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. (C) 2012 Elsevier B.V. All rights reserved.