Constructing volumetric parameterization based on directed graph simplification of ℓ1 polycube structure from complex shapes

Volumetric spline parameterization of complex geometry plays a key role in isogeometric analysis (IGA). In this paper, we propose a general framework to construct volumetric parameterization from complex shapes based on directed graph simplification of the ℓ1 polycube structure. By minimizing the ℓ1-norm of the normals on the input triangular meshes, a polycube structure can be generated with robustness, efficiency, and controllability. Then an algorithm is proposed for ℓ1 polycube structure simplification with a directed graph, which is an abstract structure generated from the polycube. After simplification, the vertex number of the directed graph will decrease, and the polycube structure will become more simple. From the simplified polycube structure, we construct a segmented surface by spline fitting techniques, and finally we fill each block with a trivariate B-spline volume with C0-constraints. The proposed method can generate volumetric parameterization without an interior extraordinary vertex, and it has very high potential for constructing volumetric parameterization in IGA simulations. Some volume parameterization examples from complex shapes are presented to show the robustness and efficiency of the proposed method.