A specialisation of the 'separable' form of the KBKZ equation [M.H. Wagner, J. Non-Newtonian Fluid Mech. 4 (1978) 39-55] is presented that allows strain-hardening response to both planar and uniaxial elongation, while simultaneously giving shear-thinning behaviour. The specialisation expresses the strain-damping function in terms of a single invariant measure of strain. It is demonstrated that comparable planar and uniaxial elongational viscosities are predicted for IUPAC LDPE, with an over-prediction of the first normal stress difference in shear flow. It is shown that it is possible to fit transient data, for an LDPE melt that exhibits strain-hardening, for both planar and uniaxial elongational data simultaneously. It is shown that the new damping model gives vortex growth for simulations of both planar and axisymmetric contraction flows of LDPE, in contrast to a popular existing model that has been demonstrated to fail to predict vortex growth in planar contraction flow [P. Olley, R. Spares, P.D. Coates, J. Non-Newtonian Fluid Mech. 86 (1999) 337-357]. The finite-element based method used to find the flow solution is described including a novel method for implementing streamline tracking. Details are given of the iterative method used to compute flow fields, according to the stress field, and for the associated method to implement under-relaxation. Two new vortex measures are defined in order to quantify simulated vortices that can have an 'hourglass' shape (a shape that has been reported experimentally for LDPE contraction flows). To allow a broad assessment, comparisons are given as geometry type, contraction ratio, damping model, and the number of available strain-damping parameters are varied. Results for planar and axisymmetric contractions are compared, and contraction ratios of 8:1 and 4:1 are studied for planar flow. Results obtained using the new damping model are compared with well-established results obtained using an existing damping model for 4:1 axisymmetric contraction flow. The quantitative effects on flow simulations from using 'single-beta' and 'multiple-beta' damping functions are assessed.