The cutting thickness of ultra-precision machining is generally very small, only a few micrometer or even down to the order of a few nanometer. In such case, a basic understanding of the mechanism on the micro-machining process is is necessary to produce a high quality surface. When machining at very small depths of cut, metal flow near a rounded tool edge become important. In this paper a finite element analysis is presented to calculate the stagnation point on the tool edge or critical depth of cut below which no cutting occurs. From the simulation, the effects of the cutting speed on the critical depths of cut were calculated and discussed. Also the transition of the stagnation point according to the increase of the depths of cut was observed.