A new kinematically admissible velocity field for a generalized three-dimensional flow is introduced, in which the flow is bounded by an analytic die-profile function. Then, by applying the upper-bound method th the velocity field, the flow patterns as the upper-bound method are obtained. Extrusion of elliptic sections from round billets is chosen as a computational example. Computation and experiments are carried out for work-hardening material such as aluminum alloy 2024. In order to visualize the plastic flow, the grid marking technique is employed. The theoretical predictions both in extrusion load and deformed pattern are in good agreement with the experimental data.