In this paper, we introduce a volumetric partitioning strategy based on a generalized sweeping framework to seamlessly partition the volume of an input triangle mesh into a collection of deformed cuboids. This is achieved by a user-designed volumetric harmonic function that guides the decomposition of the input volume into a sequence of two-manifold level sets. A skeletal structure whose corners correspond to corner vertices of a 2D parameterization is extracted for each level set. Corners are placed so that the skeletal structure aligns with features of the input object. Then, a skeletal surface is constructed by matching the skeletal structures of adjacent level sets. The surface sheets of this skeletal surface partition the input volume into the deformed cuboids. The collection of cuboids does not exhibit T-junctions, significantly simplifying the hexahedral mesh generation process, and in particular, it simplifies fitting trivariate B-splines to the deformed cuboids. Intersections of the surface sheets of the skeletal surface correspond to the singular edges of the generated hex-meshes. We apply our technique to a variety of 3D objects and demonstrate the benefit of the structure decomposition in data fitting.