Tetrahedral meshes are widely used in scientific computing for representing three-dimensional scalar, vector, and tensor fields. The size and complexity of some of these meshes can limit the performance of many visualization algorithms, making it hard to achieve interactive visualization. The use of simplified models is one way to enable the real-time exploration of these datasets. In this paper, we propose a novel technique for simplifying large unstructured meshes. Most current techniques simplify the geometry of the mesh using edge collapses. Our technique simplifies an underlying scalar field directly by segmenting the original scalar field into two pieces: the boundary of the original domain and the interior samples of the scalar field. We then simplify each piece separately, taking into account proper error bounds. Finally, we combine the simplified domain boundary and scalar field into a complete, simplified mesh that can be visualized with standard unstructured-data visualization tools. Our technique is much faster than edge-collapse-based simplification approaches. Furthermore, it is particularly suitable for aggressive simplification. Experiments show that isosurfaces and volume renderings of meshes produced by our technique have few noticeable visual artifacts.