The numerical simulation of extrudate swell is significant in extrusion processing.Precise prediction of extrudate swell is propitious to the control of melt flow and the quality of final products.A mathematical model of three-dimensional(3D)viscoelastic flow through elliptical ring die for polymer extrusion was investigated.The penalty function formulation of viscoelastic incompressible fluid was introduced to the finite element model to analyze 3D extrusion problem.The discrete elastic viscous split stress(DEVSS)and streamline-upwind PetrovGalerkin(SUPG)technology were used to obtain stable simulation results.Free surface was updated by updating the streamlines which needs less memory space.According to numerical simulation results,the effect of zero-shear viscosity and elongation parameter on extrudate swell was slight,but with the increase of volumetric flow rate and relax time the extrudate swell ratio increased markedly.Finally,the numerical simulation of extrudate swell flow for low-density polyethylene(LDPE)melts was investigated and the results agreed well with others’work.These conclusions provided quantitative basis for the forecasting extrudate swell ratio and the controlling of extrusion productivity shape. The numerical simulation of extrudate swell is significant in extrusion processing. Precise prediction of extrudate swell is propitious to the control of melt flow and the quality of final products. A mathematical model of three-dimensional (3D) viscoelastic flow through elliptical ring die for polymer extrusion was investigated. The penalty function formulation of viscoelastic incompressible fluid was introduced to the finite element model to analyze 3D extrusion problem. The discrete elastic viscous split stress (DEVSS) and streamline-upwind Petrov Galerkin (SUPG) technology were used to obtain stable simulation results .Free surface was updated by updating the streamlines which needs less memory space. According to numerical simulation results, the effect of zero-shear viscosity and elongation parameter on extrudate swell was slight, but with the increase of volumetric flow rate and relax time of the extrudate swell ratio increased markedly. Finaally, the numerical simulation of extrudate swell flow for low-density polyethylene (LDPE) melts was investigated and the results agreed well with others’work. Theses concludes provide quantitative basis for the forecasting extrudate swell ratio and the controlling of extrusion productivity shape.