

[13]  A. Keller.  Mathematical and Physical Sciences , 1933, Nature.

[14]  J. Tsamopoulos,et al.  Steady extrusion of viscoelastic materials from an annular die , 2008 .

[15]  Marek Behr,et al.  Stabilized finite element methods for incompressible flows with emphasis on moving boundaries and interfaces , 1992 .

[16]  H. Giesekus.  A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility , 1982 .

[17]  B. Gautham,et al.  Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method , 2009 .

[18]  T. McLeish,et al.  Predicting low density polyethylene melt rheology in elongational and shear flows with , 1999 .

[19]  Tayfun E. Tezduyar,et al.  Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces , 1994 .

[20]  M. Powell.  The BOBYQA algorithm for bound constrained optimization without derivatives , 2009 .

[21]  Marek Behr,et al.  Free-surface flow simulations in the presence of inclined walls , 2002 .

[22]  Fritz Ebert.  Strömung nicht-newtonscher Medien , 1980 .

[23]  Roland Keunings,et al.  Simulation of linear polymer melts in transient complex flow , 2000 .

[24]  R. Keunings,et al.  Finite-element Analysis of Die Swell of a Highly Elastic Fluid , 1982 .

[25]  T N Phillips,et al.  Contemporary Topics in Computational Rheology , 2002 .

[26]  J. Oldroyd.  On the formulation of rheological equations of state , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[27]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[28]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[29]  Raanan Fattal,et al.  Flow of viscoelastic fluids past a cylinder at high Weissenberg number : stabilized simulations using matrix logarithms , 2005 .

[30]  Raanan Fattal,et al.  Constitutive laws for the matrix-logarithm of the conformation tensor , 2004 .

[31]  E. Hinton,et al.  Computer Aided Optimisation of Profile Extrusion Dies , 2000 .

[32]  K. J. Bathe,et al.  GLS-type finite element methods for viscoelastic fluid flow simulation , 2005 .

[33]  Alfred Gray,et al.  Modern differential geometry of curves and surfaces with Mathematica (2. ed.) , 1998 .

[34]  Marek Behr,et al.  STABILIZED FINITE ELEMENT METHODS OF GLS TYPE FOR OLDROYD-B VISCOELASTIC FLUID , 2004 .

[35]  Pierre J. Carreau,et al.  A constitutive equation derived from Lodge's network theory , 1979 .

[36]  Christopher W. Macosko,et al.  Rheology: Principles, Measurements, and Applications , 1994 .

[37]  Marek Behr,et al.  Finite element solution strategies for large-scale flow simulations☆ , 1994 .

[38]  F. Shakib.  Finite element analysis of the compressible Euler and Navier-Stokes equations , 1989 .

[39]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[40]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[41]  E. Mitsoulis.  Annular extrudate swell of pseudoplastic and viscoplastic fluids , 2007 .

[42]  M. Ahmeda,et al.  Calculation of fully developed flows of complex fluids in pipes of arbitrary shape, using a mapped circular domain , 1995 .

[43]  B. Gautham,et al.  Simulation of viscoelastic flows of polymer solutions in abrupt contractions using an arbitrary Lagrangian Eulerian (ALE) based finite element method , 2007 .