We propose a fast intersection-free remeshing of triangular meshes that robustly and efficiently generates high-quality non-intersecting meshes. Conducting intersection checks on all local operations during remeshing to prevent intersections represents the principal efficiency bottleneck. Our method is based on a key observation: intersections primarily occur in structurally complex regions. Accordingly, we develop an adaptive method to identify these key regions and perform intersection checks only for local operations within these regions during remeshing, significantly improving the algorithmic efficiency. Our method is an order of magnitude faster than traditional approaches that perform intersection checks on all local operations. Furthermore, we introduce a flip-aware extension mechanism that effectively avoids triangle flipping by constraining the optimization space of local operations, thereby avoiding the formation of irregular sharp edges. We also employ an adaptive iterative size field to eliminate banding phenomenon and propose a quasi-geometric size field adjustment method to quickly achieve smooth size transitions, thereby improving mesh quality. Compared to state-of-the-art methods, our method consistently and quickly generates higher quality non-intersecting meshes. In addition, we have validated the robustness and efficiency of our method, using all 5,469 non-intersecting valid models from the Thingi10K dataset.