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Parallel streamline placement is still an open problem in flow visualization. In this paper, we propose an innovative method to place streamlines in parallel for 2D flow fields. This method is based on our proposed concept of local tracing areas (LTAs). An LTA is defined as a subdomain enclosed by streamlines and/or field borders, where the tracing of streamlines are localized. Given a flow field, it is initialized as an LTA, which is later recursively partitioned into hierarchical LTAs. Streamlines are placed within different LTAs simultaneously and independently. At the same time, to control the density of streamlines, each streamline is associated with an isolation zone and a saturation zone, both of which are center aligned with the streamline but have different widths. None of streamlines can trace into isolation zones of others. And new streamlines are only seeded within valid seeding areas (VSAs) that are enclosed by saturation zones and/or field borders. To implement the parallel strategy and the density control, a cell-based modeling is devised to describe isolation zones and LTAs as well as saturation zones and VSAs. With the help of these cell-based models, a heuristic seeding strategy is proposed to seed streamlines within irregular LTAs, and a cell-marking technique is used to control the seeding and tracing of streamlines. Test results show that the placement method can achieve highly parallel performance on shared memory systems without losing the quality of placements.