Volumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the Catmull–Clark subdivision scheme interpolates vertices of the original hexahedral mesh. The new control lattice is derived by the local push-back operation from one Catmull–Clark subdivision step with modified geometric rules. This interpolating method is simple and efficient, and several shape parameters are involved in adjusting the shape of the limit volume. The second method is based on progressive-iterative approximation using limit point formula. At each iteration step, we progressively modify vertices of an original hexahedral mesh to generate a new control lattice whose limit volume interpolates all vertices in the original hexahedral mesh. The convergence proof of the iterative process is also given. The interpolatory subdivision volume has $C^2$-smoothness at the regular region except around extraordinary vertices and edges. Furthermore, the proposed interpolatory volumetric subdivision methods can be used not only for geometry interpolation, but also for material attribute interpolation in the field of volumetric material modeling. The application of the proposed volumetric subdivision approaches on isogeometric analysis is also given with several examples.

体积建模是材料建模和等几何模拟的重要主题。本文基于Catmull-Clark细分体积的极限点公式，提出了两种非结构六面体网格插值Catmull-Clark体积细分方法。第一种方法的基本思想是构造一个新的控制格，其控制量通过Catmull-Clark细分方案进行插值，以插入原始六面体网格的顶点。新的控制晶格是通过对Catmull-Clark细分步骤的局部后推操作和修改后的几何规则得出的。这种插值方法简单有效，并且在调整极限体积的形状时涉及多个形状参数。第二种方法基于使用极限点公式的渐进迭代逼近。在每个迭代步骤中，我们逐步修改原始六面体网格的顶点，以生成一个新的控制晶格，其极限体积会插值原始六面体网格中的所有顶点。还给出了迭代过程的收敛性证明。内插细分量有$C^2$-正常区域的平滑度，非凡的顶点和边缘除外。此外，所提出的插值体积细分方法不仅可以用于几何插值，而且可以用于体积材料建模领域的材料属性插值。提出的体积细分方法在等几何分析中的应用也给出了几个示例。