Domain parameterization, i.e., constructing a map from a parameter domain to a computational domain, is a key step in isogeometric analysis. Before parameterizing the interior of the computational domain, the boundary correspondence between the parametric domain and the computational domain is required by most domain parameterization methods, and the quality of boundary correspondence has a great effect on the quality of subsequent interior parameterization and analysis. Previous methods manually fulfill this task in general, which is tedious and subject to trial and error. In this paper, we propose an automatic method to compute such a correspondence between the boundary of a unit cube and the boundary of a volumetric computational domain based on the theory of unbalanced optimal transport. Given the boundary of a volumetric computational domain, the main task is to select 8 corner points and 12 curves connecting the 8 corner points on the boundary to divide the boundary into six surface patches (corresponding to the six faces of a unit cube), such that the difference between the Gaussian and mean curvature measures of the input boundary and those of the unit cube is minimized. We formulate this problem as an optimal mass transport problem, which is subject to some restrictions on the areas of the 6 surface patches and the lengths of the 12 boundary curves. To simplify the problem, a spherical intermediate domain is introduced by spherical parameterization of the computational domain in order to reduce the problem to be solved on a sphere. Riemannian L-BFGS method is adopted to solve the optimization efficiently. Experimental examples demonstrate that the proposed approach can produce satisfactory results which are competitive with the manually designed method.

域参数化，即构建从参数域到计算域的映射，是等几何分析的关键步骤。在对计算域内部进行参数化之前，大多数域参数化方法都需要参数域与计算域的边界对应关系，边界对应关系的好坏对后续内部参数化和分析的质量有很大影响。以前的方法通常手动完成这项任务，这很乏味并且需要反复试验。在本文中，我们提出了一种基于不平衡最优传输理论的自动方法来计算单位立方体边界和体积计算域边界之间的这种对应关系。给定体积计算域的边界，主要任务是选择边界上的8个角点和连接8个角点的12条曲线，将边界划分为六个面片（对应一个单位立方体的六个面），使得高斯曲率和平均曲率之差输入边界的度量和单位立方体的度量被最小化。我们将这个问题表述为一个最优质量传输问题，它受到 6 个表面补丁的面积和 12 条边界曲线的长度的一些限制。为了简化问题，通过计算域的球面参数化引入了一个球面中间域，以减少在球面上要解决的问题。采用黎曼L-BFGS方法有效求解优化问题。