Figure 1: Our algorithm produces smooth frame fields in volumes. Frames (a) are represented by spherical harmonic functions (b), attached to each vertex of a tetrahedral mesh. Streamlines and singularities of the field are shown in yellow and red, respectively. Abstract Given a tetrahedral mesh, the algorithm described in this article produces a smooth 3D frame field, i.e. a set of three orthogonal directions associated with each vertex of the input mesh. The field varies smoothly inside the volume, and matches the normals of the volume boundary. Such a 3D frame field is a key component for some hexahedral meshing algorithms, where it is used to steer the placement of the generated elements. We improve the state-of-the art in terms of quality, efficiency and reproducibility. Our main contribution is a non-trivial extension in 3D of the existing least-squares approach used for optimizing a 2D frame field. Our algorithm is inspired by the method proposed by Huang et al. [2011], improved with an initialization that directly enforces boundary conditions. Our initialization alone is a fast and easy way to generate frames fields that are suitable for remeshing applications. For better robustness and quality, the field can be further optimized using nonlinear optimization as in Li et al [2012]. We make the remark that sampling the field on vertices instead of tetrahedra significantly improves both performance and quality.